Fractal basins of attraction in the restricted four-body problem when the primaries are triaxial rigid bodies

NASA Astrophysics Data System (ADS)

Suraj, Md Sanam; Asique, Md Chand; Prasad, Umakant; Hassan, M. R.; Shalini, Kumari

2017-11-01

The planar equilateral restricted four-body problem, formulated on the basis of Lagrange's triangular solutions is used to determine the existence and locations of libration points and the Newton-Raphson basins of convergence associated with these libration points. We have supposed that all the three primaries situated on the vertices of an equilateral triangle are triaxial rigid bodies. This paper also deals with the effect of these triaxiality parameters on the regions of motion where the test particle is free to move. Further, the regions on the configuration plane filled by the basins of attraction are determined by using the multivariate version of the Newton-Raphson iterative system. The numerical study reveals that the triaxiality of the primaries is one of the most influential parameters in the four-body problem.

Computing Maximum Likelihood Estimates of Loglinear Models from Marginal Sums with Special Attention to Loglinear Item Response Theory.

ERIC Educational Resources Information Center

Kelderman, Henk

1992-01-01

Describes algorithms used in the computer program LOGIMO for obtaining maximum likelihood estimates of the parameters in loglinear models. These algorithms are also useful for the analysis of loglinear item-response theory models. Presents modified versions of the iterative proportional fitting and Newton-Raphson algorithms. Simulated data…

Development of an autonomous video rendezvous and docking system, phase 3

NASA Technical Reports Server (NTRS)

Tietz, J. C.

1984-01-01

Field-of-view limitations proved troublesome. Higher resolution was required. Side thrusters were too weak. The strategy logic was improved and the Kalman filter was augmented to estimate target attitude and tumble rate. Two separate filters were used. The new filter estimates target attitude and angular momentum. The Newton-Raphson iteration improves image interpretation.

MHOST: An efficient finite element program for inelastic analysis of solids and structures

NASA Technical Reports Server (NTRS)

Nakazawa, S.

1988-01-01

An efficient finite element program for 3-D inelastic analysis of gas turbine hot section components was constructed and validated. A novel mixed iterative solution strategy is derived from the augmented Hu-Washizu variational principle in order to nodally interpolate coordinates, displacements, deformation, strains, stresses and material properties. A series of increasingly sophisticated material models incorporated in MHOST include elasticity, secant plasticity, infinitesimal and finite deformation plasticity, creep and unified viscoplastic constitutive model proposed by Walker. A library of high performance elements is built into this computer program utilizing the concepts of selective reduced integrations and independent strain interpolations. A family of efficient solution algorithms is implemented in MHOST for linear and nonlinear equation solution including the classical Newton-Raphson, modified, quasi and secant Newton methods with optional line search and the conjugate gradient method.

Input-output-controlled nonlinear equation solvers

NASA Technical Reports Server (NTRS)

Padovan, Joseph

1988-01-01

To upgrade the efficiency and stability of the successive substitution (SS) and Newton-Raphson (NR) schemes, the concept of input-output-controlled solvers (IOCS) is introduced. By employing the formal properties of the constrained version of the SS and NR schemes, the IOCS algorithm can handle indefiniteness of the system Jacobian, can maintain iterate monotonicity, and provide for separate control of load incrementation and iterate excursions, as well as having other features. To illustrate the algorithmic properties, the results for several benchmark examples are presented. These define the associated numerical efficiency and stability of the IOCS.

Nonlinear Transient Problems Using Structure Compatible Heat Transfer Code

NASA Technical Reports Server (NTRS)

Hou, Gene

2000-01-01

The report documents the recent effort to enhance a transient linear heat transfer code so as to solve nonlinear problems. The linear heat transfer code was originally developed by Dr. Kim Bey of NASA Largely and called the Structure-Compatible Heat Transfer (SCHT) code. The report includes four parts. The first part outlines the formulation of the heat transfer problem of concern. The second and the third parts give detailed procedures to construct the nonlinear finite element equations and the required Jacobian matrices for the nonlinear iterative method, Newton-Raphson method. The final part summarizes the results of the numerical experiments on the newly enhanced SCHT code.

Real-time Adaptive Control Using Neural Generalized Predictive Control

NASA Technical Reports Server (NTRS)

Haley, Pam; Soloway, Don; Gold, Brian

1999-01-01

The objective of this paper is to demonstrate the feasibility of a Nonlinear Generalized Predictive Control algorithm by showing real-time adaptive control on a plant with relatively fast time-constants. Generalized Predictive Control has classically been used in process control where linear control laws were formulated for plants with relatively slow time-constants. The plant of interest for this paper is a magnetic levitation device that is nonlinear and open-loop unstable. In this application, the reference model of the plant is a neural network that has an embedded nominal linear model in the network weights. The control based on the linear model provides initial stability at the beginning of network training. In using a neural network the control laws are nonlinear and online adaptation of the model is possible to capture unmodeled or time-varying dynamics. Newton-Raphson is the minimization algorithm. Newton-Raphson requires the calculation of the Hessian, but even with this computational expense the low iteration rate make this a viable algorithm for real-time control.

Tension Cutoff and Parameter Identification for the Viscoplastic Cap Model.

DTIC Science & Technology

1983-04-01

computer program "VPDRVR" which employs a Crank-Nicolson time integration scheme and a Newton-Raphson iterative solution procedure. Numerical studies were...parameters was illustrated for triaxial stress and uniaxial strain loading for a well- studied sand material (McCormick Ranch Sand). Lastly, a finite element...viscoplastic tension-cutoff cri- terion and to establish parameter identification techniques with experimental data. Herein lies the impetus of this study

A Perturbation Analysis of Harmonics Generation from Saturated Elements in Power Systems

NASA Astrophysics Data System (ADS)

Kumano, Teruhisa

Nonlinear phenomena such as saturation in magnetic flux give considerable effects in power system analysis. It is reported that a failure in a real 500kV system triggered islanding operation, where resultant even harmonics caused malfunctions in protective relays. It is also reported that the major origin of this wave distortion is nothing but unidirectional magnetization of the transformer iron core. Time simulation is widely used today to analyze this type of phenomena, but it has basically two shortcomings. One is that the time simulation takes two much computing time in the vicinity of inflection points in the saturation characteristic curve because certain iterative procedure such as N-R (Newton-Raphson) should be used and such methods tend to be caught in an ill conditioned numerical hunting. The other is that such simulation methods sometimes do not help intuitive understanding of the studied phenomenon because the whole nonlinear equations are treated in a matrix form and not properly divided into understandable parts as done in linear systems. This paper proposes a new computation scheme which is based on so called perturbation method. Magnetic saturation in iron cores in a generator and a transformer are taken into account. The proposed method has a special feature against the first shortcoming of the N-R based time simulation method stated above. In the proposed method no iterative process is used to reduce the equation residue but uses perturbation series, which means free from the ill condition problem. Users have only to calculate each perturbation terms one by one until he reaches necessary accuracy. In a numerical example treated in the present paper the first order perturbation can make reasonably high accuracy, which means very fast computing. In numerical study three nonlinear elements are considered. Calculated results are almost identical to the conventional Newton-Raphson based time simulation, which shows the validity of the method. The proposed method would be effectively used in a screening where many case studies are needed.

Finite element model correlation of a composite UAV wing using modal frequencies

NASA Astrophysics Data System (ADS)

Oliver, Joseph A.; Kosmatka, John B.; Hemez, François M.; Farrar, Charles R.

2007-04-01

The current work details the implementation of a meta-model based correlation technique on a composite UAV wing test piece and associated finite element (FE) model. This method involves training polynomial models to emulate the FE input-output behavior and then using numerical optimization to produce a set of correlated parameters which can be returned to the FE model. After discussions about the practical implementation, the technique is validated on a composite plate structure and then applied to the UAV wing structure, where it is furthermore compared to a more traditional Newton-Raphson technique which iteratively uses first-order Taylor-series sensitivity. The experimental testpiece wing comprises two graphite/epoxy prepreg and Nomex honeycomb co-cured skins and two prepreg spars bonded together in a secondary process. MSC.Nastran FE models of the four structural components are correlated independently, using modal frequencies as correlation features, before being joined together into the assembled structure and compared to experimentally measured frequencies from the assembled wing in a cantilever configuration. Results show that significant improvements can be made to the assembled model fidelity, with the meta-model procedure producing slightly superior results to Newton-Raphson iteration. Final evaluation of component correlation using the assembled wing comparison showed worse results for each correlation technique, with the meta-model technique worse overall. This can be most likely be attributed to difficultly in correlating the open-section spars; however, there is also some question about non-unique update variable combinations in the current configuration, which lead correlation away from physically probably values.

Computing Maximum Likelihood Estimates of Loglinear Models from Marginal Sums with Special Attention to Loglinear Item Response Theory. [Project Psychometric Aspects of Item Banking No. 53.] Research Report 91-1.

ERIC Educational Resources Information Center

Kelderman, Henk

In this paper, algorithms are described for obtaining the maximum likelihood estimates of the parameters in log-linear models. Modified versions of the iterative proportional fitting and Newton-Raphson algorithms are described that work on the minimal sufficient statistics rather than on the usual counts in the full contingency table. This is…

Numerical solutions of 2-D multi-stage rotor/stator unsteady flow interactions

NASA Astrophysics Data System (ADS)

Yang, R.-J.; Lin, S.-J.

1991-01-01

The Rai method of single-stage rotor/stator flow interaction is extended to handle multistage configurations. In this study, a two-dimensional Navier-Stokes multi-zone approach was used to investigate unsteady flow interactions within two multistage axial turbines. The governing equations are solved by an iterative, factored, implicit finite-difference, upwind algorithm. Numerical accuracy is checked by investigating the effect of time step size, the effect of subiteration in the Newton-Raphson technique, and the effect of full viscous versus thin-layer approximation. Computer results compared well with experimental data. Unsteady flow interactions, wake cutting, and the associated evolution of vortical entities are discussed.

An hp symplectic pseudospectral method for nonlinear optimal control

NASA Astrophysics Data System (ADS)

Peng, Haijun; Wang, Xinwei; Li, Mingwu; Chen, Biaosong

2017-01-01

An adaptive symplectic pseudospectral method based on the dual variational principle is proposed and is successfully applied to solving nonlinear optimal control problems in this paper. The proposed method satisfies the first order necessary conditions of continuous optimal control problems, also the symplectic property of the original continuous Hamiltonian system is preserved. The original optimal control problem is transferred into a set of nonlinear equations which can be solved easily by Newton-Raphson iterations, and the Jacobian matrix is found to be sparse and symmetric. The proposed method, on one hand, exhibits exponent convergence rates when the number of collocation points are increasing with the fixed number of sub-intervals; on the other hand, exhibits linear convergence rates when the number of sub-intervals is increasing with the fixed number of collocation points. Furthermore, combining with the hp method based on the residual error of dynamic constraints, the proposed method can achieve given precisions in a few iterations. Five examples highlight the high precision and high computational efficiency of the proposed method.

Unstructured Finite Volume Computational Thermo-Fluid Dynamic Method for Multi-Disciplinary Analysis and Design Optimization

NASA Technical Reports Server (NTRS)

Majumdar, Alok; Schallhorn, Paul

1998-01-01

This paper describes a finite volume computational thermo-fluid dynamics method to solve for Navier-Stokes equations in conjunction with energy equation and thermodynamic equation of state in an unstructured coordinate system. The system of equations have been solved by a simultaneous Newton-Raphson method and compared with several benchmark solutions. Excellent agreements have been obtained in each case and the method has been found to be significantly faster than conventional Computational Fluid Dynamic(CFD) methods and therefore has the potential for implementation in Multi-Disciplinary analysis and design optimization in fluid and thermal systems. The paper also describes an algorithm of design optimization based on Newton-Raphson method which has been recently tested in a turbomachinery application.

A robust direct-integration method for rotorcraft maneuver and periodic response

NASA Technical Reports Server (NTRS)

Panda, Brahmananda

1992-01-01

The Newmark-Beta method and the Newton-Raphson iteration scheme are combined to develop a direct-integration method for evaluating the maneuver and periodic-response expressions for rotorcraft. The method requires the generation of Jacobians and includes higher derivatives in the formulation of the geometric stiffness matrix to enhance the convergence of the system. The method leads to effective convergence with nonlinear structural dynamics and aerodynamic terms. Singularities in the matrices can be addressed with the method as they arise from a Lagrange multiplier approach for coupling equations with nonlinear constraints. The method is also shown to be general enough to handle singularities from quasisteady control-system models. The method is shown to be more general and robust than the similar 2GCHAS method for analyzing rotorcraft dynamics.

Performance analysis of improved iterated cubature Kalman filter and its application to GNSS/INS.

PubMed

Cui, Bingbo; Chen, Xiyuan; Xu, Yuan; Huang, Haoqian; Liu, Xiao

2017-01-01

In order to improve the accuracy and robustness of GNSS/INS navigation system, an improved iterated cubature Kalman filter (IICKF) is proposed by considering the state-dependent noise and system uncertainty. First, a simplified framework of iterated Gaussian filter is derived by using damped Newton-Raphson algorithm and online noise estimator. Then the effect of state-dependent noise coming from iterated update is analyzed theoretically, and an augmented form of CKF algorithm is applied to improve the estimation accuracy. The performance of IICKF is verified by field test and numerical simulation, and results reveal that, compared with non-iterated filter, iterated filter is less sensitive to the system uncertainty, and IICKF improves the accuracy of yaw, roll and pitch by 48.9%, 73.1% and 83.3%, respectively, compared with traditional iterated KF. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Allocation of Transaction Cost to Market Participants Using an Analytical Method in Deregulated Market

NASA Astrophysics Data System (ADS)

Jeyasankari, S.; Jeslin Drusila Nesamalar, J.; Charles Raja, S.; Venkatesh, P.

2014-04-01

Transmission cost allocation is one of the major challenges in transmission open access faced by the electric power sector. The purpose of this work is to provide an analytical method for allocating transmission transaction cost in deregulated market. This research work provides a usage based transaction cost allocation method based on line-flow impact factor (LIF) which relates the power flow in each line with respect to transacted power for the given transaction. This method provides the impact of line flows without running iterative power flow solution and is well suited for real time applications. The proposed method is compared with the Newton-Raphson (NR) method of cost allocation on sample six bus and practical Indian utility 69 bus systems by considering multilateral transaction.

Finite element implementation of state variable-based viscoplasticity models

NASA Technical Reports Server (NTRS)

Iskovitz, I.; Chang, T. Y. P.; Saleeb, A. F.

1991-01-01

The implementation of state variable-based viscoplasticity models is made in a general purpose finite element code for structural applications of metals deformed at elevated temperatures. Two constitutive models, Walker's and Robinson's models, are studied in conjunction with two implicit integration methods: the trapezoidal rule with Newton-Raphson iterations and an asymptotic integration algorithm. A comparison is made between the two integration methods, and the latter method appears to be computationally more appealing in terms of numerical accuracy and CPU time. However, in order to make the asymptotic algorithm robust, it is necessary to include a self adaptive scheme with subincremental step control and error checking of the Jacobian matrix at the integration points. Three examples are given to illustrate the numerical aspects of the integration methods tested.

Self-adaptive Solution Strategies

NASA Technical Reports Server (NTRS)

Padovan, J.

1984-01-01

The development of enhancements to current generation nonlinear finite element algorithms of the incremental Newton-Raphson type was overviewed. Work was introduced on alternative formulations which lead to improve algorithms that avoid the need for global level updating and inversion. To quantify the enhanced Newton-Raphson scheme and the new alternative algorithm, the results of several benchmarks are presented.

Kinematics and dynamics of a six-degree-of-freedom robot manipulator with closed kinematic chain mechanism

NASA Technical Reports Server (NTRS)

Nguyen, Charles C.; Pooran, Farhad J.

1989-01-01

This paper deals with a class of robot manipulators built based on the kinematic chain mechanism (CKCM). This class of CKCM manipulators consists of a fixed and a moving platform coupled together via a number of in-parallel actuators. A closed-form solution is derived for the inverse kinematic problem of a six-degre-of-freedom CKCM manipulator designed to study robotic applications in space. Iterative Newton-Raphson method is employed to solve the forward kinematic problem. Dynamics of the above manipulator is derived using the Lagrangian approach. Computer simulation of the dynamical equations shows that the actuating forces are strongly dependent on the mass and centroid of the robot links.

TAP 1: A Finite Element Program for Steady-State Thermal Analysis of Convectively Cooled Structures

NASA Technical Reports Server (NTRS)

Thornton, E. A.

1976-01-01

The program has a finite element library of six elements: two conduction/convection elements to model heat transfer in a solid, two convection elements to model heat transfer in a fluid, and two integrated conduction/convection elements to represent combined heat transfer in tubular and plate/fin fluid passages. Nonlinear thermal analysis due to temperature dependent thermal parameters is performed using the Newton-Raphson iteration method. Program output includes nodal temperatures and element heat fluxes. Pressure drops in fluid passages may be computed as an option. A companion plotting program for displaying the finite element model and predicted temperature distributions is presented. User instructions and sample problems are presented in appendixes.

Equilibrium paths of an imperfect plate with respect to its aspect ratio

NASA Astrophysics Data System (ADS)

Psotny, Martin

2017-07-01

The stability analysis of a rectangular plate loaded in compression is presented, a specialized code based on FEM has been created. Special finite element with 48 degrees of freedom has been used for analysis. The nonlinear finite element method equations are derived from the variational principle of minimum of total potential energy. To trace the complete nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used, load versus displacement control was changed during the calculation process. The peculiarities of the effects of the initial imperfections on the load-deflection paths are investigated with respect to aspect ratio of the plate. Special attention is paid to the influence of imperfections on the post-critical buckling mode.

On nonlinear finite element analysis in single-, multi- and parallel-processors

NASA Technical Reports Server (NTRS)

Utku, S.; Melosh, R.; Islam, M.; Salama, M.

1982-01-01

Numerical solution of nonlinear equilibrium problems of structures by means of Newton-Raphson type iterations is reviewed. Each step of the iteration is shown to correspond to the solution of a linear problem, therefore the feasibility of the finite element method for nonlinear analysis is established. Organization and flow of data for various types of digital computers, such as single-processor/single-level memory, single-processor/two-level-memory, vector-processor/two-level-memory, and parallel-processors, with and without sub-structuring (i.e. partitioning) are given. The effect of the relative costs of computation, memory and data transfer on substructuring is shown. The idea of assigning comparable size substructures to parallel processors is exploited. Under Cholesky type factorization schemes, the efficiency of parallel processing is shown to decrease due to the occasional shared data, just as that due to the shared facilities.

A General Program for Item-Response Analysis That Employs the Stabilized Newton-Raphson Algorithm. Research Report. ETS RR-13-32

ERIC Educational Resources Information Center

Haberman, Shelby J.

2013-01-01

A general program for item-response analysis is described that uses the stabilized Newton-Raphson algorithm. This program is written to be compliant with Fortran 2003 standards and is sufficiently general to handle independent variables, multidimensional ability parameters, and matrix sampling. The ability variables may be either polytomous or…

Reduce beam hardening artifacts of polychromatic X-ray computed tomography by an iterative approximation approach.

PubMed

Shi, Hongli; Yang, Zhi; Luo, Shuqian

2017-01-01

The beam hardening artifact is one of most important modalities of metal artifact for polychromatic X-ray computed tomography (CT), which can impair the image quality seriously. An iterative approach is proposed to reduce beam hardening artifact caused by metallic components in polychromatic X-ray CT. According to Lambert-Beer law, the (detected) projections can be expressed as monotonic nonlinear functions of element geometry projections, which are the theoretical projections produced only by the pixel intensities (image grayscale) of certain element (component). With help of a prior knowledge on spectrum distribution of X-ray beam source and energy-dependent attenuation coefficients, the functions have explicit expressions. Newton-Raphson algorithm is employed to solve the functions. The solutions are named as the synthetical geometry projections, which are the nearly linear weighted sum of element geometry projections with respect to mean of each attenuation coefficient. In this process, the attenuation coefficients are modified to make Newton-Raphson iterative functions satisfy the convergence conditions of fixed pointed iteration(FPI) so that the solutions will approach the true synthetical geometry projections stably. The underlying images are obtained using the projections by general reconstruction algorithms such as the filtered back projection (FBP). The image gray values are adjusted according to the attenuation coefficient means to obtain proper CT numbers. Several examples demonstrate the proposed approach is efficient in reducing beam hardening artifacts and has satisfactory performance in the term of some general criteria. In a simulation example, the normalized root mean square difference (NRMSD) can be reduced 17.52% compared to a newest algorithm. Since the element geometry projections are free from the effect of beam hardening, the nearly linear weighted sum of them, the synthetical geometry projections, are almost free from the effect of beam hardening. By working out the synthetical geometry projections, the proposed approach becomes quite efficient in reducing beam hardening artifacts.

A triangular thin shell finite element: Nonlinear analysis. [structural analysis

NASA Technical Reports Server (NTRS)

Thomas, G. R.; Gallagher, R. H.

1975-01-01

Aspects of the formulation of a triangular thin shell finite element which pertain to geometrically nonlinear (small strain, finite displacement) behavior are described. The procedure for solution of the resulting nonlinear algebraic equations combines a one-step incremental (tangent stiffness) approach with one iteration in the Newton-Raphson mode. A method is presented which permits a rational estimation of step size in this procedure. Limit points are calculated by means of a superposition scheme coupled to the incremental side of the solution procedure while bifurcation points are calculated through a process of interpolation of the determinants of the tangent-stiffness matrix. Numerical results are obtained for a flat plate and two curved shell problems and are compared with alternative solutions.

The Beta-Geometric Model Applied to Fecundability in a Sample of Married Women

NASA Astrophysics Data System (ADS)

Adekanmbi, D. B.; Bamiduro, T. A.

2006-10-01

The time required to achieve pregnancy among married couples termed fecundability has been proposed to follow a beta-geometric distribution. The accuracy of the method used in estimating the parameters of the model has an implication on the goodness of fit of the model. In this study, the parameters of the model are estimated using the Method of Moments and Newton-Raphson estimation procedure. The goodness of fit of the model was considered, using estimates from the two methods of estimation, as well as the asymptotic relative efficiency of the estimates. A noticeable improvement in the fit of the model to the data on time to conception was observed, when the parameters are estimated by Newton-Raphson procedure, and thereby estimating reasonable expectations of fecundability for married female population in the country.

Fisher Scoring Method for Parameter Estimation of Geographically Weighted Ordinal Logistic Regression (GWOLR) Model

NASA Astrophysics Data System (ADS)

Widyaningsih, Purnami; Retno Sari Saputro, Dewi; Nugrahani Putri, Aulia

2017-06-01

GWOLR model combines geographically weighted regression (GWR) and (ordinal logistic reression) OLR models. Its parameter estimation employs maximum likelihood estimation. Such parameter estimation, however, yields difficult-to-solve system of nonlinear equations, and therefore numerical approximation approach is required. The iterative approximation approach, in general, uses Newton-Raphson (NR) method. The NR method has a disadvantage—its Hessian matrix is always the second derivatives of each iteration so it does not always produce converging results. With regard to this matter, NR model is modified by substituting its Hessian matrix into Fisher information matrix, which is termed Fisher scoring (FS). The present research seeks to determine GWOLR model parameter estimation using Fisher scoring method and apply the estimation on data of the level of vulnerability to Dengue Hemorrhagic Fever (DHF) in Semarang. The research concludes that health facilities give the greatest contribution to the probability of the number of DHF sufferers in both villages. Based on the number of the sufferers, IR category of DHF in both villages can be determined.

Close Encounters of a Sparse Kind.

ERIC Educational Resources Information Center

Westerberg, Arthur W.

1980-01-01

By providing an example problem in solving sets of nonlinear algebraic equations, the advantages and disadvantages of two methods for its solution, the tearing approach v the Newton-Raphson approach, are elucidated. (CS)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Goodwin, D. L.; Kuprov, Ilya, E-mail: i.kuprov@soton.ac.uk

Quadratic convergence throughout the active space is achieved for the gradient ascent pulse engineering (GRAPE) family of quantum optimal control algorithms. We demonstrate in this communication that the Hessian of the GRAPE fidelity functional is unusually cheap, having the same asymptotic complexity scaling as the functional itself. This leads to the possibility of using very efficient numerical optimization techniques. In particular, the Newton-Raphson method with a rational function optimization (RFO) regularized Hessian is shown in this work to require fewer system trajectory evaluations than any other algorithm in the GRAPE family. This communication describes algebraic and numerical implementation aspects (matrixmore » exponential recycling, Hessian regularization, etc.) for the RFO Newton-Raphson version of GRAPE and reports benchmarks for common spin state control problems in magnetic resonance spectroscopy.« less

Broad-search algorithms for the spacecraft trajectory design of Callisto-Ganymede-Io triple flyby sequences from 2024 to 2040, Part II: Lambert pathfinding and trajectory solutions

NASA Astrophysics Data System (ADS)

Lynam, Alfred E.

2014-01-01

Triple-satellite-aided capture employs gravity-assist flybys of three of the Galilean moons of Jupiter in order to decrease the amount of ΔV required to capture a spacecraft into Jupiter orbit. Similarly, triple flybys can be used within a Jupiter satellite tour to rapidly modify the orbital parameters of a Jovicentric orbit, or to increase the number of science flybys. In order to provide a nearly comprehensive search of the solution space of Callisto-Ganymede-Io triple flybys from 2024 to 2040, a third-order, Chebyshev's method variant of the p-iteration solution to Lambert's problem is paired with a second-order, Newton-Raphson method, time of flight iteration solution to the V∞-matching problem. The iterative solutions of these problems provide the orbital parameters of the Callisto-Ganymede transfer, the Ganymede flyby, and the Ganymede-Io transfer, but the characteristics of the Callisto and Io flybys are unconstrained, so they are permitted to vary in order to produce an even larger number of trajectory solutions. The vast amount of solution data is searched to find the best triple-satellite-aided capture window between 2024 and 2040.

A FORTRAN program for multivariate survival analysis on the personal computer.

PubMed

Mulder, P G

1988-01-01

In this paper a FORTRAN program is presented for multivariate survival or life table regression analysis in a competing risks' situation. The relevant failure rate (for example, a particular disease or mortality rate) is modelled as a log-linear function of a vector of (possibly time-dependent) explanatory variables. The explanatory variables may also include the variable time itself, which is useful for parameterizing piecewise exponential time-to-failure distributions in a Gompertz-like or Weibull-like way as a more efficient alternative to Cox's proportional hazards model. Maximum likelihood estimates of the coefficients of the log-linear relationship are obtained from the iterative Newton-Raphson method. The program runs on a personal computer under DOS; running time is quite acceptable, even for large samples.

Hierarchically partitioned nonlinear equation solvers

NASA Technical Reports Server (NTRS)

Padovan, Joseph

1987-01-01

By partitioning solution space into a number of subspaces, a new multiply constrained partitioned Newton-Raphson nonlinear equation solver is developed. Specifically, for a given iteration, each of the various separate partitions are individually and simultaneously controlled. Due to the generality of the scheme, a hierarchy of partition levels can be employed. For finite-element-type applications, this includes the possibility of degree-of-freedom, nodal, elemental, geometric substructural, material and kinematically nonlinear group controls. It is noted that such partitioning can be continuously updated, depending on solution conditioning. In this context, convergence is ascertained at the individual partition level.

Harmonic analysis of spacecraft power systems using a personal computer

NASA Technical Reports Server (NTRS)

Williamson, Frank; Sheble, Gerald B.

1989-01-01

The effects that nonlinear devices such as ac/dc converters, HVDC transmission links, and motor drives have on spacecraft power systems are discussed. The nonsinusoidal currents, along with the corresponding voltages, are calculated by a harmonic power flow which decouples and solves for each harmonic component individually using an iterative Newton-Raphson algorithm. The sparsity of the harmonic equations and the overall Jacobian matrix is used to an advantage in terms of saving computer memory space and in terms of reducing computation time. The algorithm could also be modified to analyze each harmonic separately instead of all at the same time.

Nonlinear modelling of high-speed catenary based on analytical expressions of cable and truss elements

NASA Astrophysics Data System (ADS)

Song, Yang; Liu, Zhigang; Wang, Hongrui; Lu, Xiaobing; Zhang, Jing

2015-10-01

Due to the intrinsic nonlinear characteristics and complex structure of the high-speed catenary system, a modelling method is proposed based on the analytical expressions of nonlinear cable and truss elements. The calculation procedure for solving the initial equilibrium state is proposed based on the Newton-Raphson iteration method. The deformed configuration of the catenary system as well as the initial length of each wire can be calculated. Its accuracy and validity of computing the initial equilibrium state are verified by comparison with the separate model method, absolute nodal coordinate formulation and other methods in the previous literatures. Then, the proposed model is combined with a lumped pantograph model and a dynamic simulation procedure is proposed. The accuracy is guaranteed by the multiple iterative calculations in each time step. The dynamic performance of the proposed model is validated by comparison with EN 50318, the results of the finite element method software and SIEMENS simulation report, respectively. At last, the influence of the catenary design parameters (such as the reserved sag and pre-tension) on the dynamic performance is preliminarily analysed by using the proposed model.

Comparison between iteration schemes for three-dimensional coordinate-transformed saturated-unsaturated flow model

NASA Astrophysics Data System (ADS)

An, Hyunuk; Ichikawa, Yutaka; Tachikawa, Yasuto; Shiiba, Michiharu

2012-11-01

SummaryThree different iteration methods for a three-dimensional coordinate-transformed saturated-unsaturated flow model are compared in this study. The Picard and Newton iteration methods are the common approaches for solving Richards' equation. The Picard method is simple to implement and cost-efficient (on an individual iteration basis). However it converges slower than the Newton method. On the other hand, although the Newton method converges faster, it is more complex to implement and consumes more CPU resources per iteration than the Picard method. The comparison of the two methods in finite-element model (FEM) for saturated-unsaturated flow has been well evaluated in previous studies. However, two iteration methods might exhibit different behavior in the coordinate-transformed finite-difference model (FDM). In addition, the Newton-Krylov method could be a suitable alternative for the coordinate-transformed FDM because it requires the evaluation of a 19-point stencil matrix. The formation of a 19-point stencil is quite a complex and laborious procedure. Instead, the Newton-Krylov method calculates the matrix-vector product, which can be easily approximated by calculating the differences of the original nonlinear function. In this respect, the Newton-Krylov method might be the most appropriate iteration method for coordinate-transformed FDM. However, this method involves the additional cost of taking an approximation at each Krylov iteration in the Newton-Krylov method. In this paper, we evaluated the efficiency and robustness of three iteration methods—the Picard, Newton, and Newton-Krylov methods—for simulating saturated-unsaturated flow through porous media using a three-dimensional coordinate-transformed FDM.

Evaluation of an S-system root-finding method for estimating parameters in a metabolic reaction model.

PubMed

Iwata, Michio; Miyawaki-Kuwakado, Atsuko; Yoshida, Erika; Komori, Soichiro; Shiraishi, Fumihide

2018-02-02

In a mathematical model, estimation of parameters from time-series data of metabolic concentrations in cells is a challenging task. However, it seems that a promising approach for such estimation has not yet been established. Biochemical Systems Theory (BST) is a powerful methodology to construct a power-law type model for a given metabolic reaction system and to then characterize it efficiently. In this paper, we discuss the use of an S-system root-finding method (S-system method) to estimate parameters from time-series data of metabolite concentrations. We demonstrate that the S-system method is superior to the Newton-Raphson method in terms of the convergence region and iteration number. We also investigate the usefulness of a translocation technique and a complex-step differentiation method toward the practical application of the S-system method. The results indicate that the S-system method is useful to construct mathematical models for a variety of metabolic reaction networks. Copyright © 2018 Elsevier Inc. All rights reserved.

TAP 2: A finite element program for thermal analysis of convectively cooled structures

NASA Technical Reports Server (NTRS)

Thornton, E. A.

1980-01-01

A finite element computer program (TAP 2) for steady-state and transient thermal analyses of convectively cooled structures is presented. The program has a finite element library of six elements: two conduction/convection elements to model heat transfer in a solid, two convection elements to model heat transfer in a fluid, and two integrated conduction/convection elements to represent combined heat transfer in tubular and plate/fin fluid passages. Nonlinear thermal analysis due to temperature-dependent thermal parameters is performed using the Newton-Raphson iteration method. Transient analyses are performed using an implicit Crank-Nicolson time integration scheme with consistent or lumped capacitance matrices as an option. Program output includes nodal temperatures and element heat fluxes. Pressure drops in fluid passages may be computed as an option. User instructions and sample problems are presented in appendixes.

Estimating the Parameters of the Beta-Binomial Distribution.

ERIC Educational Resources Information Center

Wilcox, Rand R.

1979-01-01

For some situations the beta-binomial distribution might be used to describe the marginal distribution of test scores for a particular population of examinees. Several different methods of approximating the maximum likelihood estimate were investigated, and it was found that the Newton-Raphson method should be used when it yields admissable…

Flux-vector splitting algorithm for chain-rule conservation-law form

NASA Technical Reports Server (NTRS)

Shih, T. I.-P.; Nguyen, H. L.; Willis, E. A.; Steinthorsson, E.; Li, Z.

1991-01-01

A flux-vector splitting algorithm with Newton-Raphson iteration was developed for the 'full compressible' Navier-Stokes equations cast in chain-rule conservation-law form. The algorithm is intended for problems with deforming spatial domains and for problems whose governing equations cannot be cast in strong conservation-law form. The usefulness of the algorithm for such problems was demonstrated by applying it to analyze the unsteady, two- and three-dimensional flows inside one combustion chamber of a Wankel engine under nonfiring conditions. Solutions were obtained to examine the algorithm in terms of conservation error, robustness, and ability to handle complex flows on time-dependent grid systems.

Voltage profile program for the Kennedy Space Center electric power distribution system

NASA Technical Reports Server (NTRS)

1976-01-01

The Kennedy Space Center voltage profile program computes voltages at all busses greater than 1 Kv in the network under various conditions of load. The computation is based upon power flow principles and utilizes a Newton-Raphson iterative load flow algorithm. Power flow conditions throughout the network are also provided. The computer program is designed for both steady state and transient operation. In the steady state mode, automatic tap changing of primary distribution transformers is incorporated. Under transient conditions, such as motor starts etc., it is assumed that tap changing is not accomplished so that transformer secondary voltage is allowed to sag.

What Information Theory Says about Bounded Rational Best Response

NASA Technical Reports Server (NTRS)

Wolpert, David H.

2005-01-01

Probability Collectives (PC) provides the information-theoretic extension of conventional full-rationality game theory to bounded rational games. Here an explicit solution to the equations giving the bounded rationality equilibrium of a game is presented. Then PC is used to investigate games in which the players use bounded rational best-response strategies. Next it is shown that in the continuum-time limit, bounded rational best response games result in a variant of the replicator dynamics of evolutionary game theory. It is then shown that for team (shared-payoff) games, this variant of replicator dynamics is identical to Newton-Raphson iterative optimization of the shared utility function.

The Application of Simulation Method in Isothermal Elastic Natural Gas Pipeline

NASA Astrophysics Data System (ADS)

Xing, Chunlei; Guan, Shiming; Zhao, Yue; Cao, Jinggang; Chu, Yanji

2018-02-01

This Elastic pipeline mathematic model is of crucial importance in natural gas pipeline simulation because of its compliance with the practical industrial cases. The numerical model of elastic pipeline will bring non-linear complexity to the discretized equations. Hence the Newton-Raphson method cannot achieve fast convergence in this kind of problems. Therefore A new Newton Based method with Powell-Wolfe Condition to simulate the Isothermal elastic pipeline flow is presented. The results obtained by the new method aregiven based on the defined boundary conditions. It is shown that the method converges in all cases and reduces significant computational cost.

New Method of Calibrating IRT Models.

ERIC Educational Resources Information Center

Jiang, Hai; Tang, K. Linda

This discussion of new methods for calibrating item response theory (IRT) models looks into new optimization procedures, such as the Genetic Algorithm (GA) to improve on the use of the Newton-Raphson procedure. The advantages of using a global optimization procedure like GA is that this kind of procedure is not easily affected by local optima and…

Energy minimization in medical image analysis: Methodologies and applications.

PubMed

Zhao, Feng; Xie, Xianghua

2016-02-01

Energy minimization is of particular interest in medical image analysis. In the past two decades, a variety of optimization schemes have been developed. In this paper, we present a comprehensive survey of the state-of-the-art optimization approaches. These algorithms are mainly classified into two categories: continuous method and discrete method. The former includes Newton-Raphson method, gradient descent method, conjugate gradient method, proximal gradient method, coordinate descent method, and genetic algorithm-based method, while the latter covers graph cuts method, belief propagation method, tree-reweighted message passing method, linear programming method, maximum margin learning method, simulated annealing method, and iterated conditional modes method. We also discuss the minimal surface method, primal-dual method, and the multi-objective optimization method. In addition, we review several comparative studies that evaluate the performance of different minimization techniques in terms of accuracy, efficiency, or complexity. These optimization techniques are widely used in many medical applications, for example, image segmentation, registration, reconstruction, motion tracking, and compressed sensing. We thus give an overview on those applications as well. Copyright © 2015 John Wiley & Sons, Ltd.

A new constitutive analysis of hexagonal close-packed metal in equal channel angular pressing by crystal plasticity finite element method

NASA Astrophysics Data System (ADS)

Li, Hejie; Öchsner, Andreas; Yarlagadda, Prasad K. D. V.; Xiao, Yin; Furushima, Tsuyoshi; Wei, Dongbin; Jiang, Zhengyi; Manabe, Ken-ichi

2018-01-01

Most of hexagonal close-packed (HCP) metals are lightweight metals. With the increasing application of light metal products, the production of light metal is increasingly attracting the attentions of researchers worldwide. To obtain a better understanding of the deformation mechanism of HCP metals (especially for Mg and its alloys), a new constitutive analysis was carried out based on previous research. In this study, combining the theories of strain gradient and continuum mechanics, the equal channel angular pressing process is analyzed and a HCP crystal plasticity constitutive model is developed especially for Mg and its alloys. The influence of elevated temperature on the deformation mechanism of the Mg alloy (slip and twin) is novelly introduced into a crystal plasticity constitutive model. The solution for the new developed constitutive model is established on the basis of the Lagrangian iterations and Newton Raphson simplification.

Thermal Model of a Current-Carrying Wire in a Vacuum

NASA Technical Reports Server (NTRS)

Border, James

2006-01-01

A computer program implements a thermal model of an insulated wire carrying electric current and surrounded by a vacuum. The model includes the effects of Joule heating, conduction of heat along the wire, and radiation of heat from the outer surface of the insulation on the wire. The model takes account of the temperature dependences of the thermal and electrical properties of the wire, the emissivity of the insulation, and the possibility that not only can temperature vary along the wire but, in addition, the ends of the wire can be thermally grounded at different temperatures. The resulting second-order differential equation for the steady-state temperature as a function of position along the wire is highly nonlinear. The wire is discretized along its length, and the equation is solved numerically by use of an iterative algorithm that utilizes a multidimensional version of the Newton-Raphson method.

Development of kinematic equations and determination of workspace of a 6 DOF end-effector with closed-kinematic chain mechanism

NASA Technical Reports Server (NTRS)

Nguyen, Charles C.; Pooran, Farhad J.

1989-01-01

This report presents results from the research grant entitled Active Control of Robot Manipulators, funded by the Goddard Space Flight Center, under Grant NAG5-780, for the period July 1, 1988 to January 1, 1989. An analysis is presented of a 6 degree-of-freedom robot end-effector built to study telerobotic assembly of NASA hardware in space. Since the end-effector is required to perform high precision motion in a limited workspace, closed-kinematic mechanisms are chosen for its design. A closed-form solution is obtained for the inverse kinematic problem and an iterative procedure employing Newton-Raphson method is proposed to solve the forward kinematic problem. A study of the end-effector workspace results in a general procedure for the workspace determination based on link constraints. Computer simulation results are presented.

Optimization of Time-Dependent Particle Tracing Using Tetrahedral Decomposition

NASA Technical Reports Server (NTRS)

Kenwright, David; Lane, David

1995-01-01

An efficient algorithm is presented for computing particle paths, streak lines and time lines in time-dependent flows with moving curvilinear grids. The integration, velocity interpolation and step-size control are all performed in physical space which avoids the need to transform the velocity field into computational space. This leads to higher accuracy because there are no Jacobian matrix approximations or expensive matrix inversions. Integration accuracy is maintained using an adaptive step-size control scheme which is regulated by the path line curvature. The problem of cell-searching, point location and interpolation in physical space is simplified by decomposing hexahedral cells into tetrahedral cells. This enables the point location to be done analytically and substantially faster than with a Newton-Raphson iterative method. Results presented show this algorithm is up to six times faster than particle tracers which operate on hexahedral cells yet produces almost identical particle trajectories.

Sometimes "Newton's Method" Always "Cycles"

ERIC Educational Resources Information Center

Latulippe, Joe; Switkes, Jennifer

2012-01-01

Are there functions for which Newton's method cycles for all non-trivial initial guesses? We construct and solve a differential equation whose solution is a real-valued function that two-cycles under Newton iteration. Higher-order cycles of Newton's method iterates are explored in the complex plane using complex powers of "x." We find a class of…

On the Latent Regression Model of Item Response Theory. Research Report. ETS RR-07-12

ERIC Educational Resources Information Center

Antal, Tamás

2007-01-01

Full account of the latent regression model for the National Assessment of Educational Progress is given. The treatment includes derivation of the EM algorithm, Newton-Raphson method, and the asymptotic standard errors. The paper also features the use of the adaptive Gauss-Hermite numerical integration method as a basic tool to evaluate…

A Newton-Raphson Method Approach to Adjusting Multi-Source Solar Simulators

NASA Technical Reports Server (NTRS)

Snyder, David B.; Wolford, David S.

2012-01-01

NASA Glenn Research Center has been using an in house designed X25 based multi-source solar simulator since 2003. The simulator is set up for triple junction solar cells prior to measurements b y adjusting the three sources to produce the correct short circuit current, lsc, in each of three AM0 calibrated sub-cells. The past practice has been to adjust one source on one sub-cell at a time, iterating until all the sub-cells have the calibrated Isc. The new approach is to create a matrix of measured lsc for small source changes on each sub-cell. A matrix, A, is produced. This is normalized to unit changes in the sources so that Ax(delta)s = (delta)isc. This matrix can now be inverted and used with the known Isc differences from the AM0 calibrated values to indicate changes in the source settings, (delta)s = A ·'x.(delta)isc This approach is still an iterative one, but all sources are changed during each iteration step. It typically takes four to six steps to converge on the calibrated lsc values. Even though the source lamps may degrade over time, the initial matrix evaluation i s not performed each time, since measurement matrix needs to be only approximate. Because an iterative approach is used the method will still continue to be valid. This method may become more important as state-of-the-art solar cell junction responses overlap the sources of the simulator. Also, as the number of cell junctions and sources increase, this method should remain applicable.

Efficient iterative methods applied to the solution of transonic flows

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wissink, A.M.; Lyrintzis, A.S.; Chronopoulos, A.T.

1996-02-01

We investigate the use of an inexact Newton`s method to solve the potential equations in the transonic regime. As a test case, we solve the two-dimensional steady transonic small disturbance equation. Approximate factorization/ADI techniques have traditionally been employed for implicit solutions of this nonlinear equation. Instead, we apply Newton`s method using an exact analytical determination of the Jacobian with preconditioned conjugate gradient-like iterative solvers for solution of the linear systems in each Newton iteration. Two iterative solvers are tested; a block s-step version of the classical Orthomin(k) algorithm called orthogonal s-step Orthomin (OSOmin) and the well-known GIVIRES method. The preconditionermore » is a vectorizable and parallelizable version of incomplete LU (ILU) factorization. Efficiency of the Newton-Iterative method on vector and parallel computer architectures is the main issue addressed. In vectorized tests on a single processor of the Cray C-90, the performance of Newton-OSOmin is superior to Newton-GMRES and a more traditional monotone AF/ADI method (MAF) for a variety of transonic Mach numbers and mesh sizes. Newton- GIVIRES is superior to MAF for some cases. The parallel performance of the Newton method is also found to be very good on multiple processors of the Cray C-90 and on the massively parallel thinking machine CM-5, where very fast execution rates (up to 9 Gflops) are found for large problems. 38 refs., 14 figs., 7 tabs.« less

Smoothing spline ANOVA frailty model for recurrent event data.

PubMed

Du, Pang; Jiang, Yihua; Wang, Yuedong

2011-12-01

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data. © 2011, The International Biometric Society.

Analysis of a closed-kinematic chain robot manipulator

NASA Technical Reports Server (NTRS)

Nguyen, Charles C.; Pooran, Farhad J.

1988-01-01

Presented are the research results from the research grant entitled: Active Control of Robot Manipulators, sponsored by the Goddard Space Flight Center (NASA) under grant number NAG-780. This report considers a class of robot manipulators based on the closed-kinematic chain mechanism (CKCM). This type of robot manipulators mainly consists of two platforms, one is stationary and the other moving, and they are coupled together through a number of in-parallel actuators. Using spatial geometry and homogeneous transformation, a closed-form solution is derived for the inverse kinematic problem of the six-degree-of-freedom manipulator, built to study robotic assembly in space. Iterative Newton Raphson method is employed to solve the forward kinematic problem. Finally, the equations of motion of the above manipulators are obtained by employing the Lagrangian method. Study of the manipulator dynamics is performed using computer simulation whose results show that the robot actuating forces are strongly dependent on the mass and centroid locations of the robot links.

Linear homotopy solution of nonlinear systems of equations in geodesy

NASA Astrophysics Data System (ADS)

Paláncz, Béla; Awange, Joseph L.; Zaletnyik, Piroska; Lewis, Robert H.

2010-01-01

A fundamental task in geodesy is solving systems of equations. Many geodetic problems are represented as systems of multivariate polynomials. A common problem in solving such systems is improper initial starting values for iterative methods, leading to convergence to solutions with no physical meaning, or to convergence that requires global methods. Though symbolic methods such as Groebner bases or resultants have been shown to be very efficient, i.e., providing solutions for determined systems such as 3-point problem of 3D affine transformation, the symbolic algebra can be very time consuming, even with special Computer Algebra Systems (CAS). This study proposes the Linear Homotopy method that can be implemented easily in high-level computer languages like C++ and Fortran that are faster than CAS by at least two orders of magnitude. Using Mathematica, the power of Homotopy is demonstrated in solving three nonlinear geodetic problems: resection, GPS positioning, and affine transformation. The method enlarging the domain of convergence is found to be efficient, less sensitive to rounding of numbers, and has lower complexity compared to other local methods like Newton-Raphson.

A New Newton-Like Iterative Method for Roots of Analytic Functions

ERIC Educational Resources Information Center

Otolorin, Olayiwola

2005-01-01

A new Newton-like iterative formula for the solution of non-linear equations is proposed. To derive the formula, the convergence criteria of the one-parameter iteration formula, and also the quasilinearization in the derivation of Newton's formula are reviewed. The result is a new formula which eliminates the limitations of other methods. There is…

Efficient Iterative Methods Applied to the Solution of Transonic Flows

NASA Astrophysics Data System (ADS)

Wissink, Andrew M.; Lyrintzis, Anastasios S.; Chronopoulos, Anthony T.

1996-02-01

We investigate the use of an inexact Newton's method to solve the potential equations in the transonic regime. As a test case, we solve the two-dimensional steady transonic small disturbance equation. Approximate factorization/ADI techniques have traditionally been employed for implicit solutions of this nonlinear equation. Instead, we apply Newton's method using an exact analytical determination of the Jacobian with preconditioned conjugate gradient-like iterative solvers for solution of the linear systems in each Newton iteration. Two iterative solvers are tested; a block s-step version of the classical Orthomin(k) algorithm called orthogonal s-step Orthomin (OSOmin) and the well-known GMRES method. The preconditioner is a vectorizable and parallelizable version of incomplete LU (ILU) factorization. Efficiency of the Newton-Iterative method on vector and parallel computer architectures is the main issue addressed. In vectorized tests on a single processor of the Cray C-90, the performance of Newton-OSOmin is superior to Newton-GMRES and a more traditional monotone AF/ADI method (MAF) for a variety of transonic Mach numbers and mesh sizes. Newton-GMRES is superior to MAF for some cases. The parallel performance of the Newton method is also found to be very good on multiple processors of the Cray C-90 and on the massively parallel thinking machine CM-5, where very fast execution rates (up to 9 Gflops) are found for large problems.

High-order Newton-penalty algorithms

NASA Astrophysics Data System (ADS)

Dussault, Jean-Pierre

2005-10-01

Recent efforts in differentiable non-linear programming have been focused on interior point methods, akin to penalty and barrier algorithms. In this paper, we address the classical equality constrained program solved using the simple quadratic loss penalty function/algorithm. The suggestion to use extrapolations to track the differentiable trajectory associated with penalized subproblems goes back to the classic monograph of Fiacco & McCormick. This idea was further developed by Gould who obtained a two-steps quadratically convergent algorithm using prediction steps and Newton correction. Dussault interpreted the prediction step as a combined extrapolation with respect to the penalty parameter and the residual of the first order optimality conditions. Extrapolation with respect to the residual coincides with a Newton step.We explore here higher-order extrapolations, thus higher-order Newton-like methods. We first consider high-order variants of the Newton-Raphson method applied to non-linear systems of equations. Next, we obtain improved asymptotic convergence results for the quadratic loss penalty algorithm by using high-order extrapolation steps.

A novel speckle pattern—Adaptive digital image correlation approach with robust strain calculation

NASA Astrophysics Data System (ADS)

Cofaru, Corneliu; Philips, Wilfried; Van Paepegem, Wim

2012-02-01

Digital image correlation (DIC) has seen widespread acceptance and usage as a non-contact method for the determination of full-field displacements and strains in experimental mechanics. The advances of imaging hardware in the last decades led to high resolution and speed cameras being more affordable than in the past making large amounts of data image available for typical DIC experimental scenarios. The work presented in this paper is aimed at maximizing both the accuracy and speed of DIC methods when employed with such images. A low-level framework for speckle image partitioning which replaces regularly shaped blocks with image-adaptive cells in the displacement calculation is introduced. The Newton-Raphson DIC method is modified to use the image pixels of the cells and to perform adaptive regularization to increase the spatial consistency of the displacements. Furthermore, a novel robust framework for strain calculation based also on the Newton-Raphson algorithm is introduced. The proposed methods are evaluated in five experimental scenarios, out of which four use numerically deformed images and one uses real experimental data. Results indicate that, as the desired strain density increases, significant computational gains can be obtained while maintaining or improving accuracy and rigid-body rotation sensitivity.

A Differential Evolution Based Approach to Estimate the Shape and Size of Complex Shaped Anomalies Using EIT Measurements

NASA Astrophysics Data System (ADS)

Rashid, Ahmar; Khambampati, Anil Kumar; Kim, Bong Seok; Liu, Dong; Kim, Sin; Kim, Kyung Youn

EIT image reconstruction is an ill-posed problem, the spatial resolution of the estimated conductivity distribution is usually poor and the external voltage measurements are subject to variable noise. Therefore, EIT conductivity estimation cannot be used in the raw form to correctly estimate the shape and size of complex shaped regional anomalies. An efficient algorithm employing a shape based estimation scheme is needed. The performance of traditional inverse algorithms, such as the Newton Raphson method, used for this purpose is below par and depends upon the initial guess and the gradient of the cost functional. This paper presents the application of differential evolution (DE) algorithm to estimate complex shaped region boundaries, expressed as coefficients of truncated Fourier series, using EIT. DE is a simple yet powerful population-based, heuristic algorithm with the desired features to solve global optimization problems under realistic conditions. The performance of the algorithm has been tested through numerical simulations, comparing its results with that of the traditional modified Newton Raphson (mNR) method.

Bayesian Atmospheric Radiative Transfer (BART)Thermochemical Equilibrium Abundance (TEA) Code and Application to WASP-43b

NASA Astrophysics Data System (ADS)

Blecic, Jasmina; Harrington, Joseph; Bowman, Matthew O.; Cubillos, Patricio E.; Stemm, Madison; Foster, Andrew

2014-11-01

We present a new, open-source, Thermochemical Equilibrium Abundances (TEA) code that calculates the abundances of gaseous molecular species. TEA uses the Gibbs-free-energy minimization method with an iterative Lagrangian optimization scheme. It initializes the radiative-transfer calculation in our Bayesian Atmospheric Radiative Transfer (BART) code. Given elemental abundances, TEA calculates molecular abundances for a particular temperature and pressure or a list of temperature-pressure pairs. The code is tested against the original method developed by White at al. (1958), the analytic method developed by Burrows and Sharp (1999), and the Newton-Raphson method implemented in the open-source Chemical Equilibrium with Applications (CEA) code. TEA is written in Python and is available to the community via the open-source development site GitHub.com. We also present BART applied to eclipse depths of WASP-43b exoplanet, constraining atmospheric thermal and chemical parameters. This work was supported by NASA Planetary Atmospheres grant NNX12AI69G and NASA Astrophysics Data Analysis Program grant NNX13AF38G. JB holds a NASA Earth and Space Science Fellowship.

New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lipnikov, Konstantin; Moulton, David; Svyatskiy, Daniil

2016-04-29

We develop a new approach for solving the nonlinear Richards’ equation arising in variably saturated flow modeling. The growing complexity of geometric models for simulation of subsurface flows leads to the necessity of using unstructured meshes and advanced discretization methods. Typically, a numerical solution is obtained by first discretizing PDEs and then solving the resulting system of nonlinear discrete equations with a Newton-Raphson-type method. Efficiency and robustness of the existing solvers rely on many factors, including an empiric quality control of intermediate iterates, complexity of the employed discretization method and a customized preconditioner. We propose and analyze a new preconditioningmore » strategy that is based on a stable discretization of the continuum Jacobian. We will show with numerical experiments for challenging problems in subsurface hydrology that this new preconditioner improves convergence of the existing Jacobian-free solvers 3-20 times. Furthermore, we show that the Picard method with this preconditioner becomes a more efficient nonlinear solver than a few widely used Jacobian-free solvers.« less

Airbreathing engine selection criteria for SSTO propulsion system

NASA Astrophysics Data System (ADS)

Ohkami, Yoshiaki; Maita, Masataka

1995-02-01

This paper presents airbreathing engine selection criteria to be applied to the propulsion system of a Single Stage To Orbit (SSTO). To establish the criteria, a relation among three major parameters, i.e., delta-V capability, weight penalty, and effective specific impulse of the engine subsystem, is derived as compared to these parameters of the LH2/LOX rocket engine. The effective specific impulse is a function of the engine I(sub sp) and vehicle thrust-to-drag ratio which is approximated by a function of the vehicle velocity. The weight penalty includes the engine dry weight, cooling subsystem weight. The delta-V capability is defined by the velocity region starting from the minimum operating velocity up to the maximum velocity. The vehicle feasibility is investigated in terms of the structural and propellant weights, which requires an iteration process adjusting the system parameters. The system parameters are computed by iteration based on the Newton-Raphson method. It has been concluded that performance in the higher velocity region is extremely important so that the airbreathing engines are required to operate beyond the velocity equivalent to the rocket engine exhaust velocity (approximately 4500 m/s).

Linear stability analysis of scramjet unstart

NASA Astrophysics Data System (ADS)

Jang, Ik; Nichols, Joseph; Moin, Parviz

2015-11-01

We investigate the bifurcation structure of unstart and restart events in a dual-mode scramjet using the Reynolds-averaged Navier-Stokes equations. The scramjet of interest (HyShot II, Laurence et al., AIAA2011-2310) operates at a free-stream Mach number of approximately 8, and the length of the combustor chamber is 300mm. A heat-release model is applied to mimic the combustion process. Pseudo-arclength continuation with Newton-Raphson iteration is used to calculate multiple solution branches. Stability analysis based on linearized dynamics about the solution curves reveals a metric that optimally forewarns unstart. By combining direct and adjoint eigenmodes, structural sensitivity analysis suggests strategies for unstart mitigation, including changing the isolator length. This work is supported by DOE/NNSA and AFOSR.

Commercial Non-Dispersive Infrared Spectroscopy Sensors for Sub-Ambient Carbon Dioxide Detection

NASA Technical Reports Server (NTRS)

Swickrath, Michael J.; Anderson, Molly S.; McMillin, Summer; Broerman, Craig

2013-01-01

Carbon dioxide produced through respiration can accumulate rapidly within closed spaces. If not managed, a crew's respiratory rate increases, headaches and hyperventilation occur, vision and hearing are affected, and cognitive abilities decrease. Consequently, development continues on a number of CO2 removal technologies for human spacecraft and spacesuits. Terrestrially, technology development requires precise performance characterization to qualify promising air revitalization equipment. On-orbit, instrumentation is required to identify and eliminate unsafe conditions. This necessitates accurate in situ CO2 detection. Recursive compensation algorithms were developed for sub-ambient detection of CO2 with commercial off-the-shelf (COTS) non-dispersive infrared (NDIR) sensors. In addition, the source of the exponential loss in accuracy is developed theoretically. The basis of the loss can be explained through thermal, Doppler, and Lorentz broadening effects that arise as a result of the temperature, pressure, and composition of the gas mixture under analysis. The objective was to develop a mathematical routine to compensate COTS CO2 sensors relying on NDIR over pressures, temperatures, and compositions far from calibration conditions. The routine relies on a power-law relationship for the pressure dependency of the sensors along with an equivalent pressure to account for the composition dependency. A Newton-Raphson iterative technique solves for actual carbon dioxide concentration based on the reported concentration. Moreover, first principles routines were established to predict mixed-gas spectra based on sensor specifications (e.g., optical path length). The first principles model can be used to parametrically optimize sensors or sensor arrays across a wide variety of pressures/temperatures/ compositions. In this work, heuristic scaling arguments were utilized to develop reasonable compensation techniques. Experimental results confirmed this approach and provided evidence that composition broadening significantly alters spectra when pressure is reduced. Consequently, a recursive compensation technique was developed with the Newton-Raphson method, which was subsequently verified through experimentation.

Optimization design combined with coupled structural-electrostatic analysis for the electrostatically controlled deployable membrane reflector

NASA Astrophysics Data System (ADS)

Liu, Chao; Yang, Guigeng; Zhang, Yiqun

2015-01-01

The electrostatically controlled deployable membrane reflector (ECDMR) is a promising scheme to construct large size and high precision space deployable reflector antennas. This paper presents a novel design method for the large size and small F/D ECDMR considering the coupled structure-electrostatic problem. First, the fully coupled structural-electrostatic system is described by a three field formulation, in which the structure and passive electrical field is modeled by finite element method, and the deformation of the electrostatic domain is predicted by a finite element formulation of a fictitious elastic structure. A residual formulation of the structural-electrostatic field finite element model is established and solved by Newton-Raphson method. The coupled structural-electrostatic analysis procedure is summarized. Then, with the aid of this coupled analysis procedure, an integrated optimization method of membrane shape accuracy and stress uniformity is proposed, which is divided into inner and outer iterative loops. The initial state of relatively high shape accuracy and uniform stress distribution is achieved by applying the uniform prestress on the membrane design shape and optimizing the voltages, in which the optimal voltage is computed by a sensitivity analysis. The shape accuracy is further improved by the iterative prestress modification using the reposition balance method. Finally, the results of the uncoupled and coupled methods are compared and the proposed optimization method is applied to design an ECDMR. The results validate the effectiveness of this proposed methods.

Buckling and limit states of composite profiles with top-hat channel section subjected to axial compression

NASA Astrophysics Data System (ADS)

RóŻyło, Patryk; Debski, Hubert; Kral, Jan

2018-01-01

The subject of the research was a short thin-walled top-hat cross-section composite profile. The tested structure was subjected to axial compression. As part of the critical state research, critical load and the corresponding buckling mode was determined. Later in the study laminate damage areas were determined throughout numerical analysis. It was assumed that the profile is simply supported on the cross sections ends. Experimental tests were carried out on a universal testing machine Zwick Z100 and the results were compared with the results of numerical calculations. The eigenvalue problem and a non-linear problem of stability of thin-walled structures were carried out by the use of commercial software ABAQUS®. In the presented cases, it was assumed that the material is linear-elastic and non-linearity of the model results from the large displacements. Solution to the geometrically nonlinear problem was conducted by the use of the incremental-iterative Newton-Raphson method.

Inverse solutions for electrical impedance tomography based on conjugate gradients methods

NASA Astrophysics Data System (ADS)

Wang, M.

2002-01-01

A multistep inverse solution for two-dimensional electric field distribution is developed to deal with the nonlinear inverse problem of electric field distribution in relation to its boundary condition and the problem of divergence due to errors introduced by the ill-conditioned sensitivity matrix and the noise produced by electrode modelling and instruments. This solution is based on a normalized linear approximation method where the change in mutual impedance is derived from the sensitivity theorem and a method of error vector decomposition. This paper presents an algebraic solution of the linear equations at each inverse step, using a generalized conjugate gradients method. Limiting the number of iterations in the generalized conjugate gradients method controls the artificial errors introduced by the assumption of linearity and the ill-conditioned sensitivity matrix. The solution of the nonlinear problem is approached using a multistep inversion. This paper also reviews the mathematical and physical definitions of the sensitivity back-projection algorithm based on the sensitivity theorem. Simulations and discussion based on the multistep algorithm, the sensitivity coefficient back-projection method and the Newton-Raphson method are given. Examples of imaging gas-liquid mixing and a human hand in brine are presented.

Algorithms for accelerated convergence of adaptive PCA.

PubMed

Chatterjee, C; Kang, Z; Roychowdhury, V P

2000-01-01

We derive and discuss new adaptive algorithms for principal component analysis (PCA) that are shown to converge faster than the traditional PCA algorithms due to Oja, Sanger, and Xu. It is well known that traditional PCA algorithms that are derived by using gradient descent on an objective function are slow to converge. Furthermore, the convergence of these algorithms depends on appropriate choices of the gain sequences. Since online applications demand faster convergence and an automatic selection of gains, we present new adaptive algorithms to solve these problems. We first present an unconstrained objective function, which can be minimized to obtain the principal components. We derive adaptive algorithms from this objective function by using: 1) gradient descent; 2) steepest descent; 3) conjugate direction; and 4) Newton-Raphson methods. Although gradient descent produces Xu's LMSER algorithm, the steepest descent, conjugate direction, and Newton-Raphson methods produce new adaptive algorithms for PCA. We also provide a discussion on the landscape of the objective function, and present a global convergence proof of the adaptive gradient descent PCA algorithm using stochastic approximation theory. Extensive experiments with stationary and nonstationary multidimensional Gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods.We also compare the steepest descent adaptive algorithm with state-of-the-art methods on stationary and nonstationary sequences.

Nonlinear study of the parallel velocity/tearing instability using an implicit, nonlinear resistive MHD solver

NASA Astrophysics Data System (ADS)

Chacon, L.; Finn, J. M.; Knoll, D. A.

2000-10-01

Recently, a new parallel velocity instability has been found.(J. M. Finn, Phys. Plasmas), 2, 12 (1995) This mode is a tearing mode driven unstable by curvature effects and sound wave coupling in the presence of parallel velocity shear. Under such conditions, linear theory predicts that tearing instabilities will grow even in situations in which the classical tearing mode is stable. This could then be a viable seed mechanism for the neoclassical tearing mode, and hence a non-linear study is of interest. Here, the linear and non-linear stages of this instability are explored using a fully implicit, fully nonlinear 2D reduced resistive MHD code,(L. Chacon et al), ``Implicit, Jacobian-free Newton-Krylov 2D reduced resistive MHD nonlinear solver,'' submitted to J. Comput. Phys. (2000) including viscosity and particle transport effects. The nonlinear implicit time integration is performed using the Newton-Raphson iterative algorithm. Krylov iterative techniques are employed for the required algebraic matrix inversions, implemented Jacobian-free (i.e., without ever forming and storing the Jacobian matrix), and preconditioned with a ``physics-based'' preconditioner. Nonlinear results indicate that, for large total plasma beta and large parallel velocity shear, the instability results in the generation of large poloidal shear flows and large magnetic islands even in regimes when the classical tearing mode is absolutely stable. For small viscosity, the time asymptotic state can be turbulent.

Linear and nonlinear dynamic analysis of redundant load path bearingless rotor systems

NASA Technical Reports Server (NTRS)

Murthy, V. R.; Shultz, Louis A.

1994-01-01

The goal of this research is to develop the transfer matrix method to treat nonlinear autonomous boundary value problems with multiple branches. The application is the complete nonlinear aeroelastic analysis of multiple-branched rotor blades. Once the development is complete, it can be incorporated into the existing transfer matrix analyses. There are several difficulties to be overcome in reaching this objective. The conventional transfer matrix method is limited in that it is applicable only to linear branch chain-like structures, but consideration of multiple branch modeling is important for bearingless rotors. Also, hingeless and bearingless rotor blade dynamic characteristics (particularly their aeroelasticity problems) are inherently nonlinear. The nonlinear equations of motion and the multiple-branched boundary value problem are treated together using a direct transfer matrix method. First, the formulation is applied to a nonlinear single-branch blade to validate the nonlinear portion of the formulation. The nonlinear system of equations is iteratively solved using a form of Newton-Raphson iteration scheme developed for differential equations of continuous systems. The formulation is then applied to determine the nonlinear steady state trim and aeroelastic stability of a rotor blade in hover with two branches at the root. A comprehensive computer program is developed and is used to obtain numerical results for the (1) free vibration, (2) nonlinearly deformed steady state, (3) free vibration about the nonlinearly deformed steady state, and (4) aeroelastic stability tasks. The numerical results obtained by the present method agree with results from other methods.

Logistic regression for circular data

NASA Astrophysics Data System (ADS)

Al-Daffaie, Kadhem; Khan, Shahjahan

2017-05-01

This paper considers the relationship between a binary response and a circular predictor. It develops the logistic regression model by employing the linear-circular regression approach. The maximum likelihood method is used to estimate the parameters. The Newton-Raphson numerical method is used to find the estimated values of the parameters. A data set from weather records of Toowoomba city is analysed by the proposed methods. Moreover, a simulation study is considered. The R software is used for all computations and simulations.

Results of a feasibility study using the Newton-Raphson digital computer program to identify lifting body derivatives from flight data

NASA Technical Reports Server (NTRS)

Sim, A. G.

1973-01-01

A brief study was made to assess the applicability of the Newton-Raphson digital computer program as a routine technique for extracting aerodynamic derivatives from flight tests of lifting body types of vehicles. Lateral-direction flight data from flight tests of the HL-10 lifting body reserch vehicle were utilized. The results in general, show the computer program to be a reliable and expedient means for extracting derivatives for this class of vehicles as a standard procedure. This result was true even when stability augmentation was used. As a result of the study, a credible set of HL-10 lateral-directional derivatives was obtained from flight data. These derivatives are compared with results from wind-tunnel tests.

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

Numerical and Experimental Investigation of Stratified Gas-Liquid Two-Phase Flow in Horizontal Circular Pipes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Faccini, J.L.H.; Sampaio, P.A.B. de; Su, J.

This paper reports numerical and experimental investigation of stratified gas-liquid two-phase flow in horizontal circular pipes. The Reynolds averaged Navier Stokes equations (RANS) with the k-{omega} model for a fully developed stratified gas-liquid two-phase flow are solved by using the finite element method. A smooth and horizontal interface surface is assumed without considering the interfacial waves. The continuity of the shear stress across the interface is enforced with the continuity of the velocity being automatically satisfied by the variational formulation. For each given interface position and longitudinal pressure gradient, an inner iteration loop runs to solve the nonlinear equations. Themore » Newton-Raphson scheme is used to solve the transcendental equations by an outer iteration to determine the interface position and pressure gradient for a given pair of volumetric flow rates. The interface position in a 51.2 mm ID circular pipe was measured experimentally by the ultrasonic pulse-echo technique. The numerical results were also compared with experimental results in a 21 mm ID circular pipe reported by Masala [1]. The good agreement between the numerical and experimental results indicates that the k-{omega} model can be applied for the numerical simulation of stratified gas-liquid two-phase flow. (authors)« less

Application of Four-Point Newton-EGSOR iteration for the numerical solution of 2D Porous Medium Equations

NASA Astrophysics Data System (ADS)

Chew, J. V. L.; Sulaiman, J.

2017-09-01

Partial differential equations that are used in describing the nonlinear heat and mass transfer phenomena are difficult to be solved. For the case where the exact solution is difficult to be obtained, it is necessary to use a numerical procedure such as the finite difference method to solve a particular partial differential equation. In term of numerical procedure, a particular method can be considered as an efficient method if the method can give an approximate solution within the specified error with the least computational complexity. Throughout this paper, the two-dimensional Porous Medium Equation (2D PME) is discretized by using the implicit finite difference scheme to construct the corresponding approximation equation. Then this approximation equation yields a large-sized and sparse nonlinear system. By using the Newton method to linearize the nonlinear system, this paper deals with the application of the Four-Point Newton-EGSOR (4NEGSOR) iterative method for solving the 2D PMEs. In addition to that, the efficiency of the 4NEGSOR iterative method is studied by solving three examples of the problems. Based on the comparative analysis, the Newton-Gauss-Seidel (NGS) and the Newton-SOR (NSOR) iterative methods are also considered. The numerical findings show that the 4NEGSOR method is superior to the NGS and the NSOR methods in terms of the number of iterations to get the converged solutions, the time of computation and the maximum absolute errors produced by the methods.

Survey on the Performance of Source Localization Algorithms.

PubMed

Fresno, José Manuel; Robles, Guillermo; Martínez-Tarifa, Juan Manuel; Stewart, Brian G

2017-11-18

The localization of emitters using an array of sensors or antennas is a prevalent issue approached in several applications. There exist different techniques for source localization, which can be classified into multilateration, received signal strength (RSS) and proximity methods. The performance of multilateration techniques relies on measured time variables: the time of flight (ToF) of the emission from the emitter to the sensor, the time differences of arrival (TDoA) of the emission between sensors and the pseudo-time of flight (pToF) of the emission to the sensors. The multilateration algorithms presented and compared in this paper can be classified as iterative and non-iterative methods. Both standard least squares (SLS) and hyperbolic least squares (HLS) are iterative and based on the Newton-Raphson technique to solve the non-linear equation system. The metaheuristic technique particle swarm optimization (PSO) used for source localisation is also studied. This optimization technique estimates the source position as the optimum of an objective function based on HLS and is also iterative in nature. Three non-iterative algorithms, namely the hyperbolic positioning algorithms (HPA), the maximum likelihood estimator (MLE) and Bancroft algorithm, are also presented. A non-iterative combined algorithm, MLE-HLS, based on MLE and HLS, is further proposed in this paper. The performance of all algorithms is analysed and compared in terms of accuracy in the localization of the position of the emitter and in terms of computational time. The analysis is also undertaken with three different sensor layouts since the positions of the sensors affect the localization; several source positions are also evaluated to make the comparison more robust. The analysis is carried out using theoretical time differences, as well as including errors due to the effect of digital sampling of the time variables. It is shown that the most balanced algorithm, yielding better results than the other algorithms in terms of accuracy and short computational time, is the combined MLE-HLS algorithm.

Deployment of a multi-link flexible structure

NASA Astrophysics Data System (ADS)

Na, Kyung-Su; Kim, Ji-Hwan

2006-06-01

Deployment of a multi-link beam structure undergoing locking is analyzed in the Timoshenko beam theory. In the modeling of the system, dynamic forces are assumed to be torques and restoring forces due to the torsion spring at each joint. Hamilton's principle is used to determine the equations of motion and the finite element method is adopted to analyze the system. Newmark time integration and Newton-Raphson iteration methods are used to solve for the non-linear equations of motion at each time step. The locking at the joints of the multi-link flexible structure is analyzed by the momentum balance method. Numerical results are compared with the previous experimental data. The angles and angular velocities of each joint, tip displacement, and velocity of each link are investigated to study the motions of the links at each time step. To analyze the effect of thickness on the motion of the link, the angle and the tip displacement of each link are compared according to the various slenderness ratios. Additionally, in order to investigate the effect of shear, the tip displacements of a Timoshenko beam are compared with those of an Euler-Bernoulli beam.

Mixed formulation for frictionless contact problems

NASA Technical Reports Server (NTRS)

Noor, Ahmed K.; Kim, Kyun O.

1989-01-01

Simple mixed finite element models and a computational precedure are presented for the solution of frictionless contact problems. The analytical formulation is based on a form of Reissner's large rotation theory of the structure with the effects of transverse shear deformation included. The contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the internal forces (stress resultants), the generalized displacements, and the Lagrange multipliers associated with the contact conditions. The element characteristic array are obtained by using a modified form of the two-field Hellinger-Reissner mixed variational principle. The internal forces and the Lagrange multipliers are allowed to be discontinuous at interelement boundaries. The Newton-Raphson iterative scheme is used for the solution of the nonlinear algebraic equations, and the determination of the contact area and the contact pressures.

Computing maximum-likelihood estimates for parameters of the National Descriptive Model of Mercury in Fish

USGS Publications Warehouse

Donato, David I.

2012-01-01

This report presents the mathematical expressions and the computational techniques required to compute maximum-likelihood estimates for the parameters of the National Descriptive Model of Mercury in Fish (NDMMF), a statistical model used to predict the concentration of methylmercury in fish tissue. The expressions and techniques reported here were prepared to support the development of custom software capable of computing NDMMF parameter estimates more quickly and using less computer memory than is currently possible with available general-purpose statistical software. Computation of maximum-likelihood estimates for the NDMMF by numerical solution of a system of simultaneous equations through repeated Newton-Raphson iterations is described. This report explains the derivation of the mathematical expressions required for computational parameter estimation in sufficient detail to facilitate future derivations for any revised versions of the NDMMF that may be developed.

Chaos, Fractals, and Polynomials.

ERIC Educational Resources Information Center

Tylee, J. Louis; Tylee, Thomas B.

1996-01-01

Discusses chaos theory; linear algebraic equations and the numerical solution of polynomials, including the use of the Newton-Raphson technique to find polynomial roots; fractals; search region and coordinate systems; convergence; and generating color fractals on a computer. (LRW)

A comparison of several methods of solving nonlinear regression groundwater flow problems

USGS Publications Warehouse

Cooley, Richard L.

1985-01-01

Computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear-regression steady state groundwater flow problems. The fastest methods were the Marquardt and quasi-linearization methods, which required almost identical computer times and numbers of iterations; the next fastest was the quasi-Newton method, and last was the Fletcher-Reeves method, which did not converge in 100 iterations for two of the problems. The fastest method per iteration was the Fletcher-Reeves method, and this was followed closely by the quasi-Newton method. The Marquardt and quasi-linearization methods were slower. For all four methods the speed per iteration was directly related to the number of parameters in the model. However, this effect was much more pronounced for the Marquardt and quasi-linearization methods than for the other two. Hence the quasi-Newton (and perhaps Fletcher-Reeves) method might be more efficient than either the Marquardt or quasi-linearization methods if the number of parameters in a particular model were large, although this remains to be proven. The Marquardt method required somewhat less central memory than the quasi-linearization metilod for three of the four problems. For all four problems the quasi-Newton method required roughly two thirds to three quarters of the memory required by the Marquardt method, and the Fletcher-Reeves method required slightly less memory than the quasi-Newton method. Memory requirements were not excessive for any of the four methods.

Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination

NASA Astrophysics Data System (ADS)

Citro, V.; Luchini, P.; Giannetti, F.; Auteri, F.

2017-09-01

The study of the stability of a dynamical system described by a set of partial differential equations (PDEs) requires the computation of unstable states as the control parameter exceeds its critical threshold. Unfortunately, the discretization of the governing equations, especially for fluid dynamic applications, often leads to very large discrete systems. As a consequence, matrix based methods, like for example the Newton-Raphson algorithm coupled with a direct inversion of the Jacobian matrix, lead to computational costs too large in terms of both memory and execution time. We present a novel iterative algorithm, inspired by Krylov-subspace methods, which is able to compute unstable steady states and/or accelerate the convergence to stable configurations. Our new algorithm is based on the minimization of the residual norm at each iteration step with a projection basis updated at each iteration rather than at periodic restarts like in the classical GMRES method. The algorithm is able to stabilize any dynamical system without increasing the computational time of the original numerical procedure used to solve the governing equations. Moreover, it can be easily inserted into a pre-existing relaxation (integration) procedure with a call to a single black-box subroutine. The procedure is discussed for problems of different sizes, ranging from a small two-dimensional system to a large three-dimensional problem involving the Navier-Stokes equations. We show that the proposed algorithm is able to improve the convergence of existing iterative schemes. In particular, the procedure is applied to the subcritical flow inside a lid-driven cavity. We also discuss the application of Boostconv to compute the unstable steady flow past a fixed circular cylinder (2D) and boundary-layer flow over a hemispherical roughness element (3D) for supercritical values of the Reynolds number. We show that Boostconv can be used effectively with any spatial discretization, be it a finite-difference, finite-volume, finite-element or spectral method.

Testing of FTS fingers and interface using a passive compliant robot manipulator. [flight telerobot servicer

NASA Technical Reports Server (NTRS)

Nguyen, Charles C.; Antrazi, Sami S.

1992-01-01

This report deals with testing of a pair of robot fingers designed for the Flight Telerobotic Servicer (FTS) to grasp a cylinder type of Orbital Replaceable Unit (ORU) interface. The report first describes the objectives of the study and then the testbed consisting of a Stewart Platform-based manipulator equipped with a passive compliant platform which also serves as a force/torque sensor. Kinematic analysis is then performed to provide a closed-form solution for the force inverse kinematics and iterative solution for the force forward kinematics using the Newton's Raphson Method. Mathematical expressions are then derived to compute force/torques applied to the FTS fingers during the mating/demating with the interface. The report then presents the three parts of the experimental study on the feasibility and characteristics of the fingers. The first part obtains data of forces applied by the fingers to the interface under various misalignments, the second part determines the maximum allowable capture angles for mating, and the third part processes and interprets the obtained force/torque data.

A nonlinear dynamic finite element approach for simulating muscular hydrostats.

PubMed

Vavourakis, V; Kazakidi, A; Tsakiris, D P; Ekaterinaris, J A

2014-01-01

An implicit nonlinear finite element model for simulating biological muscle mechanics is developed. The numerical method is suitable for dynamic simulations of three-dimensional, nonlinear, nearly incompressible, hyperelastic materials that undergo large deformations. These features characterise biological muscles, which consist of fibres and connective tissues. It can be assumed that the stress distribution inside the muscles is the superposition of stresses along the fibres and the connective tissues. The mechanical behaviour of the surrounding tissues is determined by adopting a Mooney-Rivlin constitutive model, while the mechanical description of fibres is considered to be the sum of active and passive stresses. Due to the nonlinear nature of the problem, evaluation of the Jacobian matrix is carried out in order to subsequently utilise the standard Newton-Raphson iterative procedure and to carry out time integration with an implicit scheme. The proposed methodology is implemented into our in-house, open source, finite element software, which is validated by comparing numerical results with experimental measurements and other numerical results. Finally, the numerical procedure is utilised to simulate primitive octopus arm manoeuvres, such as bending and reaching.

Convergence of Newton's method for a single real equation

NASA Technical Reports Server (NTRS)

Campbell, C. W.

1985-01-01

Newton's method for finding the zeroes of a single real function is investigated in some detail. Convergence is generally checked using the Contraction Mapping Theorem which yields sufficient but not necessary conditions for convergence of the general single point iteration method. The resulting convergence intervals are frequently considerably smaller than actual convergence zones. For a specific single point iteration method, such as Newton's method, better estimates of regions of convergence should be possible. A technique is described which, under certain conditions (frequently satisfied by well behaved functions) gives much larger zones where convergence is guaranteed.

An approximate block Newton method for coupled iterations of nonlinear solvers: Theory and conjugate heat transfer applications

NASA Astrophysics Data System (ADS)

Yeckel, Andrew; Lun, Lisa; Derby, Jeffrey J.

2009-12-01

A new, approximate block Newton (ABN) method is derived and tested for the coupled solution of nonlinear models, each of which is treated as a modular, black box. Such an approach is motivated by a desire to maintain software flexibility without sacrificing solution efficiency or robustness. Though block Newton methods of similar type have been proposed and studied, we present a unique derivation and use it to sort out some of the more confusing points in the literature. In particular, we show that our ABN method behaves like a Newton iteration preconditioned by an inexact Newton solver derived from subproblem Jacobians. The method is demonstrated on several conjugate heat transfer problems modeled after melt crystal growth processes. These problems are represented by partitioned spatial regions, each modeled by independent heat transfer codes and linked by temperature and flux matching conditions at the boundaries common to the partitions. Whereas a typical block Gauss-Seidel iteration fails about half the time for the model problem, quadratic convergence is achieved by the ABN method under all conditions studied here. Additional performance advantages over existing methods are demonstrated and discussed.

Characterization and Implementation of a Real-World Target Tracking Algorithm on Field Programmable Gate Arrays with Kalman Filter Test Case

DTIC Science & Technology

2008-03-01

to predict its exact position. To locate Ceres, Carl Friedrich Gauss , a mere 24 years old at the time, developed a method called least-squares...dividend to produce the quotient. This method converges to the reciprocal quadratically [11]. For the special case of: 1 H × P (:, :, k)×H ′ + R (3.9) the...high-speed computation of reciprocals within the overall system. The Newton-Raphson method is also expanded for use in calculat- ing square-roots in

Robust iterative method for nonlinear Helmholtz equation

NASA Astrophysics Data System (ADS)

Yuan, Lijun; Lu, Ya Yan

2017-08-01

A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr nonlinearity. In the strongly nonlinear regime, the nonlinear Helmholtz equation could have multiple solutions related to phenomena such as optical bistability and symmetry breaking. The new method exhibits a much more robust convergence behavior than existing iterative methods, such as frozen-nonlinearity iteration, Newton's method and damped Newton's method, and it can be used to find solutions when good initial guesses are unavailable. Numerical results are presented for the scattering of light by a nonlinear circular cylinder based on the exact nonlocal boundary condition and a pseudospectral method in the polar coordinate system.

Spectral-luminosity evolution of active galactic nuclei (AGN)

NASA Technical Reports Server (NTRS)

Leiter, Darryl; Boldt, Elihu

1992-01-01

The origin of the cosmic X-ray and gamma-ray backgrounds is explained via the mechanism of AGN spectral-luminosity evolution. The spectral evolution of precursor active galaxies into AGN, and Newton-Raphson input and output parameters are discussed.

SINFAC - SYSTEMS IMPROVED NUMERICAL FLUIDS ANALYSIS CODE

NASA Technical Reports Server (NTRS)

Costello, F. A.

1994-01-01

The Systems Improved Numerical Fluids Analysis Code, SINFAC, consists of additional routines added to the April 1983 revision of SINDA, a general thermal analyzer program. The purpose of the additional routines is to allow for the modeling of active heat transfer loops. The modeler can simulate the steady-state and pseudo-transient operations of 16 different heat transfer loop components including radiators, evaporators, condensers, mechanical pumps, reservoirs and many types of valves and fittings. In addition, the program contains a property analysis routine that can be used to compute the thermodynamic properties of 20 different refrigerants. SINFAC can simulate the response to transient boundary conditions. SINFAC was first developed as a method for computing the steady-state performance of two phase systems. It was then modified using CNFRWD, SINDA's explicit time-integration scheme, to accommodate transient thermal models. However, SINFAC cannot simulate pressure drops due to time-dependent fluid acceleration, transient boil-out, or transient fill-up, except in the accumulator. SINFAC also requires the user to be familiar with SINDA. The solution procedure used by SINFAC is similar to that which an engineer would use to solve a system manually. The solution to a system requires the determination of all of the outlet conditions of each component such as the flow rate, pressure, and enthalpy. To obtain these values, the user first estimates the inlet conditions to the first component of the system, then computes the outlet conditions from the data supplied by the manufacturer of the first component. The user then estimates the temperature at the outlet of the third component and computes the corresponding flow resistance of the second component. With the flow resistance of the second component, the user computes the conditions down stream, namely the inlet conditions of the third. The computations follow for the rest of the system, back to the first component. On the first pass, the user finds that the calculated outlet conditions of the last component do not match the estimated inlet conditions of the first. The user then modifies the estimated inlet conditions of the first component in an attempt to match the calculated values. The user estimated values are called State Variables. The differences between the user estimated values and calculated values are called the Error Variables. The procedure systematically changes the State Variables until all of the Error Variables are less than the user-specified iteration limits. The solution procedure is referred to as SCX. It consists of two phases, the Systems phase and the Controller phase. The X is to imply experimental. SCX computes each next set of State Variables in two phases. In the first phase, SCX fixes the controller positions and modifies the other State Variables by the Newton-Raphson method. This first phase is the Systems phase. Once the Newton-Raphson method has solved the problem for the fixed controller positions, SCX next calculates new controller positions based on Newton's method while treating each sensor-controller pair independently but allowing all to change in one iteration. This phase is the Controller phase. SINFAC is available by license for a period of ten (10) years to approved licensees. The licenced program product includes the source code for the additional routines to SINDA, the SINDA object code, command procedures, sample data and supporting documentation. Additional documentation may be purchased at the price below. SINFAC was created for use on a DEC VAX under VMS. Source code is written in FORTRAN 77, requires 180k of memory, and should be fully transportable. The program was developed in 1988.

A fast algorithm to compute precise type-2 centroids for real-time control applications.

PubMed

Chakraborty, Sumantra; Konar, Amit; Ralescu, Anca; Pal, Nikhil R

2015-02-01

An interval type-2 fuzzy set (IT2 FS) is characterized by its upper and lower membership functions containing all possible embedded fuzzy sets, which together is referred to as the footprint of uncertainty (FOU). The FOU results in a span of uncertainty measured in the defuzzified space and is determined by the positional difference of the centroids of all the embedded fuzzy sets taken together. This paper provides a closed-form formula to evaluate the span of uncertainty of an IT2 FS. The closed-form formula offers a precise measurement of the degree of uncertainty in an IT2 FS with a runtime complexity less than that of the classical iterative Karnik-Mendel algorithm and other formulations employing the iterative Newton-Raphson algorithm. This paper also demonstrates a real-time control application using the proposed closed-form formula of centroids with reduced root mean square error and computational overhead than those of the existing methods. Computer simulations for this real-time control application indicate that parallel realization of the IT2 defuzzification outperforms its competitors with respect to maximum overshoot even at high sampling rates. Furthermore, in the presence of measurement noise in system (plant) states, the proposed IT2 FS based scheme outperforms its type-1 counterpart with respect to peak overshoot and root mean square error in plant response.

A multigrid LU-SSOR scheme for approximate Newton iteration applied to the Euler equations

NASA Technical Reports Server (NTRS)

Yoon, Seokkwan; Jameson, Antony

1986-01-01

A new efficient relaxation scheme in conjunction with a multigrid method is developed for the Euler equations. The LU SSOR scheme is based on a central difference scheme and does not need flux splitting for Newton iteration. Application to transonic flow shows that the new method surpasses the performance of the LU implicit scheme.

An improved computer program for calculating the theoretical performance parameters of a propeller type wind turbine. An appendix to the final report on feasibility of using wind power to pump irrigation water (Texas). [PROP Code

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barieau, R.E.

1977-03-01

The PROP Program of Wilson and Lissaman has been modified by adding the Newton-Raphson Method and a Step Wise Search Method, as options for the method of solution. In addition, an optimization method is included. Twist angles, tip speed ratio and the pitch angle may be varied to produce maximum power coefficient. The computer program listing is presented along with sample input and output data. Further improvements to the program are discussed.

Cubic spline anchored grid pattern algorithm for high-resolution detection of subsurface cavities by the IR-CAT method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kassab, A.J.; Pollard, J.E.

An algorithm is presented for the high-resolution detection of irregular-shaped subsurface cavities within irregular-shaped bodies by the IR-CAT method. The theoretical basis of the algorithm is rooted in the solution of an inverse geometric steady-state heat conduction problem. A Cauchy boundary condition is prescribed at the exposed surface, and the inverse geometric heat conduction problem is formulated by specifying the thermal condition at the inner cavities walls, whose unknown geometries are to be detected. The location of the inner cavities is initially estimated, and the domain boundaries are discretized. Linear boundary elements are used in conjunction with cubic splines formore » high resolution of the cavity walls. An anchored grid pattern (AGP) is established to constrain the cubic spline knots that control the inner cavity geometry to evolve along the AGP at each iterative step. A residual is defined measuring the difference between imposed and computed boundary conditions. A Newton-Raphson method with a Broyden update is used to automate the detection of inner cavity walls. During the iterative procedure, the movement of the inner cavity walls is restricted to physically realistic intermediate solutions. Numerical simulation demonstrates the superior resolution of the cubic spline AGP algorithm over the linear spline-based AGP in the detection of an irregular-shaped cavity. Numerical simulation is also used to test the sensitivity of the linear and cubic spline AGP algorithms by simulating bias and random error in measured surface temperature. The proposed AGP algorithm is shown to satisfactorily detect cavities with these simulated data.« less

Improved Quasi-Newton method via PSB update for solving systems of nonlinear equations

NASA Astrophysics Data System (ADS)

Mamat, Mustafa; Dauda, M. K.; Waziri, M. Y.; Ahmad, Fadhilah; Mohamad, Fatma Susilawati

2016-10-01

The Newton method has some shortcomings which includes computation of the Jacobian matrix which may be difficult or even impossible to compute and solving the Newton system in every iteration. Also, the common setback with some quasi-Newton methods is that they need to compute and store an n × n matrix at each iteration, this is computationally costly for large scale problems. To overcome such drawbacks, an improved Method for solving systems of nonlinear equations via PSB (Powell-Symmetric-Broyden) update is proposed. In the proposed method, the approximate Jacobian inverse Hk of PSB is updated and its efficiency has improved thereby require low memory storage, hence the main aim of this paper. The preliminary numerical results show that the proposed method is practically efficient when applied on some benchmark problems.

Vibrations of an Euler-Bernoulli beam with hysteretic damping arising from dispersed frictional microcracks

NASA Astrophysics Data System (ADS)

Maiti, Soumyabrata; Bandyopadhyay, Ritwik; Chatterjee, Anindya

2018-01-01

We study free and harmonically forced vibrations of an Euler-Bernoulli beam with rate-independent hysteretic dissipation. The dissipation follows a model proposed elsewhere for materials with randomly dispersed frictional microcracks. The virtual work of distributed dissipative moments is approximated using Gaussian quadrature, yielding a few discrete internal hysteretic states. Lagrange's equations are obtained for the modal coordinates. Differential equations for the modal coordinates and internal states are integrated together. Free vibrations decay exponentially when a single mode dominates. With multiple modes active, higher modes initially decay rapidly while lower modes decay relatively slowly. Subsequently, lower modes show their own characteristic modal damping, while small amplitude higher modes show more erratic decay. Large dissipation, for the adopted model, leads mathematically to fast and damped oscillations in the limit, unlike viscously overdamped systems. Next, harmonically forced, lightly damped responses of the beam are studied using both a slow frequency sweep and a shooting-method based search for periodic solutions along with numerical continuation. Shooting method and frequency sweep results match for large ranges of frequency. The shooting method struggles near resonances, where internal states collapse into lower dimensional behavior and Newton-Raphson iterations fail. Near the primary resonances, simple numerically-aided harmonic balance gives excellent results. Insights are also obtained into the harmonic content of secondary resonances.

Computational aspects of helicopter trim analysis and damping levels from Floquet theory

NASA Technical Reports Server (NTRS)

Gaonkar, Gopal H.; Achar, N. S.

1992-01-01

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated.

A second-order unconstrained optimization method for canonical-ensemble density-functional methods

NASA Astrophysics Data System (ADS)

Nygaard, Cecilie R.; Olsen, Jeppe

2013-03-01

A second order converging method of ensemble optimization (SOEO) in the framework of Kohn-Sham Density-Functional Theory is presented, where the energy is minimized with respect to an ensemble density matrix. It is general in the sense that the number of fractionally occupied orbitals is not predefined, but rather it is optimized by the algorithm. SOEO is a second order Newton-Raphson method of optimization, where both the form of the orbitals and the occupation numbers are optimized simultaneously. To keep the occupation numbers between zero and two, a set of occupation angles is defined, from which the occupation numbers are expressed as trigonometric functions. The total number of electrons is controlled by a built-in second order restriction of the Newton-Raphson equations, which can be deactivated in the case of a grand-canonical ensemble (where the total number of electrons is allowed to change). To test the optimization method, dissociation curves for diatomic carbon are produced using different functionals for the exchange-correlation energy. These curves show that SOEO favors symmetry broken pure-state solutions when using functionals with exact exchange such as Hartree-Fock and Becke three-parameter Lee-Yang-Parr. This is explained by an unphysical contribution to the exact exchange energy from interactions between fractional occupations. For functionals without exact exchange, such as local density approximation or Becke Lee-Yang-Parr, ensemble solutions are favored at interatomic distances larger than the equilibrium distance. Calculations on the chromium dimer are also discussed. They show that SOEO is able to converge to ensemble solutions for systems that are more complicated than diatomic carbon.

Comparison of three newton-like nonlinear least-squares methods for estimating parameters of ground-water flow models

USGS Publications Warehouse

Cooley, R.L.; Hill, M.C.

1992-01-01

Three methods of solving nonlinear least-squares problems were compared for robustness and efficiency using a series of hypothetical and field problems. A modified Gauss-Newton/full Newton hybrid method (MGN/FN) and an analogous method for which part of the Hessian matrix was replaced by a quasi-Newton approximation (MGN/QN) solved some of the problems with appreciably fewer iterations than required using only a modified Gauss-Newton (MGN) method. In these problems, model nonlinearity and a large variance for the observed data apparently caused MGN to converge more slowly than MGN/FN or MGN/QN after the sum of squared errors had almost stabilized. Other problems were solved as efficiently with MGN as with MGN/FN or MGN/QN. Because MGN/FN can require significantly more computer time per iteration and more computer storage for transient problems, it is less attractive for a general purpose algorithm than MGN/QN.

A new Newton-like method for solving nonlinear equations.

PubMed

Saheya, B; Chen, Guo-Qing; Sui, Yun-Kang; Wu, Cai-Ying

2016-01-01

This paper presents an iterative scheme for solving nonline ar equations. We establish a new rational approximation model with linear numerator and denominator which has generalizes the local linear model. We then employ the new approximation for nonlinear equations and propose an improved Newton's method to solve it. The new method revises the Jacobian matrix by a rank one matrix each iteration and obtains the quadratic convergence property. The numerical performance and comparison show that the proposed method is efficient.

HICOV - Newton-Raphson calculus of variation with automatic transversalities

NASA Technical Reports Server (NTRS)

Heintschel, T. J.

1968-01-01

Computer program generates trajectories that are optimum with respect to payload placed in an earth orbit. It uses a subroutine package which produces the terminal and transversality conditions and their partial derivatives. This program is written in FORTRAN 4 and FORMAC for the IBM 7094 computer.

Impedance computed tomography using an adaptive smoothing coefficient algorithm.

PubMed

Suzuki, A; Uchiyama, A

2001-01-01

In impedance computed tomography, a fixed coefficient regularization algorithm has been frequently used to improve the ill-conditioning problem of the Newton-Raphson algorithm. However, a lot of experimental data and a long period of computation time are needed to determine a good smoothing coefficient because a good smoothing coefficient has to be manually chosen from a number of coefficients and is a constant for each iteration calculation. Thus, sometimes the fixed coefficient regularization algorithm distorts the information or fails to obtain any effect. In this paper, a new adaptive smoothing coefficient algorithm is proposed. This algorithm automatically calculates the smoothing coefficient from the eigenvalue of the ill-conditioned matrix. Therefore, the effective images can be obtained within a short computation time. Also the smoothing coefficient is automatically adjusted by the information related to the real resistivity distribution and the data collection method. In our impedance system, we have reconstructed the resistivity distributions of two phantoms using this algorithm. As a result, this algorithm only needs one-fifth the computation time compared to the fixed coefficient regularization algorithm. When compared to the fixed coefficient regularization algorithm, it shows that the image is obtained more rapidly and applicable in real-time monitoring of the blood vessel.

A numerical simulation of peristaltic motion in the ureter using fluid structure interactions.

PubMed

Vahidi, Bahman; Fatouraee, Nasser

2007-01-01

An axisymmetric model with fluid-structure interactions (FSI) is introduced and solved to perform ureter flow and stress analysis. The Navier-Stokes equations are solved for the fluid and a linear elastic model for ureter is used. The finite element equations for both the structure and the fluid were solved by the Newton-Raphson iterative method. Our results indicated that shear stresses were high around the throat of moving contracted wall. The pressure gradient magnitude along the ureter wall and the symmetry line had the maximum value around the throat of moving contracted wall which decreased as the peristalsis propagates toward the bladder. The flow rate at the ureter outlet at the end of the peristaltic motion was about 650 mm3/s. During propagation of the peristalsis toward the bladder, the inlet backward flow region was limited to the areas near symmetry line but the inner ureter backward flow regions extended to the whole ureter contraction part. The backward flow was vanished after 1.5 seconds of peristalsis propagation start up and after that time the urine flow was forward in the whole ureter length, so reflux is more probable to be present at the beginning of the wall peristaltic motion.

Development of iterative techniques for the solution of unsteady compressible viscous flows

NASA Technical Reports Server (NTRS)

Sankar, Lakshmi N.; Hixon, Duane

1991-01-01

Efficient iterative solution methods are being developed for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. Thus, the extra work required by iterative schemes can also be designed to perform efficiently on current and future generation scalable, missively parallel machines. An obvious candidate for iteratively solving the system of coupled nonlinear algebraic equations arising in CFD applications is the Newton method. Newton's method was implemented in existing finite difference and finite volume methods. Depending on the complexity of the problem, the number of Newton iterations needed per step to solve the discretized system of equations can, however, vary dramatically from a few to several hundred. Another popular approach based on the classical conjugate gradient method, known as the GMRES (Generalized Minimum Residual) algorithm is investigated. The GMRES algorithm was used in the past by a number of researchers for solving steady viscous and inviscid flow problems with considerable success. Here, the suitability of this algorithm is investigated for solving the system of nonlinear equations that arise in unsteady Navier-Stokes solvers at each time step. Unlike the Newton method which attempts to drive the error in the solution at each and every node down to zero, the GMRES algorithm only seeks to minimize the L2 norm of the error. In the GMRES algorithm the changes in the flow properties from one time step to the next are assumed to be the sum of a set of orthogonal vectors. By choosing the number of vectors to a reasonably small value N (between 5 and 20) the work required for advancing the solution from one time step to the next may be kept to (N+1) times that of a noniterative scheme. Many of the operations required by the GMRES algorithm such as matrix-vector multiplies, matrix additions and subtractions can all be vectorized and parallelized efficiently.

Study on the variable cycle engine modeling techniques based on the component method

NASA Astrophysics Data System (ADS)

Zhang, Lihua; Xue, Hui; Bao, Yuhai; Li, Jijun; Yan, Lan

2016-01-01

Based on the structure platform of the gas turbine engine, the components of variable cycle engine were simulated by using the component method. The mathematical model of nonlinear equations correspondeing to each component of the gas turbine engine was established. Based on Matlab programming, the nonlinear equations were solved by using Newton-Raphson steady-state algorithm, and the performance of the components for engine was calculated. The numerical simulation results showed that the model bulit can describe the basic performance of the gas turbine engine, which verified the validity of the model.

A Note on the Computation of the Second-Order Derivatives of the Elementary Symmetric Functions in the Rasch Model.

ERIC Educational Resources Information Center

Formann, Anton K.

1986-01-01

It is shown that for equal parameters explicit formulas exist, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle. (Author/LMO)

An Improved Computational Method for the Calculation of Mixture Liquid-Vapor Critical Points

NASA Astrophysics Data System (ADS)

Dimitrakopoulos, Panagiotis; Jia, Wenlong; Li, Changjun

2014-05-01

Knowledge of critical points is important to determine the phase behavior of a mixture. This work proposes a reliable and accurate method in order to locate the liquid-vapor critical point of a given mixture. The theoretical model is developed from the rigorous definition of critical points, based on the SRK equation of state (SRK EoS) or alternatively, on the PR EoS. In order to solve the resulting system of nonlinear equations, an improved method is introduced into an existing Newton-Raphson algorithm, which can calculate all the variables simultaneously in each iteration step. The improvements mainly focus on the derivatives of the Jacobian matrix, on the convergence criteria, and on the damping coefficient. As a result, all equations and related conditions required for the computation of the scheme are illustrated in this paper. Finally, experimental data for the critical points of 44 mixtures are adopted in order to validate the method. For the SRK EoS, average absolute errors of the predicted critical-pressure and critical-temperature values are 123.82 kPa and 3.11 K, respectively, whereas the commercial software package Calsep PVTSIM's prediction errors are 131.02 kPa and 3.24 K. For the PR EoS, the two above mentioned average absolute errors are 129.32 kPa and 2.45 K, while the PVTSIM's errors are 137.24 kPa and 2.55 K, respectively.

The SPAR thermal analyzer: Present and future

NASA Astrophysics Data System (ADS)

Marlowe, M. B.; Whetstone, W. D.; Robinson, J. C.

The SPAR thermal analyzer, a system of finite-element processors for performing steady-state and transient thermal analyses, is described. The processors communicate with each other through the SPAR random access data base. As each processor is executed, all pertinent source data is extracted from the data base and results are stored in the data base. Steady state temperature distributions are determined by a direct solution method for linear problems and a modified Newton-Raphson method for nonlinear problems. An explicit and several implicit methods are available for the solution of transient heat transfer problems. Finite element plotting capability is available for model checkout and verification.

The SPAR thermal analyzer: Present and future

NASA Technical Reports Server (NTRS)

Marlowe, M. B.; Whetstone, W. D.; Robinson, J. C.

1982-01-01

The SPAR thermal analyzer, a system of finite-element processors for performing steady-state and transient thermal analyses, is described. The processors communicate with each other through the SPAR random access data base. As each processor is executed, all pertinent source data is extracted from the data base and results are stored in the data base. Steady state temperature distributions are determined by a direct solution method for linear problems and a modified Newton-Raphson method for nonlinear problems. An explicit and several implicit methods are available for the solution of transient heat transfer problems. Finite element plotting capability is available for model checkout and verification.

An historical survey of computational methods in optimal control.

NASA Technical Reports Server (NTRS)

Polak, E.

1973-01-01

Review of some of the salient theoretical developments in the specific area of optimal control algorithms. The first algorithms for optimal control were aimed at unconstrained problems and were derived by using first- and second-variation methods of the calculus of variations. These methods have subsequently been recognized as gradient, Newton-Raphson, or Gauss-Newton methods in function space. A much more recent addition to the arsenal of unconstrained optimal control algorithms are several variations of conjugate-gradient methods. At first, constrained optimal control problems could only be solved by exterior penalty function methods. Later algorithms specifically designed for constrained problems have appeared. Among these are methods for solving the unconstrained linear quadratic regulator problem, as well as certain constrained minimum-time and minimum-energy problems. Differential-dynamic programming was developed from dynamic programming considerations. The conditional-gradient method, the gradient-projection method, and a couple of feasible directions methods were obtained as extensions or adaptations of related algorithms for finite-dimensional problems. Finally, the so-called epsilon-methods combine the Ritz method with penalty function techniques.

Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data

NASA Astrophysics Data System (ADS)

Courdurier, M.; Monard, F.; Osses, A.; Romero, F.

2015-09-01

In medical single-photon emission computed tomography (SPECT) imaging, we seek to simultaneously obtain the internal radioactive sources and the attenuation map using not only ballistic measurements but also first-order scattering measurements and assuming a very specific scattering regime. The problem is modeled using the radiative transfer equation by means of an explicit non-linear operator that gives the ballistic and scattering measurements as a function of the radioactive source and attenuation distributions. First, by differentiating this non-linear operator we obtain a linearized inverse problem. Then, under regularity hypothesis for the source distribution and attenuation map and considering small attenuations, we rigorously prove that the linear operator is invertible and we compute its inverse explicitly. This allows proof of local uniqueness for the non-linear inverse problem. Finally, using the previous inversion result for the linear operator, we propose a new type of iterative algorithm for simultaneous source and attenuation recovery for SPECT based on the Neumann series and a Newton-Raphson algorithm.

Steady-state configuration and tension calculations of marine cables under complex currents via separated particle swarm optimization

NASA Astrophysics Data System (ADS)

Xu, Xue-song

2014-12-01

Under complex currents, the motion governing equations of marine cables are complex and nonlinear, and the calculations of cable configuration and tension become difficult compared with those under the uniform or simple currents. To obtain the numerical results, the usual Newton-Raphson iteration is often adopted, but its stability depends on the initial guessed solution to the governing equations. To improve the stability of numerical calculation, this paper proposed separated the particle swarm optimization, in which the variables are separated into several groups, and the dimension of search space is reduced to facilitate the particle swarm optimization. Via the separated particle swarm optimization, these governing nonlinear equations can be solved successfully with any initial solution, and the process of numerical calculation is very stable. For the calculations of cable configuration and tension of marine cables under complex currents, the proposed separated swarm particle optimization is more effective than the other particle swarm optimizations.

Global Asymptotic Behavior of Iterative Implicit Schemes

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sweby, P. K.

1994-01-01

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Galerkin-collocation domain decomposition method for arbitrary binary black holes

NASA Astrophysics Data System (ADS)

Barreto, W.; Clemente, P. C. M.; de Oliveira, H. P.; Rodriguez-Mueller, B.

2018-05-01

We present a new computational framework for the Galerkin-collocation method for double domain in the context of ADM 3 +1 approach in numerical relativity. This work enables us to perform high resolution calculations for initial sets of two arbitrary black holes. We use the Bowen-York method for binary systems and the puncture method to solve the Hamiltonian constraint. The nonlinear numerical code solves the set of equations for the spectral modes using the standard Newton-Raphson method, LU decomposition and Gaussian quadratures. We show convergence of our code for the conformal factor and the ADM mass. Thus, we display features of the conformal factor for different masses, spins and linear momenta.

A methodology for airplane parameter estimation and confidence interval determination in nonlinear estimation problems. Ph.D. Thesis - George Washington Univ., Apr. 1985

NASA Technical Reports Server (NTRS)

Murphy, P. C.

1986-01-01

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. With the fitted surface, sensitivity information can be updated at each iteration with less computational effort than that required by either a finite-difference method or integration of the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, and thus provides flexibility to use model equations in any convenient format. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. The degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels and to predict the degree of agreement between CR bounds and search estimates.

ASCAL: A Microcomputer Program for Estimating Logistic IRT Item Parameters.

ERIC Educational Resources Information Center

Vale, C. David; Gialluca, Kathleen A.

ASCAL is a microcomputer-based program for calibrating items according to the three-parameter logistic model of item response theory. It uses a modified multivariate Newton-Raphson procedure for estimating item parameters. This study evaluated this procedure using Monte Carlo Simulation Techniques. The current version of ASCAL was then compared to…

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

NASA Technical Reports Server (NTRS)

Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Globally convergent techniques in nonlinear Newton-Krylov

NASA Technical Reports Server (NTRS)

Brown, Peter N.; Saad, Youcef

1989-01-01

Some convergence theory is presented for nonlinear Krylov subspace methods. The basic idea of these methods is to use variants of Newton's iteration in conjunction with a Krylov subspace method for solving the Jacobian linear systems. These methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimensions. The main focus is to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms. Most of the convergence results are formulated for projection onto general subspaces rather than just Krylov subspaces.

Multilevel Iterative Methods in Nonlinear Computational Plasma Physics

NASA Astrophysics Data System (ADS)

Knoll, D. A.; Finn, J. M.

1997-11-01

Many applications in computational plasma physics involve the implicit numerical solution of coupled systems of nonlinear partial differential equations or integro-differential equations. Such problems arise in MHD, systems of Vlasov-Fokker-Planck equations, edge plasma fluid equations. We have been developing matrix-free Newton-Krylov algorithms for such problems and have applied these algorithms to the edge plasma fluid equations [1,2] and to the Vlasov-Fokker-Planck equation [3]. Recently we have found that with increasing grid refinement, the number of Krylov iterations required per Newton iteration has grown unmanageable [4]. This has led us to the study of multigrid methods as a means of preconditioning matrix-free Newton-Krylov methods. In this poster we will give details of the general multigrid preconditioned Newton-Krylov algorithm, as well as algorithm performance details on problems of interest in the areas of magnetohydrodynamics and edge plasma physics. Work supported by US DoE 1. Knoll and McHugh, J. Comput. Phys., 116, pg. 281 (1995) 2. Knoll and McHugh, Comput. Phys. Comm., 88, pg. 141 (1995) 3. Mousseau and Knoll, J. Comput. Phys. (1997) (to appear) 4. Knoll and McHugh, SIAM J. Sci. Comput. 19, (1998) (to appear)

Design of optimally normal minimum gain controllers by continuation method

NASA Technical Reports Server (NTRS)

Lim, K. B.; Juang, J.-N.; Kim, Z. C.

1989-01-01

A measure of the departure from normality is investigated for system robustness. An attractive feature of the normality index is its simplicity for pole placement designs. To allow a tradeoff between system robustness and control effort, a cost function consisting of the sum of a norm of weighted gain matrix and a normality index is minimized. First- and second-order necessary conditions for the constrained optimization problem are derived and solved by a Newton-Raphson algorithm imbedded into a one-parameter family of neighboring zero problems. The method presented allows the direct computation of optimal gains in terms of robustness and control effort for pole placement problems.

On a class of Newton-like methods for solving nonlinear equations

NASA Astrophysics Data System (ADS)

Argyros, Ioannis K.

2009-06-01

We provide a semilocal convergence analysis for a certain class of Newton-like methods considered also in [I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374-397; I.K. Argyros, Computational theory of iterative methods, in: C.K. Chui, L. Wuytack (Eds.), Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co, New York, USA, 2007; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971], in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions, instead of only Lipschitz conditions [F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985) 71-84], we provide an analysis with the following advantages over the work in [F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985) 71-84] which improved the works in [W.E. Bosarge, P.L. Falb, A multipoint method of third order, J. Optimiz. Theory Appl. 4 (1969) 156-166; W.E. Bosarge, P.L. Falb, Infinite dimensional multipoint methods and the solution of two point boundary value problems, Numer. Math. 14 (1970) 264-286; J.E. Dennis, On the Kantorovich hypothesis for Newton's method, SIAM J. Numer. Anal. 6 (3) (1969) 493-507; J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971; H.J. Kornstaedt, Ein allgemeiner Konvergenzstaz fü r verschä rfte Newton-Verfahrem, in: ISNM, vol. 28, Birkhaü ser Verlag, Basel and Stuttgart, 1975, pp. 53-69; P. Laasonen, Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser I 450 (1969) 1-10; F.A. Potra, On a modified secant method, L'analyse numérique et la theorie de l'approximation 8 (2) (1979) 203-214; F.A. Potra, An application of the induction method of V. Pták to the study of Regula Falsi, Aplikace Matematiky 26 (1981) 111-120; F.A. Potra, On the convergence of a class of Newton-like methods, in: Iterative Solution of Nonlinear Systems of Equations, in: Lecture Notes in Mathematics, vol. 953, Springer-Verlag, New York, 1982; F.A. Potra, V. Pták, Nondiscrete induction and double step secant method, Math. Scand. 46 (1980) 236-250; F.A. Potra, V. Pták, On a class of modified Newton processes, Numer. Funct. Anal. Optim. 2 (1) (1980) 107-120; F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985) 71-84; J.W. Schmidt, Untere Fehlerschranken für Regula-Falsi Verfahren, Period. Math. Hungar. 9 (3) (1978) 241-247; J.W. Schmidt, H. Schwetlick, Ableitungsfreie Verfhren mit höherer Konvergenzgeschwindifkeit, Computing 3 (1968) 215-226; J.F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1964; M.A. Wolfe, Extended iterative methods for the solution of operator equations, Numer. Math. 31 (1978) 153-174]: larger convergence domain and weaker sufficient convergence conditions. Numerical examples further validating the results are also provided.

How Can Multivariate Item Response Theory Be Used in Reporting of Susbcores? Research Report. ETS RR-10-09

ERIC Educational Resources Information Center

Haberman, Shelby J.; Sinharay, Sandip

2010-01-01

Recently, there has been increasing interest in reporting diagnostic scores. This paper examines reporting of subscores using multidimensional item response theory (MIRT) models. An MIRT model is fitted using a stabilized Newton-Raphson algorithm (Haberman, 1974, 1988) with adaptive Gauss-Hermite quadrature (Haberman, von Davier, & Lee, 2008).…

Applying a Weighted Maximum Likelihood Latent Trait Estimator to the Generalized Partial Credit Model

ERIC Educational Resources Information Center

Penfield, Randall D.; Bergeron, Jennifer M.

2005-01-01

This article applies a weighted maximum likelihood (WML) latent trait estimator to the generalized partial credit model (GPCM). The relevant equations required to obtain the WML estimator using the Newton-Raphson algorithm are presented, and a simulation study is described that compared the properties of the WML estimator to those of the maximum…

Tables Of Gaussian-Type Orbital Basis Functions

NASA Technical Reports Server (NTRS)

Partridge, Harry

1992-01-01

NASA technical memorandum contains tables of estimated Hartree-Fock wave functions for atoms lithium through neon and potassium through krypton. Sets contain optimized Gaussian-type orbital exponents and coefficients, and near Hartree-Fock quality. Orbital exponents optimized by minimizing restricted Hartree-Fock energy via scaled Newton-Raphson scheme in which Hessian evaluated numerically by use of analytically determined gradients.

Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of Newton's iteration

NASA Astrophysics Data System (ADS)

Zhang, Fei; Huang, Weizhang; Li, Xianping; Zhang, Shicheng

2018-03-01

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often fails to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without compromising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems.

Aircraft automatic-flight-control system with inversion of the model in the feed-forward path using a Newton-Raphson technique for the inversion

NASA Technical Reports Server (NTRS)

Smith, G. A.; Meyer, G.; Nordstrom, M.

1986-01-01

A new automatic flight control system concept suitable for aircraft with highly nonlinear aerodynamic and propulsion characteristics and which must operate over a wide flight envelope was investigated. This exact model follower inverts a complete nonlinear model of the aircraft as part of the feed-forward path. The inversion is accomplished by a Newton-Raphson trim of the model at each digital computer cycle time of 0.05 seconds. The combination of the inverse model and the actual aircraft in the feed-forward path alloys the translational and rotational regulators in the feedback path to be easily designed by linear methods. An explanation of the model inversion procedure is presented. An extensive set of simulation data for essentially the full flight envelope for a vertical attitude takeoff and landing aircraft (VATOL) is presented. These data demonstrate the successful, smooth, and precise control that can be achieved with this concept. The trajectory includes conventional flight from 200 to 900 ft/sec with path accelerations and decelerations, altitude changes of over 6000 ft and 2g and 3g turns. Vertical attitude maneuvering as a tail sitter along all axes is demonstrated. A transition trajectory from 200 ft/sec in conventional flight to stationary hover in the vertical attitude includes satisfactory operation through lift-cure slope reversal as attitude goes from horizontal to vertical at constant altitude. A vertical attitude takeoff from stationary hover to conventional flight is also demonstrated.

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.

PubMed

Leszczynski, Henryk; Wrzosek, Monika

2017-02-01

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newton's method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

The terminal area automated path generation problem

NASA Technical Reports Server (NTRS)

Hsin, C.-C.

1977-01-01

The automated terminal area path generation problem in the advanced Air Traffic Control System (ATC), has been studied. Definitions, input, output and the interrelationships with other ATC functions have been discussed. Alternatives in modeling the problem have been identified. Problem formulations and solution techniques are presented. In particular, the solution of a minimum effort path stretching problem (path generation on a given schedule) has been carried out using the Newton-Raphson trajectory optimization method. Discussions are presented on the effect of different delivery time, aircraft entry position, initial guess on the boundary conditions, etc. Recommendations are made on real-world implementations.

A robust return-map algorithm for general multisurface plasticity

DOE PAGES

Adhikary, Deepak P.; Jayasundara, Chandana T.; Podgorney, Robert K.; ...

2016-06-16

Three new contributions to the field of multisurface plasticity are presented for general situations with an arbitrary number of nonlinear yield surfaces with hardening or softening. A method for handling linearly dependent flow directions is described. A residual that can be used in a line search is defined. An algorithm that has been implemented and comprehensively tested is discussed in detail. Examples are presented to illustrate the computational cost of various components of the algorithm. The overall result is that a single Newton-Raphson iteration of the algorithm costs between 1.5 and 2 times that of an elastic calculation. Examples alsomore » illustrate the successful convergence of the algorithm in complicated situations. For example, without using the new contributions presented here, the algorithm fails to converge for approximately 50% of the trial stresses for a common geomechanical model of sedementary rocks, while the current algorithm results in complete success. Since it involves no approximations, the algorithm is used to quantify the accuracy of an efficient, pragmatic, but approximate, algorithm used for sedimentary-rock plasticity in a commercial software package. Furthermore, the main weakness of the algorithm is identified as the difficulty of correctly choosing the set of initially active constraints in the general setting.« less

Analysis of an arched outer-race ball bearing considering centrifugal forces

NASA Technical Reports Server (NTRS)

Hamrock, B. J.; Anderson, W. J.

1972-01-01

A Newton-Raphson method of iteration was used in evaluating the radial and axial projection of the distance between the ball center and the outer raceway groove curvature center (V and W). Fatigue life evaluations were made. The similar analysis of a conventional bearing can be directly obtained from the arched bearing analysis by simply letting the amount of arching be zero (g = 0) and not considering equations related to the unloaded half of the outer race. The analysis was applied to a 150-mm angular contact ball bearing. Results for life, contact loads, and angles are shown for a conventional bearing (g = 0) and two arched bearings (g = 0.127 mm (0.005 in.), and 0.254 mm (0.010 in.)). The results indicate that an arched bearing is highly desirable for high speed applications. In particular, for a DN value of 3 million (20,000 rpm) and an applied axial load of 4448 N (1000 lb), an arched bearing shows an improvement in life of 306 percent over that of a conventional bearing. At 4.2 million DN (28,000 rpm), the corresponding improvement is 340 percent. It was also found for low speeds, the arched bearing does not offer the advantages that it does for high speed applications.

Molecular dynamics simulation of a needle-sphere binary mixture

NASA Astrophysics Data System (ADS)

Raghavan, Karthik

This paper investigates the dynamic behaviour of a hard needle-sphere binary system using a novel numerical technique called the Newton homotopy continuation (NHC) method. This mixture is representative of a polymer melt where both long chain molecules and monomers coexist. Since the intermolecular forces are generated from hard body interactions, the consequence of missed collisions or incorrect collision sequences have a significant bearing on the dynamic properties of the fluid. To overcome this problem, in earlier work NHC was chosen over traditional Newton-Raphson methods to solve the hard body dynamics of a needle fluid in random media composed of overlapping spheres. Furthermore, the simplicity of interactions and dynamics allows us to focus our research directly on the effects of particle shape and density on the transport behaviour of the mixture. These studies are also compared with earlier works that examined molecular chains in porous media primarily to understand the differences in molecular transport in the bulk versus porous systems.

Analysis of the iteratively regularized Gauss-Newton method under a heuristic rule

NASA Astrophysics Data System (ADS)

Jin, Qinian; Wang, Wei

2018-03-01

The iteratively regularized Gauss-Newton method is one of the most prominent regularization methods for solving nonlinear ill-posed inverse problems when the data is corrupted by noise. In order to produce a useful approximate solution, this iterative method should be terminated properly. The existing a priori and a posteriori stopping rules require accurate information on the noise level, which may not be available or reliable in practical applications. In this paper we propose a heuristic selection rule for this regularization method, which requires no information on the noise level. By imposing certain conditions on the noise, we derive a posteriori error estimates on the approximate solutions under various source conditions. Furthermore, we establish a convergence result without using any source condition. Numerical results are presented to illustrate the performance of our heuristic selection rule.

An improved Newton iteration for the generalized inverse of a matrix, with applications

NASA Technical Reports Server (NTRS)

Pan, Victor; Schreiber, Robert

1990-01-01

The purpose here is to clarify and illustrate the potential for the use of variants of Newton's method of solving problems of practical interest on highly personal computers. The authors show how to accelerate the method substantially and how to modify it successfully to cope with ill-conditioned matrices. The authors conclude that Newton's method can be of value for some interesting computations, especially in parallel and other computing environments in which matrix products are especially easy to work with.

Subsampled Hessian Newton Methods for Supervised Learning.

PubMed

Wang, Chien-Chih; Huang, Chun-Heng; Lin, Chih-Jen

2015-08-01

Newton methods can be applied in many supervised learning approaches. However, for large-scale data, the use of the whole Hessian matrix can be time-consuming. Recently, subsampled Newton methods have been proposed to reduce the computational time by using only a subset of data for calculating an approximation of the Hessian matrix. Unfortunately, we find that in some situations, the running speed is worse than the standard Newton method because cheaper but less accurate search directions are used. In this work, we propose some novel techniques to improve the existing subsampled Hessian Newton method. The main idea is to solve a two-dimensional subproblem per iteration to adjust the search direction to better minimize the second-order approximation of the function value. We prove the theoretical convergence of the proposed method. Experiments on logistic regression, linear SVM, maximum entropy, and deep networks indicate that our techniques significantly reduce the running time of the subsampled Hessian Newton method. The resulting algorithm becomes a compelling alternative to the standard Newton method for large-scale data classification.

A Newton method for the magnetohydrodynamic equilibrium equations

NASA Astrophysics Data System (ADS)

Oliver, Hilary James

We have developed and implemented a (J, B) space Newton method to solve the full nonlinear three dimensional magnetohydrodynamic equilibrium equations in toroidal geometry. Various cases have been run successfully, demonstrating significant improvement over Picard iteration, including a 3D stellarator equilibrium at β = 2%. The algorithm first solves the equilibrium force balance equation for the current density J, given a guess for the magnetic field B. This step is taken from the Picard-iterative PIES 3D equilibrium code. Next, we apply Newton's method to Ampere's Law by expansion of the functional J(B), which is defined by the first step. An analytic calculation in magnetic coordinates, of how the Pfirsch-Schlüter currents vary in the plasma in response to a small change in the magnetic field, yields the Newton gradient term (analogous to ∇f . δx in Newton's method for f(x) = 0). The algorithm is computationally feasible because we do this analytically, and because the gradient term is flux surface local when expressed in terms of a vector potential in an Ar=0 gauge. The equations are discretized by a hybrid spectral/offset grid finite difference technique, and leading order radial dependence is factored from Fourier coefficients to improve finite- difference accuracy near the polar-like origin. After calculating the Newton gradient term we transfer the equation from the magnetic grid to a fixed background grid, which greatly improves the code's performance.

Increasing dimension of structures by 4D printing shape memory polymers via fused deposition modeling

NASA Astrophysics Data System (ADS)

Hu, G. F.; Damanpack, A. R.; Bodaghi, M.; Liao, W. H.

2017-12-01

The main objective of this paper is to introduce a 4D printing method to program shape memory polymers (SMPs) during fabrication process. Fused deposition modeling (FDM) as a filament-based printing method is employed to program SMPs during depositing the material. This method is implemented to fabricate complicated polymeric structures by self-bending features without need of any post-programming. Experiments are conducted to demonstrate feasibility of one-dimensional (1D)-to 2D and 2D-to-3D self-bending. It is shown that 3D printed plate structures can transform into masonry-inspired 3D curved shell structures by simply heating. Good reliability of SMP programming during printing process is also demonstrated. A 3D macroscopic constitutive model is established to simulate thermo-mechanical features of the printed SMPs. Governing equations are also derived to simulate programming mechanism during printing process and shape change of self-bending structures. In this respect, a finite element formulation is developed considering von-Kármán geometric nonlinearity and solved by implementing iterative Newton-Raphson scheme. The accuracy of the computational approach is checked with experimental results. It is demonstrated that the theoretical model is able to replicate the main characteristics observed in the experiments. This research is likely to advance the state of the art FDM 4D printing, and provide pertinent results and computational tool that are instrumental in design of smart materials and structures with self-bending features.

What Information Theory Says About Best Response and About Binding Contracts

NASA Technical Reports Server (NTRS)

Wolpert, David H.

2004-01-01

Product Distribution (PD) theory is the information-theoretic extension of conventional full- rationality game theory to bounded rational games. Here PD theory is used to investigate games in which the players use bounded rational best-response strategies. This investigation illuminates how to determine the optimal organization chart for a corporation, or more generally how to order the sequence of moves of the players / employees so as to optimize an overall objective function. It is then shown that in the continuum-time limit, bounded rational best response games result in a variant of the replicator dynamics of evolutionary game theory. This variant is then investigated for team games, in which the players share the same utility function, by showing that such continuum- limit bounded rational best response is identical to Newton-Raphson iterative optimization of the shared utility function. Next PD theory is used to investigate changing the coordinate system of the game, i.e., changing the mapping from the joint move of the players to the arguments in the utility functions. Such a change couples those arguments, essentially by making each players move be an offered binding contract.

An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions

NASA Technical Reports Server (NTRS)

Peters, B. C., Jr.; Walker, H. F.

1975-01-01

A general iterative procedure is given for determining the consistent maximum likelihood estimates of normal distributions. In addition, a local maximum of the log-likelihood function, Newtons's method, a method of scoring, and modifications of these procedures are discussed.

Three-dimensional flow measurements in a vaneless radial turbine scroll

NASA Technical Reports Server (NTRS)

Tabakoff, W.; Wood, B.; Vittal, B. V. R.

1982-01-01

The flow behavior in a vaneless radial turbine scroll was examined experimentally. The data was obtained using the slant sensor technique of hot film anemometry. This method used the unsymmetric heat transfer characteristics of a constant temperature hot film sensor to detect the flow direction and magnitude. This was achieved by obtaining a velocity vector measurement at three sensor positions with respect to the flow. The true magnitude and direction of the velocity vector was then found using these values and a Newton-Raphson numerical technique. The through flow and secondary flow velocity components are measured at various points in three scroll sections.

Designing stellarator coils by a modified Newton method using FOCUS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhu, Caoxiang; Hudson, Stuart R.; Song, Yuntao

To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

Designing stellarator coils by a modified Newton method using FOCUS

NASA Astrophysics Data System (ADS)

Zhu, Caoxiang; Hudson, Stuart R.; Song, Yuntao; Wan, Yuanxi

2018-06-01

To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

Designing stellarator coils by a modified Newton method using FOCUS

DOE PAGES

Zhu, Caoxiang; Hudson, Stuart R.; Song, Yuntao; ...

2018-03-22

To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

Method and Apparatus for Predicting Unsteady Pressure and Flow Rate Distribution in a Fluid Network

NASA Technical Reports Server (NTRS)

Majumdar, Alok K. (Inventor)

2009-01-01

A method and apparatus for analyzing steady state and transient flow in a complex fluid network, modeling phase changes, compressibility, mixture thermodynamics, external body forces such as gravity and centrifugal force and conjugate heat transfer. In some embodiments, a graphical user interface provides for the interactive development of a fluid network simulation having nodes and branches. In some embodiments, mass, energy, and specific conservation equations are solved at the nodes, and momentum conservation equations are solved in the branches. In some embodiments, contained herein are data objects for computing thermodynamic and thermophysical properties for fluids. In some embodiments, the systems of equations describing the fluid network are solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution methods.

Solving coupled groundwater flow systems using a Jacobian Free Newton Krylov method

NASA Astrophysics Data System (ADS)

Mehl, S.

2012-12-01

Jacobian Free Newton Kyrlov (JFNK) methods can have several advantages for simulating coupled groundwater flow processes versus conventional methods. Conventional methods are defined here as those based on an iterative coupling (rather than a direct coupling) and/or that use Picard iteration rather than Newton iteration. In an iterative coupling, the systems are solved separately, coupling information is updated and exchanged between the systems, and the systems are re-solved, etc., until convergence is achieved. Trusted simulators, such as Modflow, are based on these conventional methods of coupling and work well in many cases. An advantage of the JFNK method is that it only requires calculation of the residual vector of the system of equations and thus can make use of existing simulators regardless of how the equations are formulated. This opens the possibility of coupling different process models via augmentation of a residual vector by each separate process, which often requires substantially fewer changes to the existing source code than if the processes were directly coupled. However, appropriate perturbation sizes need to be determined for accurate approximations of the Frechet derivative, which is not always straightforward. Furthermore, preconditioning is necessary for reasonable convergence of the linear solution required at each Kyrlov iteration. Existing preconditioners can be used and applied separately to each process which maximizes use of existing code and robust preconditioners. In this work, iteratively coupled parent-child local grid refinement models of groundwater flow and groundwater flow models with nonlinear exchanges to streams are used to demonstrate the utility of the JFNK approach for Modflow models. Use of incomplete Cholesky preconditioners with various levels of fill are examined on a suite of nonlinear and linear models to analyze the effect of the preconditioner. Comparisons of convergence and computer simulation time are made using conventional iteratively coupled methods and those based on Picard iteration to those formulated with JFNK to gain insights on the types of nonlinearities and system features that make one approach advantageous. Results indicate that nonlinearities associated with stream/aquifer exchanges are more problematic than those resulting from unconfined flow.

Some modifications of Newton's method for the determination of the steady-state response of nonlinear oscillatory circuits

NASA Astrophysics Data System (ADS)

Grosz, F. B., Jr.; Trick, T. N.

1982-07-01

It is proposed that nondominant states should be eliminated from the Newton algorithm in the steady-state analysis of nonlinear oscillatory systems. This technique not only improves convergence, but also reduces the size of the sensitivity matrix so that less computation is required for each iteration. One or more periods of integration should be performed after each periodic state estimation before the sensitivity computations are made for the next periodic state estimation. These extra periods of integration between Newton iterations are found to allow the fast states due to parasitic effects to settle, which enables the Newton algorithm to make a better prediction. In addition, the reliability of the algorithm is improved in high Q oscillator circuits by both local and global damping in which the amount of damping is proportional to the difference between the initial and final state values.

Newton-Krylov-Schwarz: An implicit solver for CFD

NASA Technical Reports Server (NTRS)

Cai, Xiao-Chuan; Keyes, David E.; Venkatakrishnan, V.

1995-01-01

Newton-Krylov methods and Krylov-Schwarz (domain decomposition) methods have begun to become established in computational fluid dynamics (CFD) over the past decade. The former employ a Krylov method inside of Newton's method in a Jacobian-free manner, through directional differencing. The latter employ an overlapping Schwarz domain decomposition to derive a preconditioner for the Krylov accelerator that relies primarily on local information, for data-parallel concurrency. They may be composed as Newton-Krylov-Schwarz (NKS) methods, which seem particularly well suited for solving nonlinear elliptic systems in high-latency, distributed-memory environments. We give a brief description of this family of algorithms, with an emphasis on domain decomposition iterative aspects. We then describe numerical simulations with Newton-Krylov-Schwarz methods on aerodynamics applications emphasizing comparisons with a standard defect-correction approach, subdomain preconditioner consistency, subdomain preconditioner quality, and the effect of a coarse grid.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

NASA Astrophysics Data System (ADS)

Whiteley, J. P.

2017-10-01

Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton's method. On each iteration of Newton's method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems.

Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

NASA Technical Reports Server (NTRS)

Hou, Gene

2004-01-01

The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

Dynamic Failure of Materials. Volume 1 - Experiments and Analyses

DTIC Science & Technology

1998-11-01

initial increments of voids do not lead to substantial relaxation of stress. In this case, the condition (8.1) gives Vv = Ao- aVv * ß=— (8.10) pc...enough strain steps to define the process accurately. At each strain step, a combined Newton-Raphson and regulafalsi solution technique (multiple trials ...laser surgery. Clinical studies have demonstrated that, for some applications, surgical lasers are superior to conventional surgical procedures

Numerical modeling evapotranspiration flux components in shrub-encroached grassland in Inner Mongolia, China

NASA Astrophysics Data System (ADS)

Wang, Pei; Li, Xiao-Yan; Huang, Jie-Yu; Yang, Wen-Xin; Wang, Qi-Dan; Xu, Kun; Zheng, Xiao-Ran

2016-04-01

Shrub encroachment into arid grasslands occurs around the world. However, few works on shrub encroachment has been conducted in China. Moreover, its hydrological implications remain poorly investigated in arid and semiarid regions. This study combined a two-source energy balanced model and Newton-Raphson iteration scheme to simulate the evapotranspiration (ET) and their components of shrub-encroached(with 15.4% shrub coverage) grassland in Inner Mongolia. Good agreements of ET flux between modelled and measured by Bowen ratio method with relatively insensitive to uncertainties/errors in the assigned models parameters or in measured input variables for its components illustrated that our model was feasible for simulating evapotranspiration flux components in shrub-encroached grassland. The transpiration fraction(T /ET)account for 58±17%during the growing season. With the designed shrub encroachment extreme scenarios (maximum and minimum coverage),the contribution of shrub to local plant transpiration (Tshrub/T) was 20.06±7%during the growing season. Canopy conductance was the main controlling factor of T /ET. In diurnal scale short wave solar radiation was the direct influential factor while in seasonal scale leaf area index (LAI) and soil water content were the direct influential factors. We find that the seasonal variation of Tshrub/T has a good relationship with ratio of LAIshrub/LAI, and rainfall characteristics widened the difference of contribution of shrub and herbs to ecosystem evapotranspiration.

Self-expanding/shrinking structures by 4D printing

NASA Astrophysics Data System (ADS)

Bodaghi, M.; Damanpack, A. R.; Liao, W. H.

2016-10-01

The aim of this paper is to create adaptive structures capable of self-expanding and self-shrinking by means of four-dimensional printing technology. An actuator unit is designed and fabricated directly by printing fibers of shape memory polymers (SMPs) in flexible beams with different arrangements. Experiments are conducted to determine thermo-mechanical material properties of the fabricated part revealing that the printing process introduced a strong anisotropy into the printed parts. The feasibility of the actuator unit with self-expanding and self-shrinking features is demonstrated experimentally. A phenomenological constitutive model together with analytical closed-form solutions are developed to replicate thermo-mechanical behaviors of SMPs. Governing equations of equilibrium are developed for printed structures based on the non-linear Green-Lagrange strain tensor and solved implementing a finite element method along with an iterative incremental Newton-Raphson scheme. The material-structural model is then applied to digitally design and print SMP adaptive lattices in planar and tubular shapes comprising a periodic arrangement of SMP actuator units that expand and then recover their original shape automatically. Numerical and experimental results reveal that the proposed planar lattice as meta-materials can be employed for plane actuators with self-expanding/shrinking features or as structural switches providing two different dynamic characteristics. It is also shown that the proposed tubular lattice with a self-expanding/shrinking mechanism can serve as tubular stents and grippers for bio-medical or piping applications.

Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation

NASA Astrophysics Data System (ADS)

Su, Yong; Zhang, Qingchuan; Xu, Xiaohai; Gao, Zeren; Wu, Shangquan

2018-01-01

It is believed that the classic forward additive Newton-Raphson (FA-NR) algorithm and the recently introduced inverse compositional Gauss-Newton (IC-GN) algorithm give rise to roughly equal interpolation bias. Questioning the correctness of this statement, this paper presents a thorough analysis of interpolation bias for the IC-GN algorithm. A theoretical model is built to analytically characterize the dependence of interpolation bias upon speckle image, target image interpolation, and reference image gradient estimation. The interpolation biases of the FA-NR algorithm and the IC-GN algorithm can be significantly different, whose relative difference can exceed 80%. For the IC-GN algorithm, the gradient estimator can strongly affect the interpolation bias; the relative difference can reach 178%. Since the mean bias errors are insensitive to image noise, the theoretical model proposed remains valid in the presence of noise. To provide more implementation details, source codes are uploaded as a supplement.

Simulation of the hot rolling of steel with direct iteration

NASA Astrophysics Data System (ADS)

Hanoglu, Umut; Šarler, Božidar

2017-10-01

In this study a simulation system based on the meshless Local Radial Basis Function Collocation Method (LRBFCM) is applied for the hot rolling of steel. Rolling is a complex, 3D, thermo-mechanical problem; however, 2D cross-sectional slices are used as computational domains that are aligned with the rolling direction and no heat flow or strain is considered in the direction that is orthogonal to the slices. For each predefined position with respect to the rolling direction, the solution procedure is repeated until the slice reaches the final rolling position. Collocation nodes are initially distributed over the domain and boundaries of the initial slice. A local solution is achieved by considering the overlapping influence domains with either 5 or 7 nodes. Radial Basis Functions (RBFs) are used for the temperature discretization in the thermal model and displacement discretization in the mechanical model. The meshless solution procedure does not require a mesh-generation algorithm in the classic sense. Strong-form mechanical and thermal models are run for each slice regarding the contact with the roll's surface. Ideal plastic material behavior is considered for the mechanical results, where the nonlinear stress-strain relation is solved with a direct iteration. The majority of the Finite Element Model (FEM) simulations, including commercial software, use a conventional Newton-Raphson algorithm. However, direct iteration is chosen here due to its better compatibility with meshless methods. In order to overcome any unforeseen stability issues, the redistribution of the nodes by Elliptic Node Generation (ENG) is applied to one or more slices throughout the simulation. The rolling simulation presented here helps the user to design, test and optimize different rolling schedules. The results can be seen minutes after the simulation's start in terms of temperature, displacement, stress and strain fields as well as important technological parameters, like the roll-separating forces, roll toque, etc. An example of a rolling simulation, in which an initial size of 110x110 mm steel is rolled to a round bar with 80 mm diameter, is shown in Fig. 3. A user-friendly computer application for industrial use is created by using the C# and .NET frameworks.

3D modelling of non-linear visco-elasto-plastic crustal and lithospheric processes using LaMEM

NASA Astrophysics Data System (ADS)

Popov, Anton; Kaus, Boris

2016-04-01

LaMEM (Lithosphere and Mantle Evolution Model) is a three-dimensional thermo-mechanical numerical code to simulate crustal and lithospheric deformation. The code is based on a staggered finite difference (FDSTAG) discretization in space, which is a stable and very efficient technique to solve the (nearly) incompressible Stokes equations that does not suffer from spurious pressure modes or artificial compressibility (a typical feature of low-order finite element techniques). Higher order finite element methods are more accurate than FDSTAG methods under idealized test cases where the jump in viscosity is exactly aligned with the boundaries of the elements. Yet, geodynamically more realistic cases involve evolving subduction zones, nonlinear rheologies or localized plastic shear bands. In these cases, the viscosity pattern evolves spontaneously during a simulation or even during nonlinear iterations, and the advantages of higher order methods disappear and they all converge with approximately first order accuracy, similar to that of FDSTAG [1]. Yet, since FDSTAG methods have considerably less degrees of freedom than quadratic finite element methods, they require about an order of magnitude less memory for the same number of nodes in 3D which also implies that every matrix-vector multiplication is significantly faster. LaMEM is build on top of the PETSc library and uses the particle-in-cell technique to track material properties, history variables which makes it straightforward to incorporate effects like phase changes or chemistry. An internal free surface is present, together with (simple) erosion and sedimentation processes, and a number of methods are available to import complex geometries into the code (e.g, http://geomio.bitbucket.org). Customized Galerkin coupled geometric multigrid preconditioners are implemented which resulted in a good parallel scalability of the code (we have tested LaMEM on 458'752 cores [2]). Yet, the drawback of using FDSTAG discretizations is that the Jacobian, which is a key component for fast and robust convergence of Newton-Raphson nonlinear iterative solvers, is more difficult to implement than in FE codes and actually results in a larger stencil. Rather than discretizing it explicitly, we therefore developed a matrix-free analytical Jacobian implementation for the coupled sets of momentum, mass, and energy conservation equations, combined with visco-elasto-plastic rheologies. Tests show that for simple nonlinear viscous rheologies there is little advantage of the MF approach over the standard MFFD PETSc approach, but that iterations converge slightly faster if plasticity is present. Results also show that the Newton solver usually converges in a quadratic manner even for pressure-dependent Drucker-Prager rheologies and without harmonic viscosity averaging of plastic and viscous rheologies. Yet, if the timestep is too large (and the model becomes effectively viscoplastic), or if the shear band pattern changes dramatically, stagnation of iterations might occur. This can be remedied with an appropriate regularization, which we discuss. LaMEM is available as open source software. [1] Thielmann, M., May, D.A., and Kaus, B., 2014, Discretization Errors in the Hybrid Finite Element Particle-in-cell Method: Pure and Applied Geophysics,, doi: 10.1007/s00024-014-0808-9. [2] Kaus B.J.P., Popov A.A., Baumann T.S., Püsök A.E., Bauville A., Fernandez N., Collignon M. (2015) Forward and inverse modelling of lithospheric deformation on geological timescales. NIC Symposium 2016 - Proceedings. NIC Series. Vol. 48.

OGS#PETSc approach for robust and efficient simulations of strongly coupled hydrothermal processes in EGS reservoirs

NASA Astrophysics Data System (ADS)

Watanabe, Norihiro; Blucher, Guido; Cacace, Mauro; Kolditz, Olaf

2016-04-01

A robust and computationally efficient solution is important for 3D modelling of EGS reservoirs. This is particularly the case when the reservoir model includes hydraulic conduits such as induced or natural fractures, fault zones, and wellbore open-hole sections. The existence of such hydraulic conduits results in heterogeneous flow fields and in a strengthened coupling between fluid flow and heat transport processes via temperature dependent fluid properties (e.g. density and viscosity). A commonly employed partitioned solution (or operator-splitting solution) may not robustly work for such strongly coupled problems its applicability being limited by small time step sizes (e.g. 5-10 days) whereas the processes have to be simulated for 10-100 years. To overcome this limitation, an alternative approach is desired which can guarantee a robust solution of the coupled problem with minor constraints on time step sizes. In this work, we present a Newton-Raphson based monolithic coupling approach implemented in the OpenGeoSys simulator (OGS) combined with the Portable, Extensible Toolkit for Scientific Computation (PETSc) library. The PETSc library is used for both linear and nonlinear solvers as well as MPI-based parallel computations. The suggested method has been tested by application to the 3D reservoir site of Groß Schönebeck, in northern Germany. Results show that the exact Newton-Raphson approach can also be limited to small time step sizes (e.g. one day) due to slight oscillations in the temperature field. The usage of a line search technique and modification of the Jacobian matrix were necessary to achieve robust convergence of the nonlinear solution. For the studied example, the proposed monolithic approach worked even with a very large time step size of 3.5 years.

Modèle tridimensionnel pour coupler les équations magnétiques et électriques dans le cas de la magnétostatique

NASA Astrophysics Data System (ADS)

Piriou, F.; Razek, A.

1991-03-01

In this paper a 3D model for coupling of magnetic and electric equations is presented. The magnetic equations are solved with the help of finite element method using the magnetic vector potential formulation. To take into account the effects of magnetic saturation we use the Newton-Raphson algorithm. We develop the analysis permitting the coupling of magnetic and electric equations to obtain a difrerential system equations which can be solved with numerical integration. As example we model an iron core coil and the validity of our model is verified by a comparison of the obtained results with an analytical solution and a 2D code calculation. Dans cet article est présenté un modèle 3D qui permet de coupler les équations magnétiques et électriques. Les équations magnétiques sont résolues à l'aide de la méthode des éléments finis en utilisant une formulation en potentiel vecteur magnétique. Dans le modèle proposé les effets de la saturation du circuit magnétique sont pris en compte en utilisant l'algorithme de Newton-Raphson. On montre comment relier les équations magnétiques avec celles du circuit électrique pour aboutir à un système d'équations différentielles que l'on résout avec une intégration numérique. A titre d'exemple on modélise une bobine à noyau ferromagnétique et pour montrer la validité du modèle on compare les résultats obtenus avec une solution analytique et un code de calcul 2D.

Numerical Solutions for Nonlinear High Damping Rubber Bearing Isolators: Newmark's Method with Netwon-Raphson Iteration Revisited

NASA Astrophysics Data System (ADS)

Markou, A. A.; Manolis, G. D.

2018-03-01

Numerical methods for the solution of dynamical problems in engineering go back to 1950. The most famous and widely-used time stepping algorithm was developed by Newmark in 1959. In the present study, for the first time, the Newmark algorithm is developed for the case of the trilinear hysteretic model, a model that was used to describe the shear behaviour of high damping rubber bearings. This model is calibrated against free-vibration field tests implemented on a hybrid base isolated building, namely the Solarino project in Italy, as well as against laboratory experiments. A single-degree-of-freedom system is used to describe the behaviour of a low-rise building isolated with a hybrid system comprising high damping rubber bearings and low friction sliding bearings. The behaviour of the high damping rubber bearings is simulated by the trilinear hysteretic model, while the description of the behaviour of the low friction sliding bearings is modeled by a linear Coulomb friction model. In order to prove the effectiveness of the numerical method we compare the analytically solved trilinear hysteretic model calibrated from free-vibration field tests (Solarino project) against the same model solved with the Newmark method with Netwon-Raphson iteration. Almost perfect agreement is observed between the semi-analytical solution and the fully numerical solution with Newmark's time integration algorithm. This will allow for extension of the trilinear mechanical models to bidirectional horizontal motion, to time-varying vertical loads, to multi-degree-of-freedom-systems, as well to generalized models connected in parallel, where only numerical solutions are possible.

Four-body trajectory optimization

NASA Technical Reports Server (NTRS)

Pu, C. L.; Edelbaum, T. N.

1974-01-01

A comprehensive optimization program has been developed for computing fuel-optimal trajectories between the earth and a point in the sun-earth-moon system. It presents methods for generating fuel optimal two-impulse trajectories which may originate at the earth or a point in space and fuel optimal three-impulse trajectories between two points in space. The extrapolation of the state vector and the computation of the state transition matrix are accomplished by the Stumpff-Weiss method. The cost and constraint gradients are computed analytically in terms of the terminal state and the state transition matrix. The 4-body Lambert problem is solved by using the Newton-Raphson method. An accelerated gradient projection method is used to optimize a 2-impulse trajectory with terminal constraint. The Davidon's Variance Method is used both in the accelerated gradient projection method and the outer loop of a 3-impulse trajectory optimization problem.

Development of an integrated BEM approach for hot fluid structure interaction

NASA Technical Reports Server (NTRS)

Dargush, G. F.; Banerjee, P. K.; Shi, Y.

1991-01-01

The development of a comprehensive fluid-structure interaction capability within a boundary element computer code is described. This new capability is implemented in a completely general manner, so that quite arbitrary geometry, material properties and boundary conditions may be specified. Thus, a single analysis code can be used to run structures-only problems, fluids-only problems, or the combined fluid-structure problem. In all three cases, steady or transient conditions can be selected, with or without thermal effects. Nonlinear analyses can be solved via direct iteration or by employing a modified Newton-Raphson approach. A number of detailed numerical examples are included at the end of these two sections to validate the formulations and to emphasize both the accuracy and generality of the computer code. A brief review of the recent applicable boundary element literature is included for completeness. The fluid-structure interaction facility is discussed. Once again, several examples are provided to highlight this unique capability. A collection of potential boundary element applications that have been uncovered as a result of work related to the present grant is given. For most of those problems, satisfactory analysis techniques do not currently exist.

Multiple environment single system quantum mechanical/molecular mechanical (MESS-QM/MM) calculations. 1. Estimation of polarization energies.

PubMed

Sodt, Alexander J; Mei, Ye; König, Gerhard; Tao, Peng; Steele, Ryan P; Brooks, Bernard R; Shao, Yihan

2015-03-05

In combined quantum mechanical/molecular mechanical (QM/MM) free energy calculations, it is often advantageous to have a frozen geometry for the quantum mechanical (QM) region. For such multiple-environment single-system (MESS) cases, two schemes are proposed here for estimating the polarization energy: the first scheme, termed MESS-E, involves a Roothaan step extrapolation of the self-consistent field (SCF) energy; whereas the other scheme, termed MESS-H, employs a Newton-Raphson correction using an approximate inverse electronic Hessian of the QM region (which is constructed only once). Both schemes are extremely efficient, because the expensive Fock updates and SCF iterations in standard QM/MM calculations are completely avoided at each configuration. They produce reasonably accurate QM/MM polarization energies: MESS-E can predict the polarization energy within 0.25 kcal/mol in terms of the mean signed error for two of our test cases, solvated methanol and solvated β-alanine, using the M06-2X or ωB97X-D functionals; MESS-H can reproduce the polarization energy within 0.2 kcal/mol for these two cases and for the oxyluciferin-luciferase complex, if the approximate inverse electronic Hessians are constructed with sufficient accuracy.

RIO: a new computational framework for accurate initial data of binary black holes

NASA Astrophysics Data System (ADS)

Barreto, W.; Clemente, P. C. M.; de Oliveira, H. P.; Rodriguez-Mueller, B.

2018-06-01

We present a computational framework ( Rio) in the ADM 3+1 approach for numerical relativity. This work enables us to carry out high resolution calculations for initial data of two arbitrary black holes. We use the transverse conformal treatment, the Bowen-York and the puncture methods. For the numerical solution of the Hamiltonian constraint we use the domain decomposition and the spectral decomposition of Galerkin-Collocation. The nonlinear numerical code solves the set of equations for the spectral modes using the standard Newton-Raphson method, LU decomposition and Gaussian quadratures. We show the convergence of the Rio code. This code allows for easy deployment of large calculations. We show how the spin of one of the black holes is manifest in the conformal factor.

Real-Time Adaptive Least-Squares Drag Minimization for Performance Adaptive Aeroelastic Wing

NASA Technical Reports Server (NTRS)

Ferrier, Yvonne L.; Nguyen, Nhan T.; Ting, Eric

2016-01-01

This paper contains a simulation study of a real-time adaptive least-squares drag minimization algorithm for an aeroelastic model of a flexible wing aircraft. The aircraft model is based on the NASA Generic Transport Model (GTM). The wing structures incorporate a novel aerodynamic control surface known as the Variable Camber Continuous Trailing Edge Flap (VCCTEF). The drag minimization algorithm uses the Newton-Raphson method to find the optimal VCCTEF deflections for minimum drag in the context of an altitude-hold flight control mode at cruise conditions. The aerodynamic coefficient parameters used in this optimization method are identified in real-time using Recursive Least Squares (RLS). The results demonstrate the potential of the VCCTEF to improve aerodynamic efficiency for drag minimization for transport aircraft.

Application of Newton's method to the postbuckling of rings under pressure loadings

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1989-01-01

The postbuckling response of circular rings (or long cylinders) is examined. The rings are subjected to four types of external pressure loadings; each type of pressure is defined by its magnitude and direction at points on the buckled ring. Newton's method is applied to the nonlinear differential equations of the exact inextensional theory for the ring problem. A zeroth approximation for the solution of the nonlinear equations, based on the mode shape corresponding to the first buckling pressure, is derived in closed form for each of the four types of pressure. The zeroth approximation is used to start the iteration cycle in Newton's method to compute numerical solutions of the nonlinear equations. The zeroth approximations for the postbuckling pressure-deflection curves are compared with the converged solutions from Newton's method and with similar results reported in the literature.

NLSCIDNT user's guide maximum likehood parameter identification computer program with nonlinear rotorcraft model

NASA Technical Reports Server (NTRS)

1979-01-01

A nonlinear, maximum likelihood, parameter identification computer program (NLSCIDNT) is described which evaluates rotorcraft stability and control coefficients from flight test data. The optimal estimates of the parameters (stability and control coefficients) are determined (identified) by minimizing the negative log likelihood cost function. The minimization technique is the Levenberg-Marquardt method, which behaves like the steepest descent method when it is far from the minimum and behaves like the modified Newton-Raphson method when it is nearer the minimum. Twenty-one states and 40 measurement variables are modeled, and any subset may be selected. States which are not integrated may be fixed at an input value, or time history data may be substituted for the state in the equations of motion. Any aerodynamic coefficient may be expressed as a nonlinear polynomial function of selected 'expansion variables'.

A system-approach to the elastohydrodynamic lubrication point-contact problem

NASA Technical Reports Server (NTRS)

Lim, Sang Gyu; Brewe, David E.

1991-01-01

The classical EHL (elastohydrodynamic lubrication) point contact problem is solved using a new system-approach, similar to that introduced by Houpert and Hamrock for the line-contact problem. Introducing a body-fitted coordinate system, the troublesome free-boundary is transformed to a fixed domain. The Newton-Raphson method can then be used to determine the pressure distribution and the cavitation boundary subject to the Reynolds boundary condition. This method provides an efficient and rigorous way of solving the EHL point contact problem with the aid of a supercomputer and a promising method to deal with the transient EHL point contact problem. A typical pressure distribution and film thickness profile are presented and the minimum film thicknesses are compared with the solution of Hamrock and Dowson. The details of the cavitation boundaries for various operating parameters are discussed.

Domain decomposition methods for the parallel computation of reacting flows

NASA Technical Reports Server (NTRS)

Keyes, David E.

1988-01-01

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors.

Simulation of fluid-structure interaction in micropumps by coupling of two commercial finite element programs

NASA Astrophysics Data System (ADS)

Klein, Andreas; Gerlach, Gerald

1998-09-01

This paper deals with the simulation of the fluid-structure interaction phenomena in micropumps. The proposed solution approach is based on external coupling of two different solvers, which are considered here as `black boxes'. Therefore, no specific intervention is necessary into the program code, and solvers can be exchanged arbitrarily. For the realization of the external iteration loop, two algorithms are considered: the relaxation-based Gauss-Seidel method and the computationally more extensive Newton method. It is demonstrated in terms of a simplified test case, that for rather weak coupling, the Gauss-Seidel method is sufficient. However, by simply changing the considered fluid from air to water, the two physical domains become strongly coupled, and the Gauss-Seidel method fails to converge in this case. The Newton iteration scheme must be used instead.

Anderson acceleration and application to the three-temperature energy equations

NASA Astrophysics Data System (ADS)

An, Hengbin; Jia, Xiaowei; Walker, Homer F.

2017-10-01

The Anderson acceleration method is an algorithm for accelerating the convergence of fixed-point iterations, including the Picard method. Anderson acceleration was first proposed in 1965 and, for some years, has been used successfully to accelerate the convergence of self-consistent field iterations in electronic-structure computations. Recently, the method has attracted growing attention in other application areas and among numerical analysts. Compared with a Newton-like method, an advantage of Anderson acceleration is that there is no need to form the Jacobian matrix. Thus the method is easy to implement. In this paper, an Anderson-accelerated Picard method is employed to solve the three-temperature energy equations, which are a type of strong nonlinear radiation-diffusion equations. Two strategies are used to improve the robustness of the Anderson acceleration method. One strategy is to adjust the iterates when necessary to satisfy the physical constraint. Another strategy is to monitor and, if necessary, reduce the matrix condition number of the least-squares problem in the Anderson-acceleration implementation so that numerical stability can be guaranteed. Numerical results show that the Anderson-accelerated Picard method can solve the three-temperature energy equations efficiently. Compared with the Picard method without acceleration, Anderson acceleration can reduce the number of iterations by at least half. A comparison between a Jacobian-free Newton-Krylov method, the Picard method, and the Anderson-accelerated Picard method is conducted in this paper.

Iterative Methods for Solving Nonlinear Parabolic Problem in Pension Saving Management

NASA Astrophysics Data System (ADS)

Koleva, M. N.

2011-11-01

In this work we consider a nonlinear parabolic equation, obtained from Riccati like transformation of the Hamilton-Jacobi-Bellman equation, arising in pension saving management. We discuss two numerical iterative methods for solving the model problem—fully implicit Picard method and mixed Picard-Newton method, which preserves the parabolic characteristics of the differential problem. Numerical experiments for comparison the accuracy and effectiveness of the algorithms are discussed. Finally, observations are given.

Finite element method for viscoelastic medium with damage and the application to structural analysis of solid rocket motor grain

NASA Astrophysics Data System (ADS)

Deng, Bin; Shen, ZhiBin; Duan, JingBo; Tang, GuoJin

2014-05-01

This paper studies the damage-viscoelastic behavior of composite solid propellants of solid rocket motors (SRM). Based on viscoelastic theories and strain equivalent hypothesis in damage mechanics, a three-dimensional (3-D) nonlinear viscoelastic constitutive model incorporating with damage is developed. The resulting viscoelastic constitutive equations are numerically discretized by integration algorithm, and a stress-updating method is presented by solving nonlinear equations according to the Newton-Raphson method. A material subroutine of stress-updating is made up and embedded into commercial code of Abaqus. The material subroutine is validated through typical examples. Our results indicate that the finite element results are in good agreement with the analytical ones and have high accuracy, and the suggested method and designed subroutine are efficient and can be further applied to damage-coupling structural analysis of practical SRM grain.

Direct numerical simulation of laminar-turbulent flow over a flat plate at hypersonic flow speeds

NASA Astrophysics Data System (ADS)

Egorov, I. V.; Novikov, A. V.

2016-06-01

A method for direct numerical simulation of a laminar-turbulent flow around bodies at hypersonic flow speeds is proposed. The simulation is performed by solving the full three-dimensional unsteady Navier-Stokes equations. The method of calculation is oriented to application of supercomputers and is based on implicit monotonic approximation schemes and a modified Newton-Raphson method for solving nonlinear difference equations. By this method, the development of three-dimensional perturbations in the boundary layer over a flat plate and in a near-wall flow in a compression corner is studied at the Mach numbers of the free-stream of M = 5.37. In addition to pulsation characteristic, distributions of the mean coefficients of the viscous flow in the transient section of the streamlined surface are obtained, which enables one to determine the beginning of the laminar-turbulent transition and estimate the characteristics of the turbulent flow in the boundary layer.

Application of an enriched FEM technique in thermo-mechanical contact problems

NASA Astrophysics Data System (ADS)

Khoei, A. R.; Bahmani, B.

2018-02-01

In this paper, an enriched FEM technique is employed for thermo-mechanical contact problem based on the extended finite element method. A fully coupled thermo-mechanical contact formulation is presented in the framework of X-FEM technique that takes into account the deformable continuum mechanics and the transient heat transfer analysis. The Coulomb frictional law is applied for the mechanical contact problem and a pressure dependent thermal contact model is employed through an explicit formulation in the weak form of X-FEM method. The equilibrium equations are discretized by the Newmark time splitting method and the final set of non-linear equations are solved based on the Newton-Raphson method using a staggered algorithm. Finally, in order to illustrate the capability of the proposed computational model several numerical examples are solved and the results are compared with those reported in literature.

Advanced computational techniques for incompressible/compressible fluid-structure interactions

NASA Astrophysics Data System (ADS)

Kumar, Vinod

2005-07-01

Fluid-Structure Interaction (FSI) problems are of great importance to many fields of engineering and pose tremendous challenges to numerical analyst. This thesis addresses some of the hurdles faced for both 2D and 3D real life time-dependent FSI problems with particular emphasis on parachute systems. The techniques developed here would help improve the design of parachutes and are of direct relevance to several other FSI problems. The fluid system is solved using the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation for the Navier-Stokes equations of incompressible and compressible flows. The structural dynamics solver is based on a total Lagrangian finite element formulation. Newton-Raphson method is employed to linearize the otherwise nonlinear system resulting from the fluid and structure formulations. The fluid and structural systems are solved in decoupled fashion at each nonlinear iteration. While rigorous coupling methods are desirable for FSI simulations, the decoupled solution techniques provide sufficient convergence in the time-dependent problems considered here. In this thesis, common problems in the FSI simulations of parachutes are discussed and possible remedies for a few of them are presented. Further, the effects of the porosity model on the aerodynamic forces of round parachutes are analyzed. Techniques for solving compressible FSI problems are also discussed. Subsequently, a better stabilization technique is proposed to efficiently capture and accurately predict the shocks in supersonic flows. The numerical examples simulated here require high performance computing. Therefore, numerical tools using distributed memory supercomputers with message passing interface (MPI) libraries were developed.

Simulation on Natural Convection of a Nanofluid along an Isothermal Inclined Plate

NASA Astrophysics Data System (ADS)

Mitra, Asish

2017-08-01

A numerical algorithm is presented for studying laminar natural convection flow of a nanofluid along an isothermal inclined plate. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a set of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Dimensionless velocity, temperature profiles and nanoparticle concentration for various angles of inclination are illustrated graphically. The effects of Prandtl number, Brownian motion parameter and thermophoresis parameter on Nusselt number are also discussed. The results of the present simulation are then compared with previous one available in literature with good agreement.

Vortex breakdown simulation

NASA Technical Reports Server (NTRS)

Hafez, M.; Ahmad, J.; Kuruvila, G.; Salas, M. D.

1987-01-01

In this paper, steady, axisymmetric inviscid, and viscous (laminar) swirling flows representing vortex breakdown phenomena are simulated using a stream function-vorticity-circulation formulation and two numerical methods. The first is based on an inverse iteration, where a norm of the solution is prescribed and the swirling parameter is calculated as a part of the output. The second is based on direct Newton iterations, where the linearized equations, for all the unknowns, are solved simultaneously by an efficient banded Gaussian elimination procedure. Several numerical solutions for inviscid and viscous flows are demonstrated, followed by a discussion of the results. Some improvements on previous work have been achieved: first order upwind differences are replaced by second order schemes, line relaxation procedure (with linear convergence rate) is replaced by Newton's iterations (which converge quadratically), and Reynolds numbers are extended from 200 up to 1000.

Low Angle-of-Attack Longitudinal Aerodynamic Parameters of Navy T-2 Trainer Aircraft Extracted from Flight Data: A Comparison of Identification Techniques. Volume I. Data Acquisition and Modified Newton-Raphson Analysis

DTIC Science & Technology

1975-06-23

SYSTEM • The numbunng o» tec^nic«! pioject (ep<.iii muert My the N<»v-«l AN Development Center is ariaogert (or specific identitf.ition onrposti E«i.h...chord (W.P. + 73,92) Av Sweepback (257. chord) Airfoil Section lv Tall length (.25 cw to .25 cv) VERTICAL FIN Sf Area (including 2.14 ft 2

Picard Trajectory Approximation Iteration for Efficient Orbit Propagation

DTIC Science & Technology

2015-07-21

Eqn (8)) for an iterative ap- proximation of eccentric anomaly, and is transformed back to . The Lambert/ Kepler time – eccentric anomaly relationship...of Kepler motion based on spinor regulari- zation, Journal fur die Reine und Angewandt Mathematik 218, 204-219, 1965. [3] Levi-Civita T., Sur la...transformed back to  , ,x y z . The Lambert/ Kepler time – eccentric anomaly relationship is iterated by a Newton/Secant method to converge on the

Calculation of symmetric and asymmetric vortex seperation on cones and tangent ogives based on discrete vortex models

NASA Technical Reports Server (NTRS)

Chin, S.; Lan, C. Edward

1988-01-01

An inviscid discrete vortex model, with newly derived expressions for the tangential velocity imposed at the separation points, is used to investigate the symmetric and asymmetric vortex separation on cones and tangent ogives. The circumferential locations of separation are taken from experimental data. Based on a slender body theory, the resulting simultaneous nonlinear algebraic equations in a cross-flow plane are solved with Broyden's modified Newton-Raphson method. Total force coefficients are obtained through momentum principle with new expressions for nonconical flow. It is shown through the method of function deflation that multiple solutions exist at large enough angles of attack, even with symmetric separation points. These additional solutions are asymmetric in vortex separation and produce side force coefficients which agree well with data for cones and tangent ogives.

Optimum Suction Distribution for Transition Control

NASA Technical Reports Server (NTRS)

Balakumar, P.; Hall, P.

1996-01-01

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method to find the best suction distribution subject to some overall constraint applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting non-linear system of equations is solved using the Newton-Raphson technique. The computations are performed for a Blasius boundary layer on a flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero.

Implementation of an improved adaptive-implicit method in a thermal compositional simulator

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tan, T.B.

1988-11-01

A multicomponent thermal simulator with an adaptive-implicit-method (AIM) formulation/inexact-adaptive-Newton (IAN) method is presented. The final coefficient matrix retains the original banded structure so that conventional iterative methods can be used. Various methods for selection of the eliminated unknowns are tested. AIM/IAN method has a lower work count per Newtonian iteration than fully implicit methods, but a wrong choice of unknowns will result in excessive Newtonian iterations. For the problems tested, the residual-error method described in the paper for selecting implicit unknowns, together with the IAN method, had an improvement of up to 28% of the CPU time over the fullymore » implicit method.« less

An Algorithm for Efficient Maximum Likelihood Estimation and Confidence Interval Determination in Nonlinear Estimation Problems

NASA Technical Reports Server (NTRS)

Murphy, Patrick Charles

1985-01-01

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The algorithm was developed for airplane parameter estimation problems but is well suited for most nonlinear, multivariable, dynamic systems. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. The fitted surface allows sensitivity information to be updated at each iteration with a significant reduction in computational effort. MNRES determines the sensitivities with less computational effort than using either a finite-difference method or integrating the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, thus eliminating algorithm reformulation with each new model and providing flexibility to use model equations in any format that is convenient. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. It is observed that the degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. The CR bounds were found to be close to the bounds determined by the search when the degree of nonlinearity was small. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels for the parameter confidence limits. The primary utility of the measure, however, was found to be in predicting the degree of agreement between Cramer-Rao bounds and search estimates.

Computer-implemented method and apparatus for autonomous position determination using magnetic field data

NASA Technical Reports Server (NTRS)

Ketchum, Eleanor A. (Inventor)

2000-01-01

A computer-implemented method and apparatus for determining position of a vehicle within 100 km autonomously from magnetic field measurements and attitude data without a priori knowledge of position. An inverted dipole solution of two possible position solutions for each measurement of magnetic field data are deterministically calculated by a program controlled processor solving the inverted first order spherical harmonic representation of the geomagnetic field for two unit position vectors 180 degrees apart and a vehicle distance from the center of the earth. Correction schemes such as a successive substitutions and a Newton-Raphson method are applied to each dipole. The two position solutions for each measurement are saved separately. Velocity vectors for the position solutions are calculated so that a total energy difference for each of the two resultant position paths is computed. The position path with the smaller absolute total energy difference is chosen as the true position path of the vehicle.

Nonlinear and parallel algorithms for finite element discretizations of the incompressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Arteaga, Santiago Egido

1998-12-01

The steady-state Navier-Stokes equations are of considerable interest because they are used to model numerous common physical phenomena. The applications encountered in practice often involve small viscosities and complicated domain geometries, and they result in challenging problems in spite of the vast attention that has been dedicated to them. In this thesis we examine methods for computing the numerical solution of the primitive variable formulation of the incompressible equations on distributed memory parallel computers. We use the Galerkin method to discretize the differential equations, although most results are stated so that they apply also to stabilized methods. We also reformulate some classical results in a single framework and discuss some issues frequently dismissed in the literature, such as the implementation of pressure space basis and non- homogeneous boundary values. We consider three nonlinear methods: Newton's method, Oseen's (or Picard) iteration, and sequences of Stokes problems. All these iterative nonlinear methods require solving a linear system at every step. Newton's method has quadratic convergence while that of the others is only linear; however, we obtain theoretical bounds showing that Oseen's iteration is more robust, and we confirm it experimentally. In addition, although Oseen's iteration usually requires more iterations than Newton's method, the linear systems it generates tend to be simpler and its overall costs (in CPU time) are lower. The Stokes problems result in linear systems which are easier to solve, but its convergence is much slower, so that it is competitive only for large viscosities. Inexact versions of these methods are studied, and we explain why the best timings are obtained using relatively modest error tolerances in solving the corresponding linear systems. We also present a new damping optimization strategy based on the quadratic nature of the Navier-Stokes equations, which improves the robustness of all the linearization strategies considered and whose computational cost is negligible. The algebraic properties of these systems depend on both the discretization and nonlinear method used. We study in detail the positive definiteness and skewsymmetry of the advection submatrices (essentially, convection-diffusion problems). We propose a discretization based on a new trilinear form for Newton's method. We solve the linear systems using three Krylov subspace methods, GMRES, QMR and TFQMR, and compare the advantages of each. Our emphasis is on parallel algorithms, and so we consider preconditioners suitable for parallel computers such as line variants of the Jacobi and Gauss- Seidel methods, alternating direction implicit methods, and Chebyshev and least squares polynomial preconditioners. These work well for moderate viscosities (moderate Reynolds number). For small viscosities we show that effective parallel solution of the advection subproblem is a critical factor to improve performance. Implementation details on a CM-5 are presented.

Krylov Deferred Correction Accelerated Method of Lines Transpose for Parabolic Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jia, Jun; Jingfang, Huang

2008-01-01

In this paper, a new class of numerical methods for the accurate and efficient solutions of parabolic partial differential equations is presented. Unlike traditional method of lines (MoL), the new {\\bf \\it Krylov deferred correction (KDC) accelerated method of lines transpose (MoL^T)} first discretizes the temporal direction using Gaussian type nodes and spectral integration, and symbolically applies low-order time marching schemes to form a preconditioned elliptic system, which is then solved iteratively using Newton-Krylov techniques such as Newton-GMRES or Newton-BiCGStab method. Each function evaluation in the Newton-Krylov method is simply one low-order time-stepping approximation of the error by solving amore » decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated MoL^T technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time-step sizes in long-time simulations.« less

Discrete-Time Zhang Neural Network for Online Time-Varying Nonlinear Optimization With Application to Manipulator Motion Generation.

PubMed

Jin, Long; Zhang, Yunong

2015-07-01

In this brief, a discrete-time Zhang neural network (DTZNN) model is first proposed, developed, and investigated for online time-varying nonlinear optimization (OTVNO). Then, Newton iteration is shown to be derived from the proposed DTZNN model. In addition, to eliminate the explicit matrix-inversion operation, the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is introduced, which can effectively approximate the inverse of Hessian matrix. A DTZNN-BFGS model is thus proposed and investigated for OTVNO, which is the combination of the DTZNN model and the quasi-Newton BFGS method. In addition, theoretical analyses show that, with step-size h=1 and/or with zero initial error, the maximal residual error of the DTZNN model has an O(τ(2)) pattern, whereas the maximal residual error of the Newton iteration has an O(τ) pattern, with τ denoting the sampling gap. Besides, when h ≠ 1 and h ∈ (0,2) , the maximal steady-state residual error of the DTZNN model has an O(τ(2)) pattern. Finally, an illustrative numerical experiment and an application example to manipulator motion generation are provided and analyzed to substantiate the efficacy of the proposed DTZNN and DTZNN-BFGS models for OTVNO.

AZTEC: A parallel iterative package for the solving linear systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hutchinson, S.A.; Shadid, J.N.; Tuminaro, R.S.

1996-12-31

We describe a parallel linear system package, AZTEC. The package incorporates a number of parallel iterative methods (e.g. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss-Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within Newton schemes for nonlinear methods. Currently, a number of different users are using this package to solve a variety of PDE applications.

First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems

DTIC Science & Technology

2014-03-01

accuracy, with rapid convergence over each physical time step, typically less than five Newton iter - ations. 1 Contents 1 Introduction 3 2 Hyperbolic...however, we employ the Gauss - Seidel (GS) relaxation, which is also an O(N) method for the discretization arising from hyperbolic advection-diffusion system...advection-diffusion scheme. The linear dependency of the iterations on Table 1: Boundary layer problem ( Convergence criteria: Residuals < 10−8.) log10Re

Thermochemical nonequilibrium in atomic hydrogen at elevated temperatures

NASA Technical Reports Server (NTRS)

Scott, R. K.

1972-01-01

A numerical study of the nonequilibrium flow of atomic hydrogen in a cascade arc was performed to obtain insight into the physics of the hydrogen cascade arc. A rigorous mathematical model of the flow problem was formulated, incorporating the important nonequilibrium transport phenomena and atomic processes which occur in atomic hydrogen. Realistic boundary conditions, including consideration of the wall electrostatic sheath phenomenon, were included in the model. The governing equations of the asymptotic region of the cascade arc were obtained by writing conservation of mass and energy equations for the electron subgas, an energy conservation equation for heavy particles and an equation of state. Finite-difference operators for variable grid spacing were applied to the governing equations and the resulting system of strongly coupled, stiff equations were solved numerically by the Newton-Raphson method.

Modelling of hydrogen transport in silicon solar cell structures under equilibrium conditions

NASA Astrophysics Data System (ADS)

Hamer, P.; Hallam, B.; Bonilla, R. S.; Altermatt, P. P.; Wilshaw, P.; Wenham, S.

2018-01-01

This paper presents a model for the introduction and redistribution of hydrogen in silicon solar cells at temperatures between 300 and 700 °C based on a second order backwards difference formula evaluated using a single Newton-Raphson iteration. It includes the transport of hydrogen and interactions with impurities such as ionised dopants. The simulations lead to three primary conclusions: (1) hydrogen transport across an n-type emitter is heavily temperature dependent; (2) under equilibrium conditions, hydrogen is largely driven by its charged species, with the switch from a dominance of negatively charged hydrogen (H-) to positively charged hydrogen (H+) within the emitter region critical to significant transport across the junction; and (3) hydrogen transport across n-type emitters is critically dependent upon the doping profile within the emitter, and, in particular, the peak doping concentration. It is also observed that during thermal processes after an initial high temperature step, hydrogen preferentially migrates to the surface of a phosphorous doped emitter, drawing hydrogen out of the p-type bulk. This may play a role in several effects observed during post-firing anneals in relation to the passivation of recombination active defects and even the elimination of hydrogen-related defects in the bulk of silicon solar cells.

Adaptive control of turbulence intensity is accelerated by frugal flow sampling.

PubMed

Quinn, Daniel B; van Halder, Yous; Lentink, David

2017-11-01

The aerodynamic performance of vehicles and animals, as well as the productivity of turbines and energy harvesters, depends on the turbulence intensity of the incoming flow. Previous studies have pointed at the potential benefits of active closed-loop turbulence control. However, it is unclear what the minimal sensory and algorithmic requirements are for realizing this control. Here we show that very low-bandwidth anemometers record sufficient information for an adaptive control algorithm to converge quickly. Our online Newton-Raphson algorithm tunes the turbulence in a recirculating wind tunnel by taking readings from an anemometer in the test section. After starting at 9% turbulence intensity, the algorithm converges on values ranging from 10% to 45% in less than 12 iterations within 1% accuracy. By down-sampling our measurements, we show that very-low-bandwidth anemometers record sufficient information for convergence. Furthermore, down-sampling accelerates convergence by smoothing gradients in turbulence intensity. Our results explain why low-bandwidth anemometers in engineering and mechanoreceptors in biology may be sufficient for adaptive control of turbulence intensity. Finally, our analysis suggests that, if certain turbulent eddy sizes are more important to control than others, frugal adaptive control schemes can be particularly computationally effective for improving performance. © 2017 The Author(s).

Development of an integrated BEM approach for hot fluid structure interaction

NASA Technical Reports Server (NTRS)

Dargush, G. F.; Banerjee, P. K.

1989-01-01

The progress made toward the development of a boundary element formulation for the study of hot fluid-structure interaction in Earth-to-Orbit engine hot section components is reported. The convective viscous integral formulation was derived and implemented in the general purpose computer program GP-BEST. The new convective kernel functions, in turn, necessitated the development of refined integration techniques. As a result, however, since the physics of the problem is embedded in these kernels, boundary element solutions can now be obtained at very high Reynolds number. Flow around obstacles can be solved approximately with an efficient linearized boundary-only analysis or, more exactly, by including all of the nonlinearities present in the neighborhood of the obstacle. The other major accomplishment was the development of a comprehensive fluid-structure interaction capability within GP-BEST. This new facility is implemented in a completely general manner, so that quite arbitrary geometry, material properties and boundary conditions may be specified. Thus, a single analysis code (GP-BEST) can be used to run structures-only problems, fluids-only problems, or the combined fluid-structure problem. In all three cases, steady or transient conditions can be selected, with or without thermal effects. Nonlinear analyses can be solved via direct iteration or by employing a modified Newton-Raphson approach.

Computational modeling of chemo-electro-mechanical coupling: A novel implicit monolithic finite element approach

PubMed Central

Wong, J.; Göktepe, S.; Kuhl, E.

2014-01-01

Summary Computational modeling of the human heart allows us to predict how chemical, electrical, and mechanical fields interact throughout a cardiac cycle. Pharmacological treatment of cardiac disease has advanced significantly over the past decades, yet it remains unclear how the local biochemistry of an individual heart cell translates into global cardiac function. Here we propose a novel, unified strategy to simulate excitable biological systems across three biological scales. To discretize the governing chemical, electrical, and mechanical equations in space, we propose a monolithic finite element scheme. We apply a highly efficient and inherently modular global-local split, in which the deformation and the transmembrane potential are introduced globally as nodal degrees of freedom, while the chemical state variables are treated locally as internal variables. To ensure unconditional algorithmic stability, we apply an implicit backward Euler finite difference scheme to discretize the resulting system in time. To increase algorithmic robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton-Raphson scheme. The proposed algorithm allows us to simulate the interaction of chemical, electrical, and mechanical fields during a representative cardiac cycle on a patient-specific geometry, robust and stable, with calculation times on the order of four days on a standard desktop computer. PMID:23798328

Optimizing the inner loop of the gravitational force interaction on modern processors

DOE Office of Scientific and Technical Information (OSTI.GOV)

Warren, Michael S

2010-12-08

We have achieved superior performance on multiple generations of the fastest supercomputers in the world with our hashed oct-tree N-body code (HOT), spanning almost two decades and garnering multiple Gordon Bell Prizes for significant achievement in parallel processing. Execution time for our N-body code is largely influenced by the force calculation in the inner loop. Improvements to the inner loop using SSE3 instructions has enabled the calculation of over 200 million gravitational interactions per second per processor on a 2.6 GHz Opteron, for a computational rate of over 7 Gflops in single precision (700/0 of peak). We obtain optimal performancemore » some processors (including the Cell) by decomposing the reciprocal square root function required for a gravitational interaction into a table lookup, Chebychev polynomial interpolation, and Newton-Raphson iteration, using the algorithm of Karp. By unrolling the loop by a factor of six, and using SPU intrinsics to compute on vectors, we obtain performance of over 16 Gflops on a single Cell SPE. Aggregated over the 8 SPEs on a Cell processor, the overall performance is roughly 130 Gflops. In comparison, the ordinary C version of our inner loop only obtains 1.6 Gflops per SPE with the spuxlc compiler.« less

A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries.

PubMed

Asgharzadeh, Hafez; Borazjani, Iman

2017-02-15

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 - 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80-90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries

NASA Astrophysics Data System (ADS)

Asgharzadeh, Hafez; Borazjani, Iman

2017-02-01

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for non-linear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form a preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42-74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal and full Jacobian, respectivley, when the stretching factor was increased. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80-90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

MODFLOW–USG version 1: An unstructured grid version of MODFLOW for simulating groundwater flow and tightly coupled processes using a control volume finite-difference formulation

USGS Publications Warehouse

Panday, Sorab; Langevin, Christian D.; Niswonger, Richard G.; Ibaraki, Motomu; Hughes, Joseph D.

2013-01-01

A new version of MODFLOW, called MODFLOW–USG (for UnStructured Grid), was developed to support a wide variety of structured and unstructured grid types, including nested grids and grids based on prismatic triangles, rectangles, hexagons, and other cell shapes. Flexibility in grid design can be used to focus resolution along rivers and around wells, for example, or to subdiscretize individual layers to better represent hydrostratigraphic units. MODFLOW–USG is based on an underlying control volume finite difference (CVFD) formulation in which a cell can be connected to an arbitrary number of adjacent cells. To improve accuracy of the CVFD formulation for irregular grid-cell geometries or nested grids, a generalized Ghost Node Correction (GNC) Package was developed, which uses interpolated heads in the flow calculation between adjacent connected cells. MODFLOW–USG includes a Groundwater Flow (GWF) Process, based on the GWF Process in MODFLOW–2005, as well as a new Connected Linear Network (CLN) Process to simulate the effects of multi-node wells, karst conduits, and tile drains, for example. The CLN Process is tightly coupled with the GWF Process in that the equations from both processes are formulated into one matrix equation and solved simultaneously. This robustness results from using an unstructured grid with unstructured matrix storage and solution schemes. MODFLOW–USG also contains an optional Newton-Raphson formulation, based on the formulation in MODFLOW–NWT, for improving solution convergence and avoiding problems with the drying and rewetting of cells. Because the existing MODFLOW solvers were developed for structured and symmetric matrices, they were replaced with a new Sparse Matrix Solver (SMS) Package developed specifically for MODFLOW–USG. The SMS Package provides several methods for resolving nonlinearities and multiple symmetric and asymmetric linear solution schemes to solve the matrix arising from the flow equations and the Newton-Raphson formulation, respectively.

On the numerical solution of the dynamically loaded hydrodynamic lubrication of the point contact problem

NASA Technical Reports Server (NTRS)

Lim, Sang G.; Brewe, David E.; Prahl, Joseph M.

1990-01-01

The transient analysis of hydrodynamic lubrication of a point-contact is presented. A body-fitted coordinate system is introduced to transform the physical domain to a rectangular computational domain, enabling the use of the Newton-Raphson method for determining pressures and locating the cavitation boundary, where the Reynolds boundary condition is specified. In order to obtain the transient solution, an explicit Euler method is used to effect a time march. The transient dynamic load is a sinusoidal function of time with frequency, fractional loading, and mean load as parameters. Results include the variation of the minimum film thickness and phase-lag with time as functions of excitation frequency. The results are compared with the analytic solution to the transient step bearing problem with the same dynamic loading function. The similarities of the results suggest an approximate model of the point contact minimum film thickness solution.

Nonlinear analysis of 0-3 polarized PLZT microplate based on the new modified couple stress theory

NASA Astrophysics Data System (ADS)

Wang, Liming; Zheng, Shijie

2018-02-01

In this study, based on the new modified couple stress theory, the size- dependent model for nonlinear bending analysis of a pure 0-3 polarized PLZT plate is developed for the first time. The equilibrium equations are derived from a variational formulation based on the potential energy principle and the new modified couple stress theory. The Galerkin method is adopted to derive the nonlinear algebraic equations from governing differential equations. And then the nonlinear algebraic equations are solved by using Newton-Raphson method. After simplification, the new model includes only a material length scale parameter. In addition, numerical examples are carried out to study the effect of material length scale parameter on the nonlinear bending of a simply supported pure 0-3 polarized PLZT plate subjected to light illumination and uniform distributed load. The results indicate the new model is able to capture the size effect and geometric nonlinearity.

Methods of computing steady-state voltage stability margins of power systems

DOEpatents

Chow, Joe Hong; Ghiocel, Scott Gordon

2018-03-20

In steady-state voltage stability analysis, as load increases toward a maximum, conventional Newton-Raphson power flow Jacobian matrix becomes increasingly ill-conditioned so power flow fails to converge before reaching maximum loading. A method to directly eliminate this singularity reformulates the power flow problem by introducing an AQ bus with specified bus angle and reactive power consumption of a load bus. For steady-state voltage stability analysis, the angle separation between the swing bus and AQ bus can be varied to control power transfer to the load, rather than specifying the load power itself. For an AQ bus, the power flow formulation is only made up of a reactive power equation, thus reducing the size of the Jacobian matrix by one. This reduced Jacobian matrix is nonsingular at the critical voltage point, eliminating a major difficulty in voltage stability analysis for power system operations.

Numerical Modeling of Flow Distribution in Micro-Fluidics Systems

NASA Technical Reports Server (NTRS)

Majumdar, Alok; Cole, Helen; Chen, C. P.

2005-01-01

This paper describes an application of a general purpose computer program, GFSSP (Generalized Fluid System Simulation Program) for calculating flow distribution in a network of micro-channels. GFSSP employs a finite volume formulation of mass and momentum conservation equations in a network consisting of nodes and branches. Mass conservation equation is solved for pressures at the nodes while the momentum conservation equation is solved at the branches to calculate flowrate. The system of equations describing the fluid network is solved by a numerical method that is a combination of the Newton-Raphson and successive substitution methods. The numerical results have been compared with test data and detailed CFD (computational Fluid Dynamics) calculations. The agreement between test data and predictions is satisfactory. The discrepancies between the predictions and test data can be attributed to the frictional correlation which does not include the effect of surface tension or electro-kinetic effect.

Real-time absorption and scattering characterization of slab-shaped turbid samples obtained by a combination of angular and spatially resolved measurements.

PubMed

Dam, Jan S; Yavari, Nazila; Sørensen, Søren; Andersson-Engels, Stefan

2005-07-10

We present a fast and accurate method for real-time determination of the absorption coefficient, the scattering coefficient, and the anisotropy factor of thin turbid samples by using simple continuous-wave noncoherent light sources. The three optical properties are extracted from recordings of angularly resolved transmittance in addition to spatially resolved diffuse reflectance and transmittance. The applied multivariate calibration and prediction techniques are based on multiple polynomial regression in combination with a Newton--Raphson algorithm. The numerical test results based on Monte Carlo simulations showed mean prediction errors of approximately 0.5% for all three optical properties within ranges typical for biological media. Preliminary experimental results are also presented yielding errors of approximately 5%. Thus the presented methods show a substantial potential for simultaneous absorption and scattering characterization of turbid media.

Direct numerical simulation of the laminar-turbulent transition at hypersonic flow speeds on a supercomputer

NASA Astrophysics Data System (ADS)

Egorov, I. V.; Novikov, A. V.; Fedorov, A. V.

2017-08-01

A method for direct numerical simulation of three-dimensional unsteady disturbances leading to a laminar-turbulent transition at hypersonic flow speeds is proposed. The simulation relies on solving the full three-dimensional unsteady Navier-Stokes equations. The computational technique is intended for multiprocessor supercomputers and is based on a fully implicit monotone approximation scheme and the Newton-Raphson method for solving systems of nonlinear difference equations. This approach is used to study the development of three-dimensional unstable disturbances in a flat-plate and compression-corner boundary layers in early laminar-turbulent transition stages at the free-stream Mach number M = 5.37. The three-dimensional disturbance field is visualized in order to reveal and discuss features of the instability development at the linear and nonlinear stages. The distribution of the skin friction coefficient is used to detect laminar and transient flow regimes and determine the onset of the laminar-turbulent transition.

Conversion from Engineering Units to Telemetry Counts on Dryden Flight Simulators

NASA Technical Reports Server (NTRS)

Fantini, Jay A.

1998-01-01

Dryden real-time flight simulators encompass the simulation of pulse code modulation (PCM) telemetry signals. This paper presents a new method whereby the calibration polynomial (from first to sixth order), representing the conversion from counts to engineering units (EU), is numerically inverted in real time. The result is less than one-count error for valid EU inputs. The Newton-Raphson method is used to numerically invert the polynomial. A reverse linear interpolation between the EU limits is used to obtain an initial value for the desired telemetry count. The method presented here is not new. What is new is how classical numerical techniques are optimized to take advantage of modem computer power to perform the desired calculations in real time. This technique makes the method simple to understand and implement. There are no interpolation tables to store in memory as in traditional methods. The NASA F-15 simulation converts and transmits over 1000 parameters at 80 times/sec. This paper presents algorithm development, FORTRAN code, and performance results.

Proceedings of Workshop on Atmospheric Density and Aerodynamic Drag Models for Air Force Operations Held at Air Force Geophysics Laboratory on 20-22 October 1987. Volume 1

DTIC Science & Technology

1990-02-13

considered with these production processes in a simple photochemical equilibrium calculation , we are able to determine the contribution each makes to the...Hessian matrix of second derivatives (which is required in the Newton-Raphson procedure) by the vector product of the gradient (VJ) and its transpose...was focused on the altitude region 80-250 Km. Papers were presented in the folowing areas: Air Force requirements , physics of density and drag

A Comparison of Two Algorithms for the Simulation of Non-Homogeneous Poisson Processes with Degree-Two Exponential Polynomial Intensity Function.

DTIC Science & Technology

1977-09-01

process with an event streaa intensity (rate) function that is of degree-two exponential pclyncaial foru. (The use of exponential pclynoaials is...4 \\v 01 ^3 C \\ \\ •r- S_ \\ \\ O \\ \\ a \\ \\ V IA C 4-> \\ \\ •«- c \\ 1 <— 3 • o \\ \\ Ol (J \\ \\ O U —1 <o \\ I...would serve as a good initial approxiaation t* , f-r the Newton-Raphson aethod. However, for the purpose of this implementation, the end point which

Flight instrumentation specification for parameter identification: Program user's guide. [instrument errors/error analysis

NASA Technical Reports Server (NTRS)

Mohr, R. L.

1975-01-01

A set of four digital computer programs is presented which can be used to investigate the effects of instrumentation errors on the accuracy of aircraft and helicopter stability-and-control derivatives identified from flight test data. The programs assume that the differential equations of motion are linear and consist of small perturbations about a quasi-steady flight condition. It is also assumed that a Newton-Raphson optimization technique is used for identifying the estimates of the parameters. Flow charts and printouts are included.

Use of Picard and Newton iteration for solving nonlinear ground water flow equations

USGS Publications Warehouse

Mehl, S.

2006-01-01

This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems.

An analysis of the convergence of Newton iterations for solving elliptic Kepler's equation

NASA Astrophysics Data System (ADS)

Elipe, A.; Montijano, J. I.; Rández, L.; Calvo, M.

2017-12-01

In this note a study of the convergence properties of some starters E_0 = E_0(e,M) in the eccentricity-mean anomaly variables for solving the elliptic Kepler's equation (KE) by Newton's method is presented. By using a Wang Xinghua's theorem (Xinghua in Math Comput 68(225):169-186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton's method, we obtain for each starter E_0(e,M) a set of values (e,M) \\in [0, 1) × [0, π ] that lead to the q-convergence in the sense that Newton's sequence (E_n)_{n ≥ 0} generated from E_0 = E_0(e,M) is well defined, converges to the exact solution E^* = E^*(e,M) of KE and further \\vert E_n - E^* \\vert ≤ q^{2^n -1} \\vert E_0 - E^* \\vert holds for all n ≥ 0. This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27-44, 2014) by using Smale's α -test with q=1/2. Also since in KE the convergence rate of Newton's method tends to zero as e → 0, we show that the error estimates given in the Wang Xinghua's theorem for KE can also be used to determine sets of q-convergence with q = e^k \\widetilde{q} for all e \\in [0,1) and a fixed \\widetilde{q} ≤ 1. Some remarks on the use of this theorem to derive a priori estimates of the error \\vert E_n - E^* \\vert after n Kepler's iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.

Interior-Point Methods for Linear Programming: A Challenge to the Simplex Method

DTIC Science & Technology

1988-07-01

subsequently found that the method was first proposed by Dikin in 1967 [6]. Search directions are generated by the same system (5). Any hint of quadratic...1982). Inexact Newton methods, SIAM Journal on Numerical Analysis 19, 400-408. [6] I. I. Dikin (1967). Iterative solution of problems of linear and

A two-dimensional hydrodynamic model of a tidal estuary

USGS Publications Warehouse

Walters, Roy A.; Cheng, Ralph T.

1979-01-01

A finite element model is described which is used in the computation of tidal currents in an estuary. This numerical model is patterned after an existing algorithm and has been carefully tested in rectangular and curve-sided channels with constant and variable depth. One of the common uncertainties in this class of two-dimensional hydrodynamic models is the treatment of the lateral boundary conditions. Special attention is paid specifically to addressing this problem. To maintain continuity within the domain of interest, ‘smooth’ curve-sided elements must be used at all shoreline boundaries. The present model uses triangular, isoparametric elements with quadratic basis functions for the two velocity components and a linear basis function for water surface elevation. An implicit time integration is used and the model is unconditionally stable. The resultant governing equations are nonlinear owing to the advective and the bottom friction terms and are solved iteratively at each time step by the Newton-Raphson method. Model test runs have been made in the southern portion of San Francisco Bay, California (South Bay) as well as in the Bay west of Carquinez Strait. Owing to the complex bathymetry, the hydrodynamic characteristics of the Bay system are dictated by the generally shallow basins which contain deep, relict river channels. Great care must be exercised to ensure that the conservation equations remain locally as well as globally accurate. Simulations have been made over several representative tidal cycles using this finite element model, and the results compare favourably with existing data. In particular, the standing wave in South Bay and the progressive wave in the northern reach are well represented.

Conservative tightly-coupled simulations of stochastic multiscale systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Taverniers, Søren; Pigarov, Alexander Y.; Tartakovsky, Daniel M., E-mail: dmt@ucsd.edu

2016-05-15

Multiphysics problems often involve components whose macroscopic dynamics is driven by microscopic random fluctuations. The fidelity of simulations of such systems depends on their ability to propagate these random fluctuations throughout a computational domain, including subdomains represented by deterministic solvers. When the constituent processes take place in nonoverlapping subdomains, system behavior can be modeled via a domain-decomposition approach that couples separate components at the interfaces between these subdomains. Its coupling algorithm has to maintain a stable and efficient numerical time integration even at high noise strength. We propose a conservative domain-decomposition algorithm in which tight coupling is achieved by employingmore » either Picard's or Newton's iterative method. Coupled diffusion equations, one of which has a Gaussian white-noise source term, provide a computational testbed for analysis of these two coupling strategies. Fully-converged (“implicit”) coupling with Newton's method typically outperforms its Picard counterpart, especially at high noise levels. This is because the number of Newton iterations scales linearly with the amplitude of the Gaussian noise, while the number of Picard iterations can scale superlinearly. At large time intervals between two subsequent inter-solver communications, the solution error for single-iteration (“explicit”) Picard's coupling can be several orders of magnitude higher than that for implicit coupling. Increasing the explicit coupling's communication frequency reduces this difference, but the resulting increase in computational cost can make it less efficient than implicit coupling at similar levels of solution error, depending on the communication frequency of the latter and the noise strength. This trend carries over into higher dimensions, although at high noise strength explicit coupling may be the only computationally viable option.« less

Iterative algorithms for computing the feedback Nash equilibrium point for positive systems

NASA Astrophysics Data System (ADS)

Ivanov, I.; Imsland, Lars; Bogdanova, B.

2017-03-01

The paper studies N-player linear quadratic differential games on an infinite time horizon with deterministic feedback information structure. It introduces two iterative methods (the Newton method as well as its accelerated modification) in order to compute the stabilising solution of a set of generalised algebraic Riccati equations. The latter is related to the Nash equilibrium point of the considered game model. Moreover, we derive the sufficient conditions for convergence of the proposed methods. Finally, we discuss two numerical examples so as to illustrate the performance of both of the algorithms.

Parallel Newton-Krylov-Schwarz algorithms for the transonic full potential equation

NASA Technical Reports Server (NTRS)

Cai, Xiao-Chuan; Gropp, William D.; Keyes, David E.; Melvin, Robin G.; Young, David P.

1996-01-01

We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The overall algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, is robust and, economical for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report their effect on numerical convergence rate, overall execution time, and parallel efficiency on a distributed-memory parallel computer.

Calibration and compensation method of three-axis geomagnetic sensor based on pre-processing total least square iteration

NASA Astrophysics Data System (ADS)

Zhou, Y.; Zhang, X.; Xiao, W.

2018-04-01

As the geomagnetic sensor is susceptible to interference, a pre-processing total least square iteration method is proposed for calibration compensation. Firstly, the error model of the geomagnetic sensor is analyzed and the correction model is proposed, then the characteristics of the model are analyzed and converted into nine parameters. The geomagnetic data is processed by Hilbert transform (HHT) to improve the signal-to-noise ratio, and the nine parameters are calculated by using the combination of Newton iteration method and the least squares estimation method. The sifter algorithm is used to filter the initial value of the iteration to ensure that the initial error is as small as possible. The experimental results show that this method does not need additional equipment and devices, can continuously update the calibration parameters, and better than the two-step estimation method, it can compensate geomagnetic sensor error well.

3-D minimum-structure inversion of magnetotelluric data using the finite-element method and tetrahedral grids

NASA Astrophysics Data System (ADS)

Jahandari, H.; Farquharson, C. G.

2017-11-01

Unstructured grids enable representing arbitrary structures more accurately and with fewer cells compared to regular structured grids. These grids also allow more efficient refinements compared to rectilinear meshes. In this study, tetrahedral grids are used for the inversion of magnetotelluric (MT) data, which allows for the direct inclusion of topography in the model, for constraining an inversion using a wireframe-based geological model and for local refinement at the observation stations. A minimum-structure method with an iterative model-space Gauss-Newton algorithm for optimization is used. An iterative solver is employed for solving the normal system of equations at each Gauss-Newton step and the sensitivity matrix-vector products that are required by this solver are calculated using pseudo-forward problems. This method alleviates the need to explicitly form the Hessian or Jacobian matrices which significantly reduces the required computation memory. Forward problems are formulated using an edge-based finite-element approach and a sparse direct solver is used for the solutions. This solver allows saving and re-using the factorization of matrices for similar pseudo-forward problems within a Gauss-Newton iteration which greatly minimizes the computation time. Two examples are presented to show the capability of the algorithm: the first example uses a benchmark model while the second example represents a realistic geological setting with topography and a sulphide deposit. The data that are inverted are the full-tensor impedance and the magnetic transfer function vector. The inversions sufficiently recovered the models and reproduced the data, which shows the effectiveness of unstructured grids for complex and realistic MT inversion scenarios. The first example is also used to demonstrate the computational efficiency of the presented model-space method by comparison with its data-space counterpart.

Implementing a Matrix-free Analytical Jacobian to Handle Nonlinearities in Models of 3D Lithospheric Deformation

NASA Astrophysics Data System (ADS)

Kaus, B.; Popov, A.

2015-12-01

The analytical expression for the Jacobian is a key component to achieve fast and robust convergence of the nonlinear Newton-Raphson iterative solver. Accomplishing this task in practice often requires a significant algebraic effort. Therefore it is quite common to use a cheap alternative instead, for example by approximating the Jacobian with a finite difference estimation. Despite its simplicity it is a relatively fragile and unreliable technique that is sensitive to the scaling of the residual and unknowns, as well as to the perturbation parameter selection. Unfortunately no universal rule can be applied to provide both a robust scaling and a perturbation. The approach we use here is to derive the analytical Jacobian for the coupled set of momentum, mass, and energy conservation equations together with the elasto-visco-plastic rheology and a marker in cell/staggered finite difference method. The software project LaMEM (Lithosphere and Mantle Evolution Model) is primarily developed for the thermo-mechanically coupled modeling of the 3D lithospheric deformation. The code is based on a staggered grid finite difference discretization in space, and uses customized scalable solvers form PETSc library to efficiently run on the massively parallel machines (such as IBM Blue Gene/Q). Currently LaMEM relies on the Jacobian-Free Newton-Krylov (JFNK) nonlinear solver, which approximates the Jacobian-vector product using a simple finite difference formula. This approach never requires an assembled Jacobian matrix and uses only the residual computation routine. We use an approximate Jacobian (Picard) matrix to precondition the Krylov solver with the Galerkin geometric multigrid. Because of the inherent problems of the finite difference Jacobian estimation, this approach doesn't always result in stable convergence. In this work we present and discuss a matrix-free technique in which the Jacobian-vector product is replaced by analytically-derived expressions and compare results with those obtained with a finite difference approximation of the Jacobian. This project is funded by ERC Starting Grant 258830 and computer facilities were provided by Jülich supercomputer center (Germany).

Self adaptive solution strategies: Locally bound constrained Newton Raphson solution algorithms

NASA Technical Reports Server (NTRS)

Padovan, Joe

1991-01-01

A summary is given of strategies which enable the automatic adjustment of the constraint surfaces recently used to extend the range and numerical stability/efficiency of nonlinear finite element equation solvers. In addition to handling kinematic and material induced nonlinearity, both pre-and postbuckling behavior can be treated. The scheme employs localized bounds on various hierarchical partitions of the field variables. These are used to resize, shape, and orient the global constraint surface, thereby enabling essentially automatic load/deflection incrementation. Due to the generality of the approach taken, it can be implemented in conjunction with the constraints of an arbitrary functional type. To benchmark the method, several numerical experiments are presented. These include problems involving kinematic and material nonlinearity, as well as pre- and postbuckling characteristics. Also included is a list of papers published in the course of the work.

A unified convergence theory of a numerical method, and applications to the replenishment policies.

PubMed

Mi, Xiang-jiang; Wang, Xing-hua

2004-01-01

In determining the replenishment policy for an inventory system, some researchers advocated that the iterative method of Newton could be applied to the derivative of the total cost function in order to get the optimal solution. But this approach requires calculation of the second derivative of the function. Avoiding this complex computation we use another iterative method presented by the second author. One of the goals of this paper is to present a unified convergence theory of this method. Then we give a numerical example to show the application of our theory.

Cross-entropy embedding of high-dimensional data using the neural gas model.

PubMed

Estévez, Pablo A; Figueroa, Cristián J; Saito, Kazumi

2005-01-01

A cross-entropy approach to mapping high-dimensional data into a low-dimensional space embedding is presented. The method allows to project simultaneously the input data and the codebook vectors, obtained with the Neural Gas (NG) quantizer algorithm, into a low-dimensional output space. The aim of this approach is to preserve the relationship defined by the NG neighborhood function for each pair of input and codebook vectors. A cost function based on the cross-entropy between input and output probabilities is minimized by using a Newton-Raphson method. The new approach is compared with Sammon's non-linear mapping (NLM) and the hierarchical approach of combining a vector quantizer such as the self-organizing feature map (SOM) or NG with the NLM recall algorithm. In comparison with these techniques, our method delivers a clear visualization of both data points and codebooks, and it achieves a better mapping quality in terms of the topology preservation measure q(m).

Global convergence of inexact Newton methods for transonic flow

NASA Technical Reports Server (NTRS)

Young, David P.; Melvin, Robin G.; Bieterman, Michael B.; Johnson, Forrester T.; Samant, Satish S.

1990-01-01

In computational fluid dynamics, nonlinear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these nonlinear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modeled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.

A Least-Squares Commutator in the Iterative Subspace Method for Accelerating Self-Consistent Field Convergence.

PubMed

Li, Haichen; Yaron, David J

2016-11-08

A least-squares commutator in the iterative subspace (LCIIS) approach is explored for accelerating self-consistent field (SCF) calculations. LCIIS is similar to direct inversion of the iterative subspace (DIIS) methods in that the next iterate of the density matrix is obtained as a linear combination of past iterates. However, whereas DIIS methods find the linear combination by minimizing a sum of error vectors, LCIIS minimizes the Frobenius norm of the commutator between the density matrix and the Fock matrix. This minimization leads to a quartic problem that can be solved iteratively through a constrained Newton's method. The relationship between LCIIS and DIIS is discussed. Numerical experiments suggest that LCIIS leads to faster convergence than other SCF convergence accelerating methods in a statistically significant sense, and in a number of cases LCIIS leads to stable SCF solutions that are not found by other methods. The computational cost involved in solving the quartic minimization problem is small compared to the typical cost of SCF iterations and the approach is easily integrated into existing codes. LCIIS can therefore serve as a powerful addition to SCF convergence accelerating methods in computational quantum chemistry packages.

Observations and Thermochemical Calculations for Hot-Jupiter Atmospheres

NASA Astrophysics Data System (ADS)

Blecic, Jasmina; Harrington, Joseph; Bowman, M. Oliver; Cubillos, Patricio; Stemm, Madison

2015-01-01

I present Spitzer eclipse observations for WASP-14b and WASP-43b, an open source tool for thermochemical equilibrium calculations, and components of an open source tool for atmospheric parameter retrieval from spectroscopic data. WASP-14b is a planet that receives high irradiation from its host star, yet, although theory does not predict it, the planet hosts a thermal inversion. The WASP-43b eclipses have signal-to-noise ratios of ~25, one of the largest among exoplanets. To assess these planets' atmospheric composition and thermal structure, we developed an open-source Bayesian Atmospheric Radiative Transfer (BART) code. My dissertation tasks included developing a Thermochemical Equilibrium Abundances (TEA) code, implementing the eclipse geometry calculation in BART's radiative transfer module, and generating parameterized pressure and temperature profiles so the radiative-transfer module can be driven by the statistical module.To initialize the radiative-transfer calculation in BART, TEA calculates the equilibrium abundances of gaseous molecular species at a given temperature and pressure. It uses the Gibbs-free-energy minimization method with an iterative Lagrangian optimization scheme. Given elemental abundances, TEA calculates molecular abundances for a particular temperature and pressure or a list of temperature-pressure pairs. The code is tested against the original method developed by White at al. (1958), the analytic method developed by Burrows and Sharp (1999), and the Newton-Raphson method implemented in the open-source Chemical Equilibrium with Applications (CEA) code. TEA, written in Python, is modular, documented, and available to the community via the open-source development site GitHub.com.Support for this work was provided by NASA Headquarters under the NASA Earth and Space Science Fellowship Program, grant NNX12AL83H, by NASA through an award issued by JPL/Caltech, and through the Science Mission Directorate's Planetary Atmospheres Program, grant NNX12AI69G.

Limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) method for the parameter estimation on geographically weighted ordinal logistic regression model (GWOLR)

NASA Astrophysics Data System (ADS)

Saputro, Dewi Retno Sari; Widyaningsih, Purnami

2017-08-01

In general, the parameter estimation of GWOLR model uses maximum likelihood method, but it constructs a system of nonlinear equations, making it difficult to find the solution. Therefore, an approximate solution is needed. There are two popular numerical methods: the methods of Newton and Quasi-Newton (QN). Newton's method requires large-scale time in executing the computation program since it contains Jacobian matrix (derivative). QN method overcomes the drawback of Newton's method by substituting derivative computation into a function of direct computation. The QN method uses Hessian matrix approach which contains Davidon-Fletcher-Powell (DFP) formula. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is categorized as the QN method which has the DFP formula attribute of having positive definite Hessian matrix. The BFGS method requires large memory in executing the program so another algorithm to decrease memory usage is needed, namely Low Memory BFGS (LBFGS). The purpose of this research is to compute the efficiency of the LBFGS method in the iterative and recursive computation of Hessian matrix and its inverse for the GWOLR parameter estimation. In reference to the research findings, we found out that the BFGS and LBFGS methods have arithmetic operation schemes, including O(n2) and O(nm).

A Flexible CUDA LU-based Solver for Small, Batched Linear Systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tumeo, Antonino; Gawande, Nitin A.; Villa, Oreste

This chapter presents the implementation of a batched CUDA solver based on LU factorization for small linear systems. This solver may be used in applications such as reactive flow transport models, which apply the Newton-Raphson technique to linearize and iteratively solve the sets of non linear equations that represent the reactions for ten of thousands to millions of physical locations. The implementation exploits somewhat counterintuitive GPGPU programming techniques: it assigns the solution of a matrix (representing a system) to a single CUDA thread, does not exploit shared memory and employs dynamic memory allocation on the GPUs. These techniques enable ourmore » implementation to simultaneously solve sets of systems with over 100 equations and to employ LU decomposition with complete pivoting, providing the higher numerical accuracy required by certain applications. Other currently available solutions for batched linear solvers are limited by size and only support partial pivoting, although they may result faster in certain conditions. We discuss the code of our implementation and present a comparison with the other implementations, discussing the various tradeoffs in terms of performance and flexibility. This work will enable developers that need batched linear solvers to choose whichever implementation is more appropriate to the features and the requirements of their applications, and even to implement dynamic switching approaches that can choose the best implementation depending on the input data.« less

Multiple zeros of polynomials

NASA Technical Reports Server (NTRS)

Wood, C. A.

1974-01-01

For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.

A fast, time-accurate unsteady full potential scheme

NASA Technical Reports Server (NTRS)

Shankar, V.; Ide, H.; Gorski, J.; Osher, S.

1985-01-01

The unsteady form of the full potential equation is solved in conservation form by an implicit method based on approximate factorization. At each time level, internal Newton iterations are performed to achieve time accuracy and computational efficiency. A local time linearization procedure is introduced to provide a good initial guess for the Newton iteration. A novel flux-biasing technique is applied to generate proper forms of the artificial viscosity to treat hyperbolic regions with shocks and sonic lines present. The wake is properly modeled by accounting not only for jumps in phi, but also for jumps in higher derivatives of phi, obtained by imposing the density to be continuous across the wake. The far field is modeled using the Riemann invariants to simulate nonreflecting boundary conditions. The resulting unsteady method performs well which, even at low reduced frequency levels of 0.1 or less, requires fewer than 100 time steps per cycle at transonic Mach numbers. The code is fully vectorized for the CRAY-XMP and the VPS-32 computers.

Generalized Gaussian wave packet dynamics: Integrable and chaotic systems.

PubMed

Pal, Harinder; Vyas, Manan; Tomsovic, Steven

2016-01-01

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.

Peak-Seeking Optimization of Spanwise Lift Distribution for Wings in Formation Flight

NASA Technical Reports Server (NTRS)

Hanson, Curtis E.; Ryan, Jack

2012-01-01

A method is presented for the in-flight optimization of the lift distribution across the wing for minimum drag of an aircraft in formation flight. The usual elliptical distribution that is optimal for a given wing with a given span is no longer optimal for the trailing wing in a formation due to the asymmetric nature of the encountered flow field. Control surfaces along the trailing edge of the wing can be configured to obtain a non-elliptical profile that is more optimal in terms of minimum combined induced and profile drag. Due to the difficult-to-predict nature of formation flight aerodynamics, a Newton-Raphson peak-seeking controller is used to identify in real time the best aileron and flap deployment scheme for minimum total drag. Simulation results show that the peak-seeking controller correctly identifies an optimal trim configuration that provides additional drag savings above those achieved with conventional anti-symmetric aileron trim.

Experimental study of trajectory planning and control of a high precision robot manipulator

NASA Technical Reports Server (NTRS)

Nguyen, Charles C.; Antrazi, Sami S.

1991-01-01

The kinematic and trajectory planning is presented for a 6 DOF end-effector whose design was based on the Stewart Platform mechanism. The end-effector was used as a testbed for studying robotic assembly of NASA hardware with passive compliance. Vector analysis was employed to derive a closed-form solution for the end-effector inverse kinematic transformation. A computationally efficient numerical solution was obtained for the end-effector forward kinematic transformation using Newton-Raphson method. Three trajectory planning schemes, two for fine motion and one for gross motion, were developed for the end-effector. Experiments conducted to evaluate the performance of the trajectory planning schemes showed excellent tracking quality with minimal errors. Current activities focus on implementing the developed trajectory planning schemes on mating and demating space-rated connectors and using the compliant platform to acquire forces/torques applied on the end-effector during the assembly task.

Transparent binary-thickness coatings on metal substrates that produce binary patterns of orthogonal elliptical polarization states in reflected light

NASA Astrophysics Data System (ADS)

Azzam, Rasheed M. A.; Angel, Wade W.

1992-12-01

A reflective division-of-wavefront polarizing beam splitter is described that uses a dual- thickness transparent thin-film coating on a metal substrate. A previous design that used a partially clad substrate at the principal angle of the metal [Azzam, JOSA A 5, 1576 (1988)] is replaced by a more general one in which the substrate is coated throughout and the film thickness alternates between two non-zero levels. The incident linear polarization azimuth is chosen near, but not restricted to, 45 degree(s) (measured from the plane of incidence), and the angle of incidence may be selected over a range of values. The design procedure, which uses the two-dimensional Newton-Raphson method, is applied to the SiO2-Au film- substrate system at 633 nm wavelength, as an example, and the characteristics of the various possible coatings are presented.

[Poverty profile regarding households participating in a food assistance program].

PubMed

Álvarez-Uribe, Martha C; Aguirre-Acevedo, Daniel C

2012-06-01

This study was aimed at establishing subgroups having specific socioeconomic characteristics by using latent class analysis as a method for segmenting target population members of the MANA-ICBF supplementary food program in the Antioquia department of Colombia and determine their differences regarding poverty and health conditions in efficiently addressing pertinent resources, programs and policies. The target population consisted of 200,000 children and their households involved in the MANA food assistance program; a representative sample by region was used. Latent class analysis was used, as were the expectation-maximization and Newton Raphson algorithms for identifying the appropriate number of classes. The final model classified the households into four clusters or classes, differing according to well-defined socio-demographic conditions affecting children's health. Some homes had a greater depth of poverty, therefore lowering the families' quality of life and affecting the health of the children in this age group.

A theoretical study on tunneling based biosensor having a redox-active monolayer using physics based simulation

NASA Astrophysics Data System (ADS)

Kim, Kyoung Yeon; Lee, Won Cheol; Yun, Jun Yeon; Lee, Youngeun; Choi, Seoungwook; Jin, Seonghoon; Park, Young June

2018-01-01

We developed a numerical simulator to model the operation of a tunneling based biosensor which has a redox-active monolayer. The simulator takes a realistic device structure as a simulation domain, and it employs the drift-diffusion equation for ion transport, the non-equilibrium Green's function formalism for electron tunneling, and the Ramo-Shockley theorem for accurate calculation of non-faradaic current. We also accounted for the buffer reaction and the immobilized peptide layer. For efficient transient simulation, the implicit time integration scheme is employed where the solution at each time step is obtained from the coupled Newton-Raphson method. As an application, we studied the operation of a recently fabricated reference-electrode free biosensor in various bias conditions and confirmed the effect of buffer reaction and the current flowing mechanism. Using the simulator, we also found a strategy to maximize the sensitivity of the tunneling based sensor.

Numerical Modeling of Saturated Boiling in a Heated Tube

NASA Technical Reports Server (NTRS)

Majumdar, Alok; LeClair, Andre; Hartwig, Jason

2017-01-01

This paper describes a mathematical formulation and numerical solution of boiling in a heated tube. The mathematical formulation involves a discretization of the tube into a flow network consisting of fluid nodes and branches and a thermal network consisting of solid nodes and conductors. In the fluid network, the mass, momentum and energy conservation equations are solved and in the thermal network, the energy conservation equation of solids is solved. A pressure-based, finite-volume formulation has been used to solve the equations in the fluid network. The system of equations is solved by a hybrid numerical scheme which solves the mass and momentum conservation equations by a simultaneous Newton-Raphson method and the energy conservation equation by a successive substitution method. The fluid network and thermal network are coupled through heat transfer between the solid and fluid nodes which is computed by Chen's correlation of saturated boiling heat transfer. The computer model is developed using the Generalized Fluid System Simulation Program and the numerical predictions are compared with test data.

Demonstration of the Dynamic Flowgraph Methodology using the Titan 2 Space Launch Vehicle Digital Flight Control System

NASA Technical Reports Server (NTRS)

Yau, M.; Guarro, S.; Apostolakis, G.

1993-01-01

Dynamic Flowgraph Methodology (DFM) is a new approach developed to integrate the modeling and analysis of the hardware and software components of an embedded system. The objective is to complement the traditional approaches which generally follow the philosophy of separating out the hardware and software portions of the assurance analysis. In this paper, the DFM approach is demonstrated using the Titan 2 Space Launch Vehicle Digital Flight Control System. The hardware and software portions of this embedded system are modeled in an integrated framework. In addition, the time dependent behavior and the switching logic can be captured by this DFM model. In the modeling process, it is found that constructing decision tables for software subroutines is very time consuming. A possible solution is suggested. This approach makes use of a well-known numerical method, the Newton-Raphson method, to solve the equations implemented in the subroutines in reverse. Convergence can be achieved in a few steps.

Robustness of Modeling of Out-of-Service Gas Mechanical Face Seal

NASA Technical Reports Server (NTRS)

Green, Itzhak

2007-01-01

Gas lubricated mechanical face seal are ubiquitous in many high performance applications such as compressors and gas turbines. The literature contains various analyses of seals having orderly face patterns (radial taper, waves, spiral grooves, etc.). These are useful for design purposes and for performance predictions. However, seals returning from service (or from testing) inevitably contain wear tracks and warped faces that depart from the aforementioned orderly patterns. Questions then arise as to the heat generated at the interface, leakage rates, axial displacement and tilts, minimum film thickness, contact forces, etc. This work describes an analysis of seals that may inherit any (i.e., random) face pattern. A comprehensive computer code is developed, based upon the Newton- Raphson method, which solves for the equilibrium of the axial force and tilting moments that are generated by asperity contact and fluid film effects. A contact mechanics model is incorporated along with a finite volume method that solves the compressible Reynolds equation. Results are presented for a production seal that has sustained a testing cycle.

Spongiosa Primary Development: A Biochemical Hypothesis by Turing Patterns Formations

PubMed Central

López-Vaca, Oscar Rodrigo; Garzón-Alvarado, Diego Alexander

2012-01-01

We propose a biochemical model describing the formation of primary spongiosa architecture through a bioregulatory model by metalloproteinase 13 (MMP13) and vascular endothelial growth factor (VEGF). It is assumed that MMP13 regulates cartilage degradation and the VEGF allows vascularization and advances in the ossification front through the presence of osteoblasts. The coupling of this set of molecules is represented by reaction-diffusion equations with parameters in the Turing space, creating a stable spatiotemporal pattern that leads to the formation of the trabeculae present in the spongy tissue. Experimental evidence has shown that the MMP13 regulates VEGF formation, and it is assumed that VEGF negatively regulates MMP13 formation. Thus, the patterns obtained by ossification may represent the primary spongiosa formation during endochondral ossification. Moreover, for the numerical solution, we used the finite element method with the Newton-Raphson method to approximate partial differential nonlinear equations. Ossification patterns obtained may represent the primary spongiosa formation during endochondral ossification. PMID:23193429

Transmission Loss Calculation using A and B Loss Coefficients in Dynamic Economic Dispatch Problem

NASA Astrophysics Data System (ADS)

Jethmalani, C. H. Ram; Dumpa, Poornima; Simon, Sishaj P.; Sundareswaran, K.

2016-04-01

This paper analyzes the performance of A-loss coefficients while evaluating transmission losses in a Dynamic Economic Dispatch (DED) Problem. The performance analysis is carried out by comparing the losses computed using nominal A loss coefficients and nominal B loss coefficients in reference with load flow solution obtained by standard Newton-Raphson (NR) method. Density based clustering method based on connected regions with sufficiently high density (DBSCAN) is employed in identifying the best regions of A and B loss coefficients. Based on the results obtained through cluster analysis, a novel approach in improving the accuracy of network loss calculation is proposed. Here, based on the change in per unit load values between the load intervals, loss coefficients are updated for calculating the transmission losses. The proposed algorithm is tested and validated on IEEE 6 bus system, IEEE 14 bus, system IEEE 30 bus system and IEEE 118 bus system. All simulations are carried out using SCILAB 5.4 (www.scilab.org) which is an open source software.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lin, Paul T.; Shadid, John N.; Tsuji, Paul H.

Here, this study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid (AMG) preconditioned Newton- Krylov solution approach applied to a fully-implicit variational multiscale (VMS) nite element (FE) resistive magnetohydrodynamics (MHD) formulation. In this context a Newton iteration is used for the nonlinear system and a Krylov (GMRES) method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play a critical role. Krylov smoothers are considered inmore » an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. Three time dependent resistive MHD test cases are considered to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.« less

Computerized Business Calculus Using Calculators, Examples from Mathematics to Finance.

ERIC Educational Resources Information Center

Vest, Floyd

1991-01-01

After discussing the role of supercalculators within the business calculus curriculum, several examples are presented which allow the reader to examine the capabilities and codes of calculators specific to different major manufacturers. The topics examined include annuities, Newton's method, fixed point iteration, graphing, solvers, and…

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

PubMed Central

Asgharzadeh, Hafez; Borazjani, Iman

2016-01-01

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42 – 74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge-Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge-Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal Jacobian when the stretching factor was increased, respectively. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80–90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future. PMID:28042172

Robust large-scale parallel nonlinear solvers for simulations.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bader, Brett William; Pawlowski, Roger Patrick; Kolda, Tamara Gibson

2005-11-01

This report documents research to develop robust and efficient solution techniques for solving large-scale systems of nonlinear equations. The most widely used method for solving systems of nonlinear equations is Newton's method. While much research has been devoted to augmenting Newton-based solvers (usually with globalization techniques), little has been devoted to exploring the application of different models. Our research has been directed at evaluating techniques using different models than Newton's method: a lower order model, Broyden's method, and a higher order model, the tensor method. We have developed large-scale versions of each of these models and have demonstrated their usemore » in important applications at Sandia. Broyden's method replaces the Jacobian with an approximation, allowing codes that cannot evaluate a Jacobian or have an inaccurate Jacobian to converge to a solution. Limited-memory methods, which have been successful in optimization, allow us to extend this approach to large-scale problems. We compare the robustness and efficiency of Newton's method, modified Newton's method, Jacobian-free Newton-Krylov method, and our limited-memory Broyden method. Comparisons are carried out for large-scale applications of fluid flow simulations and electronic circuit simulations. Results show that, in cases where the Jacobian was inaccurate or could not be computed, Broyden's method converged in some cases where Newton's method failed to converge. We identify conditions where Broyden's method can be more efficient than Newton's method. We also present modifications to a large-scale tensor method, originally proposed by Bouaricha, for greater efficiency, better robustness, and wider applicability. Tensor methods are an alternative to Newton-based methods and are based on computing a step based on a local quadratic model rather than a linear model. The advantage of Bouaricha's method is that it can use any existing linear solver, which makes it simple to write and easily portable. However, the method usually takes twice as long to solve as Newton-GMRES on general problems because it solves two linear systems at each iteration. In this paper, we discuss modifications to Bouaricha's method for a practical implementation, including a special globalization technique and other modifications for greater efficiency. We present numerical results showing computational advantages over Newton-GMRES on some realistic problems. We further discuss a new approach for dealing with singular (or ill-conditioned) matrices. In particular, we modify an algorithm for identifying a turning point so that an increasingly ill-conditioned Jacobian does not prevent convergence.« less

Comparison and Tensorial Formulation of Inelastic Constitutive Models of Salt Rock Behaviour and Efficient Numerical Implementatio

NASA Astrophysics Data System (ADS)

Nagel, T.; Böttcher, N.; Görke, U. J.; Kolditz, O.

2014-12-01

The design process of geotechnical installations includes the application of numerical simulation tools for safety assessment, dimensioning and long term effectiveness estimations. Underground salt caverns can be used for the storage of natural gas, hydrogen, oil, waste or compressed air. For their design one has to take into account fluctuating internal pressures due to different levels of filling, the stresses imposed by the surrounding rock mass, irregular geometries and possibly heterogeneous material properties [3] in order to estimate long term cavern convergence as well as locally critical wall stresses. Constitutive models applied to rock salt are usually viscoplastic in nature and most often based on a Burgers-type rheological model extended by non-linear viscosity functions and/or plastic friction elements. Besides plastic dilatation, healing and damage are sometimes accounted for as well [2]. The scales of the geotechnical system to be simulated and the laboratory tests from which material parameters are determined are vastly different. The most common material testing modalities to determine material parameters in geoengineering are the uniaxial and the triaxial compression tests. Some constitutive formulations in widespread use are formulated based on equivalent rather than tensorial quantities valid under these specific test conditions and are subsequently applied to heterogeneous underground systems and complex 3D load cases. We show here that this procedure is inappropriate and can lead to erroneous results. We further propose alternative formulations of the constitutive models in question that restore their validity under arbitrary loading conditions. For an efficient numerical simulation, the discussed constitutive models are integrated locally with a Newton-Raphson algorithm that directly provides the algorithmically consistent tangent matrix for the global Newton iteration of the displacement based finite element formulation. Finally, the finite element implementations of the proposed constitutive formulations are employed to simulate an underground salt cavern used for compressed air energy storage with OpenGeoSys [1]. Transient convergence and stress fields are evaluated for typical fluctuating operation pressure regimes.

A numerical method for measuring capacitive soft sensors through one channel

NASA Astrophysics Data System (ADS)

Tairych, Andreas; Anderson, Iain A.

2018-03-01

Soft capacitive stretch sensors are well suited for unobtrusive wearable body motion capture. Conventional sensing methods measure sensor capacitances through separate channels. In sensing garments with many sensors, this results in high wiring complexity, and a large footprint of rigid sensing circuit boards. We have developed a more efficient sensing method that detects multiple sensors through only one channel, and one set of wires. It is based on a R-C transmission line assembled from capacitive conductive fabric stretch sensors, and external resistors. The unknown capacitances are identified by solving a system of nonlinear equations. These equations are established by modelling and continuously measuring transmission line reactances at different frequencies. Solving these equations numerically with a Newton-Raphson solver for the unknown capacitances enables real time reading of all sensors. The method was verified with a prototype comprising three sensors that is capable of detecting both individually and simultaneously stretched sensors. Instead of using three channels and six wires to detect the sensors, the task was achieved with only one channel and two wires.

PAFAC- PLASTIC AND FAILURE ANALYSIS OF COMPOSITES

NASA Technical Reports Server (NTRS)

Bigelow, C. A.

1994-01-01

The increasing number of applications of fiber-reinforced composites in industry demands a detailed understanding of their material properties and behavior. A three-dimensional finite-element computer program called PAFAC (Plastic and Failure Analysis of Composites) has been developed for the elastic-plastic analysis of fiber-reinforced composite materials and structures. The evaluation of stresses and deformations at edges, cut-outs, and joints is essential in understanding the strength and failure for metal-matrix composites since the onset of plastic yielding starts very early in the loading process as compared to the composite's ultimate strength. Such comprehensive analysis can only be achieved by a finite-element program like PAFAC. PAFAC is particularly suited for the analysis of laminated metal-matrix composites. It can model the elastic-plastic behavior of the matrix phase while the fibers remain elastic. Since the PAFAC program uses a three-dimensional element, the program can also model the individual layers of the laminate to account for thickness effects. In PAFAC, the composite is modeled as a continuum reinforced by cylindrical fibers of vanishingly small diameter which occupy a finite volume fraction of the composite. In this way, the essential axial constraint of the phases is retained. Furthermore, the local stress and strain fields are uniform. The PAFAC finite-element solution is obtained using the displacement method. Solution of the nonlinear equilibrium equations is obtained with a Newton-Raphson iteration technique. The elastic-plastic behavior of composites consisting of aligned, continuous elastic filaments and an elastic-plastic matrix is described in terms of the constituent properties, their volume fractions, and mutual constraints between phases indicated by the geometry of the microstructure. The program uses an iterative procedure to determine the overall response of the laminate, then from the overall response determines the stress state in each phase of the composite material. Failure of the fibers or matrix within an element can also be modeled by PAFAC. PAFAC is written in FORTRAN IV for batch execution and has been implemented on a CDC CYBER 170 series computer with a segmented memory requirement of approximately 66K (octal) of 60 bit words. PAFAC was developed in 1982.

Concerning an application of the method of least squares with a variable weight matrix

NASA Technical Reports Server (NTRS)

Sukhanov, A. A.

1979-01-01

An estimate of a state vector for a physical system when the weight matrix in the method of least squares is a function of this vector is considered. An iterative procedure is proposed for calculating the desired estimate. Conditions for the existence and uniqueness of the limit of this procedure are obtained, and a domain is found which contains the limit estimate. A second method for calculating the desired estimate which reduces to the solution of a system of algebraic equations is proposed. The question of applying Newton's method of tangents to solving the given system of algebraic equations is considered and conditions for the convergence of the modified Newton's method are obtained. Certain properties of the estimate obtained are presented together with an example.

Numerical methods for the design of gradient-index optical coatings.

PubMed

Anzengruber, Stephan W; Klann, Esther; Ramlau, Ronny; Tonova, Diana

2012-12-01

We formulate the problem of designing gradient-index optical coatings as the task of solving a system of operator equations. We use iterative numerical procedures known from the theory of inverse problems to solve it with respect to the coating refractive index profile and thickness. The mathematical derivations necessary for the application of the procedures are presented, and different numerical methods (Landweber, Newton, and Gauss-Newton methods, Tikhonov minimization with surrogate functionals) are implemented. Procedures for the transformation of the gradient coating designs into quasi-gradient ones (i.e., multilayer stacks of homogeneous layers with different refractive indices) are also developed. The design algorithms work with physically available coating materials that could be produced with the modern coating technologies.

On multilevel RBF collocation to solve nonlinear PDEs arising from endogenous stochastic volatility models

NASA Astrophysics Data System (ADS)

Bastani, Ali Foroush; Dastgerdi, Maryam Vahid; Mighani, Abolfazl

2018-06-01

The main aim of this paper is the analytical and numerical study of a time-dependent second-order nonlinear partial differential equation (PDE) arising from the endogenous stochastic volatility model, introduced in [Bensoussan, A., Crouhy, M. and Galai, D., Stochastic equity volatility related to the leverage effect (I): equity volatility behavior. Applied Mathematical Finance, 1, 63-85, 1994]. As the first step, we derive a consistent set of initial and boundary conditions to complement the PDE, when the firm is financed by equity and debt. In the sequel, we propose a Newton-based iteration scheme for nonlinear parabolic PDEs which is an extension of a method for solving elliptic partial differential equations introduced in [Fasshauer, G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43, 423-438, 2002]. The scheme is based on multilevel collocation using radial basis functions (RBFs) to solve the resulting locally linearized elliptic PDEs obtained at each level of the Newton iteration. We show the effectiveness of the resulting framework by solving a prototypical example from the field and compare the results with those obtained from three different techniques: (1) a finite difference discretization; (2) a naive RBF collocation and (3) a benchmark approximation, introduced for the first time in this paper. The numerical results confirm the robustness, higher convergence rate and good stability properties of the proposed scheme compared to other alternatives. We also comment on some possible research directions in this field.

A contracting-interval program for the Danilewski method. Ph.D. Thesis - Va. Univ.

NASA Technical Reports Server (NTRS)

Harris, J. D.

1971-01-01

The concept of contracting-interval programs is applied to finding the eigenvalues of a matrix. The development is a three-step process in which (1) a program is developed for the reduction of a matrix to Hessenberg form, (2) a program is developed for the reduction of a Hessenberg matrix to colleague form, and (3) the characteristic polynomial with interval coefficients is readily obtained from the interval of colleague matrices. This interval polynomial is then factored into quadratic factors so that the eigenvalues may be obtained. To develop a contracting-interval program for factoring this polynomial with interval coefficients it is necessary to have an iteration method which converges even in the presence of controlled rounding errors. A theorem is stated giving sufficient conditions for the convergence of Newton's method when both the function and its Jacobian cannot be evaluated exactly but errors can be made proportional to the square of the norm of the difference between the previous two iterates. This theorem is applied to prove the convergence of the generalization of the Newton-Bairstow method that is used to obtain quadratic factors of the characteristic polynomial.

One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure

NASA Technical Reports Server (NTRS)

Amar, Adam J.; Blackwell, Ben F.; Edwards, Jack R.

2007-01-01

The development and verification of a one-dimensional material thermal response code with ablation is presented. The implicit time integrator, control volume finite element spatial discretization, and Newton's method for nonlinear iteration on the entire system of residual equations have been implemented and verified for the thermochemical ablation of internally decomposing materials. This study is a continuation of the work presented in "One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure" (AIAA-2006-2910), which described the derivation, implementation, and verification of the constant density solid energy equation terms and boundary conditions. The present study extends the model to decomposing materials including decomposition kinetics, pyrolysis gas flow through the porous char layer, and a mixture (solid and gas) energy equation. Verification results are presented for the thermochemical ablation of a carbon-phenolic ablator which involves the solution of the entire system of governing equations.

Time domain contact model for tyre/road interaction including nonlinear contact stiffness due to small-scale roughness

NASA Astrophysics Data System (ADS)

Andersson, P. B. U.; Kropp, W.

2008-11-01

Rolling resistance, traction, wear, excitation of vibrations, and noise generation are all attributes to consider in optimisation of the interaction between automotive tyres and wearing courses of roads. The key to understand and describe the interaction is to include a wide range of length scales in the description of the contact geometry. This means including scales on the order of micrometres that have been neglected in previous tyre/road interaction models. A time domain contact model for the tyre/road interaction that includes interfacial details is presented. The contact geometry is discretised into multiple elements forming pairs of matching points. The dynamic response of the tyre is calculated by convolving the contact forces with pre-calculated Green's functions. The smaller-length scales are included by using constitutive interfacial relations, i.e. by using nonlinear contact springs, for each pair of contact elements. The method is presented for normal (out-of-plane) contact and a method for assessing the stiffness of the nonlinear springs based on detailed geometry and elastic data of the tread is suggested. The governing equations of the nonlinear contact problem are solved with the Newton-Raphson iterative scheme. Relations between force, indentation, and contact stiffness are calculated for a single tread block in contact with a road surface. The calculated results have the same character as results from measurements found in literature. Comparison to traditional contact formulations shows that the effect of the small-scale roughness is large; the contact stiffness is only up to half of the stiffness that would result if contact is made over the whole element directly to the bulk of the tread. It is concluded that the suggested contact formulation is a suitable model to include more details of the contact interface. Further, the presented result for the tread block in contact with the road is a suitable input for a global tyre/road interaction model that is also based on the presented contact formulation.

Rapid computation of chemical equilibrium composition - An application to hydrocarbon combustion

NASA Technical Reports Server (NTRS)

Erickson, W. D.; Prabhu, R. K.

1986-01-01

A scheme for rapidly computing the chemical equilibrium composition of hydrocarbon combustion products is derived. A set of ten governing equations is reduced to a single equation that is solved by the Newton iteration method. Computation speeds are approximately 80 times faster than the often used free-energy minimization method. The general approach also has application to many other chemical systems.

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sousedík, Bedřich, E-mail: sousedik@umbc.edu; Elman, Howard C., E-mail: elman@cs.umd.edu

2016-07-01

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

DOE PAGES

Sousedík, Bedřich; Elman, Howard C.

2016-04-12

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmarkmore » problems.« less

NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES

DOE Office of Scientific and Technical Information (OSTI.GOV)

Christensen, Max La Cour; Villa, Umberto E.; Engsig-Karup, Allan P.

The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, ourmore » FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.« less

Soil heating and evaporation under extreme conditions: Forest fires and slash pile burns

NASA Astrophysics Data System (ADS)

Massman, W. J.

2011-12-01

Heating any soil during a sufficiently intense wild fire or prescribed burn can alter soil irreversibly, resulting in many significant and well known, long term biological, chemical, and hydrological effects. To better understand how fire impacts soil, especially considering the increasing probability of wildfires that is being driven by climate change and the increasing use of prescribe burns by land managers, it is important to better understand the dynamics of the coupled heat and moisture transport in soil during these extreme heating events. Furthermore, improving understanding of heat and mass transport during such extreme conditions should also provide insights into the associated transport mechanisms under more normal conditions as well. Here I describe the development of a new model designed to simulate soil heat and moisture transport during fires where the surface heating often ranges between 10,000 and 100,000 Wm-2 for several minutes to several hours. Model performance is tested against laboratory measurements of soil temperature and moisture changes at several depths during controlled heating events created with an extremely intense radiant heater. The laboratory tests employed well described soils with well known physical properties. The model, on the other hand, is somewhat unusual in that it employs formulations for temperature dependencies of the soil specific heat, thermal conductivity, and the water retention curve (relation between soil moisture and soil moisture potential). It also employs a new formulation for the surface evaporation rate as a component of the upper boundary condition, as well as the Newton-Raphson method and the generalized Thomas algorithm for inverting block tri-diagonal matrices to solve for soil temperature and soil moisture potential. Model results show rapid evaporation rates with significant vapor transfer not only to the free atmosphere above the soil, but to lower depths of the soil, where the vapor re-condenses ahead of the heating front. Consequently the trajectory of the solution (soil volumetric water content versus soil temperature) is very unusual and highly nonlinear, which may explain why more traditional methods (i.e., those based on finite difference or finite element approaches) tend to show more numerical instabilities than the Newton-Raphson method when used to model these extreme conditions. But, despite the intuitive and qualitative appeal of the model's numerical solution, it underestimates the rate of soil moisture loss observed during the laboratory trials, although the soil temperatures are reasonably well simulated.

Highly Accurate Analytical Approximate Solution to a Nonlinear Pseudo-Oscillator

NASA Astrophysics Data System (ADS)

Wu, Baisheng; Liu, Weijia; Lim, C. W.

2017-07-01

A second-order Newton method is presented to construct analytical approximate solutions to a nonlinear pseudo-oscillator in which the restoring force is inversely proportional to the dependent variable. The nonlinear equation is first expressed in a specific form, and it is then solved in two steps, a predictor and a corrector step. In each step, the harmonic balance method is used in an appropriate manner to obtain a set of linear algebraic equations. With only one simple second-order Newton iteration step, a short, explicit, and highly accurate analytical approximate solution can be derived. The approximate solutions are valid for all amplitudes of the pseudo-oscillator. Furthermore, the method incorporates second-order Taylor expansion in a natural way, and it is of significant faster convergence rate.

A superlinear interior points algorithm for engineering design optimization

NASA Technical Reports Server (NTRS)

Herskovits, J.; Asquier, J.

1990-01-01

We present a quasi-Newton interior points algorithm for nonlinear constrained optimization. It is based on a general approach consisting of the iterative solution in the primal and dual spaces of the equalities in Karush-Kuhn-Tucker optimality conditions. This is done in such a way to have primal and dual feasibility at each iteration, which ensures satisfaction of those optimality conditions at the limit points. This approach is very strong and efficient, since at each iteration it only requires the solution of two linear systems with the same matrix, instead of quadratic programming subproblems. It is also particularly appropriate for engineering design optimization inasmuch at each iteration a feasible design is obtained. The present algorithm uses a quasi-Newton approximation of the second derivative of the Lagrangian function in order to have superlinear asymptotic convergence. We discuss theoretical aspects of the algorithm and its computer implementation.

Two-component mixture model: Application to palm oil and exchange rate

NASA Astrophysics Data System (ADS)

Phoong, Seuk-Yen; Ismail, Mohd Tahir; Hamzah, Firdaus Mohamad

2014-12-01

Palm oil is a seed crop which is widely adopt for food and non-food products such as cookie, vegetable oil, cosmetics, household products and others. Palm oil is majority growth in Malaysia and Indonesia. However, the demand for palm oil is getting growth and rapidly running out over the years. This phenomenal cause illegal logging of trees and destroy the natural habitat. Hence, the present paper investigates the relationship between exchange rate and palm oil price in Malaysia by using Maximum Likelihood Estimation via Newton-Raphson algorithm to fit a two components mixture model. Besides, this paper proposes a mixture of normal distribution to accommodate with asymmetry characteristics and platykurtic time series data.

Aircraft interior noise reduction by alternate resonance tuning

NASA Technical Reports Server (NTRS)

Bliss, Donald B.; Gottwald, James A.; Gustaveson, Mark B.; Burton, James R., III

1988-01-01

Model problem development and analysis continues with the Alternate Resonance Tuning (ART) concept. The various topics described are presently at different stages of completion: investigation of the effectiveness of the ART concept under an external propagating pressure field associated with propeller passage by the fuselage; analysis of ART performance with a double panel wall mounted in a flexible frame model; development of a data fitting scheme using a branch analysis with a Newton-Raphson scheme in multiple dimensions to determine values of critical parameters in the actual experimental apparatus; and investigation of the ART effect with real panels as opposed to the spring-mass-damper systems currently used in much of the theory.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cao, Yong; Chu, Yuchuan; He, Xiaoming

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

Unsteady Flow Simulation: A Numerical Challenge

DTIC Science & Technology

2003-03-01

drive to convergence the numerical unsteady term. The time marching procedure is based on the approximate implicit Newton method for systems of non...computed through analytical derivatives of S. The linear system stemming from equation (3) is solved at each integration step by the same iterative method...significant reduction of memory usage, thanks to the reduced dimensions of the linear system matrix during the implicit marching of the solution. The

Density reconstruction in multiparameter elastic full-waveform inversion

NASA Astrophysics Data System (ADS)

Sun, Min'ao; Yang, Jizhong; Dong, Liangguo; Liu, Yuzhu; Huang, Chao

2017-12-01

Elastic full-waveform inversion (EFWI) is a quantitative data fitting procedure that recovers multiple subsurface parameters from multicomponent seismic data. As density is involved in addition to P- and S-wave velocities, the multiparameter EFWI suffers from more serious tradeoffs. In addition, compared with P- and S-wave velocities, the misfit function is less sensitive to density perturbation. Thus, a robust density reconstruction remains a difficult problem in multiparameter EFWI. In this paper, we develop an improved scattering-integral-based truncated Gauss-Newton method to simultaneously recover P- and S-wave velocities and density in EFWI. In this method, the inverse Gauss-Newton Hessian has been estimated by iteratively solving the Gauss-Newton equation with a matrix-free conjugate gradient algorithm. Therefore, it is able to properly handle the parameter tradeoffs. To give a detailed illustration of the tradeoffs between P- and S-wave velocities and density in EFWI, wavefield-separated sensitivity kernels and the Gauss-Newton Hessian are numerically computed, and their distribution characteristics are analyzed. Numerical experiments on a canonical inclusion model and a modified SEG/EAGE Overthrust model have demonstrated that the proposed method can effectively mitigate the tradeoff effects, and improve multiparameter gradients. Thus, a high convergence rate and an accurate density reconstruction can be achieved.

A mixed parallel strategy for the solution of coupled multi-scale problems at finite strains

NASA Astrophysics Data System (ADS)

Lopes, I. A. Rodrigues; Pires, F. M. Andrade; Reis, F. J. P.

2018-02-01

A mixed parallel strategy for the solution of homogenization-based multi-scale constitutive problems undergoing finite strains is proposed. The approach aims to reduce the computational time and memory requirements of non-linear coupled simulations that use finite element discretization at both scales (FE^2). In the first level of the algorithm, a non-conforming domain decomposition technique, based on the FETI method combined with a mortar discretization at the interface of macroscopic subdomains, is employed. A master-slave scheme, which distributes tasks by macroscopic element and adopts dynamic scheduling, is then used for each macroscopic subdomain composing the second level of the algorithm. This strategy allows the parallelization of FE^2 simulations in computers with either shared memory or distributed memory architectures. The proposed strategy preserves the quadratic rates of asymptotic convergence that characterize the Newton-Raphson scheme. Several examples are presented to demonstrate the robustness and efficiency of the proposed parallel strategy.

Computation of the anharmonic orbits in two piecewise monotonic maps with a single discontinuity

NASA Astrophysics Data System (ADS)

Li, Yurong; Du, Zhengdong

2017-02-01

In this paper, the bifurcation values for two typical piecewise monotonic maps with a single discontinuity are computed. The variation of the parameter of those maps leads to a sequence of border-collision and period-doubling bifurcations, generating a sequence of anharmonic orbits on the boundary of chaos. The border-collision and period-doubling bifurcation values are computed by the word-lifting technique and the Maple fsolve function or the Newton-Raphson method, respectively. The scaling factors which measure the convergent rates of the bifurcation values and the width of the stable periodic windows, respectively, are investigated. We found that these scaling factors depend on the parameters of the maps, implying that they are not universal. Moreover, if one side of the maps is linear, our numerical results suggest that those quantities converge increasingly. In particular, for the linear-quadratic case, they converge to one of the Feigenbaum constants δ _F= 4.66920160\\cdots.

Implicit solvers for unstructured meshes

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.; Mavriplis, Dimitri J.

1991-01-01

Implicit methods were developed and tested for unstructured mesh computations. The approximate system which arises from the Newton linearization of the nonlinear evolution operator is solved by using the preconditioned GMRES (Generalized Minimum Residual) technique. Three different preconditioners were studied, namely, the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over relaxation (SSOR). The preconditioners were optimized to have good vectorization properties. SSOR and ILU were also studied as iterative schemes. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also studied. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.

Domain decomposition methods in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Gropp, William D.; Keyes, David E.

1991-01-01

The divide-and-conquer paradigm of iterative domain decomposition, or substructuring, has become a practical tool in computational fluid dynamic applications because of its flexibility in accommodating adaptive refinement through locally uniform (or quasi-uniform) grids, its ability to exploit multiple discretizations of the operator equations, and the modular pathway it provides towards parallelism. These features are illustrated on the classic model problem of flow over a backstep using Newton's method as the nonlinear iteration. Multiple discretizations (second-order in the operator and first-order in the preconditioner) and locally uniform mesh refinement pay dividends separately, and they can be combined synergistically. Sample performance results are included from an Intel iPSC/860 hypercube implementation.

First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems

NASA Technical Reports Server (NTRS)

Mazaheri, Alireza; Nishikawa, Hiroaki

2014-01-01

A time-dependent extension of the first-order hyperbolic system method for advection-diffusion problems is introduced. Diffusive/viscous terms are written and discretized as a hyperbolic system, which recovers the original equation in the steady state. The resulting scheme offers advantages over traditional schemes: a dramatic simplification in the discretization, high-order accuracy in the solution gradients, and orders-of-magnitude convergence acceleration. The hyperbolic advection-diffusion system is discretized by the second-order upwind residual-distribution scheme in a unified manner, and the system of implicit-residual-equations is solved by Newton's method over every physical time step. The numerical results are presented for linear and nonlinear advection-diffusion problems, demonstrating solutions and gradients produced to the same order of accuracy, with rapid convergence over each physical time step, typically less than five Newton iterations.

Efficient numerical calculation of MHD equilibria with magnetic islands, with particular application to saturated neoclassical tearing modes

NASA Astrophysics Data System (ADS)

Raburn, Daniel Louis

We have developed a preconditioned, globalized Jacobian-free Newton-Krylov (JFNK) solver for calculating equilibria with magnetic islands. The solver has been developed in conjunction with the Princeton Iterative Equilibrium Solver (PIES) and includes two notable enhancements over a traditional JFNK scheme: (1) globalization of the algorithm by a sophisticated backtracking scheme, which optimizes between the Newton and steepest-descent directions; and, (2) adaptive preconditioning, wherein information regarding the system Jacobian is reused between Newton iterations to form a preconditioner for our GMRES-like linear solver. We have developed a formulation for calculating saturated neoclassical tearing modes (NTMs) which accounts for the incomplete loss of a bootstrap current due to gradients of multiple physical quantities. We have applied the coupled PIES-JFNK solver to calculate saturated island widths on several shots from the Tokamak Fusion Test Reactor (TFTR) and have found reasonable agreement with experimental measurement.

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

NASA Astrophysics Data System (ADS)

Rangan, Aaditya V.; Cai, David; Tao, Louis

2007-02-01

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions - with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton's iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton's iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Advancing Detached-Eddy Simulation

DTIC Science & Technology

2007-01-01

fluxes leads to an improvement in the stability of the solution . This matrix is solved iteratively using a symmetric Gauss - Seidel procedure. Newtons sub...model (TLM) is a zonal approach, proposed by Balaras and Benocci (5) and Balaras et al. (4). The method involved the solution of filtered Navier...LES mesh. The method was subsequently used by Cabot (6) and Diurno et al. (7) to obtain the solution of the flow over a backward facing step and by

An efficient and general numerical method to compute steady uniform vortices

NASA Astrophysics Data System (ADS)

Luzzatto-Fegiz, Paolo; Williamson, Charles H. K.

2011-07-01

Steady uniform vortices are widely used to represent high Reynolds number flows, yet their efficient computation still presents some challenges. Existing Newton iteration methods become inefficient as the vortices develop fine-scale features; in addition, these methods cannot, in general, find solutions with specified Casimir invariants. On the other hand, available relaxation approaches are computationally inexpensive, but can fail to converge to a solution. In this paper, we overcome these limitations by introducing a new discretization, based on an inverse-velocity map, which radically increases the efficiency of Newton iteration methods. In addition, we introduce a procedure to prescribe Casimirs and remove the degeneracies in the steady vorticity equation, thus ensuring convergence for general vortex configurations. We illustrate our methodology by considering several unbounded flows involving one or two vortices. Our method enables the computation, for the first time, of steady vortices that do not exhibit any geometric symmetry. In addition, we discover that, as the limiting vortex state for each flow is approached, each family of solutions traces a clockwise spiral in a bifurcation plot consisting of a velocity-impulse diagram. By the recently introduced "IVI diagram" stability approach [Phys. Rev. Lett. 104 (2010) 044504], each turn of this spiral is associated with a loss of stability for the steady flows. Such spiral structure is suggested to be a universal feature of steady, uniform-vorticity flows.

A solution to the Navier-Stokes equations based upon the Newton Kantorovich method

NASA Technical Reports Server (NTRS)

Davis, J. E.; Gabrielsen, R. E.; Mehta, U. B.

1977-01-01

An implicit finite difference scheme based on the Newton-Kantorovich technique was developed for the numerical solution of the nonsteady, incompressible, two-dimensional Navier-Stokes equations in conservation-law form. The algorithm was second-order-time accurate, noniterative with regard to the nonlinear terms in the vorticity transport equation except at the earliest few time steps, and spatially factored. Numerical results were obtained with the technique for a circular cylinder at Reynolds number 15. Results indicate that the technique is in excellent agreement with other numerical techniques for all geometries and Reynolds numbers investigated, and indicates a potential for significant reduction in computation time over current iterative techniques.

Trust-region based return mapping algorithm for implicit integration of elastic-plastic constitutive models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lester, Brian; Scherzinger, William

2017-01-19

Here, a new method for the solution of the non-linear equations forming the core of constitutive model integration is proposed. Specifically, the trust-region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic-plastic models. Although attention here is restricted to these rate-independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non-quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, andmore » compared to other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared to existing algorithms. Through these efforts, it is shown that the utilization of a trust-region approach leads to superior performance versus a traditional closest-point projection Newton-Raphson method and comparable speed and robustness to a line search augmented scheme.« less

Trust-region based return mapping algorithm for implicit integration of elastic-plastic constitutive models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lester, Brian T.; Scherzinger, William M.

2017-01-19

A new method for the solution of the non-linear equations forming the core of constitutive model integration is proposed. Specifically, the trust-region method that has been developed in the numerical optimization community is successfully modified for use in implicit integration of elastic-plastic models. Although attention here is restricted to these rate-independent formulations, the proposed approach holds substantial promise for adoption with models incorporating complex physics, multiple inelastic mechanisms, and/or multiphysics. As a first step, the non-quadratic Hosford yield surface is used as a representative case to investigate computationally challenging constitutive models. The theory and implementation are presented, discussed, and comparedmore » to other common integration schemes. Multiple boundary value problems are studied and used to verify the proposed algorithm and demonstrate the capabilities of this approach over more common methodologies. Robustness and speed are then investigated and compared to existing algorithms. As a result through these efforts, it is shown that the utilization of a trust-region approach leads to superior performance versus a traditional closest-point projection Newton-Raphson method and comparable speed and robustness to a line search augmented scheme.« less

Harmonic Optimization in Voltage Source Inverter for PV Application using Heuristic Algorithms

NASA Astrophysics Data System (ADS)

Kandil, Shaimaa A.; Ali, A. A.; El Samahy, Adel; Wasfi, Sherif M.; Malik, O. P.

2016-12-01

Selective Harmonic Elimination (SHE) technique is the fundamental switching frequency scheme that is used to eliminate specific order harmonics. Its application to minimize low order harmonics in a three level inverter is proposed in this paper. The modulation strategy used here is SHEPWM and the nonlinear equations, that characterize the low order harmonics, are solved using Harmony Search Algorithm (HSA) to obtain the optimal switching angles that minimize the required harmonics and maintain the fundamental at the desired value. Total Harmonic Distortion (THD) of the output voltage is minimized maintaining selected harmonics within allowable limits. A comparison has been drawn between HSA, Genetic Algorithm (GA) and Newton Raphson (NR) technique using MATLAB software to determine the effectiveness of getting optimized switching angles.

Regularization and computational methods for precise solution of perturbed orbit transfer problems

NASA Astrophysics Data System (ADS)

Woollands, Robyn Michele

The author has developed a suite of algorithms for solving the perturbed Lambert's problem in celestial mechanics. These algorithms have been implemented as a parallel computation tool that has broad applicability. This tool is composed of four component algorithms and each provides unique benefits for solving a particular type of orbit transfer problem. The first one utilizes a Keplerian solver (a-iteration) for solving the unperturbed Lambert's problem. This algorithm not only provides a "warm start" for solving the perturbed problem but is also used to identify which of several perturbed solvers is best suited for the job. The second algorithm solves the perturbed Lambert's problem using a variant of the modified Chebyshev-Picard iteration initial value solver that solves two-point boundary value problems. This method converges over about one third of an orbit and does not require a Newton-type shooting method and thus no state transition matrix needs to be computed. The third algorithm makes use of regularization of the differential equations through the Kustaanheimo-Stiefel transformation and extends the domain of convergence over which the modified Chebyshev-Picard iteration two-point boundary value solver will converge, from about one third of an orbit to almost a full orbit. This algorithm also does not require a Newton-type shooting method. The fourth algorithm uses the method of particular solutions and the modified Chebyshev-Picard iteration initial value solver to solve the perturbed two-impulse Lambert problem over multiple revolutions. The method of particular solutions is a shooting method but differs from the Newton-type shooting methods in that it does not require integration of the state transition matrix. The mathematical developments that underlie these four algorithms are derived in the chapters of this dissertation. For each of the algorithms, some orbit transfer test cases are included to provide insight on accuracy and efficiency of these individual algorithms. Following this discussion, the combined parallel algorithm, known as the unified Lambert tool, is presented and an explanation is given as to how it automatically selects which of the three perturbed solvers to compute the perturbed solution for a particular orbit transfer. The unified Lambert tool may be used to determine a single orbit transfer or for generating of an extremal field map. A case study is presented for a mission that is required to rendezvous with two pieces of orbit debris (spent rocket boosters). The unified Lambert tool software developed in this dissertation is already being utilized by several industrial partners and we are confident that it will play a significant role in practical applications, including solution of Lambert problems that arise in the current applications focused on enhanced space situational awareness.

Optimal Control of Micro Grid Operation Mode Seamless Switching Based on Radau Allocation Method

NASA Astrophysics Data System (ADS)

Chen, Xiaomin; Wang, Gang

2017-05-01

The seamless switching process of micro grid operation mode directly affects the safety and stability of its operation. According to the switching process from island mode to grid-connected mode of micro grid, we establish a dynamic optimization model based on two grid-connected inverters. We use Radau allocation method to discretize the model, and use Newton iteration method to obtain the optimal solution. Finally, we implement the optimization mode in MATLAB and get the optimal control trajectory of the inverters.

Decoupling the Stationary Navier-Stokes-Darcy System with the Beavers-Joseph-Saffman Interface Condition

DOE PAGES

Cao, Yong; Chu, Yuchuan; He, Xiaoming; ...

2013-01-01

This paper proposes a domain decomposition method for the coupled stationary Navier-Stokes and Darcy equations with the Beavers-Joseph-Saffman interface condition in order to improve the efficiency of the finite element method. The physical interface conditions are directly utilized to construct the boundary conditions on the interface and then decouple the Navier-Stokes and Darcy equations. Newton iteration will be used to deal with the nonlinear systems. Numerical results are presented to illustrate the features of the proposed method.

High-Order Residual-Distribution Hyperbolic Advection-Diffusion Schemes: 3rd-, 4th-, and 6th-Order

NASA Technical Reports Server (NTRS)

Mazaheri, Alireza R.; Nishikawa, Hiroaki

2014-01-01

In this paper, spatially high-order Residual-Distribution (RD) schemes using the first-order hyperbolic system method are proposed for general time-dependent advection-diffusion problems. The corresponding second-order time-dependent hyperbolic advection- diffusion scheme was first introduced in [NASA/TM-2014-218175, 2014], where rapid convergences over each physical time step, with typically less than five Newton iterations, were shown. In that method, the time-dependent hyperbolic advection-diffusion system (linear and nonlinear) was discretized by the second-order upwind RD scheme in a unified manner, and the system of implicit-residual-equations was solved efficiently by Newton's method over every physical time step. In this paper, two techniques for the source term discretization are proposed; 1) reformulation of the source terms with their divergence forms, and 2) correction to the trapezoidal rule for the source term discretization. Third-, fourth, and sixth-order RD schemes are then proposed with the above techniques that, relative to the second-order RD scheme, only cost the evaluation of either the first derivative or both the first and the second derivatives of the source terms. A special fourth-order RD scheme is also proposed that is even less computationally expensive than the third-order RD schemes. The second-order Jacobian formulation was used for all the proposed high-order schemes. The numerical results are then presented for both steady and time-dependent linear and nonlinear advection-diffusion problems. It is shown that these newly developed high-order RD schemes are remarkably efficient and capable of producing the solutions and the gradients to the same order of accuracy of the proposed RD schemes with rapid convergence over each physical time step, typically less than ten Newton iterations.

Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation

NASA Astrophysics Data System (ADS)

Agarwal, P.; El-Sayed, A. A.

2018-06-01

In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton's iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples.

Dynamics of a new family of iterative processes for quadratic polynomials

NASA Astrophysics Data System (ADS)

Gutiérrez, J. M.; Hernández, M. A.; Romero, N.

2010-03-01

In this work we show the presence of the well-known Catalan numbers in the study of the convergence and the dynamical behavior of a family of iterative methods for solving nonlinear equations. In fact, we introduce a family of methods, depending on a parameter . These methods reach the order of convergence m+2 when they are applied to quadratic polynomials with different roots. Newton's and Chebyshev's methods appear as particular choices of the family appear for m=0 and m=1, respectively. We make both analytical and graphical studies of these methods, which give rise to rational functions defined in the extended complex plane. Firstly, we prove that the coefficients of the aforementioned family of iterative processes can be written in terms of the Catalan numbers. Secondly, we make an incursion into its dynamical behavior. In fact, we show that the rational maps related to these methods can be written in terms of the entries of the Catalan triangle. Next we analyze its general convergence, by including some computer plots showing the intricate structure of the Universal Julia sets associated with the methods.

Multivariable frequency domain identification via 2-norm minimization

NASA Technical Reports Server (NTRS)

Bayard, David S.

1992-01-01

The author develops a computational approach to multivariable frequency domain identification, based on 2-norm minimization. In particular, a Gauss-Newton (GN) iteration is developed to minimize the 2-norm of the error between frequency domain data and a matrix fraction transfer function estimate. To improve the global performance of the optimization algorithm, the GN iteration is initialized using the solution to a particular sequentially reweighted least squares problem, denoted as the SK iteration. The least squares problems which arise from both the SK and GN iterations are shown to involve sparse matrices with identical block structure. A sparse matrix QR factorization method is developed to exploit the special block structure, and to efficiently compute the least squares solution. A numerical example involving the identification of a multiple-input multiple-output (MIMO) plant having 286 unknown parameters is given to illustrate the effectiveness of the algorithm.

Survey on the Performance of Source Localization Algorithms

PubMed Central

2017-01-01

The localization of emitters using an array of sensors or antennas is a prevalent issue approached in several applications. There exist different techniques for source localization, which can be classified into multilateration, received signal strength (RSS) and proximity methods. The performance of multilateration techniques relies on measured time variables: the time of flight (ToF) of the emission from the emitter to the sensor, the time differences of arrival (TDoA) of the emission between sensors and the pseudo-time of flight (pToF) of the emission to the sensors. The multilateration algorithms presented and compared in this paper can be classified as iterative and non-iterative methods. Both standard least squares (SLS) and hyperbolic least squares (HLS) are iterative and based on the Newton–Raphson technique to solve the non-linear equation system. The metaheuristic technique particle swarm optimization (PSO) used for source localisation is also studied. This optimization technique estimates the source position as the optimum of an objective function based on HLS and is also iterative in nature. Three non-iterative algorithms, namely the hyperbolic positioning algorithms (HPA), the maximum likelihood estimator (MLE) and Bancroft algorithm, are also presented. A non-iterative combined algorithm, MLE-HLS, based on MLE and HLS, is further proposed in this paper. The performance of all algorithms is analysed and compared in terms of accuracy in the localization of the position of the emitter and in terms of computational time. The analysis is also undertaken with three different sensor layouts since the positions of the sensors affect the localization; several source positions are also evaluated to make the comparison more robust. The analysis is carried out using theoretical time differences, as well as including errors due to the effect of digital sampling of the time variables. It is shown that the most balanced algorithm, yielding better results than the other algorithms in terms of accuracy and short computational time, is the combined MLE-HLS algorithm. PMID:29156565

Marine Controlled-Source Electromagnetic 2D Inversion for synthetic models.

NASA Astrophysics Data System (ADS)

Liu, Y.; Li, Y.

2016-12-01

We present a 2D inverse algorithm for frequency domain marine controlled-source electromagnetic (CSEM) data, which is based on the regularized Gauss-Newton approach. As a forward solver, our parallel adaptive finite element forward modeling program is employed. It is a self-adaptive, goal-oriented grid refinement algorithm in which a finite element analysis is performed on a sequence of refined meshes. The mesh refinement process is guided by a dual error estimate weighting to bias refinement towards elements that affect the solution at the EM receiver locations. With the use of the direct solver (MUMPS), we can effectively compute the electromagnetic fields for multi-sources and parametric sensitivities. We also implement the parallel data domain decomposition approach of Key and Ovall (2011), with the goal of being able to compute accurate responses in parallel for complicated models and a full suite of data parameters typical of offshore CSEM surveys. All minimizations are carried out by using the Gauss-Newton algorithm and model perturbations at each iteration step are obtained by using the Inexact Conjugate Gradient iteration method. Synthetic test inversions are presented.

Effects of ionic strength and ion pairing on (plant-wide) modelling of anaerobic digestion.

PubMed

Solon, Kimberly; Flores-Alsina, Xavier; Mbamba, Christian Kazadi; Volcke, Eveline I P; Tait, Stephan; Batstone, Damien; Gernaey, Krist V; Jeppsson, Ulf

2015-03-01

Plant-wide models of wastewater treatment (such as the Benchmark Simulation Model No. 2 or BSM2) are gaining popularity for use in holistic virtual studies of treatment plant control and operations. The objective of this study is to show the influence of ionic strength (as activity corrections) and ion pairing on modelling of anaerobic digestion processes in such plant-wide models of wastewater treatment. Using the BSM2 as a case study with a number of model variants and cationic load scenarios, this paper presents the effects of an improved physico-chemical description on model predictions and overall plant performance indicators, namely effluent quality index (EQI) and operational cost index (OCI). The acid-base equilibria implemented in the Anaerobic Digestion Model No. 1 (ADM1) are modified to account for non-ideal aqueous-phase chemistry. The model corrects for ionic strength via the Davies approach to consider chemical activities instead of molar concentrations. A speciation sub-routine based on a multi-dimensional Newton-Raphson (NR) iteration method is developed to address algebraic interdependencies. The model also includes ion pairs that play an important role in wastewater treatment. The paper describes: 1) how the anaerobic digester performance is affected by physico-chemical corrections; 2) the effect on pH and the anaerobic digestion products (CO2, CH4 and H2); and, 3) how these variations are propagated from the sludge treatment to the water line. Results at high ionic strength demonstrate that corrections to account for non-ideal conditions lead to significant differences in predicted process performance (up to 18% for effluent quality and 7% for operational cost) but that for pH prediction, activity corrections are more important than ion pairing effects. Both are likely to be required when precipitation is to be modelled. Copyright © 2014 Elsevier Ltd. All rights reserved.

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

NASA Astrophysics Data System (ADS)

Woollands, Robyn M.; Read, Julie L.; Probe, Austin B.; Junkins, John L.

2017-12-01

We present a new method for solving the multiple revolution perturbed Lambert problem using the method of particular solutions and modified Chebyshev-Picard iteration. The method of particular solutions differs from the well-known Newton-shooting method in that integration of the state transition matrix (36 additional differential equations) is not required, and instead it makes use of a reference trajectory and a set of n particular solutions. Any numerical integrator can be used for solving two-point boundary problems with the method of particular solutions, however we show that using modified Chebyshev-Picard iteration affords an avenue for increased efficiency that is not available with other step-by-step integrators. We take advantage of the path approximation nature of modified Chebyshev-Picard iteration (nodes iteratively converge to fixed points in space) and utilize a variable fidelity force model for propagating the reference trajectory. Remarkably, we demonstrate that computing the particular solutions with only low fidelity function evaluations greatly increases the efficiency of the algorithm while maintaining machine precision accuracy. Our study reveals that solving the perturbed Lambert's problem using the method of particular solutions with modified Chebyshev-Picard iteration is about an order of magnitude faster compared with the classical shooting method and a tenth-twelfth order Runge-Kutta integrator. It is well known that the solution to Lambert's problem over multiple revolutions is not unique and to ensure that all possible solutions are considered we make use of a reliable preexisting Keplerian Lambert solver to warm start our perturbed algorithm.

Polynomiography and Chaos

NASA Astrophysics Data System (ADS)

Kalantari, Bahman

Polynomiography is the algorithmic visualization of iterative systems for computing roots of a complex polynomial. It is well known that iterations of a rational function in the complex plane result in chaotic behavior near its Julia set. In one scheme of computing polynomiography for a given polynomial p(z), we select an individual member from the Basic Family, an infinite fundamental family of rational iteration functions that in particular include Newton's. Polynomiography is an excellent means for observing, understanding, and comparing chaotic behavior for variety of iterative systems. Other iterative schemes in polynomiography are possible and result in chaotic behavior of different kinds. In another scheme, the Basic Family is collectively applied to p(z) and the iterates for any seed in the Voronoi cell of a root converge to that root. Polynomiography reveals chaotic behavior of another kind near the boundary of the Voronoi diagram of the roots. We also describe a novel Newton-Ellipsoid iterative system with its own chaos and exhibit images demonstrating polynomiographies of chaotic behavior of different kinds. Finally, we consider chaos for the more general case of polynomiography of complex analytic functions. On the one hand polynomiography is a powerful medium capable of demonstrating chaos in different forms, it is educationally instructive to students and researchers, also it gives rise to numerous research problems. On the other hand, it is a medium resulting in images with enormous aesthetic appeal to general audiences.

A Strassen-Newton algorithm for high-speed parallelizable matrix inversion

NASA Technical Reports Server (NTRS)

Bailey, David H.; Ferguson, Helaman R. P.

1988-01-01

Techniques are described for computing matrix inverses by algorithms that are highly suited to massively parallel computation. The techniques are based on an algorithm suggested by Strassen (1969). Variations of this scheme use matrix Newton iterations and other methods to improve the numerical stability while at the same time preserving a very high level of parallelism. One-processor Cray-2 implementations of these schemes range from one that is up to 55 percent faster than a conventional library routine to one that is slower than a library routine but achieves excellent numerical stability. The problem of computing the solution to a single set of linear equations is discussed, and it is shown that this problem can also be solved efficiently using these techniques.

3D CSEM data inversion using Newton and Halley class methods

NASA Astrophysics Data System (ADS)

Amaya, M.; Hansen, K. R.; Morten, J. P.

2016-05-01

For the first time in 3D controlled source electromagnetic data inversion, we explore the use of the Newton and the Halley optimization methods, which may show their potential when the cost function has a complex topology. The inversion is formulated as a constrained nonlinear least-squares problem which is solved by iterative optimization. These methods require the derivatives up to second order of the residuals with respect to model parameters. We show how Green's functions determine the high-order derivatives, and develop a diagrammatical representation of the residual derivatives. The Green's functions are efficiently calculated on-the-fly, making use of a finite-difference frequency-domain forward modelling code based on a multi-frontal sparse direct solver. This allow us to build the second-order derivatives of the residuals keeping the memory cost in the same order as in a Gauss-Newton (GN) scheme. Model updates are computed with a trust-region based conjugate-gradient solver which does not require the computation of a stabilizer. We present inversion results for a synthetic survey and compare the GN, Newton, and super-Halley optimization schemes, and consider two different approaches to set the initial trust-region radius. Our analysis shows that the Newton and super-Halley schemes, using the same regularization configuration, add significant information to the inversion so that the convergence is reached by different paths. In our simple resistivity model examples, the convergence speed of the Newton and the super-Halley schemes are either similar or slightly superior with respect to the convergence speed of the GN scheme, close to the minimum of the cost function. Due to the current noise levels and other measurement inaccuracies in geophysical investigations, this advantageous behaviour is at present of low consequence, but may, with the further improvement of geophysical data acquisition, be an argument for more accurate higher-order methods like those applied in this paper.

Ambiguity resolution in systems using Omega for position location

NASA Technical Reports Server (NTRS)

Frenkel, G.; Gan, D. G.

1974-01-01

The lane ambiguity problem prevents the utilization of the Omega system for many applications such as locating buoys and balloons. The method of multiple lines of position introduced herein uses signals from four or more Omega stations for ambiguity resolution. The coordinates of the candidate points are determined first through the use of the Newton iterative procedure. Subsequently, a likelihood function is generated for each point, and the ambiguity is resolved by selecting the most likely point. The method was tested through simulation.

A SEMI-LAGRANGIAN TWO-LEVEL PRECONDITIONED NEWTON-KRYLOV SOLVER FOR CONSTRAINED DIFFEOMORPHIC IMAGE REGISTRATION.

PubMed

Mang, Andreas; Biros, George

2017-01-01

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20 × speedup for a two dimensional, real world multi-subject medical image registration problem.

Nonlinear damage identification of breathing cracks in Truss system

NASA Astrophysics Data System (ADS)

Zhao, Jie; DeSmidt, Hans

2014-03-01

The breathing cracks in truss system are detected by Frequency Response Function (FRF) based damage identification method. This method utilizes damage-induced changes of frequency response functions to estimate the severity and location of structural damage. This approach enables the possibility of arbitrary interrogation frequency and multiple inputs/outputs which greatly enrich the dataset for damage identification. The dynamical model of truss system is built using the finite element method and the crack model is based on fracture mechanics. Since the crack is driven by tensional and compressive forces of truss member, only one damage parameter is needed to represent the stiffness reduction of each truss member. Assuming that the crack constantly breathes with the exciting frequency, the linear damage detection algorithm is developed in frequency/time domain using Least Square and Newton Raphson methods. Then, the dynamic response of the truss system with breathing cracks is simulated in the time domain and meanwhile the crack breathing status for each member is determined by the feedback from real-time displacements of member's nodes. Harmonic Fourier Coefficients (HFCs) of dynamical response are computed by processing the data through convolution and moving average filters. Finally, the results show the effectiveness of linear damage detection algorithm in identifying the nonlinear breathing cracks using different combinations of HFCs and sensors.

Differentially private distributed logistic regression using private and public data.

PubMed

Ji, Zhanglong; Jiang, Xiaoqian; Wang, Shuang; Xiong, Li; Ohno-Machado, Lucila

2014-01-01

Privacy protecting is an important issue in medical informatics and differential privacy is a state-of-the-art framework for data privacy research. Differential privacy offers provable privacy against attackers who have auxiliary information, and can be applied to data mining models (for example, logistic regression). However, differentially private methods sometimes introduce too much noise and make outputs less useful. Given available public data in medical research (e.g. from patients who sign open-consent agreements), we can design algorithms that use both public and private data sets to decrease the amount of noise that is introduced. In this paper, we modify the update step in Newton-Raphson method to propose a differentially private distributed logistic regression model based on both public and private data. We try our algorithm on three different data sets, and show its advantage over: (1) a logistic regression model based solely on public data, and (2) a differentially private distributed logistic regression model based on private data under various scenarios. Logistic regression models built with our new algorithm based on both private and public datasets demonstrate better utility than models that trained on private or public datasets alone without sacrificing the rigorous privacy guarantee.

Effect of initial strain and material nonlinearity on the nonlinear static and dynamic response of graphene sheets

NASA Astrophysics Data System (ADS)

Singh, Sandeep; Patel, B. P.

2018-06-01

Computationally efficient multiscale modelling based on Cauchy-Born rule in conjunction with finite element method is employed to study static and dynamic characteristics of graphene sheets, with/without considering initial strain, involving Green-Lagrange geometric and material nonlinearities. The strain energy density function at continuum level is established by coupling the deformation at continuum level to that at atomic level through Cauchy-Born rule. The atomic interactions between carbon atoms are modelled through Tersoff-Brenner potential. The governing equation of motion obtained using Hamilton's principle is solved through standard Newton-Raphson method for nonlinear static response and Newmark's time integration technique to obtain nonlinear transient response characteristics. Effect of initial strain on the linear free vibration frequencies, nonlinear static and dynamic response characteristics is investigated in detail. The present multiscale modelling based results are found to be in good agreement with those obtained through molecular mechanics simulation. Two different types of boundary constraints generally used in MM simulation are explored in detail and few interesting findings are brought out. The effect of initial strain is found to be greater in linear response when compared to that in nonlinear response.

Cascade Optimization for Aircraft Engines With Regression and Neural Network Analysis - Approximators

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Guptill, James D.; Hopkins, Dale A.; Lavelle, Thomas M.

2000-01-01

The NASA Engine Performance Program (NEPP) can configure and analyze almost any type of gas turbine engine that can be generated through the interconnection of a set of standard physical components. In addition, the code can optimize engine performance by changing adjustable variables under a set of constraints. However, for engine cycle problems at certain operating points, the NEPP code can encounter difficulties: nonconvergence in the currently implemented Powell's optimization algorithm and deficiencies in the Newton-Raphson solver during engine balancing. A project was undertaken to correct these deficiencies. Nonconvergence was avoided through a cascade optimization strategy, and deficiencies associated with engine balancing were eliminated through neural network and linear regression methods. An approximation-interspersed cascade strategy was used to optimize the engine's operation over its flight envelope. Replacement of Powell's algorithm by the cascade strategy improved the optimization segment of the NEPP code. The performance of the linear regression and neural network methods as alternative engine analyzers was found to be satisfactory. This report considers two examples-a supersonic mixed-flow turbofan engine and a subsonic waverotor-topped engine-to illustrate the results, and it discusses insights gained from the improved version of the NEPP code.

A Gauss-Newton full-waveform inversion in PML-truncated domains using scalar probing waves

NASA Astrophysics Data System (ADS)

Pakravan, Alireza; Kang, Jun Won; Newtson, Craig M.

2017-12-01

This study considers the characterization of subsurface shear wave velocity profiles in semi-infinite media using scalar waves. Using surficial responses caused by probing waves, a reconstruction of the material profile is sought using a Gauss-Newton full-waveform inversion method in a two-dimensional domain truncated by perfectly matched layer (PML) wave-absorbing boundaries. The PML is introduced to limit the semi-infinite extent of the half-space and to prevent reflections from the truncated boundaries. A hybrid unsplit-field PML is formulated in the inversion framework to enable more efficient wave simulations than with a fully mixed PML. The full-waveform inversion method is based on a constrained optimization framework that is implemented using Karush-Kuhn-Tucker (KKT) optimality conditions to minimize the objective functional augmented by PML-endowed wave equations via Lagrange multipliers. The KKT conditions consist of state, adjoint, and control problems, and are solved iteratively to update the shear wave velocity profile of the PML-truncated domain. Numerical examples show that the developed Gauss-Newton inversion method is accurate enough and more efficient than another inversion method. The algorithm's performance is demonstrated by the numerical examples including the case of noisy measurement responses and the case of reduced number of sources and receivers.

An Iterative Decambering Approach for Post-Stall Prediction of Wing Characteristics using known Section Data

NASA Technical Reports Server (NTRS)

Mukherjee, Rinku; Gopalarathnam, Ashok; Kim, Sung Wan

2003-01-01

An iterative decambering approach for the post stall prediction of wings using known section data as inputs is presented. The method can currently be used for incompressible .ow and can be extended to compressible subsonic .ow using Mach number correction schemes. A detailed discussion of past work on this topic is presented first. Next, an overview of the decambering approach is presented and is illustrated by applying the approach to the prediction of the two-dimensional C(sub l) and C(sub m) curves for an airfoil. The implementation of the approach for iterative decambering of wing sections is then discussed. A novel feature of the current e.ort is the use of a multidimensional Newton iteration for taking into consideration the coupling between the di.erent sections of the wing. The approach lends itself to implementation in a variety of finite-wing analysis methods such as lifting-line theory, discrete-vortex Weissinger's method, and vortex lattice codes. Results are presented for a rectangular wing for a from 0 to 25 deg. The results are compared for both increasing and decreasing directions of a, and they show that a hysteresis loop can be predicted for post-stall angles of attack.

Development of computer program NAS3D using Vector processing for geometric nonlinear analysis of structures

NASA Technical Reports Server (NTRS)

Mangalgiri, P. D.; Prabhakaran, R.

1986-01-01

An algorithm for vectorized computation of stiffness matrices of an 8 noded isoparametric hexahedron element for geometric nonlinear analysis was developed. This was used in conjunction with the earlier 2-D program GAMNAS to develop the new program NAS3D for geometric nonlinear analysis. A conventional, modified Newton-Raphson process is used for the nonlinear analysis. New schemes for the computation of stiffness and strain energy release rates is presented. The organization the program is explained and some results on four sample problems are given. The study of CPU times showed that savings by a factor of 11 to 13 were achieved when vectorized computation was used for the stiffness instead of the conventional scalar one. Finally, the scheme of inputting data is explained.

Approaches to the simulation of unconfined flow and perched groundwater flow in MODFLOW

USGS Publications Warehouse

Bedekar, Vivek; Niswonger, Richard G.; Kipp, Kenneth; Panday, Sorab; Tonkin, Matthew

2012-01-01

Various approaches have been proposed to manage the nonlinearities associated with the unconfined flow equation and to simulate perched groundwater conditions using the MODFLOW family of codes. The approaches comprise a variety of numerical techniques to prevent dry cells from becoming inactive and to achieve a stable solution focused on formulations of the unconfined, partially-saturated, groundwater flow equation. Keeping dry cells active avoids a discontinuous head solution which in turn improves the effectiveness of parameter estimation software that relies on continuous derivatives. Most approaches implement an upstream weighting of intercell conductance and Newton-Raphson linearization to obtain robust convergence. In this study, several published approaches were implemented in a stepwise manner into MODFLOW for comparative analysis. First, a comparative analysis of the methods is presented using synthetic examples that create convergence issues or difficulty in handling perched conditions with the more common dry-cell simulation capabilities of MODFLOW. Next, a field-scale three-dimensional simulation is presented to examine the stability and performance of the discussed approaches in larger, practical, simulation settings.

Performance Analysis of a CO2 Heat Pump Water Heating System Under a Daily Change in a Simulated Demand

NASA Astrophysics Data System (ADS)

Yokoyama, Ryohei; Kohno, Yasuhiro; Wakui, Tetsuya; Takemura, Kazuhisa

Air-to-water heat pumps using CO2 as a refrigerant have been developed. In addition, water heating systems each of which combines a CO2 heat pump with a hot water storage tank have been commercialized and widespread. They are expected to contribute to energy saving in residential hot water supply. It has become more and more important to enhance the system performance. In this paper, the performance of a CO2 heat pump water heating system is analyzed under a daily change in a simulated hot water demand by numerical simulation. A static model of a CO2 heat pump and a dynamic model of a storage tank result in a set of differential algebraic equations, and it is solved numerically by a hierarchical combination of Runge-Kutta and Newton-Raphson methods. Daily changes in the temperature distributions in the storage tank and the system performance criteria such as volumes of stored and unused hot water, coefficient of performance, and storage and system efficiencies are clarified under a series of daily hot water demands during a month.

Likelihood-based confidence intervals for estimating floods with given return periods

NASA Astrophysics Data System (ADS)

Martins, Eduardo Sávio P. R.; Clarke, Robin T.

1993-06-01

This paper discusses aspects of the calculation of likelihood-based confidence intervals for T-year floods, with particular reference to (1) the two-parameter gamma distribution; (2) the Gumbel distribution; (3) the two-parameter log-normal distribution, and other distributions related to the normal by Box-Cox transformations. Calculation of the confidence limits is straightforward using the Nelder-Mead algorithm with a constraint incorporated, although care is necessary to ensure convergence either of the Nelder-Mead algorithm, or of the Newton-Raphson calculation of maximum-likelihood estimates. Methods are illustrated using records from 18 gauging stations in the basin of the River Itajai-Acu, State of Santa Catarina, southern Brazil. A small and restricted simulation compared likelihood-based confidence limits with those given by use of the central limit theorem; for the same confidence probability, the confidence limits of the simulation were wider than those of the central limit theorem, which failed more frequently to contain the true quantile being estimated. The paper discusses possible applications of likelihood-based confidence intervals in other areas of hydrological analysis.

Kinematics of an in-parallel actuated manipulator based on the Stewart platform mechanism

NASA Technical Reports Server (NTRS)

Williams, Robert L., II

1992-01-01

This paper presents kinematic equations and solutions for an in-parallel actuated robotic mechanism based on Stewart's platform. These equations are required for inverse position and resolved rate (inverse velocity) platform control. NASA LaRC has a Vehicle Emulator System (VES) platform designed by MIT which is based on Stewart's platform. The inverse position solution is straight-forward and computationally inexpensive. Given the desired position and orientation of the moving platform with respect to the base, the lengths of the prismatic leg actuators are calculated. The forward position solution is more complicated and theoretically has 16 solutions. The position and orientation of the moving platform with respect to the base is calculated given the leg actuator lengths. Two methods are pursued in this paper to solve this problem. The resolved rate (inverse velocity) solution is derived. Given the desired Cartesian velocity of the end-effector, the required leg actuator rates are calculated. The Newton-Raphson Jacobian matrix resulting from the second forward position kinematics solution is a modified inverse Jacobian matrix. Examples and simulations are given for the VES.

A nonlinear relaxation/quasi-Newton algorithm for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Edwards, Jack R.; Mcrae, D. S.

1992-01-01

A highly efficient implicit method for the computation of steady, two-dimensional compressible Navier-Stokes flowfields is presented. The discretization of the governing equations is hybrid in nature, with flux-vector splitting utilized in the streamwise direction and central differences with flux-limited artificial dissipation used for the transverse fluxes. Line Jacobi relaxation is used to provide a suitable initial guess for a new nonlinear iteration strategy based on line Gauss-Seidel sweeps. The applicability of quasi-Newton methods as convergence accelerators for this and other line relaxation algorithms is discussed, and efficient implementations of such techniques are presented. Convergence histories and comparisons with experimental data are presented for supersonic flow over a flat plate and for several high-speed compression corner interactions. Results indicate a marked improvement in computational efficiency over more conventional upwind relaxation strategies, particularly for flowfields containing large pockets of streamwise subsonic flow.

Preconditioning strategies for nonlinear conjugate gradient methods, based on quasi-Newton updates

NASA Astrophysics Data System (ADS)

Andrea, Caliciotti; Giovanni, Fasano; Massimo, Roma

2016-10-01

This paper reports two proposals of possible preconditioners for the Nonlinear Conjugate Gradient (NCG) method, in large scale unconstrained optimization. On one hand, the common idea of our preconditioners is inspired to L-BFGS quasi-Newton updates, on the other hand we aim at explicitly approximating in some sense the inverse of the Hessian matrix. Since we deal with large scale optimization problems, we propose matrix-free approaches where the preconditioners are built using symmetric low-rank updating formulae. Our distinctive new contributions rely on using information on the objective function collected as by-product of the NCG, at previous iterations. Broadly speaking, our first approach exploits the secant equation, in order to impose interpolation conditions on the objective function. In the second proposal we adopt and ad hoc modified-secant approach, in order to possibly guarantee some additional theoretical properties.

Building a flood hazard map due to magma effusion into the caldera lake of the Baekdusan Volcano

NASA Astrophysics Data System (ADS)

Lee, K.; Kim, S.; Yun, S.; Yu, S.; Kim, I.

2013-12-01

Many volcanic craters and calderas are filled with large amounts of water that can pose significant flood hazards to downstream communities due to their high elevation and the potential for catastrophic releases of water. Recent reports pointed out the Baekdusan volcano that is located between the border of China and North Korea as a potential active volcano. Since Millennium Eruption around 1000 AD, smaller eruptions have occurred at roughly 100-year intervals, with the last one in 1903. The volcano is showing signs of waking from a century-long slumber recently and the volcanic ash may spread up to the northeastern of Japan. The development of various forecasting techniques to prevent and minimize economic and social damage is in urgent need. Floods from lake-filled calderas may be particularly large and high. Volcanic flood may cause significant hydrologic hazards for this reason. This study focuses on constructing a flood hazard map triggered by the uplift of lake bottom due to magma effusion in the Baekdusan volcano. A physically-based uplift model was developed to compute the amount of water and time to peak flow. The ordinary differential equation was numerically solved using the finite difference method and Newton-Raphson iteration method was used to solve nonlinear equation. The magma effusion rate into the caldera lake is followed by the past record from other volcanic activities. As a result, the hydrograph serves as an upper boundary condition when hydrodynamic model (Flo-2D) runs to simulate channel routing downstream. The final goal of the study stresses the potential flood hazard represented by the huge volume of water in the caldera lake, the unique geography, and the limited control capability. he study will contribute to build a geohazard map for the decision-makers and practitioners. Keywords: Effusion rate, Volcanic flood, Caldera lake, Uplift, Flood hazard map Acknowledgement This research was supported by a grant [NEMA-BAEKDUSAN-2012-1-2] from the Volcanic Disaster Preparedness Research Center sponsored by National Emergency Management Agency of Korea. Inundation map triggered by magma effusion simulated by Flo-2D

A comparison between Gauss-Newton and Markov chain Monte Carlo basedmethods for inverting spectral induced polarization data for Cole-Coleparameters

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Jinsong; Kemna, Andreas; Hubbard, Susan S.

2008-05-15

We develop a Bayesian model to invert spectral induced polarization (SIP) data for Cole-Cole parameters using Markov chain Monte Carlo (MCMC) sampling methods. We compare the performance of the MCMC based stochastic method with an iterative Gauss-Newton based deterministic method for Cole-Cole parameter estimation through inversion of synthetic and laboratory SIP data. The Gauss-Newton based method can provide an optimal solution for given objective functions under constraints, but the obtained optimal solution generally depends on the choice of initial values and the estimated uncertainty information is often inaccurate or insufficient. In contrast, the MCMC based inversion method provides extensive globalmore » information on unknown parameters, such as the marginal probability distribution functions, from which we can obtain better estimates and tighter uncertainty bounds of the parameters than with the deterministic method. Additionally, the results obtained with the MCMC method are independent of the choice of initial values. Because the MCMC based method does not explicitly offer single optimal solution for given objective functions, the deterministic and stochastic methods can complement each other. For example, the stochastic method can first be used to obtain the means of the unknown parameters by starting from an arbitrary set of initial values and the deterministic method can then be initiated using the means as starting values to obtain the optimal estimates of the Cole-Cole parameters.« less

An advanced dissymmetric rolling model for online regulation

NASA Astrophysics Data System (ADS)

Cao, Trong-Son

2017-10-01

Roll-bite model is employed to predict the rolling force, torque as well as to estimate the forward slip for preset or online regulation at industrial rolling mills. The rolling process is often dissymmetric in terms of work-rolls rotation speeds and diameters as well as the friction conditions at upper and lower contact surfaces between work-rolls and the strip. The roll-bite model thus must be able to account for these dissymmetries and in the same time has to be accurate and fast enough for online applications. In the present study, a new method, namely Adapted Discretization Slab Method (ADSM) is proposed to obtain a robust roll-bite model, which can take into account the aforementioned dissymmetries and has a very short response time, lower than one millisecond. This model is based on the slab method, with an adaptive discretization and a global Newton-Raphson procedure to improve the convergence speed. The model was validated by comparing with other dissymmetric models proposed in the literature, as well as Finite Element simulations and industrial pilot trials. Furthermore, back-calculation tool was also constructed for friction management for both offline and online applications. With very short CPU time, the ADSM-based model is thus attractive for all online applications, both for cold and hot rolling.

Localization of source with unknown amplitude using IPMC sensor arrays

NASA Astrophysics Data System (ADS)

Abdulsadda, Ahmad T.; Zhang, Feitian; Tan, Xiaobo

2011-04-01

The lateral line system, consisting of arrays of neuromasts functioning as flow sensors, is an important sensory organ for fish that enables them to detect predators, locate preys, perform rheotaxis, and coordinate schooling. Creating artificial lateral line systems is of significant interest since it will provide a new sensing mechanism for control and coordination of underwater robots and vehicles. In this paper we propose recursive algorithms for localizing a vibrating sphere, also known as a dipole source, based on measurements from an array of flow sensors. A dipole source is frequently used in the study of biological lateral lines, as a surrogate for underwater motion sources such as a flapping fish fin. We first formulate a nonlinear estimation problem based on an analytical model for the dipole-generated flow field. Two algorithms are presented to estimate both the source location and the vibration amplitude, one based on the least squares method and the other based on the Newton-Raphson method. Simulation results show that both methods deliver comparable performance in source localization. A prototype of artificial lateral line system comprising four ionic polymer-metal composite (IPMC) sensors is built, and experimental results are further presented to demonstrate the effectiveness of IPMC lateral line systems and the proposed estimation algorithms.

Combined magnetic vector-scalar potential finite element computation of 3D magnetic field and performance of modified Lundell alternators in Space Station applications. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Wang, Ren H.

1991-01-01

A method of combined use of magnetic vector potential (MVP) based finite element (FE) formulations and magnetic scalar potential (MSP) based FE formulations for computation of three-dimensional (3D) magnetostatic fields is developed. This combined MVP-MSP 3D-FE method leads to considerable reduction by nearly a factor of 3 in the number of unknowns in comparison to the number of unknowns which must be computed in global MVP based FE solutions. This method allows one to incorporate portions of iron cores sandwiched in between coils (conductors) in current-carrying regions. Thus, it greatly simplifies the geometries of current carrying regions (in comparison with the exclusive MSP based methods) in electric machinery applications. A unique feature of this approach is that the global MSP solution is single valued in nature, that is, no branch cut is needed. This is again a superiority over the exclusive MSP based methods. A Newton-Raphson procedure with a concept of an adaptive relaxation factor was developed and successfully used in solving the 3D-FE problem with magnetic material anisotropy and nonlinearity. Accordingly, this combined MVP-MSP 3D-FE method is most suited for solution of large scale global type magnetic field computations in rotating electric machinery with very complex magnetic circuit geometries, as well as nonlinear and anisotropic material properties.

Adaptive eigenspace method for inverse scattering problems in the frequency domain

NASA Astrophysics Data System (ADS)

Grote, Marcus J.; Kray, Marie; Nahum, Uri

2017-02-01

A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method.

Optimal sixteenth order convergent method based on quasi-Hermite interpolation for computing roots.

PubMed

Zafar, Fiza; Hussain, Nawab; Fatimah, Zirwah; Kharal, Athar

2014-01-01

We have given a four-step, multipoint iterative method without memory for solving nonlinear equations. The method is constructed by using quasi-Hermite interpolation and has order of convergence sixteen. As this method requires four function evaluations and one derivative evaluation at each step, it is optimal in the sense of the Kung and Traub conjecture. The comparisons are given with some other newly developed sixteenth-order methods. Interval Newton's method is also used for finding the enough accurate initial approximations. Some figures show the enclosure of finitely many zeroes of nonlinear equations in an interval. Basins of attractions show the effectiveness of the method.

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

NASA Astrophysics Data System (ADS)

Cartarius, Holger; Musslimani, Ziad H.; Schwarz, Lukas; Wunner, Günter

2018-03-01

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrödinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed-point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode, and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) it is easy to implement especially at higher dimensions, and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian PT -symmetric models.

Bending and buckling formulation of graphene sheets based on nonlocal simple first-order shear deformation theory

NASA Astrophysics Data System (ADS)

Golmakani, M. E.; Malikan, M.; Sadraee Far, M. N.; Majidi, H. R.

2018-06-01

This paper presents a formulation based on simple first-order shear deformation theory (S-FSDT) for large deflection and buckling of orthotropic single-layered graphene sheets (SLGSs). The S-FSDT has many advantages compared to the classical plate theory (CPT) and conventional FSDT such as needless of shear correction factor, containing less number of unknowns than the existing FSDT and strong similarities with the CPT. Governing equations and boundary conditions are derived based on Hamilton’s principle using the nonlocal differential constitutive relations of Eringen and von Kármán geometrical model. Numerical results are obtained using differential quadrature (DQ) method and the Newton–Raphson iterative scheme. Finally, some comparison studies are carried out to show the high accuracy and reliability of the present formulations compared to the nonlocal CPT and FSDT for different thicknesses, elastic foundations and nonlocal parameters.

Quickprop method to speed up learning process of Artificial Neural Network in money's nominal value recognition case

NASA Astrophysics Data System (ADS)

Swastika, Windra

2017-03-01

A money's nominal value recognition system has been developed using Artificial Neural Network (ANN). ANN with Back Propagation has one disadvantage. The learning process is very slow (or never reach the target) in the case of large number of iteration, weight and samples. One way to speed up the learning process is using Quickprop method. Quickprop method is based on Newton's method and able to speed up the learning process by assuming that the weight adjustment (E) is a parabolic function. The goal is to minimize the error gradient (E'). In our system, we use 5 types of money's nominal value, i.e. 1,000 IDR, 2,000 IDR, 5,000 IDR, 10,000 IDR and 50,000 IDR. One of the surface of each nominal were scanned and digitally processed. There are 40 patterns to be used as training set in ANN system. The effectiveness of Quickprop method in the ANN system was validated by 2 factors, (1) number of iterations required to reach error below 0.1; and (2) the accuracy to predict nominal values based on the input. Our results shows that the use of Quickprop method is successfully reduce the learning process compared to Back Propagation method. For 40 input patterns, Quickprop method successfully reached error below 0.1 for only 20 iterations, while Back Propagation method required 2000 iterations. The prediction accuracy for both method is higher than 90%.

Computational simulation of laser heat processing of materials

NASA Astrophysics Data System (ADS)

Shankar, Vijaya; Gnanamuthu, Daniel

1987-04-01

A computational model simulating the laser heat treatment of AISI 4140 steel plates with a CW CO2 laser beam has been developed on the basis of the three-dimensional, time-dependent heat equation (subject to the appropriate boundary conditions). The solution method is based on Newton iteration applied to a triple-approximate factorized form of the equation. The method is implicit and time-accurate; the maintenance of time-accuracy in the numerical formulation is noted to be critical for the simulation of finite length workpieces with a finite laser beam dwell time.

Formulation for Simultaneous Aerodynamic Analysis and Design Optimization

NASA Technical Reports Server (NTRS)

Hou, G. W.; Taylor, A. C., III; Mani, S. V.; Newman, P. A.

1993-01-01

An efficient approach for simultaneous aerodynamic analysis and design optimization is presented. This approach does not require the performance of many flow analyses at each design optimization step, which can be an expensive procedure. Thus, this approach brings us one step closer to meeting the challenge of incorporating computational fluid dynamic codes into gradient-based optimization techniques for aerodynamic design. An adjoint-variable method is introduced to nullify the effect of the increased number of design variables in the problem formulation. The method has been successfully tested on one-dimensional nozzle flow problems, including a sample problem with a normal shock. Implementations of the above algorithm are also presented that incorporate Newton iterations to secure a high-quality flow solution at the end of the design process. Implementations with iterative flow solvers are possible and will be required for large, multidimensional flow problems.

A Numerical-Analytical Approach Based on Canonical Transformations for Computing Optimal Low-Thrust Transfers

NASA Astrophysics Data System (ADS)

da Silva Fernandes, S.; das Chagas Carvalho, F.; Bateli Romão, J. V.

2018-04-01

A numerical-analytical procedure based on infinitesimal canonical transformations is developed for computing optimal time-fixed low-thrust limited power transfers (no rendezvous) between coplanar orbits with small eccentricities in an inverse-square force field. The optimization problem is formulated as a Mayer problem with a set of non-singular orbital elements as state variables. Second order terms in eccentricity are considered in the development of the maximum Hamiltonian describing the optimal trajectories. The two-point boundary value problem of going from an initial orbit to a final orbit is solved by means of a two-stage Newton-Raphson algorithm which uses an infinitesimal canonical transformation. Numerical results are presented for some transfers between circular orbits with moderate radius ratio, including a preliminary analysis of Earth-Mars and Earth-Venus missions.

Adaptive [theta]-methods for pricing American options

NASA Astrophysics Data System (ADS)

Khaliq, Abdul Q. M.; Voss, David A.; Kazmi, Kamran

2008-12-01

We develop adaptive [theta]-methods for solving the Black-Scholes PDE for American options. By adding a small, continuous term, the Black-Scholes PDE becomes an advection-diffusion-reaction equation on a fixed spatial domain. Standard implementation of [theta]-methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which [theta]-methods satisfy a discrete version of the positivity constraint characteristic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one- and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency.

Fully coupled simulation of multiple hydraulic fractures to propagate simultaneously from a perforated horizontal wellbore

NASA Astrophysics Data System (ADS)

Zeng, Qinglei; Liu, Zhanli; Wang, Tao; Gao, Yue; Zhuang, Zhuo

2018-02-01

In hydraulic fracturing process in shale rock, multiple fractures perpendicular to a horizontal wellbore are usually driven to propagate simultaneously by the pumping operation. In this paper, a numerical method is developed for the propagation of multiple hydraulic fractures (HFs) by fully coupling the deformation and fracturing of solid formation, fluid flow in fractures, fluid partitioning through a horizontal wellbore and perforation entry loss effect. The extended finite element method (XFEM) is adopted to model arbitrary growth of the fractures. Newton's iteration is proposed to solve these fully coupled nonlinear equations, which is more efficient comparing to the widely adopted fixed-point iteration in the literatures and avoids the need to impose fluid pressure boundary condition when solving flow equations. A secant iterative method based on the stress intensity factor (SIF) is proposed to capture different propagation velocities of multiple fractures. The numerical results are compared with theoretical solutions in literatures to verify the accuracy of the method. The simultaneous propagation of multiple HFs is simulated by the newly proposed algorithm. The coupled influences of propagation regime, stress interaction, wellbore pressure loss and perforation entry loss on simultaneous propagation of multiple HFs are investigated.

Iterative Adaptive Dynamic Programming for Solving Unknown Nonlinear Zero-Sum Game Based on Online Data.

PubMed

Zhu, Yuanheng; Zhao, Dongbin; Li, Xiangjun

2017-03-01

H ∞ control is a powerful method to solve the disturbance attenuation problems that occur in some control systems. The design of such controllers relies on solving the zero-sum game (ZSG). But in practical applications, the exact dynamics is mostly unknown. Identification of dynamics also produces errors that are detrimental to the control performance. To overcome this problem, an iterative adaptive dynamic programming algorithm is proposed in this paper to solve the continuous-time, unknown nonlinear ZSG with only online data. A model-free approach to the Hamilton-Jacobi-Isaacs equation is developed based on the policy iteration method. Control and disturbance policies and value are approximated by neural networks (NNs) under the critic-actor-disturber structure. The NN weights are solved by the least-squares method. According to the theoretical analysis, our algorithm is equivalent to a Gauss-Newton method solving an optimization problem, and it converges uniformly to the optimal solution. The online data can also be used repeatedly, which is highly efficient. Simulation results demonstrate its feasibility to solve the unknown nonlinear ZSG. When compared with other algorithms, it saves a significant amount of online measurement time.

Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting

DOE PAGES

Carlberg, Kevin; Ray, Jaideep; van Bloemen Waanders, Bart

2015-02-14

Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of system's spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation. We propose exploiting knowledge of the model's temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equationsmore » at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. As a result, the goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.« less

Unified Lambert Tool for Massively Parallel Applications in Space Situational Awareness

NASA Astrophysics Data System (ADS)

Woollands, Robyn M.; Read, Julie; Hernandez, Kevin; Probe, Austin; Junkins, John L.

2018-03-01

This paper introduces a parallel-compiled tool that combines several of our recently developed methods for solving the perturbed Lambert problem using modified Chebyshev-Picard iteration. This tool (unified Lambert tool) consists of four individual algorithms, each of which is unique and better suited for solving a particular type of orbit transfer. The first is a Keplerian Lambert solver, which is used to provide a good initial guess (warm start) for solving the perturbed problem. It is also used to determine the appropriate algorithm to call for solving the perturbed problem. The arc length or true anomaly angle spanned by the transfer trajectory is the parameter that governs the automated selection of the appropriate perturbed algorithm, and is based on the respective algorithm convergence characteristics. The second algorithm solves the perturbed Lambert problem using the modified Chebyshev-Picard iteration two-point boundary value solver. This algorithm does not require a Newton-like shooting method and is the most efficient of the perturbed solvers presented herein, however the domain of convergence is limited to about a third of an orbit and is dependent on eccentricity. The third algorithm extends the domain of convergence of the modified Chebyshev-Picard iteration two-point boundary value solver to about 90% of an orbit, through regularization with the Kustaanheimo-Stiefel transformation. This is the second most efficient of the perturbed set of algorithms. The fourth algorithm uses the method of particular solutions and the modified Chebyshev-Picard iteration initial value solver for solving multiple revolution perturbed transfers. This method does require "shooting" but differs from Newton-like shooting methods in that it does not require propagation of a state transition matrix. The unified Lambert tool makes use of the General Mission Analysis Tool and we use it to compute thousands of perturbed Lambert trajectories in parallel on the Space Situational Awareness computer cluster at the LASR Lab, Texas A&M University. We demonstrate the power of our tool by solving a highly parallel example problem, that is the generation of extremal field maps for optimal spacecraft rendezvous (and eventual orbit debris removal). In addition we demonstrate the need for including perturbative effects in simulations for satellite tracking or data association. The unified Lambert tool is ideal for but not limited to space situational awareness applications.

Documentation of computer program VS2D to solve the equations of fluid flow in variably saturated porous media

USGS Publications Warehouse

Lappala, E.G.; Healy, R.W.; Weeks, E.P.

1987-01-01

This report documents FORTRAN computer code for solving problems involving variably saturated single-phase flow in porous media. The flow equation is written with total hydraulic potential as the dependent variable, which allows straightforward treatment of both saturated and unsaturated conditions. The spatial derivatives in the flow equation are approximated by central differences, and time derivatives are approximated either by a fully implicit backward or by a centered-difference scheme. Nonlinear conductance and storage terms may be linearized using either an explicit method or an implicit Newton-Raphson method. Relative hydraulic conductivity is evaluated at cell boundaries by using either full upstream weighting, the arithmetic mean, or the geometric mean of values from adjacent cells. Nonlinear boundary conditions treated by the code include infiltration, evaporation, and seepage faces. Extraction by plant roots that is caused by atmospheric demand is included as a nonlinear sink term. These nonlinear boundary and sink terms are linearized implicitly. The code has been verified for several one-dimensional linear problems for which analytical solutions exist and against two nonlinear problems that have been simulated with other numerical models. A complete listing of data-entry requirements and data entry and results for three example problems are provided. (USGS)

Dynamic imaging in electrical impedance tomography of the human chest with online transition matrix identification.

PubMed

Moura, Fernando Silva; Aya, Julio Cesar Ceballos; Fleury, Agenor Toledo; Amato, Marcelo Britto Passos; Lima, Raul Gonzalez

2010-02-01

One of the electrical impedance tomography objectives is to estimate the electrical resistivity distribution in a domain based only on electrical potential measurements at its boundary generated by an imposed electrical current distribution into the boundary. One of the methods used in dynamic estimation is the Kalman filter. In biomedical applications, the random walk model is frequently used as evolution model and, under this conditions, poor tracking ability of the extended Kalman filter (EKF) is achieved. An analytically developed evolution model is not feasible at this moment. The paper investigates the identification of the evolution model in parallel to the EKF and updating the evolution model with certain periodicity. The evolution model transition matrix is identified using the history of the estimated resistivity distribution obtained by a sensitivity matrix based algorithm and a Newton-Raphson algorithm. To numerically identify the linear evolution model, the Ibrahim time-domain method is used. The investigation is performed by numerical simulations of a domain with time-varying resistivity and by experimental data collected from the boundary of a human chest during normal breathing. The obtained dynamic resistivity values lie within the expected values for the tissues of a human chest. The EKF results suggest that the tracking ability is significantly improved with this approach.

Differentially private distributed logistic regression using private and public data

PubMed Central

2014-01-01

Background Privacy protecting is an important issue in medical informatics and differential privacy is a state-of-the-art framework for data privacy research. Differential privacy offers provable privacy against attackers who have auxiliary information, and can be applied to data mining models (for example, logistic regression). However, differentially private methods sometimes introduce too much noise and make outputs less useful. Given available public data in medical research (e.g. from patients who sign open-consent agreements), we can design algorithms that use both public and private data sets to decrease the amount of noise that is introduced. Methodology In this paper, we modify the update step in Newton-Raphson method to propose a differentially private distributed logistic regression model based on both public and private data. Experiments and results We try our algorithm on three different data sets, and show its advantage over: (1) a logistic regression model based solely on public data, and (2) a differentially private distributed logistic regression model based on private data under various scenarios. Conclusion Logistic regression models built with our new algorithm based on both private and public datasets demonstrate better utility than models that trained on private or public datasets alone without sacrificing the rigorous privacy guarantee. PMID:25079786

Parallel computing techniques for rotorcraft aerodynamics

NASA Astrophysics Data System (ADS)

Ekici, Kivanc

The modification of unsteady three-dimensional Navier-Stokes codes for application on massively parallel and distributed computing environments is investigated. The Euler/Navier-Stokes code TURNS (Transonic Unsteady Rotor Navier-Stokes) was chosen as a test bed because of its wide use by universities and industry. For the efficient implementation of TURNS on parallel computing systems, two algorithmic changes are developed. First, main modifications to the implicit operator, Lower-Upper Symmetric Gauss Seidel (LU-SGS) originally used in TURNS, is performed. Second, application of an inexact Newton method, coupled with a Krylov subspace iterative method (Newton-Krylov method) is carried out. Both techniques have been tried previously for the Euler equations mode of the code. In this work, we have extended the methods to the Navier-Stokes mode. Several new implicit operators were tried because of convergence problems of traditional operators with the high cell aspect ratio (CAR) grids needed for viscous calculations on structured grids. Promising results for both Euler and Navier-Stokes cases are presented for these operators. For the efficient implementation of Newton-Krylov methods to the Navier-Stokes mode of TURNS, efficient preconditioners must be used. The parallel implicit operators used in the previous step are employed as preconditioners and the results are compared. The Message Passing Interface (MPI) protocol has been used because of its portability to various parallel architectures. It should be noted that the proposed methodology is general and can be applied to several other CFD codes (e.g. OVERFLOW).

Estimation of parameters in rational reaction rates of molecular biological systems via weighted least squares

NASA Astrophysics Data System (ADS)

Wu, Fang-Xiang; Mu, Lei; Shi, Zhong-Ke

2010-01-01

The models of gene regulatory networks are often derived from statistical thermodynamics principle or Michaelis-Menten kinetics equation. As a result, the models contain rational reaction rates which are nonlinear in both parameters and states. It is challenging to estimate parameters nonlinear in a model although there have been many traditional nonlinear parameter estimation methods such as Gauss-Newton iteration method and its variants. In this article, we develop a two-step method to estimate the parameters in rational reaction rates of gene regulatory networks via weighted linear least squares. This method takes the special structure of rational reaction rates into consideration. That is, in the rational reaction rates, the numerator and the denominator are linear in parameters. By designing a special weight matrix for the linear least squares, parameters in the numerator and the denominator can be estimated by solving two linear least squares problems. The main advantage of the developed method is that it can produce the analytical solutions to the estimation of parameters in rational reaction rates which originally is nonlinear parameter estimation problem. The developed method is applied to a couple of gene regulatory networks. The simulation results show the superior performance over Gauss-Newton method.

Efficient combination of a 3D Quasi-Newton inversion algorithm and a vector dual-primal finite element tearing and interconnecting method

NASA Astrophysics Data System (ADS)

Voznyuk, I.; Litman, A.; Tortel, H.

2015-08-01

A Quasi-Newton method for reconstructing the constitutive parameters of three-dimensional (3D) penetrable scatterers from scattered field measurements is presented. This method is adapted for handling large-scale electromagnetic problems while keeping the memory requirement and the time flexibility as low as possible. The forward scattering problem is solved by applying the finite-element tearing and interconnecting full-dual-primal (FETI-FDP2) method which shares the same spirit as the domain decomposition methods for finite element methods. The idea is to split the computational domain into smaller non-overlapping sub-domains in order to simultaneously solve local sub-problems. Various strategies are proposed in order to efficiently couple the inversion algorithm with the FETI-FDP2 method: a separation into permanent and non-permanent subdomains is performed, iterative solvers are favorized for resolving the interface problem and a marching-on-in-anything initial guess selection further accelerates the process. The computational burden is also reduced by applying the adjoint state vector methodology. Finally, the inversion algorithm is confronted to measurements extracted from the 3D Fresnel database.

Inverts permittivity and conductivity with structural constraint in GPR FWI based on truncated Newton method

NASA Astrophysics Data System (ADS)

Ren, Qianci

2018-04-01

Full waveform inversion (FWI) of ground penetrating radar (GPR) is a promising technique to quantitatively evaluate the permittivity and conductivity of near subsurface. However, these two parameters are simultaneously inverted in the GPR FWI, increasing the difficulty to obtain accurate inversion results for both parameters. In this study, I present a structural constrained GPR FWI procedure to jointly invert the two parameters, aiming to force a structural relationship between permittivity and conductivity in the process of model reconstruction. The structural constraint is enforced by a cross-gradient function. In this procedure, the permittivity and conductivity models are inverted alternately at each iteration and updated with hierarchical frequency components in the frequency domain. The joint inverse problem is solved by the truncated Newton method which considering the effect of Hessian operator and using the approximated solution of Newton equation to be the perturbation model in the updating process. The joint inversion procedure is tested by three synthetic examples. The results show that jointly inverting permittivity and conductivity in GPR FWI effectively increases the structural similarities between the two parameters, corrects the structures of parameter models, and significantly improves the accuracy of conductivity model, resulting in a better inversion result than the individual inversion.

On the use of finite difference matrix-vector products in Newton-Krylov solvers for implicit climate dynamics with spectral elements

DOE PAGES

Woodward, Carol S.; Gardner, David J.; Evans, Katherine J.

2015-01-01

Efficient solutions of global climate models require effectively handling disparate length and time scales. Implicit solution approaches allow time integration of the physical system with a step size governed by accuracy of the processes of interest rather than by stability of the fastest time scales present. Implicit approaches, however, require the solution of nonlinear systems within each time step. Usually, a Newton's method is applied to solve these systems. Each iteration of the Newton's method, in turn, requires the solution of a linear model of the nonlinear system. This model employs the Jacobian of the problem-defining nonlinear residual, but thismore » Jacobian can be costly to form. If a Krylov linear solver is used for the solution of the linear system, the action of the Jacobian matrix on a given vector is required. In the case of spectral element methods, the Jacobian is not calculated but only implemented through matrix-vector products. The matrix-vector multiply can also be approximated by a finite difference approximation which may introduce inaccuracy in the overall nonlinear solver. In this paper, we review the advantages and disadvantages of finite difference approximations of these matrix-vector products for climate dynamics within the spectral element shallow water dynamical core of the Community Atmosphere Model.« less

Efficient Inversion of Mult-frequency and Multi-Source Electromagnetic Data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gary D. Egbert

2007-03-22

The project covered by this report focused on development of efficient but robust non-linear inversion algorithms for electromagnetic induction data, in particular for data collected with multiple receivers, and multiple transmitters, a situation extremely common in eophysical EM subsurface imaging methods. A key observation is that for such multi-transmitter problems each step in commonly used linearized iterative limited memory search schemes such as conjugate gradients (CG) requires solution of forward and adjoint EM problems for each of the N frequencies or sources, essentially generating data sensitivities for an N dimensional data-subspace. These multiple sensitivities allow a good approximation to themore » full Jacobian of the data mapping to be built up in many fewer search steps than would be required by application of textbook optimization methods, which take no account of the multiplicity of forward problems that must be solved for each search step. We have applied this idea to a develop a hybrid inversion scheme that combines features of the iterative limited memory type methods with a Newton-type approach using a partial calculation of the Jacobian. Initial tests on 2D problems show that the new approach produces results essentially identical to a Newton type Occam minimum structure inversion, while running more rapidly than an iterative (fixed regularization parameter) CG style inversion. Memory requirements, while greater than for something like CG, are modest enough that even in 3D the scheme should allow 3D inverse problems to be solved on a common desktop PC, at least for modest (~ 100 sites, 15-20 frequencies) data sets. A secondary focus of the research has been development of a modular system for EM inversion, using an object oriented approach. This system has proven useful for more rapid prototyping of inversion algorithms, in particular allowing initial development and testing to be conducted with two-dimensional example problems, before approaching more computationally cumbersome three-dimensional problems.« less

A Block Preconditioned Conjugate Gradient-type Iterative Solver for Linear Systems in Thermal Reservoir Simulation

NASA Astrophysics Data System (ADS)

Betté, Srinivas; Diaz, Julio C.; Jines, William R.; Steihaug, Trond

1986-11-01

A preconditioned residual-norm-reducing iterative solver is described. Based on a truncated form of the generalized-conjugate-gradient method for nonsymmetric systems of linear equations, the iterative scheme is very effective for linear systems generated in reservoir simulation of thermal oil recovery processes. As a consequence of employing an adaptive implicit finite-difference scheme to solve the model equations, the number of variables per cell-block varies dynamically over the grid. The data structure allows for 5- and 9-point operators in the areal model, 5-point in the cross-sectional model, and 7- and 11-point operators in the three-dimensional model. Block-diagonal-scaling of the linear system, done prior to iteration, is found to have a significant effect on the rate of convergence. Block-incomplete-LU-decomposition (BILU) and block-symmetric-Gauss-Seidel (BSGS) methods, which result in no fill-in, are used as preconditioning procedures. A full factorization is done on the well terms, and the cells are ordered in a manner which minimizes the fill-in in the well-column due to this factorization. The convergence criterion for the linear (inner) iteration is linked to that of the nonlinear (Newton) iteration, thereby enhancing the efficiency of the computation. The algorithm, with both BILU and BSGS preconditioners, is evaluated in the context of a variety of thermal simulation problems. The solver is robust and can be used with little or no user intervention.

Recursive inverse kinematics for robot arms via Kalman filtering and Bryson-Frazier smoothing

NASA Technical Reports Server (NTRS)

Rodriguez, G.; Scheid, R. E., Jr.

1987-01-01

This paper applies linear filtering and smoothing theory to solve recursively the inverse kinematics problem for serial multilink manipulators. This problem is to find a set of joint angles that achieve a prescribed tip position and/or orientation. A widely applicable numerical search solution is presented. The approach finds the minimum of a generalized distance between the desired and the actual manipulator tip position and/or orientation. Both a first-order steepest-descent gradient search and a second-order Newton-Raphson search are developed. The optimal relaxation factor required for the steepest descent method is computed recursively using an outward/inward procedure similar to those used typically for recursive inverse dynamics calculations. The second-order search requires evaluation of a gradient and an approximate Hessian. A Gauss-Markov approach is used to approximate the Hessian matrix in terms of products of first-order derivatives. This matrix is inverted recursively using a two-stage process of inward Kalman filtering followed by outward smoothing. This two-stage process is analogous to that recently developed by the author to solve by means of spatial filtering and smoothing the forward dynamics problem for serial manipulators.

Estimation of the longitudinal and lateral-directional aerodynamic parameters from flight data for the NASA F/A-18 HARV

NASA Technical Reports Server (NTRS)

Napolitano, Marcello R.

1996-01-01

This progress report presents the results of an investigation focused on parameter identification for the NASA F/A-18 HARV. This aircraft was used in the high alpha research program at the NASA Dryden Flight Research Center. In this study the longitudinal and lateral-directional stability derivatives are estimated from flight data using the Maximum Likelihood method coupled with a Newton-Raphson minimization technique. The objective is to estimate an aerodynamic model describing the aircraft dynamics over a range of angle of attack from 5 deg to 60 deg. The mathematical model is built using the traditional static and dynamic derivative buildup. Flight data used in this analysis were from a variety of maneuvers. The longitudinal maneuvers included large amplitude multiple doublets, optimal inputs, frequency sweeps, and pilot pitch stick inputs. The lateral-directional maneuvers consisted of large amplitude multiple doublets, optimal inputs and pilot stick and rudder inputs. The parameter estimation code pEst, developed at NASA Dryden, was used in this investigation. Results of the estimation process from alpha = 5 deg to alpha = 60 deg are presented and discussed.

A Second Law Based Unstructured Finite Volume Procedure for Generalized Flow Simulation

NASA Technical Reports Server (NTRS)

Majumdar, Alok

1998-01-01

An unstructured finite volume procedure has been developed for steady and transient thermo-fluid dynamic analysis of fluid systems and components. The procedure is applicable for a flow network consisting of pipes and various fittings where flow is assumed to be one dimensional. It can also be used to simulate flow in a component by modeling a multi-dimensional flow using the same numerical scheme. The flow domain is discretized into a number of interconnected control volumes located arbitrarily in space. The conservation equations for each control volume account for the transport of mass, momentum and entropy from the neighboring control volumes. In addition, they also include the sources of each conserved variable and time dependent terms. The source term of entropy equation contains entropy generation due to heat transfer and fluid friction. Thermodynamic properties are computed from the equation of state of a real fluid. The system of equations is solved by a hybrid numerical method which is a combination of simultaneous Newton-Raphson and successive substitution schemes. The paper also describes the application and verification of the procedure by comparing its predictions with the analytical and numerical solution of several benchmark problems.

Critical point analysis of phase envelope diagram

DOE Office of Scientific and Technical Information (OSTI.GOV)

Soetikno, Darmadi; Siagian, Ucok W. R.; Kusdiantara, Rudy, E-mail: rkusdiantara@s.itb.ac.id

2014-03-24

Phase diagram or phase envelope is a relation between temperature and pressure that shows the condition of equilibria between the different phases of chemical compounds, mixture of compounds, and solutions. Phase diagram is an important issue in chemical thermodynamics and hydrocarbon reservoir. It is very useful for process simulation, hydrocarbon reactor design, and petroleum engineering studies. It is constructed from the bubble line, dew line, and critical point. Bubble line and dew line are composed of bubble points and dew points, respectively. Bubble point is the first point at which the gas is formed when a liquid is heated. Meanwhile,more » dew point is the first point where the liquid is formed when the gas is cooled. Critical point is the point where all of the properties of gases and liquids are equal, such as temperature, pressure, amount of substance, and others. Critical point is very useful in fuel processing and dissolution of certain chemicals. Here in this paper, we will show the critical point analytically. Then, it will be compared with numerical calculations of Peng-Robinson equation by using Newton-Raphson method. As case studies, several hydrocarbon mixtures are simulated using by Matlab.« less

A study of the parallel algorithm for large-scale DC simulation of nonlinear systems

NASA Astrophysics Data System (ADS)

Cortés Udave, Diego Ernesto; Ogrodzki, Jan; Gutiérrez de Anda, Miguel Angel

Newton-Raphson DC analysis of large-scale nonlinear circuits may be an extremely time consuming process even if sparse matrix techniques and bypassing of nonlinear models calculation are used. A slight decrease in the time required for this task may be enabled on multi-core, multithread computers if the calculation of the mathematical models for the nonlinear elements as well as the stamp management of the sparse matrix entries are managed through concurrent processes. This numerical complexity can be further reduced via the circuit decomposition and parallel solution of blocks taking as a departure point the BBD matrix structure. This block-parallel approach may give a considerable profit though it is strongly dependent on the system topology and, of course, on the processor type. This contribution presents the easy-parallelizable decomposition-based algorithm for DC simulation and provides a detailed study of its effectiveness.

HADY-I, a FORTRAN program for the compressible stability analysis of three-dimensional boundary layers. [on swept and tapered wings

NASA Technical Reports Server (NTRS)

El-Hady, N. M.

1981-01-01

A computer program HADY-I for calculating the linear incompressible or compressible stability characteristics of the laminar boundary layer on swept and tapered wings is described. The eigenvalue problem and its adjoint arising from the linearized disturbance equations with the appropriate boundary conditions are solved numerically using a combination of Newton-Raphson interative scheme and a variable step size integrator based on the Runge-Kutta-Fehlburh fifth-order formulas. The integrator is used in conjunction with a modified Gram-Schmidt orthonormalization procedure. The computer program HADY-I calculates the growth rates of crossflow or streamwise Tollmien-Schlichting instabilities. It also calculates the group velocities of these disturbances. It is restricted to parallel stability calculations, where the boundary layer (meanflow) is assumed to be parallel. The meanflow solution is an input to the program.

Revisiting the factors which control the angle of shear bands in geodynamic numerical models of brittle deformation

NASA Astrophysics Data System (ADS)

Thieulot, Cedric

2017-04-01

In this work I present Finite Element numerical simulations of brittle deformation in two-dimensional Cartesian systems subjected to compressional or extensional kinematical boundary conditions with a basal velocity discontinuity. The rheology is visco-plastic and is characterised by a cohesion and an angle of internal friction (Drucker-Prager type). I will explore the influence of the following factors on the recovered shear band angles when the angle of internal friction is varied: a) element type (quadrilateral vs triangle), b) element order, c) continuous vs discontinous pressure, d) visco-plasticity model implementation, e) the nonlinear tolerance value, f) the use of markers, g) Picard vs Newton-Raphson, h) velocity discontinuity nature. I will present these results in the light of already published literature (e.g. Lemiale et al, PEPI 171, 2008; Kaus, Tectonophysics 484, 2010).

Solving regularly and singularly perturbed reaction-diffusion equations in three space dimensions

NASA Astrophysics Data System (ADS)

Moore, Peter K.

2007-06-01

In [P.K. Moore, Effects of basis selection and h-refinement on error estimator reliability and solution efficiency for higher-order methods in three space dimensions, Int. J. Numer. Anal. Mod. 3 (2006) 21-51] a fixed, high-order h-refinement finite element algorithm, Href, was introduced for solving reaction-diffusion equations in three space dimensions. In this paper Href is coupled with continuation creating an automatic method for solving regularly and singularly perturbed reaction-diffusion equations. The simple quasilinear Newton solver of Moore, (2006) is replaced by the nonlinear solver NITSOL [M. Pernice, H.F. Walker, NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. Sci. Comput. 19 (1998) 302-318]. Good initial guesses for the nonlinear solver are obtained using continuation in the small parameter ɛ. Two strategies allow adaptive selection of ɛ. The first depends on the rate of convergence of the nonlinear solver and the second implements backtracking in ɛ. Finally a simple method is used to select the initial ɛ. Several examples illustrate the effectiveness of the algorithm.

Implicit solvers for unstructured meshes

NASA Technical Reports Server (NTRS)

Venkatakrishnan, V.; Mavriplis, Dimitri J.

1991-01-01

Implicit methods for unstructured mesh computations are developed and tested. The approximate system which arises from the Newton-linearization of the nonlinear evolution operator is solved by using the preconditioned generalized minimum residual technique. These different preconditioners are investigated: the incomplete LU factorization (ILU), block diagonal factorization, and the symmetric successive over-relaxation (SSOR). The preconditioners have been optimized to have good vectorization properties. The various methods are compared over a wide range of problems. Ordering of the unknowns, which affects the convergence of these sparse matrix iterative methods, is also investigated. Results are presented for inviscid and turbulent viscous calculations on single and multielement airfoil configurations using globally and adaptively generated meshes.

Numerical modelling of thin-walled Z-columns made of general laminates subjected to uniform shortening

NASA Astrophysics Data System (ADS)

Teter, Andrzej; Kolakowski, Zbigniew

2018-01-01

The numerical modelling of a plate structure was performed with the finite element method and a one-mode approach based on Koiter's method. The first order approximation of Koiter's method enables one to solve the eigenvalue problem. The second order approximation describes post-buckling equilibrium paths. In the finite element analysis, the Lanczos method was used to solve the linear problem of buckling. Simulations of the non-linear problem were performed with the Newton-Raphson method. Detailed calculations were carried out for a short Z-column made of general laminates. Configurations of laminated layers were non-symmetric. Due to possibilities of its application, the general laminate is very interesting. The length of the samples was chosen to obtain the lowest value of local buckling load. The amplitude of initial imperfections was 10% of the wall thickness. Thin-walled structures were simply supported on both ends. The numerical results were verified in experimental tests. A strain-gauge technique was applied. A static compression test was performed on a universal testing machine and a special grip, which consisted of two rigid steel plates and clamping sleeves, was used. Specimens were obtained with an autoclave technique. Tests were performed at a constant velocity of the cross-bar equal to 2 mm/min. The compressive load was less than 150% of the bifurcation load. Additionally, soft and thin pads were used to reduce inaccuracy of the sample ends.

Multiple steady states in atmospheric chemistry

NASA Technical Reports Server (NTRS)

Stewart, Richard W.

1993-01-01

The equations describing the distributions and concentrations of trace species are nonlinear and may thus possess more than one solution. This paper develops methods for searching for multiple physical solutions to chemical continuity equations and applies these to subsets of equations describing tropospheric chemistry. The calculations are carried out with a box model and use two basic strategies. The first strategy is a 'search' method. This involves fixing model parameters at specified values, choosing a wide range of initial guesses at a solution, and using a Newton-Raphson technique to determine if different initial points converge to different solutions. The second strategy involves a set of techniques known as homotopy methods. These do not require an initial guess, are globally convergent, and are guaranteed, in principle, to find all solutions of the continuity equations. The first method is efficient but essentially 'hit or miss' in the sense that it cannot guarantee that all solutions which may exist will be found. The second method is computationally burdensome but can, in principle, determine all the solutions of a photochemical system. Multiple solutions have been found for models that contain a basic complement of photochemical reactions involving O(x), HO(x), NO(x), and CH4. In the present calculations, transitions occur between stable branches of a multiple solution set as a control parameter is varied. These transitions are manifestations of hysteresis phenomena in the photochemical system and may be triggered by increasing the NO flux or decreasing the CH4 flux from current mean tropospheric levels.

Accelerated gradient based diffuse optical tomographic image reconstruction.

PubMed

Biswas, Samir Kumar; Rajan, K; Vasu, R M

2011-01-01

Fast reconstruction of interior optical parameter distribution using a new approach called Broyden-based model iterative image reconstruction (BMOBIIR) and adjoint Broyden-based MOBIIR (ABMOBIIR) of a tissue and a tissue mimicking phantom from boundary measurement data in diffuse optical tomography (DOT). DOT is a nonlinear and ill-posed inverse problem. Newton-based MOBIIR algorithm, which is generally used, requires repeated evaluation of the Jacobian which consumes bulk of the computation time for reconstruction. In this study, we propose a Broyden approach-based accelerated scheme for Jacobian computation and it is combined with conjugate gradient scheme (CGS) for fast reconstruction. The method makes explicit use of secant and adjoint information that can be obtained from forward solution of the diffusion equation. This approach reduces the computational time many fold by approximating the system Jacobian successively through low-rank updates. Simulation studies have been carried out with single as well as multiple inhomogeneities. Algorithms are validated using an experimental study carried out on a pork tissue with fat acting as an inhomogeneity. The results obtained through the proposed BMOBIIR and ABMOBIIR approaches are compared with those of Newton-based MOBIIR algorithm. The mean squared error and execution time are used as metrics for comparing the results of reconstruction. We have shown through experimental and simulation studies that Broyden-based MOBIIR and adjoint Broyden-based methods are capable of reconstructing single as well as multiple inhomogeneities in tissue and a tissue-mimicking phantom. Broyden MOBIIR and adjoint Broyden MOBIIR methods are computationally simple and they result in much faster implementations because they avoid direct evaluation of Jacobian. The image reconstructions have been carried out with different initial values using Newton, Broyden, and adjoint Broyden approaches. These algorithms work well when the initial guess is close to the true solution. However, when initial guess is far away from true solution, Newton-based MOBIIR gives better reconstructed images. The proposed methods are found to be stable with noisy measurement data.

Behaviour of Frictional Joints in Steel Arch Yielding Supports

NASA Astrophysics Data System (ADS)

Horyl, Petr; Šňupárek, Richard; Maršálek, Pavel

2014-10-01

The loading capacity and ability of steel arch supports to accept deformations from the surrounding rock mass is influenced significantly by the function of the connections and in particular, the tightening of the bolts. This contribution deals with computer modelling of the yielding bolt connections for different torques to determine the load-bearing capacity of the connections. Another parameter that affects the loading capacity significantly is the value of the friction coefficient of the contacts between the elements of the joints. The authors investigated both the behaviour and conditions of the individual parts for three values of tightening moment and the relation between the value of screw tightening and load-bearing capacity of the connections for different friction coefficients. ANSYS software and the finite element method were used for the computer modelling. The solution is nonlinear because of the bi-linear material properties of steel and the large deformations. The geometry of the computer model was created from designs of all four parts of the structure. The calculation also defines the weakest part of the joint's structure based on stress analysis. The load was divided into two loading steps: the pre-tensioning of connecting bolts and the deformation loading corresponding to 50-mm slip of one support. The full Newton-Raphson method was chosen for the solution. The calculations were carried out on a computer at the Supercomputing Centre VSB-Technical University of Ostrava.

Size-dependent axisymmetric vibration of functionally graded circular plates in bifurcation/limit point instability

NASA Astrophysics Data System (ADS)

Ashoori, A. R.; Vanini, S. A. Sadough; Salari, E.

2017-04-01

In the present paper, vibration behavior of size-dependent functionally graded (FG) circular microplates subjected to thermal loading are carried out in pre/post-buckling of bifurcation/limit-load instability for the first time. Two kinds of frequently used thermal loading, i.e., uniform temperature rise and heat conduction across the thickness direction are considered. Thermo-mechanical material properties of FG plate are supposed to vary smoothly and continuously throughout the thickness based on power law model. Modified couple stress theory is exploited to describe the size dependency of microplate. The nonlinear governing equations of motion and associated boundary conditions are extracted through generalized form of Hamilton's principle and von-Karman geometric nonlinearity for the vibration analysis of circular FG plates including size effects. Ritz finite element method is then employed to construct the matrix representation of governing equations which are solved by two different strategies including Newton-Raphson scheme and cylindrical arc-length method. Moreover, in the following a parametric study is accompanied to examine the effects of the several parameters such as material length scale parameter, temperature distributions, type of buckling, thickness to radius ratio, boundary conditions and power law index on the dimensionless frequency of post-buckled/snapped size-dependent FG plates in detail. It is found that the material length scale parameter and thermal loading have a significant effect on vibration characteristics of size-dependent circular FG plates.

Efficient implementation of three-dimensional reference interaction site model self-consistent-field method: Application to solvatochromic shift calculations

NASA Astrophysics Data System (ADS)

Minezawa, Noriyuki; Kato, Shigeki

2007-02-01

The authors present an implementation of the three-dimensional reference interaction site model self-consistent-field (3D-RISM-SCF) method. First, they introduce a robust and efficient algorithm for solving the 3D-RISM equation. The algorithm is a hybrid of the Newton-Raphson and Picard methods. The Jacobian matrix is analytically expressed in a computationally useful form. Second, they discuss the solute-solvent electrostatic interaction. For the solute to solvent route, the electrostatic potential (ESP) map on a 3D grid is constructed directly from the electron density. The charge fitting procedure is not required to determine the ESP. For the solvent to solute route, the ESP acting on the solute molecule is derived from the solvent charge distribution obtained by solving the 3D-RISM equation. Matrix elements of the solute-solvent interaction are evaluated by the direct numerical integration. A remarkable reduction in the computational time is observed in both routes. Finally, the authors implement the first derivatives of the free energy with respect to the solute nuclear coordinates. They apply the present method to "solute" water and formaldehyde in aqueous solvent using the simple point charge model, and the results are compared with those from other methods: the six-dimensional molecular Ornstein-Zernike SCF, the one-dimensional site-site RISM-SCF, and the polarizable continuum model. The authors also calculate the solvatochromic shifts of acetone, benzonitrile, and nitrobenzene using the present method and compare them with the experimental and other theoretical results.

Efficient implementation of three-dimensional reference interaction site model self-consistent-field method: application to solvatochromic shift calculations.

PubMed

Minezawa, Noriyuki; Kato, Shigeki

2007-02-07

The authors present an implementation of the three-dimensional reference interaction site model self-consistent-field (3D-RISM-SCF) method. First, they introduce a robust and efficient algorithm for solving the 3D-RISM equation. The algorithm is a hybrid of the Newton-Raphson and Picard methods. The Jacobian matrix is analytically expressed in a computationally useful form. Second, they discuss the solute-solvent electrostatic interaction. For the solute to solvent route, the electrostatic potential (ESP) map on a 3D grid is constructed directly from the electron density. The charge fitting procedure is not required to determine the ESP. For the solvent to solute route, the ESP acting on the solute molecule is derived from the solvent charge distribution obtained by solving the 3D-RISM equation. Matrix elements of the solute-solvent interaction are evaluated by the direct numerical integration. A remarkable reduction in the computational time is observed in both routes. Finally, the authors implement the first derivatives of the free energy with respect to the solute nuclear coordinates. They apply the present method to "solute" water and formaldehyde in aqueous solvent using the simple point charge model, and the results are compared with those from other methods: the six-dimensional molecular Ornstein-Zernike SCF, the one-dimensional site-site RISM-SCF, and the polarizable continuum model. The authors also calculate the solvatochromic shifts of acetone, benzonitrile, and nitrobenzene using the present method and compare them with the experimental and other theoretical results.

SIERRA Multimechanics Module: Aria User Manual Version 4.44

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sierra Thermal /Fluid Team

2017-04-01

Aria is a Galerkin fnite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process fows via the incompressible Navier-Stokes equations specialized to a low Reynolds number ( %3C 1 ) regime. Enhanced modeling support of manufacturing processing is made possible through use of eithermore » arbitrary Lagrangian- Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newton's method with analytic or numerical sensitivities, fully-coupled Newton- Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h -adaptivity and dynamic load balancing are some of Aria's more advanced capabilities. Aria is based upon the Sierra Framework.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sierra Thermal/Fluid Team

Aria is a Galerkin fnite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process fows via the incompressible Navier-Stokes equations specialized to a low Reynolds number ( %3C 1 ) regime. Enhanced modeling support of manufacturing processing is made possible through use of eithermore » arbitrary Lagrangian- Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newton's method with analytic or numerical sensitivities, fully-coupled Newton- Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h -adaptivity and dynamic load balancing are some of Aria's more advanced capabilities. Aria is based upon the Sierra Framework.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sierra Thermal /Fluid Team

Aria is a Galerkin finite element based program for solving coupled-physics problems described by systems of PDEs and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in two and three dimensions on parallel architectures. The suite of physics currently supported by Aria includes thermal energy transport, species transport, and electrostatics as well as generalized scalar, vector and tensor transport equations. Additionally, Aria includes support for manufacturing process flows via the incompressible Navier-Stokes equations specialized to a low Reynolds number (Re %3C 1) regime. Enhanced modeling support of manufacturing processing is made possible through use of either arbitrarymore » Lagrangian- Eulerian (ALE) and level set based free and moving boundary tracking in conjunction with quasi-static nonlinear elastic solid mechanics for mesh control. Coupled physics problems are solved in several ways including fully-coupled Newton's method with analytic or numerical sensitivities, fully-coupled Newton- Krylov methods and a loosely-coupled nonlinear iteration about subsets of the system that are solved using combinations of the aforementioned methods. Error estimation, uniform and dynamic h-adaptivity and dynamic load balancing are some of Aria's more advanced capabilities. Aria is based upon the Sierra Framework.« less

A deterministic particle method for one-dimensional reaction-diffusion equations

NASA Technical Reports Server (NTRS)

Mascagni, Michael

1995-01-01

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.

Ion-dipole interactions in concentrated organic electrolytes.

PubMed

Chagnes, Alexandre; Nicolis, Stamatios; Carré, Bernard; Willmann, Patrick; Lemordant, Daniel

2003-06-16

An algorithm is proposed for calculating the energy of ion-dipole interactions in concentrated organic electrolytes. The ion-dipole interactions increase with increasing salt concentration and must be taken into account when the activation energy for the conductivity is calculated. In this case, the contribution of ion-dipole interactions to the activation energy for this transport process is of the same order of magnitude as the contribution of ion-ion interactions. The ion-dipole interaction energy was calculated for a cell of eight ions, alternatingly anions and cations, placed on the vertices of an expanded cubic lattice whose parameter is related to the mean interionic distance (pseudolattice theory). The solvent dipoles were introduced randomly into the cell by assuming a randomness compacity of 0.58. The energy of the dipole assembly in the cell was minimized by using a Newton-Raphson numerical method. The dielectric field gradient around ions was taken into account by a distance parameter and a dielectric constant of epsilon = 3 at the surfaces of the ions. A fair agreement between experimental and calculated activation energy has been found for systems composed of gamma-butyrolactone (BL) as solvent and lithium perchlorate (LiClO4), lithium tetrafluoroborate (LiBF4), lithium hexafluorophosphate (LiPF6), lithium hexafluoroarsenate (LiAsF6), and lithium bis(trifluoromethylsulfonyl)imide (LiTFSI) as salts.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rycroft, Chris H.; Bazant, Martin Z.

An advection-diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and collapse to a single point in finite time. The simulations reveal a surprising exact relationship, whereby the collapse point is the root of a non-Analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape.more » This result is subsequently derived using residue calculus. The structure of the non-Analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton-Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finite-Time singularities of both continuous and stochastic dynamical systems.« less

Theoretical study of heat transfer with moving phase-change interface in thawing of frozen food

NASA Astrophysics Data System (ADS)

Leung, M.; Ching, W. H.; Leung, D. Y. C.; Lam, G. C. K.

2005-02-01

A theoretical solution was obtained for a transient phase-change heat transfer problem in thawing of frozen food. In the physical model, a sphere originally at a uniform temperature below the phase-change temperature is suddenly immersed in a fluid at a temperature above the phase-change temperature. As the body temperature increases, the phase-change interface will be first formed on the surface. Subsequently, the interface will absorb the latent heat and move towards the centre until the whole body undergoes complete phase change. In the mathematical formulation, the nonhomogeneous problem arises from the moving phase-change interface. The solution in terms of the time-dependent temperature field was obtained by use of Green's function. A one-step Newton-Raphson method was specially designed to solve for the position of the moving interface to satisfy the interface condition. The theoretical results were compared with numerical results generated by a finite difference model and experimental measurements collected from a cold water thawing process. As a good agreement was found, the theoretical solution developed in this study was verified numerically and experimentally. Besides thawing of frozen food, there are many other practical applications of the theoretical solution, such as food freezing, soil freezing/thawing, metal casting and bath quenching heat treatment, among others.

The influence of image reconstruction algorithms on linear thorax EIT image analysis of ventilation.

PubMed

Zhao, Zhanqi; Frerichs, Inéz; Pulletz, Sven; Müller-Lisse, Ullrich; Möller, Knut

2014-06-01

Analysis methods of electrical impedance tomography (EIT) images based on different reconstruction algorithms were examined. EIT measurements were performed on eight mechanically ventilated patients with acute respiratory distress syndrome. A maneuver with step increase of airway pressure was performed. EIT raw data were reconstructed offline with (1) filtered back-projection (BP); (2) the Dräger algorithm based on linearized Newton-Raphson (DR); (3) the GREIT (Graz consensus reconstruction algorithm for EIT) reconstruction algorithm with a circular forward model (GR(C)) and (4) GREIT with individual thorax geometry (GR(T)). Individual thorax contours were automatically determined from the routine computed tomography images. Five indices were calculated on the resulting EIT images respectively: (a) the ratio between tidal and deep inflation impedance changes; (b) tidal impedance changes in the right and left lungs; (c) center of gravity; (d) the global inhomogeneity index and (e) ventilation delay at mid-dorsal regions. No significant differences were found in all examined indices among the four reconstruction algorithms (p > 0.2, Kruskal-Wallis test). The examined algorithms used for EIT image reconstruction do not influence the selected indices derived from the EIT image analysis. Indices that validated for images with one reconstruction algorithm are also valid for other reconstruction algorithms.

The Effect of Plug-in Electric Vehicles on Harmonic Analysis of Smart Grid

NASA Astrophysics Data System (ADS)

Heidarian, T.; Joorabian, M.; Reza, A.

2015-12-01

In this paper, the effect of plug-in electric vehicles is studied on the smart distribution system with a standard IEEE 30-bus network. At first, harmonic power flow analysis is performed by Newton-Raphson method and by considering distorted substation voltage. Afterward, proper sizes of capacitors is selected by cuckoo optimization algorithm to reduce the power losses and cost and by imposing acceptable limit for total harmonic distortion and RMS voltages. It is proposed that the impact of generated current harmonics by electric vehicle battery chargers should be factored into overall load control strategies of smart appliances. This study is generalized to the different hours of a day by using daily load curve, and then optimum time for charging of electric vehicles batteries in the parking lots are determined by cuckoo optimization algorithm. The results show that injecting harmonic currents of plug-in electric vehicles causes a drop in the voltage profile and increases power loss. Moreover, charging the vehicle batteries has more impact on increasing the power losses rather than the harmonic currents effect. Also, the findings showed that the current harmonics has a great influence on increasing of THD. Finally, optimum working times of all parking lots was obtained for the utilization cost reduction.

Co-rotational thermo-mechanically coupled multi-field framework and finite element for the large displacement analysis of multi-layered shape memory alloy beam-like structures

NASA Astrophysics Data System (ADS)

Solomou, Alexandros G.; Machairas, Theodoros T.; Karakalas, Anargyros A.; Saravanos, Dimitris A.

2017-06-01

A thermo-mechanically coupled finite element (FE) for the simulation of multi-layered shape memory alloy (SMA) beams admitting large displacements and rotations (LDRs) is developed to capture the geometrically nonlinear effects which are present in many SMA applications. A generalized multi-field beam theory implementing a SMA constitutive model based on small strain theory, thermo-mechanically coupled governing equations and multi-field kinematic hypotheses combining first order shear deformation assumptions with a sixth order polynomial temperature field through the thickness of the beam section are extended to admit LDRs. The co-rotational formulation is adopted, where the motion of the beam is decomposed to rigid body motion and relative small deformation in the local frame. A new generalized multi-layered SMA FE is formulated. The nonlinear transient spatial discretized equations of motion of the SMA structure are synthesized and solved using the Newton-Raphson method combined with an implicit time integration scheme. Correlations of models incorporating the present beam FE with respective results of models incorporating plane stress SMA FEs, demonstrate excellent agreement of the predicted LDRs response, temperature and phase transformation fields, as well as, significant gains in computational time.

A displacement-based finite element formulation for incompressible and nearly-incompressible cardiac mechanics.

PubMed

Hadjicharalambous, Myrianthi; Lee, Jack; Smith, Nicolas P; Nordsletten, David A

2014-06-01

The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii-Newton-Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.

Asymmetric collapse by dissolution or melting in a uniform flow

DOE PAGES

Rycroft, Chris H.; Bazant, Martin Z.

2016-01-06

An advection-diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and collapse to a single point in finite time. The simulations reveal a surprising exact relationship, whereby the collapse point is the root of a non-Analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape.more » This result is subsequently derived using residue calculus. The structure of the non-Analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton-Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). In conclusion, the model raises fundamental mathematical questions about broken symmetries in finite-Time singularities of both continuous and stochastic dynamical systems.« less

Numerical simulation of the nonlinear response of composite plates under combined thermal and acoustic loading

NASA Technical Reports Server (NTRS)

Mei, Chuh; Moorthy, Jayashree

1995-01-01

A time-domain study of the random response of a laminated plate subjected to combined acoustic and thermal loads is carried out. The features of this problem also include given uniform static inplane forces. The formulation takes into consideration a possible initial imperfection in the flatness of the plate. High decibel sound pressure levels along with high thermal gradients across thickness drive the plate response into nonlinear regimes. This calls for the analysis to use von Karman large deflection strain-displacement relationships. A finite element model that combines the von Karman strains with the first-order shear deformation plate theory is developed. The development of the analytical model can accommodate an anisotropic composite laminate built up of uniformly thick layers of orthotropic, linearly elastic laminae. The global system of finite element equations is then reduced to a modal system of equations. Numerical simulation using a single-step algorithm in the time-domain is then carried out to solve for the modal coordinates. Nonlinear algebraic equations within each time-step are solved by the Newton-Raphson method. The random gaussian filtered white noise load is generated using Monte Carlo simulation. The acoustic pressure distribution over the plate is capable of accounting for a grazing incidence wavefront. Numerical results are presented to study a variety of cases.

A nearly-linear computational-cost scheme for the forward dynamics of an N-body pendulum

NASA Technical Reports Server (NTRS)

Chou, Jack C. K.

1989-01-01

The dynamic equations of motion of an n-body pendulum with spherical joints are derived to be a mixed system of differential and algebraic equations (DAE's). The DAE's are kept in implicit form to save arithmetic and preserve the sparsity of the system and are solved by the robust implicit integration method. At each solution point, the predicted solution is corrected to its exact solution within given tolerance using Newton's iterative method. For each iteration, a linear system of the form J delta X = E has to be solved. The computational cost for solving this linear system directly by LU factorization is O(n exp 3), and it can be reduced significantly by exploring the structure of J. It is shown that by recognizing the recursive patterns and exploiting the sparsity of the system the multiplicative and additive computational costs for solving J delta X = E are O(n) and O(n exp 2), respectively. The formulation and solution method for an n-body pendulum is presented. The computational cost is shown to be nearly linearly proportional to the number of bodies.

Implicit solution of Navier-Stokes equations on staggered curvilinear grids using a Newton-Krylov method with a novel analytical Jacobian.

NASA Astrophysics Data System (ADS)

Borazjani, Iman; Asgharzadeh, Hafez

2015-11-01

Flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates with explicit and semi-implicit schemes. Implicit schemes can be used to overcome these restrictions. However, implementing implicit solver for nonlinear equations including Navier-Stokes is not straightforward. Newton-Krylov subspace methods (NKMs) are one of the most advanced iterative methods to solve non-linear equations such as implicit descritization of the Navier-Stokes equation. The efficiency of NKMs massively depends on the Jacobian formation method, e.g., automatic differentiation is very expensive, and matrix-free methods slow down as the mesh is refined. Analytical Jacobian is inexpensive method, but derivation of analytical Jacobian for Navier-Stokes equation on staggered grid is challenging. The NKM with a novel analytical Jacobian was developed and validated against Taylor-Green vortex and pulsatile flow in a 90 degree bend. The developed method successfully handled the complex geometries such as an intracranial aneurysm with multiple overset grids, and immersed boundaries. It is shown that the NKM with an analytical Jacobian is 3 to 25 times faster than the fixed-point implicit Runge-Kutta method, and more than 100 times faster than automatic differentiation depending on the grid (size) and the flow problem. The developed methods are fully parallelized with parallel efficiency of 80-90% on the problems tested.

A numerical study of steady crystal growth in a vertical Bridgman device

NASA Astrophysics Data System (ADS)

Jalics, Miklos Kalman

Electronics based on semiconductors creates an enormous demand for high quality semiconductor single crystals. The vertical Bridgman device is commonly used for growing single crystals for a variety of materials such as GaAs, InP and HgCdTe. A mathematical model is presented for steady crystal growth under conditions where crystal growth is determined strictly by heat transfer. The ends of the ampoule are chosen far away from the insulation zone to allow for steady growth. A numerical solution is sought for this mathematical model. The equations are transformed into a rectangular geometry and appropriate finite difference techniques are applied on the transformed equations. Newton's method solves the nonlinear problem. To improve efficiency GMRES with preconditioning is used to compute the Newton iterates. The numerical results are used to compare with two current asymptotic theories that assume small Biot numbers. Results indicate that one of the asymptotic theories is accurate for even moderate Biot numbers.

Two-dimensional computer simulation of EMVJ and grating solar cells under AMO illumination

NASA Technical Reports Server (NTRS)

Gray, J. L.; Schwartz, R. J.

1984-01-01

A computer program, SCAP2D (Solar Cell Analysis Program in 2-Dimensions), is used to evaluate the Etched Multiple Vertical Junction (EMVJ) and grating solar cells. The aim is to demonstrate how SCAP2D can be used to evaluate cell designs. The cell designs studied are by no means optimal designs. The SCAP2D program solves the three coupled, nonlinear partial differential equations, Poisson's Equation and the hole and electron continuity equations, simultaneously in two-dimensions using finite differences to discretize the equations and Newton's Method to linearize them. The variables solved for are the electrostatic potential and the hole and electron concentrations. Each linear system of equations is solved directly by Gaussian Elimination. Convergence of the Newton Iteration is assumed when the largest correction to the electrostatic potential or hole or electron quasi-potential is less than some predetermined error. A typical problem involves 2000 nodes with a Jacobi matrix of order 6000 and a bandwidth of 243.

Strategies for the coupling of global and local crystal growth models

NASA Astrophysics Data System (ADS)

Derby, Jeffrey J.; Lun, Lisa; Yeckel, Andrew

2007-05-01

The modular coupling of existing numerical codes to model crystal growth processes will provide for maximum effectiveness, capability, and flexibility. However, significant challenges are posed to make these coupled models mathematically self-consistent and algorithmically robust. This paper presents sample results from a coupling of the CrysVUn code, used here to compute furnace-scale heat transfer, and Cats2D, used to calculate melt fluid dynamics and phase-change phenomena, to form a global model for a Bridgman crystal growth system. However, the strategy used to implement the CrysVUn-Cats2D coupling is unreliable and inefficient. The implementation of under-relaxation within a block Gauss-Seidel iteration is shown to be ineffective for improving the coupling performance in a model one-dimensional problem representative of a melt crystal growth model. Ideas to overcome current convergence limitations using approximations to a full Newton iteration method are discussed.

Accelerating Subsurface Transport Simulation on Heterogeneous Clusters

DOE Office of Scientific and Technical Information (OSTI.GOV)

Villa, Oreste; Gawande, Nitin A.; Tumeo, Antonino

Reactive transport numerical models simulate chemical and microbiological reactions that occur along a flowpath. These models have to compute reactions for a large number of locations. They solve the set of ordinary differential equations (ODEs) that describes the reaction for each location through the Newton-Raphson technique. This technique involves computing a Jacobian matrix and a residual vector for each set of equation, and then solving iteratively the linearized system by performing Gaussian Elimination and LU decomposition until convergence. STOMP, a well known subsurface flow simulation tool, employs matrices with sizes in the order of 100x100 elements and, for numerical accuracy,more » LU factorization with full pivoting instead of the faster partial pivoting. Modern high performance computing systems are heterogeneous machines whose nodes integrate both CPUs and GPUs, exposing unprecedented amounts of parallelism. To exploit all their computational power, applications must use both the types of processing elements. For the case of subsurface flow simulation, this mainly requires implementing efficient batched LU-based solvers and identifying efficient solutions for enabling load balancing among the different processors of the system. In this paper we discuss two approaches that allows scaling STOMP's performance on heterogeneous clusters. We initially identify the challenges in implementing batched LU-based solvers for small matrices on GPUs, and propose an implementation that fulfills STOMP's requirements. We compare this implementation to other existing solutions. Then, we combine the batched GPU solver with an OpenMP-based CPU solver, and present an adaptive load balancer that dynamically distributes the linear systems to solve between the two components inside a node. We show how these approaches, integrated into the full application, provide speed ups from 6 to 7 times on large problems, executed on up to 16 nodes of a cluster with two AMD Opteron 6272 and a Tesla M2090 per node.« less

DUKSUP: A Computer Program for High Thrust Launch Vehicle Trajectory Design and Optimization

NASA Technical Reports Server (NTRS)

Williams, C. H.; Spurlock, O. F.

2014-01-01

From the late 1960's through 1997, the leadership of NASA's Intermediate and Large class unmanned expendable launch vehicle projects resided at the NASA Lewis (now Glenn) Research Center (LeRC). One of LeRC's primary responsibilities --- trajectory design and performance analysis --- was accomplished by an internally-developed analytic three dimensional computer program called DUKSUP. Because of its Calculus of Variations-based optimization routine, this code was generally more capable of finding optimal solutions than its contemporaries. A derivation of optimal control using the Calculus of Variations is summarized including transversality, intermediate, and final conditions. The two point boundary value problem is explained. A brief summary of the code's operation is provided, including iteration via the Newton-Raphson scheme and integration of variational and motion equations via a 4th order Runge-Kutta scheme. Main subroutines are discussed. The history of the LeRC trajectory design efforts in the early 1960's is explained within the context of supporting the Centaur upper stage program. How the code was constructed based on the operation of the Atlas/Centaur launch vehicle, the limits of the computers of that era, the limits of the computer programming languages, and the missions it supported are discussed. The vehicles DUKSUP supported (Atlas/Centaur, Titan/Centaur, and Shuttle/Centaur) are briefly described. The types of missions, including Earth orbital and interplanetary, are described. The roles of flight constraints and their impact on launch operations are detailed (such as jettisoning hardware on heating, Range Safety, ground station tracking, and elliptical parking orbits). The computer main frames on which the code was hosted are described. The applications of the code are detailed, including independent check of contractor analysis, benchmarking, leading edge analysis, and vehicle performance improvement assessments. Several of DUKSUP's many major impacts on launches are discussed including Intelsat, Voyager, Pioneer Venus, HEAO, Galileo, and Cassini.

DUKSUP: A Computer Program for High Thrust Launch Vehicle Trajectory Design and Optimization

NASA Technical Reports Server (NTRS)

Spurlock, O. Frank; Williams, Craig H.

2015-01-01

From the late 1960s through 1997, the leadership of NASAs Intermediate and Large class unmanned expendable launch vehicle projects resided at the NASA Lewis (now Glenn) Research Center (LeRC). One of LeRCs primary responsibilities --- trajectory design and performance analysis --- was accomplished by an internally-developed analytic three dimensional computer program called DUKSUP. Because of its Calculus of Variations-based optimization routine, this code was generally more capable of finding optimal solutions than its contemporaries. A derivation of optimal control using the Calculus of Variations is summarized including transversality, intermediate, and final conditions. The two point boundary value problem is explained. A brief summary of the codes operation is provided, including iteration via the Newton-Raphson scheme and integration of variational and motion equations via a 4th order Runge-Kutta scheme. Main subroutines are discussed. The history of the LeRC trajectory design efforts in the early 1960s is explained within the context of supporting the Centaur upper stage program. How the code was constructed based on the operation of the AtlasCentaur launch vehicle, the limits of the computers of that era, the limits of the computer programming languages, and the missions it supported are discussed. The vehicles DUKSUP supported (AtlasCentaur, TitanCentaur, and ShuttleCentaur) are briefly described. The types of missions, including Earth orbital and interplanetary, are described. The roles of flight constraints and their impact on launch operations are detailed (such as jettisoning hardware on heating, Range Safety, ground station tracking, and elliptical parking orbits). The computer main frames on which the code was hosted are described. The applications of the code are detailed, including independent check of contractor analysis, benchmarking, leading edge analysis, and vehicle performance improvement assessments. Several of DUKSUPs many major impacts on launches are discussed including Intelsat, Voyager, Pioneer Venus, HEAO, Galileo, and Cassini.

Robust integration schemes for generalized viscoplasticity with internal-state variables. Part 2: Algorithmic developments and implementation

NASA Technical Reports Server (NTRS)

Li, Wei; Saleeb, Atef F.

1995-01-01

This two-part report is concerned with the development of a general framework for the implicit time-stepping integrators for the flow and evolution equations in generalized viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its finite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, the general framework is developed on the basis of the unconditionally-stable, backward-Euler difference scheme as a starting point. Its mathematical structure is of sufficient generality to allow a unified treatment of different classes of viscoplastic models with internal variables. In particular, two specific models of this type, which are representative of the present start-of-art in metal viscoplasticity, are considered in applications reported here; i.e., fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to efficiency and robustness are emphasized in developing the (local) interative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiffness arrays are given explicitly for both GVIPS and NAV models, with their maximum sizes 'optimized' to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Significant robustness of the local iterative solution is provided by complementing the basic Newton-Raphson scheme with a line-search strategy for convergence. In the present second part of the report, we focus on the specific details of the numerical schemes, and associated computer algorithms, for the finite-element implementation of GVIPS and NAV models.

Robust integration schemes for generalized viscoplasticity with internal-state variables. Part 1: Theoretical developments and applications

NASA Technical Reports Server (NTRS)

Saleeb, Atef F.; Li, Wei

1995-01-01

This two-part report is concerned with the development of a general framework for the implicit time-stepping integrators for the flow and evolution equations in generalized viscoplastic models. The primary goal is to present a complete theoretical formulation, and to address in detail the algorithmic and numerical analysis aspects involved in its finite element implementation, as well as to critically assess the numerical performance of the developed schemes in a comprehensive set of test cases. On the theoretical side, the general framework is developed on the basis of the unconditionally-stable, backward-Euler difference scheme as a starting point. Its mathematical structure is of sufficient generality to allow a unified treatment of different classes of viscoplastic models with internal variables. In particular, two specific models of this type, which are representative of the present start-of-art in metal viscoplasticity, are considered in applications reported here; i.e., fully associative (GVIPS) and non-associative (NAV) models. The matrix forms developed for both these models are directly applicable for both initially isotropic and anisotropic materials, in general (three-dimensional) situations as well as subspace applications (i.e., plane stress/strain, axisymmetric, generalized plane stress in shells). On the computational side, issues related to efficiency and robustness are emphasized in developing the (local) interative algorithm. In particular, closed-form expressions for residual vectors and (consistent) material tangent stiffness arrays are given explicitly for both GVIPS and NAV models, with their maximum sizes 'optimized' to depend only on the number of independent stress components (but independent of the number of viscoplastic internal state parameters). Significant robustness of the local iterative solution is provided by complementing the basic Newton-Raphson scheme with a line-search strategy for convergence. In the present first part of the report, we focus on the theoretical developments, and discussions of the results of numerical-performance studies using the integration schemes for GVIPS and NAV models.

Nonlinear equation of the modes in circular slab waveguides and its application.

PubMed

Zhu, Jianxin; Zheng, Jia

2013-11-20

In this paper, circularly curved inhomogeneous waveguides are transformed into straight inhomogeneous waveguides first by a conformal mapping. Then, the differential transfer matrix method is introduced and adopted to deduce the exact dispersion relation for modes. This relation itself is complex and difficult to solve, but it can be approximated by a simpler nonlinear equation in practical applications, which is close to the exact relation and quite easy to analyze. Afterward, optimized asymptotic solutions are obtained and act as initial guesses for the following Newton's iteration. Finally, very accurate solutions are achieved in the numerical experiment.

On the primary variable switching technique for simulating unsaturated-saturated flows

NASA Astrophysics Data System (ADS)

Diersch, H.-J. G.; Perrochet, P.

Primary variable switching appears as a promising numerical technique for variably saturated flows. While the standard pressure-based form of the Richards equation can suffer from poor mass balance accuracy, the mixed form with its improved conservative properties can possess convergence difficulties for dry initial conditions. On the other hand, variable switching can overcome most of the stated numerical problems. The paper deals with variable switching for finite elements in two and three dimensions. The technique is incorporated in both an adaptive error-controlled predictor-corrector one-step Newton (PCOSN) iteration strategy and a target-based full Newton (TBFN) iteration scheme. Both schemes provide different behaviors with respect to accuracy and solution effort. Additionally, a simplified upstream weighting technique is used. Compared with conventional approaches the primary variable switching technique represents a fast and robust strategy for unsaturated problems with dry initial conditions. The impact of the primary variable switching technique is studied over a wide range of mostly 2D and partly difficult-to-solve problems (infiltration, drainage, perched water table, capillary barrier), where comparable results are available. It is shown that the TBFN iteration is an effective but error-prone procedure. TBFN sacrifices temporal accuracy in favor of accelerated convergence if aggressive time step sizes are chosen.

Newton's method

DOE Office of Scientific and Technical Information (OSTI.GOV)

More, J. J.; Sorensen, D. C.

1982-02-01

Newton's method plays a central role in the development of numerical techniques for optimization. In fact, most of the current practical methods for optimization can be viewed as variations on Newton's method. It is therefore important to understand Newton's method as an algorithm in its own right and as a key introduction to the most recent ideas in this area. One of the aims of this expository paper is to present and analyze two main approaches to Newton's method for unconstrained minimization: the line search approach and the trust region approach. The other aim is to present some of themore » recent developments in the optimization field which are related to Newton's method. In particular, we explore several variations on Newton's method which are appropriate for large scale problems, and we also show how quasi-Newton methods can be derived quite naturally from Newton's method.« less

Modified kinetic-hydraulic UASB reactor model for treatment of wastewater containing biodegradable organic substrates.

PubMed

El-Seddik, Mostafa M; Galal, Mona M; Radwan, A G; Abdel-Halim, Hisham S

2016-01-01

This paper addresses a modified kinetic-hydraulic model for up-flow anaerobic sludge blanket (UASB) reactor aimed to treat wastewater of biodegradable organic substrates as acetic acid based on Van der Meer model incorporated with biological granules inclusion. This dynamic model illustrates the biomass kinetic reaction rate for both direct and indirect growth of microorganisms coupled with the amount of biogas produced by methanogenic bacteria in bed and blanket zones of reactor. Moreover, the pH value required for substrate degradation at the peak specific growth rate of bacteria is discussed for Andrews' kinetics. The sensitivity analyses of biomass concentration with respect to fraction of volume of reactor occupied by granules and up-flow velocity are also demonstrated. Furthermore, the modified mass balance equations of reactor are applied during steady state using Newton Raphson technique to obtain a suitable degree of freedom for the modified model matching with the measured results of UASB Sanhour wastewater treatment plant in Fayoum, Egypt.

Improving sub-grid scale accuracy of boundary features in regional finite-difference models

USGS Publications Warehouse

Panday, Sorab; Langevin, Christian D.

2012-01-01

As an alternative to grid refinement, the concept of a ghost node, which was developed for nested grid applications, has been extended towards improving sub-grid scale accuracy of flow to conduits, wells, rivers or other boundary features that interact with a finite-difference groundwater flow model. The formulation is presented for correcting the regular finite-difference groundwater flow equations for confined and unconfined cases, with or without Newton Raphson linearization of the nonlinearities, to include the Ghost Node Correction (GNC) for location displacement. The correction may be applied on the right-hand side vector for a symmetric finite-difference Picard implementation, or on the left-hand side matrix for an implicit but asymmetric implementation. The finite-difference matrix connectivity structure may be maintained for an implicit implementation by only selecting contributing nodes that are a part of the finite-difference connectivity. Proof of concept example problems are provided to demonstrate the improved accuracy that may be achieved through sub-grid scale corrections using the GNC schemes.

Theoretical and experimental study on near infrared time-resolved optical diffuse tomography

NASA Astrophysics Data System (ADS)

Zhao, Huijuan; Gao, Feng; Tanikawa, Yukari; Yamada, Yukio

2006-08-01

Parts of the works of our group in the past five years on near infrared time-resolved (TR) optical tomography are summarized in this paper. The image reconstruction algorithm is based on Newton Raphson scheme with a datatype R generated from modified Generalized Pulse Spectrum Technique. Firstly, the algorithm is evaluated with simulated data from a 2-D model and the datatype R is compared with other popularly used datatypes. In this second part of the paper, the in vitro and in vivo NIR DOT imaging on a chicken leg and a human forearm, respectively are presented for evaluating both the image reconstruction algorithm and the TR measurement system. The third part of this paper is about the differential pathlength factor of human head while monitoring head activity with NIRS and applying the modified Lambert-Beer law. Benefiting from the TR system, the measured DPF maps of the three import areas of human head are presented in this paper.

A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Ito, K.

1991-01-01

A hybrid method for computing the feedback gains in linear quadratic regulator problem is proposed. The method, which combines use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite-dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantages of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed, and numerical evidence of the efficacy of these ideas is presented.

A numerical algorithm for optimal feedback gains in high dimensional LQR problems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Ito, K.

1986-01-01

A hybrid method for computing the feedback gains in linear quadratic regulator problems is proposed. The method, which combines the use of a Chandrasekhar type system with an iteration of the Newton-Kleinman form with variable acceleration parameter Smith schemes, is formulated so as to efficiently compute directly the feedback gains rather than solutions of an associated Riccati equation. The hybrid method is particularly appropriate when used with large dimensional systems such as those arising in approximating infinite dimensional (distributed parameter) control systems (e.g., those governed by delay-differential and partial differential equations). Computational advantage of the proposed algorithm over the standard eigenvector (Potter, Laub-Schur) based techniques are discussed and numerical evidence of the efficacy of our ideas presented.

Extension of transonic flow computational concepts in the analysis of cavitated bearings

NASA Technical Reports Server (NTRS)

Vijayaraghavan, D.; Keith, T. G., Jr.; Brewe, D. E.

1990-01-01

An analogy between the mathematical modeling of transonic potential flow and the flow in a cavitating bearing is described. Based on the similarities, characteristics of the cavitated region and jump conditions across the film reformation and rupture fronts are developed using the method of weak solutions. The mathematical analogy is extended by utilizing a few computational concepts of transonic flow to numerically model the cavitating bearing. Methods of shock fitting and shock capturing are discussed. Various procedures used in transonic flow computations are adapted to bearing cavitation applications, for example, type differencing, grid transformation, an approximate factorization technique, and Newton's iteration method. These concepts have proved to be successful and have vastly improved the efficiency of numerical modeling of cavitated bearings.

NPARC v3.1 User's Guide: A Companion to the NPARC v3.0 User's Guide

NASA Technical Reports Server (NTRS)

Chung, Joongkee; Slater, John W.; Suresh, Ambady; Townsend, Scott

1999-01-01

NPARC v3.1 is a modification to the NPARC v3.0 computer program which expands the capabilities for time-accurate computations through the use of a Newton iterative implicit method, time-varying boundary conditions, and planar dynamic grids. This document discusses some of the changes from the NPARC v3.0, specifically: changes to the directory structure and execution, changes to the input format. background on new methods, new boundary conditions. dynamic grids, new options for output, usage concepts, and some test cases to serve as tutorials. This document is intended to be used in conjunction with the NPARC v3.0 user's guide.

Volcanic flood simulation of magma effusion using FLO-2D for drainage of a caldera lake at the Mt. Baekdusan

NASA Astrophysics Data System (ADS)

Lee, Khil-Ha; Kim, Sung-Wook; Kim, Sang-Hyun

2014-05-01

Many volcanic craters and calderas are filled with large amounts of water that can pose significant flood hazards to downstream communities due to their high elevation and the potential for catastrophic releases of water. Recent reports pointed out the Baekdusan volcano that is located between the border of China and North Korea as a potential active volcano. Since Millennium Eruption around 1000 AD, smaller eruptions have occurred at roughly 100-year intervals, with the last one in 1903. Sudden release of huge volume of water stored in temporarily elevated caldera lakes are a recurrent feature of volcanic environments, due to the case with which outlet channels are blocked by and re-cut through, unwelded pyroclastic deposits. The volcano is showing signs of waking from a century-long slumber recently. Volcanic floods, including breakouts from volcanic lakes, can affect communities beyond the areas immediately affected by a volcanic eruption and cause significant hydrological hazards because floods from lake-filled calderas may be particularly large and high. Although a number of case studies have been presented in the literature, investigation of the underlying physical processes is required as well as a method for interpreting the process of the rapid release of water stored in a caldera lake. The development of various forecasting techniques to prevent and minimize economic and social damage is in urgent need. This study focuses on constructing a flood hazard map triggered by the magma effusion in the Baekdusan volcano. A physically-based uplift model was developed to compute the amount of water and time to peak flow. The ordinary differential equation was numerically solved using the finite difference method and Newton-Raphson iteration method was used to solve nonlinear equation. The magma effusion rate into the caldera lake is followed by examples at other volcanic activities. As a result, the hydrograph serves as an upper boundary condition when hydrodynamic model, called FLO-2D runs to simulate channel routing downstream to give the maximum water level. Once probable inundation areas are identified by the huge volume of water in the caldera lake, the unique geography, and the limited control capability, a potential hazard assessment can be represented. The study will contribute to build a geohazard map for the decision-makers and practitioners. Keywords: Volcanic flood, Caldera lake, Hazard assessment, Magma effusion Acknowledgement This research was supported by a grant [NEMA-BAEKDUSAN-2012-1-2] from the Volcanic Disaster Preparedness Research Center sponsored by National Emergency Management Agency of Korea.

Flexible parallel implicit modelling of coupled thermal-hydraulic-mechanical processes in fractured rocks

NASA Astrophysics Data System (ADS)

Cacace, Mauro; Jacquey, Antoine B.

2017-09-01

Theory and numerical implementation describing groundwater flow and the transport of heat and solute mass in fully saturated fractured rocks with elasto-plastic mechanical feedbacks are developed. In our formulation, fractures are considered as being of lower dimension than the hosting deformable porous rock and we consider their hydraulic and mechanical apertures as scaling parameters to ensure continuous exchange of fluid mass and energy within the fracture-solid matrix system. The coupled system of equations is implemented in a new simulator code that makes use of a Galerkin finite-element technique. The code builds on a flexible, object-oriented numerical framework (MOOSE, Multiphysics Object Oriented Simulation Environment) which provides an extensive scalable parallel and implicit coupling to solve for the multiphysics problem. The governing equations of groundwater flow, heat and mass transport, and rock deformation are solved in a weak sense (either by classical Newton-Raphson or by free Jacobian inexact Newton-Krylow schemes) on an underlying unstructured mesh. Nonlinear feedbacks among the active processes are enforced by considering evolving fluid and rock properties depending on the thermo-hydro-mechanical state of the system and the local structure, i.e. degree of connectivity, of the fracture system. A suite of applications is presented to illustrate the flexibility and capability of the new simulator to address problems of increasing complexity and occurring at different spatial (from centimetres to tens of kilometres) and temporal scales (from minutes to hundreds of years).

Three-Dimensional Navier-Stokes Method with Two-Equation Turbulence Models for Efficient Numerical Simulation of Hypersonic Flows

NASA Technical Reports Server (NTRS)

Bardina, J. E.

1994-01-01

A new computational efficient 3-D compressible Reynolds-averaged implicit Navier-Stokes method with advanced two equation turbulence models for high speed flows is presented. All convective terms are modeled using an entropy satisfying higher-order Total Variation Diminishing (TVD) scheme based on implicit upwind flux-difference split approximations and arithmetic averaging procedure of primitive variables. This method combines the best features of data management and computational efficiency of space marching procedures with the generality and stability of time dependent Navier-Stokes procedures to solve flows with mixed supersonic and subsonic zones, including streamwise separated flows. Its robust stability derives from a combination of conservative implicit upwind flux-difference splitting with Roe's property U to provide accurate shock capturing capability that non-conservative schemes do not guarantee, alternating symmetric Gauss-Seidel 'method of planes' relaxation procedure coupled with a three-dimensional two-factor diagonal-dominant approximate factorization scheme, TVD flux limiters of higher-order flux differences satisfying realizability, and well-posed characteristic-based implicit boundary-point a'pproximations consistent with the local characteristics domain of dependence. The efficiency of the method is highly increased with Newton Raphson acceleration which allows convergence in essentially one forward sweep for supersonic flows. The method is verified by comparing with experiment and other Navier-Stokes methods. Here, results of adiabatic and cooled flat plate flows, compression corner flow, and 3-D hypersonic shock-wave/turbulent boundary layer interaction flows are presented. The robust 3-D method achieves a better computational efficiency of at least one order of magnitude over the CNS Navier-Stokes code. It provides cost-effective aerodynamic predictions in agreement with experiment, and the capability of predicting complex flow structures in complex geometries with good accuracy.

Nonlinear Krylov and moving nodes in the method of lines

NASA Astrophysics Data System (ADS)

Miller, Keith

2005-11-01

We report on some successes and problem areas in the Method of Lines from our work with moving node finite element methods. First, we report on our "nonlinear Krylov accelerator" for the modified Newton's method on the nonlinear equations of our stiff ODE solver. Since 1990 it has been robust, simple, cheap, and automatic on all our moving node computations. We publicize further trials with it here because it should be of great general usefulness to all those solving evolutionary equations. Second, we discuss the need for reliable automatic choice of spatially variable time steps. Third, we discuss the need for robust and efficient iterative solvers for the difficult linearized equations (Jx=b) of our stiff ODE solver. Here, the 1997 thesis of Zulu Xaba has made significant progress.

Geometrical-optics code for computing the optical properties of large dielectric spheres.

PubMed

Zhou, Xiaobing; Li, Shusun; Stamnes, Knut

2003-07-20

Absorption of electromagnetic radiation by absorptive dielectric spheres such as snow grains in the near-infrared part of the solar spectrum cannot be neglected when radiative properties of snow are computed. Thus a new, to our knowledge, geometrical-optics code is developed to compute scattering and absorption cross sections of large dielectric particles of arbitrary complex refractive index. The number of internal reflections and transmissions are truncated on the basis of the ratio of the irradiance incident at the nth interface to the irradiance incident at the first interface for a specific optical ray. Thus the truncation number is a function of the angle of incidence. Phase functions for both near- and far-field absorption and scattering of electromagnetic radiation are calculated directly at any desired scattering angle by using a hybrid algorithm based on the bisection and Newton-Raphson methods. With these methods a large sphere's absorption and scattering properties of light can be calculated for any wavelength from the ultraviolet to the microwave regions. Assuming that large snow meltclusters (1-cm order), observed ubiquitously in the snow cover during summer, can be characterized as spheres, one may compute absorption and scattering efficiencies and the scattering phase function on the basis of this geometrical-optics method. A geometrical-optics method for sphere (GOMsphere) code is developed and tested against Wiscombe's Mie scattering code (MIE0) and a Monte Carlo code for a range of size parameters. GOMsphere can be combined with MIE0 to calculate the single-scattering properties of dielectric spheres of any size.

CREKID: A computer code for transient, gas-phase combustion of kinetics

NASA Technical Reports Server (NTRS)

Pratt, D. T.; Radhakrishnan, K.

1984-01-01

A new algorithm was developed for fast, automatic integration of chemical kinetic rate equations describing homogeneous, gas-phase combustion at constant pressure. Particular attention is paid to the distinguishing physical and computational characteristics of the induction, heat-release and equilibration regimes. The two-part predictor-corrector algorithm, based on an exponentially-fitted trapezoidal rule, includes filtering of ill-posed initial conditions, automatic selection of Newton-Jacobi or Newton iteration for convergence to achieve maximum computational efficiency while observing a prescribed error tolerance. The new algorithm was found to compare favorably with LSODE on two representative test problems drawn from combustion kinetics.

Methods for calculating confidence and credible intervals for the residual between-study variance in random effects meta-regression models

PubMed Central

2014-01-01

Background Meta-regression is becoming increasingly used to model study level covariate effects. However this type of statistical analysis presents many difficulties and challenges. Here two methods for calculating confidence intervals for the magnitude of the residual between-study variance in random effects meta-regression models are developed. A further suggestion for calculating credible intervals using informative prior distributions for the residual between-study variance is presented. Methods Two recently proposed and, under the assumptions of the random effects model, exact methods for constructing confidence intervals for the between-study variance in random effects meta-analyses are extended to the meta-regression setting. The use of Generalised Cochran heterogeneity statistics is extended to the meta-regression setting and a Newton-Raphson procedure is developed to implement the Q profile method for meta-analysis and meta-regression. WinBUGS is used to implement informative priors for the residual between-study variance in the context of Bayesian meta-regressions. Results Results are obtained for two contrasting examples, where the first example involves a binary covariate and the second involves a continuous covariate. Intervals for the residual between-study variance are wide for both examples. Conclusions Statistical methods, and R computer software, are available to compute exact confidence intervals for the residual between-study variance under the random effects model for meta-regression. These frequentist methods are almost as easily implemented as their established counterparts for meta-analysis. Bayesian meta-regressions are also easily performed by analysts who are comfortable using WinBUGS. Estimates of the residual between-study variance in random effects meta-regressions should be routinely reported and accompanied by some measure of their uncertainty. Confidence and/or credible intervals are well-suited to this purpose. PMID:25196829

Numerical Simulation of Illumination and Thermal Conditions at the Lunar Poles Using LOLA DTMs

NASA Technical Reports Server (NTRS)

Glaser, P.; Glaser, D.; Oberst, J.; Neumann, G. A.; Mazarico, E.; Siegler, M. A.

2017-01-01

We are interested in illumination conditions and the temperature distribution within the upper two meters of regolith near the lunar poles. Here, areas exist receiving almost constant illumination near areas in permanent shadow, which were identified as potential exploration sites for future missions. For our study a numerical simulation of the illumination and thermal environment for lunar near-polar regions is needed. Our study is based on high-resolution, twenty meters per pixel and 400 x 400 km large polar Digital Terrain Models (DTMs), which were derived from Lunar Orbiter Laser Altimeter (LOLA) data. Illumination conditions were simulated by synthetically illuminating the LOLA DTMs using the horizon method considering the Sun as an extended source. We model polar illumination for the central 50 x 50 km subset and use it as an input at each time-step (2 h) to evaluate the heating of the lunar surface and subsequent conduction in the sub-surface. At surface level we balance the incoming insolation with the subsurface conduction and radiation into space, whereas in the sub-surface we consider conduction with an additional constant radiogenic heat source at the bottom of our two-meter layer. Density is modeled as depth-dependent, the specific heat parameter as temperature-dependent and the thermal conductivity as depth- and temperature-dependent. We implemented a fully implicit finite-volume method in space and backward Euler scheme in time to solve the one-dimensional heat equation at each pixel in our 50 x 50 km DTM. Due to the non-linear dependencies of the parameters mentioned above, Newton's method is employed as the non-linear solver together with the Gauss-Seidel method as the iterative linear solver in each Newton iteration. The software is written in OpenCL and runs in parallel on the GPU cores, which allows for fast computation of large areas and long time scales.

Distorted Born iterative T-matrix method for inversion of CSEM data in anisotropic media

NASA Astrophysics Data System (ADS)

Jakobsen, Morten; Tveit, Svenn

2018-05-01

We present a direct iterative solutions to the nonlinear controlled-source electromagnetic (CSEM) inversion problem in the frequency domain, which is based on a volume integral equation formulation of the forward modelling problem in anisotropic conductive media. Our vectorial nonlinear inverse scattering approach effectively replaces an ill-posed nonlinear inverse problem with a series of linear ill-posed inverse problems, for which there already exist efficient (regularized) solution methods. The solution update the dyadic Green's function's from the source to the scattering-volume and from the scattering-volume to the receivers, after each iteration. The T-matrix approach of multiple scattering theory is used for efficient updating of all dyadic Green's functions after each linearized inversion step. This means that we have developed a T-matrix variant of the Distorted Born Iterative (DBI) method, which is often used in the acoustic and electromagnetic (medical) imaging communities as an alternative to contrast-source inversion. The main advantage of using the T-matrix approach in this context, is that it eliminates the need to perform a full forward simulation at each iteration of the DBI method, which is known to be consistent with the Gauss-Newton method. The T-matrix allows for a natural domain decomposition, since in the sense that a large model can be decomposed into an arbitrary number of domains that can be treated independently and in parallel. The T-matrix we use for efficient model updating is also independent of the source-receiver configuration, which could be an advantage when performing fast-repeat modelling and time-lapse inversion. The T-matrix is also compatible with the use of modern renormalization methods that can potentially help us to reduce the sensitivity of the CSEM inversion results on the starting model. To illustrate the performance and potential of our T-matrix variant of the DBI method for CSEM inversion, we performed a numerical experiments based on synthetic CSEM data associated with 2D VTI and 3D orthorombic model inversions. The results of our numerical experiment suggest that the DBIT method for inversion of CSEM data in anisotropic media is both accurate and efficient.

Estimation in SEM: A Concrete Example

ERIC Educational Resources Information Center

Ferron, John M.; Hess, Melinda R.

2007-01-01

A concrete example is used to illustrate maximum likelihood estimation of a structural equation model with two unknown parameters. The fitting function is found for the example, as are the vector of first-order partial derivatives, the matrix of second-order partial derivatives, and the estimates obtained from each iteration of the Newton-Raphson…

Analysis of one dimension migration law from rainfall runoff on urban roof

NASA Astrophysics Data System (ADS)

Weiwei, Chen

2017-08-01

Research was taken on the hydrology and water quality process in the natural rain condition and water samples were collected and analyzed. The pollutant were included SS, COD and TN. Based on the mass balance principle, one dimension migration model was built for the rainfall runoff pollution in surface. The difference equation was developed according to the finite difference method, by applying the Newton iteration method for solving it. The simulated pollutant concentration process was in consistent with the measured value on model, and Nash-Sutcliffe coefficient was higher than 0.80. The model had better practicability, which provided evidence for effectively utilizing urban rainfall resource, non-point source pollution of making management technologies and measures, sponge city construction, and so on.

Implementation of the SPH Procedure Within the MOOSE Finite Element Framework

NASA Astrophysics Data System (ADS)

Laurier, Alexandre

The goal of this thesis was to implement the SPH homogenization procedure within the MOOSE finite element framework at INL. Before this project, INL relied on DRAGON to do their SPH homogenization which was not flexible enough for their needs. As such, the SPH procedure was implemented for the neutron diffusion equation with the traditional, Selengut and true Selengut normalizations. Another aspect of this research was to derive the SPH corrected neutron transport equations and implement them in the same framework. Following in the footsteps of other articles, this feature was implemented and tested successfully with both the PN and S N transport calculation schemes. Although the results obtained for the power distribution in PWR assemblies show no advantages over the use of the SPH diffusion equation, we believe the inclusion of this transport correction will allow for better results in cases where either P N or SN are required. An additional aspect of this research was the implementation of a novel way of solving the non-linear SPH problem. Traditionally, this was done through a Picard, fixed-point iterative process whereas the new implementation relies on MOOSE's Preconditioned Jacobian-Free Newton Krylov (PJFNK) method to allow for a direct solution to the non-linear problem. This novel implementation showed a decrease in calculation time by a factor reaching 50 and generated SPH factors that correspond to those obtained through a fixed-point iterative process with a very tight convergence criteria: epsilon < 10-8. The use of the PJFNK SPH procedure also allows to reach convergence in problems containing important reflector regions and void boundary conditions, something that the traditional SPH method has never been able to achieve. At times when the PJFNK method cannot reach convergence to the SPH problem, a hybrid method is used where by the traditional SPH iteration forces the initial condition to be within the radius of convergence of the Newton method. This new method was tested on a simplified model of INL's TREAT reactor, a problem that includes very important graphite reflector regions as well as vacuum boundary conditions with great success. To demonstrate the power of PJFNK SPH on a more common case, the correction was applied to a simplified PWR reactor core from the BEAVRS benchmark that included 15 assemblies and the water reflector to obtain very good results. This opens up the possibility to apply the SPH correction to full reactor cores in order to reduce homogenization errors for use in transient or multi-physics calculations.

C library for topological study of the electronic charge density.

PubMed

Vega, David; Aray, Yosslen; Rodríguez, Jesús

2012-12-05

The topological study of the electronic charge density is useful to obtain information about the kinds of bonds (ionic or covalent) and the atom charges on a molecule or crystal. For this study, it is necessary to calculate, at every space point, the electronic density and its electronic density derivatives values up to second order. In this work, a grid-based method for these calculations is described. The library, implemented for three dimensions, is based on a multidimensional Lagrange interpolation in a regular grid; by differentiating the resulting polynomial, the gradient vector, the Hessian matrix and the Laplacian formulas were obtained for every space point. More complex functions such as the Newton-Raphson method (to find the critical points, where the gradient is null) and the Cash-Karp Runge-Kutta method (used to make the gradient paths) were programmed. As in some crystals, the unit cell has angles different from 90°, the described library includes linear transformations to correct the gradient and Hessian when the grid is distorted (inclined). Functions were also developed to handle grid containing files (grd from DMol® program, CUBE from Gaussian® program and CHGCAR from VASP® program). Each one of these files contains the data for a molecular or crystal electronic property (such as charge density, spin density, electrostatic potential, and others) in a three-dimensional (3D) grid. The library can be adapted to make the topological study in any regular 3D grid by modifying the code of these functions. Copyright © 2012 Wiley Periodicals, Inc.

WARP3D-Release 10.8: Dynamic Nonlinear Analysis of Solids using a Preconditioned Conjugate Gradient Software Architecture

NASA Technical Reports Server (NTRS)

Koppenhoefer, Kyle C.; Gullerud, Arne S.; Ruggieri, Claudio; Dodds, Robert H., Jr.; Healy, Brian E.

1998-01-01

This report describes theoretical background material and commands necessary to use the WARP3D finite element code. WARP3D is under continuing development as a research code for the solution of very large-scale, 3-D solid models subjected to static and dynamic loads. Specific features in the code oriented toward the investigation of ductile fracture in metals include a robust finite strain formulation, a general J-integral computation facility (with inertia, face loading), an element extinction facility to model crack growth, nonlinear material models including viscoplastic effects, and the Gurson-Tver-gaard dilatant plasticity model for void growth. The nonlinear, dynamic equilibrium equations are solved using an incremental-iterative, implicit formulation with full Newton iterations to eliminate residual nodal forces. The history integration of the nonlinear equations of motion is accomplished with Newmarks Beta method. A central feature of WARP3D involves the use of a linear-preconditioned conjugate gradient (LPCG) solver implemented in an element-by-element format to replace a conventional direct linear equation solver. This software architecture dramatically reduces both the memory requirements and CPU time for very large, nonlinear solid models since formation of the assembled (dynamic) stiffness matrix is avoided. Analyses thus exhibit the numerical stability for large time (load) steps provided by the implicit formulation coupled with the low memory requirements characteristic of an explicit code. In addition to the much lower memory requirements of the LPCG solver, the CPU time required for solution of the linear equations during each Newton iteration is generally one-half or less of the CPU time required for a traditional direct solver. All other computational aspects of the code (element stiffnesses, element strains, stress updating, element internal forces) are implemented in the element-by- element, blocked architecture. This greatly improves vectorization of the code on uni-processor hardware and enables straightforward parallel-vector processing of element blocks on multi-processor hardware.

Approximation of the Newton Step by a Defect Correction Process

NASA Technical Reports Server (NTRS)

Arian, E.; Batterman, A.; Sachs, E. W.

1999-01-01

In this paper, an optimal control problem governed by a partial differential equation is considered. The Newton step for this system can be computed by solving a coupled system of equations. To do this efficiently with an iterative defect correction process, a modifying operator is introduced into the system. This operator is motivated by local mode analysis. The operator can be used also for preconditioning in Generalized Minimum Residual (GMRES). We give a detailed convergence analysis for the defect correction process and show the derivation of the modifying operator. Numerical tests are done on the small disturbance shape optimization problem in two dimensions for the defect correction process and for GMRES.

Solution of a few nonlinear problems in aerodynamics by the finite elements and functional least squares methods. Ph.D. Thesis - Paris Univ.; [mathematical models of transonic flow using nonlinear equations

NASA Technical Reports Server (NTRS)

Periaux, J.

1979-01-01

The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.

Application of PSAT to Load Flow Analysis with STATCOM under Load Increase Scenario and Line Contingencies

NASA Astrophysics Data System (ADS)

Telang, Aparna S.; Bedekar, P. P.

2017-09-01

Load flow analysis is the initial and essential step for any power system computation. It is required for choosing better options for power system expansion to meet with ever increasing load demand. Implementation of Flexible AC Transmission System (FACTS) device like STATCOM, in the load flow, which is having fast and very flexible control, is one of the important tasks for power system researchers. This paper presents a simple and systematic approach for steady state power flow calculations with FACTS controller, static synchronous compensator (STATCOM) using command line usage of MATLAB tool-power system analysis toolbox (PSAT). The complexity of MATLAB language programming increases due to incorporation of STATCOM in an existing Newton-Raphson load flow algorithm. Thus, the main contribution of this paper is to show how command line usage of user friendly MATLAB tool, PSAT, can extensively be used for quicker and wider interpretation of the results of load flow with STATCOM. The novelty of this paper lies in the method of applying the load increase pattern, where the active and reactive loads have been changed simultaneously at all the load buses under consideration for creating stressed conditions for load flow analysis with STATCOM. The performance have been evaluated on many standard IEEE test systems and the results for standard IEEE-30 bus system, IEEE-57 bus system, and IEEE-118 bus system are presented.

Numerical Analysis on the High-Strength Concrete Beams Ultimate Behaviour

NASA Astrophysics Data System (ADS)

Smarzewski, Piotr; Stolarski, Adam

2017-10-01

Development of technologies of high-strength concrete (HSC) beams production, with the aim of creating a secure and durable material, is closely linked with the numerical models of real objects. The three-dimensional nonlinear finite element models of reinforced high-strength concrete beams with a complex geometry has been investigated in this study. The numerical analysis is performed using the ANSYS finite element package. The arc-length (A-L) parameters and the adaptive descent (AD) parameters are used with Newton-Raphson method to trace the complete load-deflection curves. Experimental and finite element modelling results are compared graphically and numerically. Comparison of these results indicates the correctness of failure criteria assumed for the high-strength concrete and the steel reinforcement. The results of numerical simulation are sensitive to the modulus of elasticity and the shear transfer coefficient for an open crack assigned to high-strength concrete. The full nonlinear load-deflection curves at mid-span of the beams, the development of strain in compressive concrete and the development of strain in tensile bar are in good agreement with the experimental results. Numerical results for smeared crack patterns are qualitatively agreeable as to the location, direction, and distribution with the test data. The model was capable of predicting the introduction and propagation of flexural and diagonal cracks. It was concluded that the finite element model captured successfully the inelastic flexural behaviour of the beams to failure.

Performance study of LMS based adaptive algorithms for unknown system identification

NASA Astrophysics Data System (ADS)

Javed, Shazia; Ahmad, Noor Atinah

2014-07-01

Adaptive filtering techniques have gained much popularity in the modeling of unknown system identification problem. These techniques can be classified as either iterative or direct. Iterative techniques include stochastic descent method and its improved versions in affine space. In this paper we present a comparative study of the least mean square (LMS) algorithm and some improved versions of LMS, more precisely the normalized LMS (NLMS), LMS-Newton, transform domain LMS (TDLMS) and affine projection algorithm (APA). The performance evaluation of these algorithms is carried out using adaptive system identification (ASI) model with random input signals, in which the unknown (measured) signal is assumed to be contaminated by output noise. Simulation results are recorded to compare the performance in terms of convergence speed, robustness, misalignment, and their sensitivity to the spectral properties of input signals. Main objective of this comparative study is to observe the effects of fast convergence rate of improved versions of LMS algorithms on their robustness and misalignment.

Performance study of LMS based adaptive algorithms for unknown system identification

DOE Office of Scientific and Technical Information (OSTI.GOV)

Javed, Shazia; Ahmad, Noor Atinah

Adaptive filtering techniques have gained much popularity in the modeling of unknown system identification problem. These techniques can be classified as either iterative or direct. Iterative techniques include stochastic descent method and its improved versions in affine space. In this paper we present a comparative study of the least mean square (LMS) algorithm and some improved versions of LMS, more precisely the normalized LMS (NLMS), LMS-Newton, transform domain LMS (TDLMS) and affine projection algorithm (APA). The performance evaluation of these algorithms is carried out using adaptive system identification (ASI) model with random input signals, in which the unknown (measured) signalmore » is assumed to be contaminated by output noise. Simulation results are recorded to compare the performance in terms of convergence speed, robustness, misalignment, and their sensitivity to the spectral properties of input signals. Main objective of this comparative study is to observe the effects of fast convergence rate of improved versions of LMS algorithms on their robustness and misalignment.« less

Hybrid parallelization of the XTOR-2F code for the simulation of two-fluid MHD instabilities in tokamaks

NASA Astrophysics Data System (ADS)

Marx, Alain; Lütjens, Hinrich

2017-03-01

A hybrid MPI/OpenMP parallel version of the XTOR-2F code [Lütjens and Luciani, J. Comput. Phys. 229 (2010) 8130] solving the two-fluid MHD equations in full tokamak geometry by means of an iterative Newton-Krylov matrix-free method has been developed. The present work shows that the code has been parallelized significantly despite the numerical profile of the problem solved by XTOR-2F, i.e. a discretization with pseudo-spectral representations in all angular directions, the stiffness of the two-fluid stability problem in tokamaks, and the use of a direct LU decomposition to invert the physical pre-conditioner at every Krylov iteration of the solver. The execution time of the parallelized version is an order of magnitude smaller than the sequential one for low resolution cases, with an increasing speedup when the discretization mesh is refined. Moreover, it allows to perform simulations with higher resolutions, previously forbidden because of memory limitations.

A novel approach to solve nonlinear Fredholm integral equations of the second kind.

PubMed

Li, Hu; Huang, Jin

2016-01-01

In this paper, we present a novel approach to solve nonlinear Fredholm integral equations of the second kind. This algorithm is constructed by the integral mean value theorem and Newton iteration. Convergence and error analysis of the numerical solutions are given. Moreover, Numerical examples show the algorithm is very effective and simple.

Camera-pose estimation via projective Newton optimization on the manifold.

PubMed

Sarkis, Michel; Diepold, Klaus

2012-04-01

Determining the pose of a moving camera is an important task in computer vision. In this paper, we derive a projective Newton algorithm on the manifold to refine the pose estimate of a camera. The main idea is to benefit from the fact that the 3-D rigid motion is described by the special Euclidean group, which is a Riemannian manifold. The latter is equipped with a tangent space defined by the corresponding Lie algebra. This enables us to compute the optimization direction, i.e., the gradient and the Hessian, at each iteration of the projective Newton scheme on the tangent space of the manifold. Then, the motion is updated by projecting back the variables on the manifold itself. We also derive another version of the algorithm that employs homeomorphic parameterization to the special Euclidean group. We test the algorithm on several simulated and real image data sets. Compared with the standard Newton minimization scheme, we are now able to obtain the full numerical formula of the Hessian with a 60% decrease in computational complexity. Compared with Levenberg-Marquardt, the results obtained are more accurate while having a rather similar complexity.

Three-dimensional full waveform inversion of short-period teleseismic wavefields based upon the SEM-DSM hybrid method

NASA Astrophysics Data System (ADS)

Monteiller, Vadim; Chevrot, Sébastien; Komatitsch, Dimitri; Wang, Yi

2015-08-01

We present a method for high-resolution imaging of lithospheric structures based on full waveform inversion of teleseismic waveforms. We model the propagation of seismic waves using our recently developed direct solution method/spectral-element method hybrid technique, which allows us to simulate the propagation of short-period teleseismic waves through a regional 3-D model. We implement an iterative quasi-Newton method based upon the L-BFGS algorithm, where the gradient of the misfit function is computed using the adjoint-state method. Compared to gradient or conjugate-gradient methods, the L-BFGS algorithm has a much faster convergence rate. We illustrate the potential of this method on a synthetic test case that consists of a crustal model with a crustal discontinuity at 25 km depth and a sharp Moho jump. This model contains short- and long-wavelength heterogeneities along the lateral and vertical directions. The iterative inversion starts from a smooth 1-D model derived from the IASP91 reference Earth model. We invert both radial and vertical component waveforms, starting from long-period signals filtered at 10 s and gradually decreasing the cut-off period down to 1.25 s. This multiscale algorithm quickly converges towards a model that is very close to the true model, in contrast to inversions involving short-period waveforms only, which always get trapped into a local minimum of the cost function.

Computation of iodine species concentrations in water

NASA Technical Reports Server (NTRS)

Schultz, John R.; Mudgett, Paul D.; Flanagan, David T.; Sauer, Richard L.

1994-01-01

During an evaluation of the use of iodine as a water disinfectant and the development of methods for measuring various iodine species in water onboard Space Freedom, it became necessary to compute the concentration of the various species based on equilibrium principles alone. Of particular concern was the case when various amounts of iodine, iodide, strong acid, and strong base are added to water. Such solutions can be used to evaluate the performance of various monitoring methods being considered. The authors of this paper present an overview of aqueous iodine chemistry, a set of nonlinear equations which can be used to model the above case, and a computer program for solving this system of equations using the Newton-Raphson method. The program was validated by comparing results over a range of concentrations and pH values with those previously presented by Gottardi for a given pH. Use of this program indicated that there are multiple roots to many cases and selecting an appropriate initial guess is important. Comparison of program results with laboratory results for the case when only iodine is added to water indicates the program gives high pH values for the iodine concentrations normally used for water disinfection. Extending the model to include the effects of iodate formation results in the computer pH values being closer to those observed, but the model with iodate does not agree well for the case in which base is added in addition to iodine to raise the pH. Potential explanations include failure to obtain equilibrium conditions in the lab, inaccuracies in published values for the equilibrium constants, and inadequate model of iodine chemistry and/or the lack of adequate analytical methods for measuring the various iodine species in water.

Peak Seeking Control for Reduced Fuel Consumption with Preliminary Flight Test Results

NASA Technical Reports Server (NTRS)

Brown, Nelson

2012-01-01

The Environmentally Responsible Aviation project seeks to accomplish the simultaneous reduction of fuel burn, noise, and emissions. A project at NASA Dryden Flight Research Center is contributing to ERAs goals by exploring the practical application of real-time trim configuration optimization for enhanced performance and reduced fuel consumption. This peak-seeking control approach is based on Newton-Raphson algorithm using a time-varying Kalman filter to estimate the gradient of the performance function. In real-time operation, deflection of symmetric ailerons, trailing-edge flaps, and leading-edge flaps of a modified F-18 are directly optimized, and the horizontal stabilators and angle of attack are indirectly optimized. Preliminary results from three research flights are presented herein. The optimization system found a trim configuration that required approximately 3.5% less fuel flow than the baseline trim at the given flight condition. The algorithm consistently rediscovered the solution from several initial conditions. These preliminary results show the algorithm has good performance and is expected to show similar results at other flight conditions and aircraft configurations.

How College Students' Conceptions of Newton's Second and Third Laws Change Through Watching Interactive Video Vignettes: A Mixed Methods Study

NASA Astrophysics Data System (ADS)

Engelman, Jonathan

Changing student conceptions in physics is a difficult process and has been a topic of research for many years. The purpose of this study was to understand what prompted students to change or not change their incorrect conceptions of Newtons Second or Third Laws in response to an intervention, Interactive Video Vignettes (IVVs), designed to overcome them. This study is based on prior research reported in the literature which has found that a curricular framework of elicit, confront, resolve, and reflect (ECRR) is important for changing student conceptions (McDermott, 2001). This framework includes four essential parts such that during an instructional event student conceptions should be elicited, incorrect conceptions confronted, these conflicts resolved, and then students should be prompted to reflect on their learning. Twenty-two undergraduate student participants who completed either or both IVVs were studied to determine whether or not they experienced components of the ECRR framework at multiple points within the IVVs. A fully integrated, mixed methods design was used to address the study purpose. Both quantitative and qualitative data were collected iteratively for each participant. Successive data collections were informed by previous data collections. All data were analyzed concurrently. The quantitative strand included a pre/post test that participants took before and after completing a given IVV and was used to measure the effect of each IVV on learning. The qualitative strand included video of each participant completing the IVV as well as an audio-recorded video elicitation interview after the post-test. The qualitative data collection was designed to describe student experiences with each IVV as well as to observe how the ECRR framework was experienced. Collecting and analyzing data using this mixed methods approach helped develop a more complete understanding of how student conceptions of Newtons Second and Third Laws changed through completion of IVVs and how the ECRR framework was experienced. In answering the research questions, two major conclusions were reached: (1) while the ECRR framework was experienced in both the Newtons 2nd Law and Newtons 3rd Law IVVs, these experiences were qualitatively different from each other and these differences help support the differences in gain scores on the post-tests for the participants; and (2) both IVVs were able to change certain misconceptions associated with either Newtons 2nd or 3rd laws more than others. Therefore, in researching student experiences while completing the Newtons 2nd Law and Newtons 3rd Law IVVs, I determined that a complete, sequential experience of the elicit, confront, resolve, reflect framework led to the greatest change in student conceptions. This dissertation adds to the field of physics education through finding the positive impact of the ECRR framework, as IVVs are still being created and disseminated. Physics educators and researchers interested in conceptual change can use these findings to provide evidence on what students think when interacting with videos designed to change their conceptions. Finally, this dissertation supports the conceptual change literature in that the full, sequential experience involving each component of the ECRR framework led to a change in student conceptions.

Optimization of flexible wing structures subject to strength and induced drag constraints

NASA Technical Reports Server (NTRS)

Haftka, R. T.

1977-01-01

An optimization procedure for designing wing structures subject to stress, strain, and drag constraints is presented. The optimization method utilizes an extended penalty function formulation for converting the constrained problem into a series of unconstrained ones. Newton's method is used to solve the unconstrained problems. An iterative analysis procedure is used to obtain the displacements of the wing structure including the effects of load redistribution due to the flexibility of the structure. The induced drag is calculated from the lift distribution. Approximate expressions for the constraints used during major portions of the optimization process enhance the efficiency of the procedure. A typical fighter wing is used to demonstrate the procedure. Aluminum and composite material designs are obtained. The tradeoff between weight savings and drag reduction is investigated.

Profiles of electrified drops and bubbles

NASA Technical Reports Server (NTRS)

Basaran, O. A.; Scriven, L. E.

1982-01-01

Axisymmetric equilibrium shapes of conducting drops and bubbles, (1) pendant or sessile on one face of a circular parallel-plate capacitor or (2) free and surface-charged, are found by solving simultaneously the free boundary problem consisting of the augmented Young-Laplace equation for surface shape and the Laplace equation for electrostatic field, given the surface potential. The problem is nonlinear and the method is a finite element algorithm employing Newton iteration, a modified frontal solver, and triangular as well as quadrilateral tessellations of the domain exterior to the drop in order to facilitate refined analysis of sharply curved drop tips seen in experiments. The stability limit predicted by this computer-aided theoretical analysis agrees well with experiments.

[The research on separating and extracting overlapping spectral feature lines in LIBS using damped least squares method].

PubMed

Wang, Yin; Zhao, Nan-jing; Liu, Wen-qing; Yu, Yang; Fang, Li; Meng, De-shuo; Hu, Li; Zhang, Da-hai; Ma, Min-jun; Xiao, Xue; Wang, Yu; Liu, Jian-guo

2015-02-01

In recent years, the technology of laser induced breakdown spectroscopy has been developed rapidly. As one kind of new material composition detection technology, laser induced breakdown spectroscopy can simultaneously detect multi elements fast and simply without any complex sample preparation and realize field, in-situ material composition detection of the sample to be tested. This kind of technology is very promising in many fields. It is very important to separate, fit and extract spectral feature lines in laser induced breakdown spectroscopy, which is the cornerstone of spectral feature recognition and subsequent elements concentrations inversion research. In order to realize effective separation, fitting and extraction of spectral feature lines in laser induced breakdown spectroscopy, the original parameters for spectral lines fitting before iteration were analyzed and determined. The spectral feature line of' chromium (Cr I : 427.480 nm) in fly ash gathered from a coal-fired power station, which was overlapped with another line(FeI: 427.176 nm), was separated from the other one and extracted by using damped least squares method. Based on Gauss-Newton iteration, damped least squares method adds damping factor to step and adjust step length dynamically according to the feedback information after each iteration, in order to prevent the iteration from diverging and make sure that the iteration could converge fast. Damped least squares method helps to obtain better results of separating, fitting and extracting spectral feature lines and give more accurate intensity values of these spectral feature lines: The spectral feature lines of chromium in samples which contain different concentrations of chromium were separated and extracted. And then, the intensity values of corresponding spectral lines were given by using damped least squares method and least squares method separately. The calibration curves were plotted, which showed the relationship between spectral line intensity values and chromium concentrations in different samples. And then their respective linear correlations were compared. The experimental results showed that the linear correlation of the intensity values of spectral feature lines and the concentrations of chromium in different samples, which was obtained by damped least squares method, was better than that one obtained by least squares method. And therefore, damped least squares method was stable, reliable and suitable for separating, fitting and extracting spectral feature lines in laser induced breakdown spectroscopy.

EIT image reconstruction based on a hybrid FE-EFG forward method and the complete-electrode model.

PubMed

Hadinia, M; Jafari, R; Soleimani, M

2016-06-01

This paper presents the application of the hybrid finite element-element free Galerkin (FE-EFG) method for the forward and inverse problems of electrical impedance tomography (EIT). The proposed method is based on the complete electrode model. Finite element (FE) and element-free Galerkin (EFG) methods are accurate numerical techniques. However, the FE technique has meshing task problems and the EFG method is computationally expensive. In this paper, the hybrid FE-EFG method is applied to take both advantages of FE and EFG methods, the complete electrode model of the forward problem is solved, and an iterative regularized Gauss-Newton method is adopted to solve the inverse problem. The proposed method is applied to compute Jacobian in the inverse problem. Utilizing 2D circular homogenous models, the numerical results are validated with analytical and experimental results and the performance of the hybrid FE-EFG method compared with the FE method is illustrated. Results of image reconstruction are presented for a human chest experimental phantom.

Development of iterative techniques for the solution of unsteady compressible viscous flows

NASA Technical Reports Server (NTRS)

Hixon, Duane; Sankar, L. N.

1993-01-01

During the past two decades, there has been significant progress in the field of numerical simulation of unsteady compressible viscous flows. At present, a variety of solution techniques exist such as the transonic small disturbance analyses (TSD), transonic full potential equation-based methods, unsteady Euler solvers, and unsteady Navier-Stokes solvers. These advances have been made possible by developments in three areas: (1) improved numerical algorithms; (2) automation of body-fitted grid generation schemes; and (3) advanced computer architectures with vector processing and massively parallel processing features. In this work, the GMRES scheme has been considered as a candidate for acceleration of a Newton iteration time marching scheme for unsteady 2-D and 3-D compressible viscous flow calculation; from preliminary calculations, this will provide up to a 65 percent reduction in the computer time requirements over the existing class of explicit and implicit time marching schemes. The proposed method has ben tested on structured grids, but is flexible enough for extension to unstructured grids. The described scheme has been tested only on the current generation of vector processor architecture of the Cray Y/MP class, but should be suitable for adaptation to massively parallel machines.

Evaluation of the diffusion coefficient for controlled release of oxytetracycline from alginate/chitosan/poly(ethylene glycol) microbeads in simulated gastrointestinal environments.

PubMed

Cruz, Maria C Pinto; Ravagnani, Sergio P; Brogna, Fabio M S; Campana, Sérgio P; Triviño, Galo Cardenas; Lisboa, Antonio C Luz; Mei, Lucia H Innocentini

2004-12-01

Diffusion studies of OTC (oxytetracycline) entrapped in microbeads of calcium alginate, calcium alginate coacervated with chitosan (of high, medium and low viscosity) and calcium alginate coacervated with chitosan of low viscosity, covered with PEG [poly(ethylene glycol) of molecular mass 2, 4.6 and 10 kDa, were carried out at 37+/-0.5 degrees C, in pH 7.4 and pH 1.2 buffer solutions - conditions similar to those found in the gastrointestinal system. The diffusion coefficient, or diffusivity (D), of OTC was calculated by equations provided by Crank [(1975) Mathematics in Diffusion, p. 85, Clarendon Press, Oxford] for diffusion, which follows Fick's [(1855) Ann. Physik (Leipzig) 170, 59] second law, considering the diffusion from the inner parts to the surface of the microbeads. The least-squares and the Newton-Raphson [Carnahan, Luther and Wilkes (1969) Applied Numerical Methods, p. 319, John Wiley & Sons, New York] methods were used to obtain the diffusion coefficients. The microbead swelling at pH 7.4 and OTC diffusion is classically Fickian, suggesting that the OTC transport, in this case, is controlled by the exchange rates of free water and relaxation of calcium alginate chains. In case of acid media, it was observed that the phenomenon did not follow Fick's law, owing, probably, to the high solubility of the OTC in this environment. It was possible to modulate the release rate of OTC in several types of microbeads. The presence of cracks formed during the process of drying the microbeads was observed by scanning electron microscopy.

Efficient Coupling of Fluid-Plasma and Monte-Carlo-Neutrals Models for Edge Plasma Transport

NASA Astrophysics Data System (ADS)

Dimits, A. M.; Cohen, B. I.; Friedman, A.; Joseph, I.; Lodestro, L. L.; Rensink, M. E.; Rognlien, T. D.; Sjogreen, B.; Stotler, D. P.; Umansky, M. V.

2017-10-01

UEDGE has been valuable for modeling transport in the tokamak edge and scrape-off layer due in part to its efficient fully implicit solution of coupled fluid neutrals and plasma models. We are developing an implicit coupling of the kinetic Monte-Carlo (MC) code DEGAS-2, as the neutrals model component, to the UEDGE plasma component, based on an extension of the Jacobian-free Newton-Krylov (JFNK) method to MC residuals. The coupling components build on the methods and coding already present in UEDGE. For the linear Krylov iterations, a procedure has been developed to ``extract'' a good preconditioner from that of UEDGE. This preconditioner may also be used to greatly accelerate the convergence rate of a relaxed fixed-point iteration, which may provide a useful ``intermediate'' algorithm. The JFNK method also requires calculation of Jacobian-vector products, for which any finite-difference procedure is inaccurate when a MC component is present. A semi-analytical procedure that retains the standard MC accuracy and fully kinetic neutrals physics is therefore being developed. Prepared for US DOE by LLNL under Contract DE-AC52-07NA27344 and LDRD project 15-ERD-059, by PPPL under Contract DE-AC02-09CH11466, and supported in part by the U.S. DOE, OFES.

A Generalized Fluid System Simulation Program to Model Flow Distribution in Fluid Networks

NASA Technical Reports Server (NTRS)

Majumdar, Alok; Bailey, John W.; Schallhorn, Paul; Steadman, Todd

1998-01-01

This paper describes a general purpose computer program for analyzing steady state and transient flow in a complex network. The program is capable of modeling phase changes, compressibility, mixture thermodynamics and external body forces such as gravity and centrifugal. The program's preprocessor allows the user to interactively develop a fluid network simulation consisting of nodes and branches. Mass, energy and specie conservation equations are solved at the nodes; the momentum conservation equations are solved in the branches. The program contains subroutines for computing "real fluid" thermodynamic and thermophysical properties for 33 fluids. The fluids are: helium, methane, neon, nitrogen, carbon monoxide, oxygen, argon, carbon dioxide, fluorine, hydrogen, parahydrogen, water, kerosene (RP-1), isobutane, butane, deuterium, ethane, ethylene, hydrogen sulfide, krypton, propane, xenon, R-11, R-12, R-22, R-32, R-123, R-124, R-125, R-134A, R-152A, nitrogen trifluoride and ammonia. The program also provides the options of using any incompressible fluid with constant density and viscosity or ideal gas. Seventeen different resistance/source options are provided for modeling momentum sources or sinks in the branches. These options include: pipe flow, flow through a restriction, non-circular duct, pipe flow with entrance and/or exit losses, thin sharp orifice, thick orifice, square edge reduction, square edge expansion, rotating annular duct, rotating radial duct, labyrinth seal, parallel plates, common fittings and valves, pump characteristics, pump power, valve with a given loss coefficient, and a Joule-Thompson device. The system of equations describing the fluid network is solved by a hybrid numerical method that is a combination of the Newton-Raphson and successive substitution methods. This paper also illustrates the application and verification of the code by comparison with Hardy Cross method for steady state flow and analytical solution for unsteady flow.

Solution Methods for 3D Tomographic Inversion Using A Highly Non-Linear Ray Tracer

NASA Astrophysics Data System (ADS)

Hipp, J. R.; Ballard, S.; Young, C. J.; Chang, M.

2008-12-01

To develop 3D velocity models to improve nuclear explosion monitoring capability, we have developed a 3D tomographic modeling system that traces rays using an implementation of the Um and Thurber ray pseudo- bending approach, with full enforcement of Snell's Law in 3D at the major discontinuities. Due to the highly non-linear nature of the ray tracer, however, we are forced to substantially damp the inversion in order to converge on a reasonable model. Unfortunately the amount of damping is not known a priori and can significantly extend the number of calls of the computationally expensive ray-tracer and the least squares matrix solver. If the damping term is too small the solution step-size produces either an un-realistic model velocity change or places the solution in or near a local minimum from which extrication is nearly impossible. If the damping term is too large, convergence can be very slow or premature convergence can occur. Standard approaches involve running inversions with a suite of damping parameters to find the best model. A better solution methodology is to take advantage of existing non-linear solution techniques such as Levenberg-Marquardt (LM) or quasi-newton iterative solvers. In particular, the LM algorithm was specifically designed to find the minimum of a multi-variate function that is expressed as the sum of squares of non-linear real-valued functions. It has become a standard technique for solving non-linear least squared problems, and is widely adopted in a broad spectrum of disciplines, including the geosciences. At each iteration, the LM approach dynamically varies the level of damping to optimize convergence. When the current estimate of the solution is far from the ultimate solution LM behaves as a steepest decent method, but transitions to Gauss- Newton behavior, with near quadratic convergence, as the estimate approaches the final solution. We show typical linear solution techniques and how they can lead to local minima if the damping is set too low. We also describe the LM technique and show how it automatically determines the appropriate damping factor as it iteratively converges on the best solution. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under Contract DE-AC04- 94AL85000.

Semismooth Newton method for gradient constrained minimization problem

NASA Astrophysics Data System (ADS)

Anyyeva, Serbiniyaz; Kunisch, Karl

2012-08-01

In this paper we treat a gradient constrained minimization problem, particular case of which is the elasto-plastic torsion problem. In order to get the numerical approximation to the solution we have developed an algorithm in an infinite dimensional space framework using the concept of the generalized (Newton) differentiation. Regularization was done in order to approximate the problem with the unconstrained minimization problem and to make the pointwise maximum function Newton differentiable. Using semismooth Newton method, continuation method was developed in function space. For the numerical implementation the variational equations at Newton steps are discretized using finite elements method.

Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods

NASA Astrophysics Data System (ADS)

Vinding, Mads S.; Maximov, Ivan I.; Tošner, Zdeněk; Nielsen, Niels Chr.

2012-08-01

The use of increasingly strong magnetic fields in magnetic resonance imaging (MRI) improves sensitivity, susceptibility contrast, and spatial or spectral resolution for functional and localized spectroscopic imaging applications. However, along with these benefits come the challenges of increasing static field (B0) and rf field (B1) inhomogeneities induced by radial field susceptibility differences and poorer dielectric properties of objects in the scanner. Increasing fields also impose the need for rf irradiation at higher frequencies which may lead to elevated patient energy absorption, eventually posing a safety risk. These reasons have motivated the use of multidimensional rf pulses and parallel rf transmission, and their combination with tailoring of rf pulses for fast and low-power rf performance. For the latter application, analytical and approximate solutions are well-established in linear regimes, however, with increasing nonlinearities and constraints on the rf pulses, numerical iterative methods become attractive. Among such procedures, optimal control methods have recently demonstrated great potential. Here, we present a Krotov-based optimal control approach which as compared to earlier approaches provides very fast, monotonic convergence even without educated initial guesses. This is essential for in vivo MRI applications. The method is compared to a second-order gradient ascent method relying on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method, and a hybrid scheme Krotov-BFGS is also introduced in this study. These optimal control approaches are demonstrated by the design of a 2D spatial selective rf pulse exciting the letters "JCP" in a water phantom.

Reconstruction of electrical impedance tomography (EIT) images based on the expectation maximum (EM) method.

PubMed

Wang, Qi; Wang, Huaxiang; Cui, Ziqiang; Yang, Chengyi

2012-11-01

Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative values in the solution. The negativity of the solution produces artifacts in reconstructed images in presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of choosing parameters. Simulation and experimental results indicate that the reconstructed images with higher quality can be obtained by the EM method, compared with the traditional Tikhonov and conjugate gradient (CG) methods, even with non-negative processing. Copyright © 2012 ISA. Published by Elsevier Ltd. All rights reserved.

POSTPROCESSING MIXED FINITE ELEMENT METHODS FOR SOLVING CAHN-HILLIARD EQUATION: METHODS AND ERROR ANALYSIS

PubMed Central

Wang, Wansheng; Chen, Long; Zhou, Jie

2015-01-01

A postprocessing technique for mixed finite element methods for the Cahn-Hilliard equation is developed and analyzed. Once the mixed finite element approximations have been computed at a fixed time on the coarser mesh, the approximations are postprocessed by solving two decoupled Poisson equations in an enriched finite element space (either on a finer grid or a higher-order space) for which many fast Poisson solvers can be applied. The nonlinear iteration is only applied to a much smaller size problem and the computational cost using Newton and direct solvers is negligible compared with the cost of the linear problem. The analysis presented here shows that this technique remains the optimal rate of convergence for both the concentration and the chemical potential approximations. The corresponding error estimate obtained in our paper, especially the negative norm error estimates, are non-trivial and different with the existing results in the literatures. PMID:27110063

Basin boundaries and focal points in a map coming from Bairstow's method.

PubMed

Gardini, Laura; Bischi, Gian-Italo; Fournier-Prunaret, Daniele

1999-06-01

This paper is devoted to the study of the global dynamical properties of a two-dimensional noninvertible map, with a denominator which can vanish, obtained by applying Bairstow's method to a cubic polynomial. It is shown that the complicated structure of the basins of attraction of the fixed points is due to the existence of singularities such as sets of nondefinition, focal points, and prefocal curves, which are specific to maps with a vanishing denominator, and have been recently introduced in the literature. Some global bifurcations that change the qualitative structure of the basin boundaries, are explained in terms of contacts among these singularities. The techniques used in this paper put in evidence some new dynamic behaviors and bifurcations, which are peculiar of maps with denominator; hence they can be applied to the analysis of other classes of maps coming from iterative algorithms (based on Newton's method, or others). (c) 1999 American Institute of Physics.

Regularization of nonlinear decomposition of spectral x-ray projection images.

PubMed

Ducros, Nicolas; Abascal, Juan Felipe Perez-Juste; Sixou, Bruno; Rit, Simon; Peyrin, Françoise

2017-09-01

Exploiting the x-ray measurements obtained in different energy bins, spectral computed tomography (CT) has the ability to recover the 3-D description of a patient in a material basis. This may be achieved solving two subproblems, namely the material decomposition and the tomographic reconstruction problems. In this work, we address the material decomposition of spectral x-ray projection images, which is a nonlinear ill-posed problem. Our main contribution is to introduce a material-dependent spatial regularization in the projection domain. The decomposition problem is solved iteratively using a Gauss-Newton algorithm that can benefit from fast linear solvers. A Matlab implementation is available online. The proposed regularized weighted least squares Gauss-Newton algorithm (RWLS-GN) is validated on numerical simulations of a thorax phantom made of up to five materials (soft tissue, bone, lung, adipose tissue, and gadolinium), which is scanned with a 120 kV source and imaged by a 4-bin photon counting detector. To evaluate the method performance of our algorithm, different scenarios are created by varying the number of incident photons, the concentration of the marker and the configuration of the phantom. The RWLS-GN method is compared to the reference maximum likelihood Nelder-Mead algorithm (ML-NM). The convergence of the proposed method and its dependence on the regularization parameter are also studied. We show that material decomposition is feasible with the proposed method and that it converges in few iterations. Material decomposition with ML-NM was very sensitive to noise, leading to decomposed images highly affected by noise, and artifacts even for the best case scenario. The proposed method was less sensitive to noise and improved contrast-to-noise ratio of the gadolinium image. Results were superior to those provided by ML-NM in terms of image quality and decomposition was 70 times faster. For the assessed experiments, material decomposition was possible with the proposed method when the number of incident photons was equal or larger than 10 5 and when the marker concentration was equal or larger than 0.03 g·cm -3 . The proposed method efficiently solves the nonlinear decomposition problem for spectral CT, which opens up new possibilities such as material-specific regularization in the projection domain and a parallelization framework, in which projections are solved in parallel. © 2017 American Association of Physicists in Medicine.

Three-dimensional forward modeling and inversion of marine CSEM data in anisotropic conductivity structures

NASA Astrophysics Data System (ADS)

Han, B.; Li, Y.

2016-12-01

We present a three-dimensional (3D) forward and inverse modeling code for marine controlled-source electromagnetic (CSEM) surveys in anisotropic media. The forward solution is based on a primary/secondary field approach, in which secondary fields are solved using a staggered finite-volume (FV) method and primary fields are solved for 1D isotropic background models analytically. It is shown that it is rather straightforward to extend the isotopic 3D FV algorithm to a triaxial anisotropic one, while additional coefficients are required to account for full tensor conductivity. To solve the linear system resulting from FV discretization of Maxwell' s equations, both iterative Krylov solvers (e.g. BiCGSTAB) and direct solvers (e.g. MUMPS) have been implemented, makes the code flexible for different computing platforms and different problems. For iterative soloutions, the linear system in terms of electromagnetic potentials (A-Phi) is used to precondition the original linear system, transforming the discretized Curl-Curl equations to discretized Laplace-like equations, thus much more favorable numerical properties can be obtained. Numerical experiments suggest that this A-Phi preconditioner can dramatically improve the convergence rate of an iterative solver and high accuracy can be achieved without divergence correction even for low frequencies. To efficiently calculate the sensitivities, i.e. the derivatives of CSEM data with respect to tensor conductivity, the adjoint method is employed. For inverse modeling, triaxial anisotropy is taken into account. Since the number of model parameters to be resolved of triaxial anisotropic medias is twice or thrice that of isotropic medias, the data-space version of the Gauss-Newton (GN) minimization method is preferred due to its lower computational cost compared with the traditional model-space GN method. We demonstrate the effectiveness of the code with synthetic examples.

Self-adaptive predictor-corrector algorithm for static nonlinear structural analysis

NASA Technical Reports Server (NTRS)

Padovan, J.

1981-01-01

A multiphase selfadaptive predictor corrector type algorithm was developed. This algorithm enables the solution of highly nonlinear structural responses including kinematic, kinetic and material effects as well as pro/post buckling behavior. The strategy involves three main phases: (1) the use of a warpable hyperelliptic constraint surface which serves to upperbound dependent iterate excursions during successive incremental Newton Ramphson (INR) type iterations; (20 uses an energy constraint to scale the generation of successive iterates so as to maintain the appropriate form of local convergence behavior; (3) the use of quality of convergence checks which enable various self adaptive modifications of the algorithmic structure when necessary. The restructuring is achieved by tightening various conditioning parameters as well as switch to different algorithmic levels to improve the convergence process. The capabilities of the procedure to handle various types of static nonlinear structural behavior are illustrated.

Fast polar decomposition of an arbitrary matrix

NASA Technical Reports Server (NTRS)

Higham, Nicholas J.; Schreiber, Robert S.

1988-01-01

The polar decomposition of an m x n matrix A of full rank, where m is greater than or equal to n, can be computed using a quadratically convergent algorithm. The algorithm is based on a Newton iteration involving a matrix inverse. With the use of a preliminary complete orthogonal decomposition the algorithm can be extended to arbitrary A. How to use the algorithm to compute the positive semi-definite square root of a Hermitian positive semi-definite matrix is described. A hybrid algorithm which adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Bjorck and Bowie is formulated. The decision when to switch is made using a condition estimator. This matrix multiplication rich algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.

Effect of carbon nano tube working electrode thickness on charge transport kinetics and photo-electrochemical characteristics of dye-sensitized solar cells

NASA Astrophysics Data System (ADS)

Gacemi, Yahia; Cheknane, Ali; Hilal, Hikmat S.

2018-02-01

Physiochemical processes at the photo-electrode and the counter electrode of dye sensitized solar cells (DSSCs) involving having carbon nanotubes (CNTs) instead of the TiO2 layer, within the working electrode, are simulated in this work. Attention is paid to find the effect of CNT layer thickness on photo-electrochemical (PEC) characteristics of the CNT-DSSCs. Comparison with other conventional TiO2-DSSC systems, taking into account the working electrode film thickness, is also described here. To achieve these goals, a model is presented to explain charge transport and electron recombination which involve electron photo-excitation in dye molecules, injection of electrons from the excited dye to CNT working electrode conduction band, diffusion of electrons inside the CNT electrode, charge transfer between oxidized dye and (I-) and recombination of electrons. The simulation is based on solving non-linear equations using the Newton-Raphson numerical method. This concept is proposed for modelling numerical Faradaic impedance at the photo-electrode and the platinum counter electrode. It then simulates the cell impedance spectrum describing the locus of the three semicircles in the Nyquist diagram. The transient equivalent circuit model is also presented based on optimizing current-voltage curves of CNT-DSSCs so as to optimize the fill factor (FF) and conversion efficiency (η). The results show that the simulated characteristics of CNT-DSSCs, with different active CNT layer thicknesses, are superior to conventional TiO2-DSSCs.

Solution des systemes de controle de grandes dimensions basee sur les boucles de retroaction dans la simulation des reseaux electriques

NASA Astrophysics Data System (ADS)

Mugombozi, Chuma Francis

The generation of electrical energy, as well as its transportation and consumption, requires complex control systems for the regulation of power and frequency. These control systems must take into account, among others, new energy sources such as wind energy and new technologies for interconnection by high voltage DC link. These control systems must be able to monitor and achieve such regulation in accordance with the dynamics of the energy source, faults and other events which may induce transients phenomena into the power network. Such transients conditions have to be analyzed using the most accurate and detailed hence, complex models of control system. In addition, in the feasibility study phase, the calibration or the setup of equipment as well as in the operation of the power network, one may require decision aid tools for engineers. This includes, for instance, knowledge of energy dissipated into the arresters in transient analysis. These tools use simulation programs data as inputs and may require that complex functions be solved with numerical methods. These functions are part of control system in computer simulator. Moreover, the simulation evolves in a broader context of the development of digital controller, distributed and parallel high performance computing and rapid evolutions in computer (multiprocessor) technology. In such context, a continuing improvement of the control equations solver is welcomed. Control systems are modelled using ax=b simultaneous system of equations. These equations are sometimes non-linear with feedback loops and thus require iterative Newton methods, including the formation of a Jacobian matrix and ordering as well as processing by graph theory tools. The proposed approach is based on the formulation of a reduced rank Jacobian matrix. The dimension is reduced up to the count of feedback loops. With this new approach, gains in computation speed are expected without compromising its accuracy when compared to classical full rank Jacobian matrix representation. A directed graph representation is adopted and a proper approach for detecting and removing cycles within the graph is introduced. A condition of all zero eigenvalues of adjacency matrix of the graph is adopted. The transformation of the graph of controls with no cycle permits a formulation of control equations for only feedback points. This yields a general feedback interconnection (GFBI) representation of control, which is the main contribution of this thesis. Methods for solving (non-linear) equations of control systems were deployed into the new GFBI approach. Five variants of the new approach were illustrated including firstly, a basic Newton method (1), a more robust one, the Dogleg method (2) and a fixed-point iterations method (3). I. The presented approach is implemented in Electromagnetic Transient program EMTP-RV and tested on practical systems of various types and levels of complexity: the PLL, an asynchronous machine with 87 blocks reduced to 23 feedback equations by GFBI, and 12 wind power plants integrated to the IEEE-39 buses system. Further analysis, which opens up avenues for future research includes comparison of the proposed approach against existing ones. With respect to the sole representation, it is shown that the proposed approach is equivalent to full classic representation of system of equations through a proper substitution process which complies with topological sequence and by skipping feedback variable identified by GFBI. Moreover, a second comparison with state space based approach, such as that in MATLAB/Simulink, shows that output evaluation step in state-space approach with algebraic constraints is similar to the GFBI. The algebraic constraints are similar to feedback variables. A difference may arise, however, when the number of algebraic constraints is not the optimal number of cuts for the GFBI method: for the PLL, for example, MATLAB/Simulink generated 3 constraints while the GFBI generated only 2. The GFBI method may offer some advantages in this case. A last analysis axis prompted further work in initialization. It is shown that GFBI method may modifies the convergence properties of iterations of the Newton method. The Newton- Kantorovich theorem, using bounds on the norms of the Jacobian, has been applied to the proposed GFBI and classic full representation of control equations. The expressions of the Jacobian norms have been established for generic cases using Coates graph. It appears from the analysis of a simple case, for the same initial conditions, the behaviour of the Newton- Kantorovich theorem differs in both cases. These differences may also be more pronounced in the non-linear case. Further work would be useful to investigate this aspect and, eventually, pave the way to new initialization approaches. Despite these limitations, not to mention areas for improvement in further work, one notes the contribution of this thesis to improve the gain of time on simulation for the solution of control systems. (Abstract shortened by UMI.).

Efficient parallel implicit methods for rotary-wing aerodynamics calculations

NASA Astrophysics Data System (ADS)

Wissink, Andrew M.

Euler/Navier-Stokes Computational Fluid Dynamics (CFD) methods are commonly used for prediction of the aerodynamics and aeroacoustics of modern rotary-wing aircraft. However, their widespread application to large complex problems is limited lack of adequate computing power. Parallel processing offers the potential for dramatic increases in computing power, but most conventional implicit solution methods are inefficient in parallel and new techniques must be adopted to realize its potential. This work proposes alternative implicit schemes for Euler/Navier-Stokes rotary-wing calculations which are robust and efficient in parallel. The first part of this work proposes an efficient parallelizable modification of the Lower Upper-Symmetric Gauss Seidel (LU-SGS) implicit operator used in the well-known Transonic Unsteady Rotor Navier Stokes (TURNS) code. The new hybrid LU-SGS scheme couples a point-relaxation approach of the Data Parallel-Lower Upper Relaxation (DP-LUR) algorithm for inter-processor communication with the Symmetric Gauss Seidel algorithm of LU-SGS for on-processor computations. With the modified operator, TURNS is implemented in parallel using Message Passing Interface (MPI) for communication. Numerical performance and parallel efficiency are evaluated on the IBM SP2 and Thinking Machines CM-5 multi-processors for a variety of steady-state and unsteady test cases. The hybrid LU-SGS scheme maintains the numerical performance of the original LU-SGS algorithm in all cases and shows a good degree of parallel efficiency. It experiences a higher degree of robustness than DP-LUR for third-order upwind solutions. The second part of this work examines use of Krylov subspace iterative solvers for the nonlinear CFD solutions. The hybrid LU-SGS scheme is used as a parallelizable preconditioner. Two iterative methods are tested, Generalized Minimum Residual (GMRES) and Orthogonal s-Step Generalized Conjugate Residual (OSGCR). The Newton method demonstrates good parallel performance on the IBM SP2, with OS-GCR giving slightly better performance than GMRES on large numbers of processors. For steady and quasi-steady calculations, the convergence rate is accelerated but the overall solution time remains about the same as the standard hybrid LU-SGS scheme. For unsteady calculations, however, the Newton method maintains a higher degree of time-accuracy which allows tbe use of larger timesteps and results in CPU savings of 20-35%.

Taylor O(h³) Discretization of ZNN Models for Dynamic Equality-Constrained Quadratic Programming With Application to Manipulators.

PubMed

Liao, Bolin; Zhang, Yunong; Jin, Long

2016-02-01

In this paper, a new Taylor-type numerical differentiation formula is first presented to discretize the continuous-time Zhang neural network (ZNN), and obtain higher computational accuracy. Based on the Taylor-type formula, two Taylor-type discrete-time ZNN models (termed Taylor-type discrete-time ZNNK and Taylor-type discrete-time ZNNU models) are then proposed and discussed to perform online dynamic equality-constrained quadratic programming. For comparison, Euler-type discrete-time ZNN models (called Euler-type discrete-time ZNNK and Euler-type discrete-time ZNNU models) and Newton iteration, with interesting links being found, are also presented. It is proved herein that the steady-state residual errors of the proposed Taylor-type discrete-time ZNN models, Euler-type discrete-time ZNN models, and Newton iteration have the patterns of O(h(3)), O(h(2)), and O(h), respectively, with h denoting the sampling gap. Numerical experiments, including the application examples, are carried out, of which the results further substantiate the theoretical findings and the efficacy of Taylor-type discrete-time ZNN models. Finally, the comparisons with Taylor-type discrete-time derivative model and other Lagrange-type discrete-time ZNN models for dynamic equality-constrained quadratic programming substantiate the superiority of the proposed Taylor-type discrete-time ZNN models once again.

Sensitivity analysis for aeroacoustic and aeroelastic design of turbomachinery blades

NASA Technical Reports Server (NTRS)

Lorence, Christopher B.; Hall, Kenneth C.

1995-01-01

A new method for computing the effect that small changes in the airfoil shape and cascade geometry have on the aeroacoustic and aeroelastic behavior of turbomachinery cascades is presented. The nonlinear unsteady flow is assumed to be composed of a nonlinear steady flow plus a small perturbation unsteady flow that is harmonic in time. First, the full potential equation is used to describe the behavior of the nonlinear mean (steady) flow through a two-dimensional cascade. The small disturbance unsteady flow through the cascade is described by the linearized Euler equations. Using rapid distortion theory, the unsteady velocity is split into a rotational part that contains the vorticity and an irrotational part described by a scalar potential. The unsteady vorticity transport is described analytically in terms of the drift and stream functions computed from the steady flow. Hence, the solution of the linearized Euler equations may be reduced to a single inhomogeneous equation for the unsteady potential. The steady flow and small disturbance unsteady flow equations are discretized using bilinear quadrilateral isoparametric finite elements. The nonlinear mean flow solution and streamline computational grid are computed simultaneously using Newton iteration. At each step of the Newton iteration, LU decomposition is used to solve the resulting set of linear equations. The unsteady flow problem is linear, and is also solved using LU decomposition. Next, a sensitivity analysis is performed to determine the effect small changes in cascade and airfoil geometry have on the mean and unsteady flow fields. The sensitivity analysis makes use of the nominal steady and unsteady flow LU decompositions so that no additional matrices need to be factored. Hence, the present method is computationally very efficient. To demonstrate how the sensitivity analysis may be used to redesign cascades, a compressor is redesigned for improved aeroelastic stability and two different fan exit guide vanes are redesigned for reduced downstream radiated noise. In addition, a framework detailing how the two-dimensional version of the method may be used to redesign three-dimensional geometries is presented.

Joint Inversion of 3d Mt/gravity/magnetic at Pisagua Fault.

NASA Astrophysics Data System (ADS)

Bascur, J.; Saez, P.; Tapia, R.; Humpire, M.

2017-12-01

This work shows the results of a joint inversion at Pisagua Fault using 3D Magnetotellurics (MT), gravity and regional magnetic data. The MT survey has a poor coverage of study area with only 21 stations; however, it allows to detect a low resistivity zone aligned with the Pisagua Fault trace that it is interpreted as a damage zone. The integration of gravity and magnetic data, which have more dense sampling and coverage, adds more detail and resolution to the detected low resistivity structure and helps to improve the structure interpretation using the resulted models (density, magnetic-susceptibility and electrical resistivity). The joint inversion process minimizes a multiple target function which includes the data misfit, model roughness and coupling norms (crossgradient and direct relations) for all geophysical methods considered (MT, gravity and magnetic). This process is solved iteratively using the Gauss-Newton method which updates the model of each geophysical method improving its individual data misfit, model roughness and the coupling with the other geophysical models. For solving the model updates of magnetic and gravity methods were developed dedicated 3D inversion software codes which include the coupling norms with additionals geophysical parameters. The model update of the 3D MT is calculated using an iterative method which sequentially filters the priority model and the output model of a single 3D MT inversion process for obtaining the resistivity model coupled solution with the gravity and magnetic methods.

A novel approach to calibrate the hemodynamic model using functional Magnetic Resonance Imaging (fMRI) measurements.

PubMed

Khoram, Nafiseh; Zayane, Chadia; Djellouli, Rabia; Laleg-Kirati, Taous-Meriem

2016-03-15

The calibration of the hemodynamic model that describes changes in blood flow and blood oxygenation during brain activation is a crucial step for successfully monitoring and possibly predicting brain activity. This in turn has the potential to provide diagnosis and treatment of brain diseases in early stages. We propose an efficient numerical procedure for calibrating the hemodynamic model using some fMRI measurements. The proposed solution methodology is a regularized iterative method equipped with a Kalman filtering-type procedure. The Newton component of the proposed method addresses the nonlinear aspect of the problem. The regularization feature is used to ensure the stability of the algorithm. The Kalman filter procedure is incorporated here to address the noise in the data. Numerical results obtained with synthetic data as well as with real fMRI measurements are presented to illustrate the accuracy, robustness to the noise, and the cost-effectiveness of the proposed method. We present numerical results that clearly demonstrate that the proposed method outperforms the Cubature Kalman Filter (CKF), one of the most prominent existing numerical methods. We have designed an iterative numerical technique, called the TNM-CKF algorithm, for calibrating the mathematical model that describes the single-event related brain response when fMRI measurements are given. The method appears to be highly accurate and effective in reconstructing the BOLD signal even when the measurements are tainted with high noise level (as high as 30%). Published by Elsevier B.V.

Inexact adaptive Newton methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bertiger, W.I.; Kelsey, F.J.

1985-02-01

The Inexact Adaptive Newton method (IAN) is a modification of the Adaptive Implicit Method/sup 1/ (AIM) with improved Newton convergence. Both methods simplify the Jacobian at each time step by zeroing coefficients in regions where saturations are changing slowly. The methods differ in how the diagonal block terms are treated. On test problems with up to 3,000 cells, IAN consistently saves approximately 30% of the CPU time when compared to the fully implicit method. AIM shows similar savings on some problems, but takes as much CPU time as fully implicit on other test problems due to poor Newton convergence.

How concept images affect students' interpretations of Newton's method

NASA Astrophysics Data System (ADS)

Engelke Infante, Nicole; Murphy, Kristen; Glenn, Celeste; Sealey, Vicki

2018-07-01

Knowing when students have the prerequisite knowledge to be able to read and understand a mathematical text is a perennial concern for instructors. Using text describing Newton's method and Vinner's notion of concept image, we exemplify how prerequisite knowledge influences understanding. Through clinical interviews with first-semester calculus students, we determined how evoked concept images of tangent lines and roots contributed to students' interpretation and application of Newton's method. Results show that some students' concept images of root and tangent line developed throughout the interview process, and most students were able to adequately interpret the text on Newton's method. However, students with insufficient concept images of tangent line and students who were unwilling or unable to modify their concept images of tangent line after reading the text were not successful in interpreting Newton's method.

Recent work on material interface reconstruction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mosso, S.J.; Swartz, B.K.

1997-12-31

For the last 15 years, many Eulerian codes have relied on a series of piecewise linear interface reconstruction algorithms developed by David Youngs. In a typical Youngs` method, the material interfaces were reconstructed based upon nearly cell values of volume fractions of each material. The interfaces were locally represented by linear segments in two dimensions and by pieces of planes in three dimensions. The first step in such reconstruction was to locally approximate an interface normal. In Youngs` 3D method, a local gradient of a cell-volume-fraction function was estimated and taken to be the local interface normal. A linear interfacemore » was moved perpendicular to the now known normal until the mass behind it matched the material volume fraction for the cell in question. But for distorted or nonorthogonal meshes, the gradient normal estimate didn`t accurately match that of linear material interfaces. Moreover, curved material interfaces were also poorly represented. The authors will present some recent work in the computation of more accurate interface normals, without necessarily increasing stencil size. Their estimate of the normal is made using an iterative process that, given mass fractions for nearby cells of known but arbitrary variable density, converges in 3 or 4 passes in practice (and quadratically--like Newton`s method--in principle). The method reproduces a linear interface in both orthogonal and nonorthogonal meshes. The local linear approximation is generally 2nd-order accurate, with a 1st-order accurate normal for curved interfaces in both two and three dimensional polyhedral meshes. Recent work demonstrating the interface reconstruction for curved surfaces will /be discussed.« less

A LAGRANGIAN GAUSS-NEWTON-KRYLOV SOLVER FOR MASS- AND INTENSITY-PRESERVING DIFFEOMORPHIC IMAGE REGISTRATION.

PubMed

Mang, Andreas; Ruthotto, Lars

2017-01-01

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves. We approximate these curves using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of the final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.

Use of Generalized Fluid System Simulation Program (GFSSP) for Teaching and Performing Senior Design Projects at the Educational Institutions

NASA Technical Reports Server (NTRS)

Majumdar, A. K.; Hedayat, A.

2015-01-01

This paper describes the experience of the authors in using the Generalized Fluid System Simulation Program (GFSSP) in teaching Design of Thermal Systems class at University of Alabama in Huntsville. GFSSP is a finite volume based thermo-fluid system network analysis code, developed at NASA/Marshall Space Flight Center, and is extensively used in NASA, Department of Defense, and aerospace industries for propulsion system design, analysis, and performance evaluation. The educational version of GFSSP is freely available to all US higher education institutions. The main purpose of the paper is to illustrate the utilization of this user-friendly code for the thermal systems design and fluid engineering courses and to encourage the instructors to utilize the code for the class assignments as well as senior design projects. The need for a generalized computer program for thermofluid analysis in a flow network has been felt for a long time in aerospace industries. Designers of thermofluid systems often need to know pressures, temperatures, flow rates, concentrations, and heat transfer rates at different parts of a flow circuit for steady state or transient conditions. Such applications occur in propulsion systems for tank pressurization, internal flow analysis of rocket engine turbopumps, chilldown of cryogenic tanks and transfer lines, and many other applications of gas-liquid systems involving fluid transients and conjugate heat and mass transfer. Computer resource requirements to perform time-dependent, three-dimensional Navier-Stokes computational fluid dynamic (CFD) analysis of such systems are prohibitive and therefore are not practical. Available commercial codes are generally suitable for steady state, single-phase incompressible flow. Because of the proprietary nature of such codes, it is not possible to extend their capability to satisfy the above-mentioned needs. Therefore, the Generalized Fluid System Simulation Program (GFSSP1) has been developed at NASA Marshall Space Flight Center (MSFC) as a general fluid flow system solver capable of handling phase changes, compressibility, mixture thermodynamics and transient operations. It also includes the capability to model external body forces such as gravity and centrifugal effects in a complex flow network. The objectives of GFSSP development are: a) to develop a robust and efficient numerical algorithm to solve a system of equations describing a flow network containing phase changes, mixing, and rotation; and b) to implement the algorithm in a structured, easy-to-use computer program. The analysis of thermofluid dynamics in a complex network requires resolution of the system into fluid nodes and branches, and solid nodes and conductors as shown in Figure 1. Figure 1 shows a schematic and GFSSP flow circuit of a counter-flow heat exchanger. Hot nitrogen gas is flowing through a pipe, colder nitrogen is flowing counter to the hot stream in the annulus pipe and heat transfer occurs through metal tubes. The problem considered is to calculate flowrates and temperature distributions in both streams. GFSSP has a unique data structure, as shown in Figure 2, that allows constructing all possible arrangements of a flow network with no limit on the number of elements. The elements of a flow network are boundary nodes where pressure and temperature are specified, internal nodes where pressure and temperature are calculated, and branches where flowrates are calculated. For conjugate heat transfer problems, there are three additional elements: solid node, ambient node, and conductor. The solid and fluid nodes are connected with solid-fluid conductors. GFSSP solves the conservation equations of mass and energy, and equation of state in internal nodes to calculate pressure, temperature and resident mass. The momentum conservation equation is solved in branches to calculate flowrate. It also solves for energy conservation equations to calculate temperatures of solid nodes. The equations are coupled and nonlinear; therefore, they are solved by an iterative numerical scheme. GFSSP employs a unique numerical scheme known as simultaneous adjustment with successive substitution (SASS), which is a combination of Newton-Raphson and successive substitution methods. The mass and momentum conservation equations and the equation of state are solved by the Newton-Raphson method while the conservation of energy and species are solved by the successive substitution method. GFSSP is linked with two thermodynamic property programs, GASP2 and WASP3 and GASPAK4, that provide thermodynamic and thermophysical properties of selected fluids. Both programs cover a range of pressure and temperature that allows fluid properties to be evaluated for liquid, liquid-vapor (saturation), and vapor region. GASP and WASP provide properties of 12 fluids. GASPAK includes a library of 36 fluids. GFSSP has three major parts. The first part is the graphical user interface (GUI), visual thermofluid analyzer of systems and components (VTASC). VTASC allows users to create a flow circuit by a 'point and click' paradigm. It creates the GFSSP input file after the completion of the model building process. GFSSP's GUI provides the users a platform to build and run their models. It also allows post-processing of results. The network flow circuit is first built using three basic elements: boundary node, internal node, and branch.

Documentation for the MODFLOW 6 Groundwater Flow Model

USGS Publications Warehouse

Langevin, Christian D.; Hughes, Joseph D.; Banta, Edward R.; Niswonger, Richard G.; Panday, Sorab; Provost, Alden M.

2017-08-10

This report documents the Groundwater Flow (GWF) Model for a new version of MODFLOW called MODFLOW 6. The GWF Model for MODFLOW 6 is based on a generalized control-volume finite-difference approach in which a cell can be hydraulically connected to any number of surrounding cells. Users can define the model grid using one of three discretization packages, including (1) a structured discretization package for defining regular MODFLOW grids consisting of layers, rows, and columns, (2) a discretization by ver­tices package for defining layered unstructured grids consisting of layers and cells, and (3) a general unstruc­tured discretization package for defining flexible grids comprised of cells and their connection properties. For layered grids, a new capability is available for removing thin cells and vertically connecting cells overlying and underlying the thin cells. For complex problems involving water-table conditions, an optional Newton-Raphson formulation, based on the formulations in MODFLOW-NWT and MODFLOW-USG, can be acti­vated. Use of the Newton-Raphson formulation will often improve model convergence and allow solutions to be obtained for difficult problems that cannot be solved using the traditional wetting and drying approach. The GWF Model is divided into “packages,” as was done in previous MODFLOW versions. A package is the part of the model that deals with a single aspect of simulation. Packages included with the GWF Model include those related to internal calculations of groundwater flow (discretization, initial conditions, hydraulic conduc­tance, and storage), stress packages (constant heads, wells, recharge, rivers, general head boundaries, drains, and evapotranspiration), and advanced stress packages (streamflow routing, lakes, multi-aquifer wells, and unsaturated zone flow). An additional package is also available for moving water available in one package into the individual features of the advanced stress packages. The GWF Model also has packages for obtaining and controlling output from the model. This report includes detailed explanations of physical and mathematical concepts on which the GWF Model and its packages are based.Like its predecessors, MODFLOW 6 is based on a highly modular structure; however, this structure has been extended into an object-oriented framework. The framework includes a robust and generalized numeri­cal solution object, which can be used to solve many different types of models. The numerical solution object has several different matrix preconditioning options as well as several methods for solving the linear system of equations. In this new framework, the GWF Model itself is an object as are each of the GWF Model packages. A benefit of the object-oriented structure is that multiple objects of the same type can be used in a single sim­ulation. Thus, a single forward run with MODFLOW 6 may contain multiple GWF Models. GWF Models can be hydraulically connected using GWF-GWF Exchange objects. Connecting GWF models in different ways permits the user to utilize a local grid refinement strategy consisting of parent and child models or to couple adjacent GWF Models. An advantage of the approach implemented in MODFLOW 6 is that multiple models and their exchanges can be incorporated into a single numerical solution object. With this design, models can be tightly coupled at the matrix level.

Damage identification of supporting structures with a moving sensory system

NASA Astrophysics Data System (ADS)

Zhu, X. Q.; Law, S. S.; Huang, L.; Zhu, S. Y.

2018-02-01

An innovative approach to identify local anomalies in a structural beam bridge with an instrumented vehicle moving as a sensory system across the bridge. Accelerations at both the axle and vehicle body are measured from which vehicle-bridge interaction force on the structure is determined. Local anomalies of the structure are estimated from this interaction force with the Newton's iterative method basing on the homotopy continuation method. Numerical results with the vehicle moving over simply supported or continuous beams show that the acceleration responses from the vehicle or the bridge structure are less sensitive to the local damages than the interaction force between the wheel and the structure. Effects of different movement patterns and moving speed of the vehicle are investigated, and the effect of measurement noise on the identified results is discussed. A heavier or slower vehicle has been shown to be less sensitive to measurement noise giving more accurate results.

Study of three-dimensional effects on vortex breakdown

NASA Technical Reports Server (NTRS)

Salas, M. D.; Kuruvila, G.

1988-01-01

The incompressible axisymmetric steady Navier-Stokes equations in primitive variables are used to simulate vortex breakdown. The equations, discretized using a second-order, central-difference scheme, are linearized and then solved using an exact LU decomposition, Gaussian elimination, and Newton iteration. Solutions are presented for Reynolds numbers, based on vortex-core radius, as high as 1500. An attempt to study the stability of the axisymmetric solutions against three-dimensional perturbations is discussed.

Numerical method for the solution of large systems of differential equations of the boundary layer type

NASA Technical Reports Server (NTRS)

Green, M. J.; Nachtsheim, P. R.

1972-01-01

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

2D Seismic Imaging of Elastic Parameters by Frequency Domain Full Waveform Inversion

NASA Astrophysics Data System (ADS)

Brossier, R.; Virieux, J.; Operto, S.

2008-12-01

Thanks to recent advances in parallel computing, full waveform inversion is today a tractable seismic imaging method to reconstruct physical parameters of the earth interior at different scales ranging from the near- surface to the deep crust. We present a massively parallel 2D frequency-domain full-waveform algorithm for imaging visco-elastic media from multi-component seismic data. The forward problem (i.e. the resolution of the frequency-domain 2D PSV elastodynamics equations) is based on low-order Discontinuous Galerkin (DG) method (P0 and/or P1 interpolations). Thanks to triangular unstructured meshes, the DG method allows accurate modeling of both body waves and surface waves in case of complex topography for a discretization of 10 to 15 cells per shear wavelength. The frequency-domain DG system is solved efficiently for multiple sources with the parallel direct solver MUMPS. The local inversion procedure (i.e. minimization of residuals between observed and computed data) is based on the adjoint-state method which allows to efficiently compute the gradient of the objective function. Applying the inversion hierarchically from the low frequencies to the higher ones defines a multiresolution imaging strategy which helps convergence towards the global minimum. In place of expensive Newton algorithm, the combined use of the diagonal terms of the approximate Hessian matrix and optimization algorithms based on quasi-Newton methods (Conjugate Gradient, LBFGS, ...) allows to improve the convergence of the iterative inversion. The distribution of forward problem solutions over processors driven by a mesh partitioning performed by METIS allows to apply most of the inversion in parallel. We shall present the main features of the parallel modeling/inversion algorithm, assess its scalability and illustrate its performances with realistic synthetic case studies.

A model reduction approach to numerical inversion for a parabolic partial differential equation

NASA Astrophysics Data System (ADS)

Borcea, Liliana; Druskin, Vladimir; Mamonov, Alexander V.; Zaslavsky, Mikhail

2014-12-01

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

An Laudau-Lifschitz theory based algorithm on calculating post-buckling configuration of a rod buckling in elastic media

NASA Astrophysics Data System (ADS)

Huang, Shicheng; Tan, Likun; Hu, Nan; Grover, Hannah; Chu, Kevin; Chen, Zi

This reserach introduces a new numerical approach of calculating the post-buckling configuration of a thin rod embedded in elastic media. The theoretical base is the governing ODEs describing the balance of forces and moments, the length conservation, and the physics of bending and twisting by Laudau and Lifschitz. The numerical methods applied in the calculation are continuation method and Newton's method of iteration in combination with spectrum method. To the authors' knowledge, it is the first trial of directly applying the L-L theory to numerically studying the phenomenon of rod buckling in elastic medium. This method accounts for nonlinearity of geometry, thus is capable of calculating large deformation. The stability of this method is another advantage achieved by expressing the governing equations in a set of first-order derivative form. The wave length, amplitude, and decay effect all agree with the experiment without any further assumptions. This program can be applied to different occasions with varying stiffness of the elastic medai and rigidity of the rod.

Newton's method applied to finite-difference approximations for the steady-state compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Bailey, Harry E.; Beam, Richard M.

1991-01-01

Finite-difference approximations for steady-state compressible Navier-Stokes equations, whose two spatial dimensions are written in generalized curvilinear coordinates and strong conservation-law form, are presently solved by means of Newton's method in order to obtain a lifting-airfoil flow field under subsonic and transonnic conditions. In addition to ascertaining the computational requirements of an initial guess ensuring convergence and the degree of computational efficiency obtainable via the approximate Newton method's freezing of the Jacobian matrices, attention is given to the need for auxiliary methods assessing the temporal stability of steady-state solutions. It is demonstrated that nonunique solutions of the finite-difference equations are obtainable by Newton's method in conjunction with a continuation method.

Fully Coupled Nonlinear Fluid Flow and Poroelasticity in Arbitrarily Fractured Porous Media: A Hybrid-Dimensional Computational Model

NASA Astrophysics Data System (ADS)

Jin, L.; Zoback, M. D.

2017-10-01

We formulate the problem of fully coupled transient fluid flow and quasi-static poroelasticity in arbitrarily fractured, deformable porous media saturated with a single-phase compressible fluid. The fractures we consider are hydraulically highly conductive, allowing discontinuous fluid flux across them; mechanically, they act as finite-thickness shear deformation zones prior to failure (i.e., nonslipping and nonpropagating), leading to "apparent discontinuity" in strain and stress across them. Local nonlinearity arising from pressure-dependent permeability of fractures is also included. Taking advantage of typically high aspect ratio of a fracture, we do not resolve transversal variations and instead assume uniform flow velocity and simple shear strain within each fracture, rendering the coupled problem numerically more tractable. Fractures are discretized as lower dimensional zero-thickness elements tangentially conforming to unstructured matrix elements. A hybrid-dimensional, equal-low-order, two-field mixed finite element method is developed, which is free from stability issues for a drained coupled system. The fully implicit backward Euler scheme is employed for advancing the fully coupled solution in time, and the Newton-Raphson scheme is implemented for linearization. We show that the fully discretized system retains a canonical form of a fracture-free poromechanical problem; the effect of fractures is translated to the modification of some existing terms as well as the addition of several terms to the capacity, conductivity, and stiffness matrices therefore allowing the development of independent subroutines for treating fractures within a standard computational framework. Our computational model provides more realistic inputs for some fracture-dominated poromechanical problems like fluid-induced seismicity.

Rheological effects of micropolar slime on the gliding motility of bacteria with slip boundary condition

NASA Astrophysics Data System (ADS)

Asghar, Z.; Ali, N.; Anwar Bég, O.; Javed, T.

2018-06-01

Gliding bacteria are virtually everywhere. These organisms are phylogenetically diverse with their hundreds of types, different shapes and several modes of motility. One possible mode of gliding motility in the rod shaped bacteria is that they propel themselves by producing undulating waves in their body. Few bacteria glides near the solid surface over the slime without any aid of flagella so the classical Navier-Stokes equations are incapable of explaining the slime rheology at the microscopic level. Micropolar fluid dynamics however provides a solid framework for mimicking bacterial physical phenomena at both micro and nano-scales, and therefore we use the micropolar fluid to characterize the rheology of a thin layer of slime and its dominant microrotation effects. It is also assumed that there is a certain degree of slip between slime and bacterial undulating surface and also between slime and solid substrate. The flow equations are formulated under long wavelength and low Reynolds number assumptions. Exact expressions for stream function and pressure gradient are obtained. The speed of the gliding bacteria is numerically calculated by using a modified Newton-Raphson method. Slip effects and effects of non-Newtonian slime parameters on bacterial speed and power are also quantified. In addition, when the glider is fixed, the effects of slip and rheological properties of micropolar slime parameters on the velocity, micro-rotation (angular velocity) of spherical slime particles, pressure rise per wavelength, pumping and trapping phenomena are also shown graphically and discussed in detail. The study is relevant to emerging biofuel cell technologies and also bacterial biophysics.

Development of an hp-version finite element method for computational optimal control

NASA Technical Reports Server (NTRS)

Hodges, Dewey H.; Warner, Michael S.

1993-01-01

The purpose of this research effort was to begin the study of the application of hp-version finite elements to the numerical solution of optimal control problems. Under NAG-939, the hybrid MACSYMA/FORTRAN code GENCODE was developed which utilized h-version finite elements to successfully approximate solutions to a wide class of optimal control problems. In that code the means for improvement of the solution was the refinement of the time-discretization mesh. With the extension to hp-version finite elements, the degrees of freedom include both nodal values and extra interior values associated with the unknown states, co-states, and controls, the number of which depends on the order of the shape functions in each element. One possible drawback is the increased computational effort within each element required in implementing hp-version finite elements. We are trying to determine whether this computational effort is sufficiently offset by the reduction in the number of time elements used and improved Newton-Raphson convergence so as to be useful in solving optimal control problems in real time. Because certain of the element interior unknowns can be eliminated at the element level by solving a small set of nonlinear algebraic equations in which the nodal values are taken as given, the scheme may turn out to be especially powerful in a parallel computing environment. A different processor could be assigned to each element. The number of processors, strictly speaking, is not required to be any larger than the number of sub-regions which are free of discontinuities of any kind.

Studies on the finite element simulation in sheet metal stamping processes

NASA Astrophysics Data System (ADS)

Huang, Ying

The sheet metal stamping process plays an important role in modern industry. With the ever-increasing demand for shape complexity, product quality and new materials, the traditional trial and error method for setting up a sheet metal stamping process is no longer efficient. As a result, the Finite Element Modeling (FEM) method has now been widely used. From a physical point of view, the formability and the quality of a product are influenced by several factors. The design of the product in the initial stage and the motion of the press during the production stage are two of these crucial factors. This thesis focuses on the numerical simulation for these two factors using FEM. Currently, there are a number of commercial FEM software systems available in the market. These software systems are based on an incremental FEM process that models the sheet metal stamping process in small incremental steps. Even though the incremental FEM is accurate, it is not suitable for the initial conceptual design for its needing of detailed design parameters and enormous calculation times. As a result, another type of FEM, called the inverse FEM method or one-step FEM method, has been proposed. While it is less accurate than that of the incremental method, this method requires much less computation and hence, has a great potential. However, it also faces a number of unsolved problems, which limits its application. This motivates the presented research. After the review of the basic theory of the inverse method, a new modified arc-length search method is proposed to find better initial solution. The methods to deal with the vertical walls are also discussed and presented. Then, a generalized multi-step inverse FEM method is proposed. It solves two key obstacles: the first one is to determine the initial solution of the intermediate three-dimensional configurations and the other is to control the movement of nodes so they could only slide on constraint surfaces during the search by Newton-Raphson iteration. The computer implementation of the generalized multi-step inverse FEM is also presented. By comparing to the simulation results using a commercial software system, the effectiveness of the new method is validated. Other than the product design, the punch motion (including punch speed and punch trajectory) of the stamping press also has significant effect on the formability and the quality of the product. In fact, this is one of the major reasons why hydraulic presses and/or servo presses are used for parts which demand high quality. In order to reveal the quantitative correlation between the punch motion and the part quality, the Cowper-Symonds strain rate constitutive model and the implicit dynamic incremental FEM are combined to conduct the research. The effects of the punch motion on the part quality, especially the plastic strain distribution and the potential springback, have been investigated for the deep drawing and the bending processes respectively. A qualitative relationship between the punch motion and the part quality is also derived. The reaction force of the punch motion causes the dynamic deformation of the press during the stamping, which in turn influences the part quality as well. This dynamic information, in the form of the strain signal, is an important basis for the on-line monitoring of the part quality. By using the actual force as the input to the press, the incremental FEM is needed to predict the strain of the press. The result is validated by means of experiments and can be used to assist the on-line monitoring.

Reading the Principia

NASA Astrophysics Data System (ADS)

Guicciardini, Niccoló

2003-10-01

1. Purpose of this book; Part I. Newton's Methods: 2. Newton's methods of series and fluxions; 3. The mathematical methods of the Principia; Part II. Three Readers: 4. Newton: between tradition and innovation; 5. Huygens: the Principia and proportion theory; 6. Leibniz: not equivalent in practice; Part III. Two Schools: 7. Britain: in the wake of the Principia; 8. Basel: challenging the Principia; 9. Conclusion: Newtonians, Leibnizians and Eulerians; References.

Segregated Methods for Two-Fluid Models

DOE Office of Scientific and Technical Information (OSTI.GOV)

Prosperetti, Andrea; Sundaresan, Sankaran; Pannala, Sreekanth

2007-01-01

The previous chapter, with its direct simulation of the fluid flow and a modeling approach to the particle phase, may be seen as a transition between the methods for a fully resolved simulation described in the first part of this book and those for a coarse grained description based on the averaging approach described in chapter ??. We now turn to the latter, which in practice are the only methods able to deal with the complex flows encountered in most situations of practical interest such as fluidized beds, pipelines, energy generation, sediment transport, and others. This chapter and the nextmore » one are devoted to numerical methods for so-called two-fluid models in which the phases are treated as inter-penetrating continua describing, e.g., a liquid and a gas, or a fluid and a suspended solid phase. These models can be extended to deal with more than two continua and, then, the denomination multi-fluid models might be more appropriate. For example, the commercial code OLGA (Bendiksen et al. 1991), widely used in the oil industry, recognizes three phases, all treated as interpenetrating continua: a continuous liquid, a gas, and a disperse liquid phase present as drops suspended in the gas phase. The more recent PeTra (Petroleum Transport, Larsen et al. 1997) also describes three phases, gas, oil, and water. Recent approaches to the description of complex boiling flows recognize four inter-penetrating phases: a liquid phase present both as a continuum and as a dispersion of droplets, and a gas/vapor phase also present as a continuum and a dispersion of bubbles. Methods for these multi-fluid models are based on those developed for the two-fluid model to which we limit ourselves. In principle, one could simply take the model equations, discretize them, and solve them by a method suitable for non-linear problems, e.g. Newton-Raphson iteration. In practice, the computational cost of such a frontal attack is nearly always prohibitive in terms of storage requirement and execution time. It is therefore necessary to devise different, less direct strategies. Two principal classes of algorithms have been developed for this purpose. The first one, described in this chapter, consists of algorithms derived from the pressure based schemes widely used in single-phase flow, such as SIMPLE and its variations (see e.g. Patankar 1980). In this approach, the model equations are solved sequentially and, therefore, these methods are often referred to as segregated algorithms to distinguish them from a second class of methods, object of the next chapter, in which a coupled or semi-coupled time-marching solution strategy is adopted. Broadly speaking, the first class of methods is suitable for relatively slow transients, such as fluidized beds, or phenomena with a long duration, such as flow in pipelines. The methods in the second group have been designed to deal principally with fast transients, such as those hypothesized in nuclear reactor safety. Since in segregated solvers the equations are solved one by one, it is possible to add equations to the mathematical model - to describe e.g. turbulence - at a later stage after the development of the initial code without major modifications of the algorithm.« less

A unified, multifidelity quasi-newton optimization method with application to aero-structural designa

NASA Astrophysics Data System (ADS)

Bryson, Dean Edward

A model's level of fidelity may be defined as its accuracy in faithfully reproducing a quantity or behavior of interest of a real system. Increasing the fidelity of a model often goes hand in hand with increasing its cost in terms of time, money, or computing resources. The traditional aircraft design process relies upon low-fidelity models for expedience and resource savings. However, the reduced accuracy and reliability of low-fidelity tools often lead to the discovery of design defects or inadequacies late in the design process. These deficiencies result either in costly changes or the acceptance of a configuration that does not meet expectations. The unknown opportunity cost is the discovery of superior vehicles that leverage phenomena unknown to the designer and not illuminated by low-fidelity tools. Multifidelity methods attempt to blend the increased accuracy and reliability of high-fidelity models with the reduced cost of low-fidelity models. In building surrogate models, where mathematical expressions are used to cheaply approximate the behavior of costly data, low-fidelity models may be sampled extensively to resolve the underlying trend, while high-fidelity data are reserved to correct inaccuracies at key locations. Similarly, in design optimization a low-fidelity model may be queried many times in the search for new, better designs, with a high-fidelity model being exercised only once per iteration to evaluate the candidate design. In this dissertation, a new multifidelity, gradient-based optimization algorithm is proposed. It differs from the standard trust region approach in several ways, stemming from the new method maintaining an approximation of the inverse Hessian, that is the underlying curvature of the design problem. Whereas the typical trust region approach performs a full sub-optimization using the low-fidelity model at every iteration, the new technique finds a suitable descent direction and focuses the search along it, reducing the number of low-fidelity evaluations required. This narrowing of the search domain also alleviates the burden on the surrogate model corrections between the low- and high-fidelity data. Rather than requiring the surrogate to be accurate in a hyper-volume bounded by the trust region, the model needs only to be accurate along the forward-looking search direction. Maintaining the approximate inverse Hessian also allows the multifidelity algorithm to revert to high-fidelity optimization at any time. In contrast, the standard approach has no memory of the previously-computed high-fidelity data. The primary disadvantage of the proposed algorithm is that it may require modifications to the optimization software, whereas standard optimizers may be used as black-box drivers in the typical trust region method. A multifidelity, multidisciplinary simulation of aeroelastic vehicle performance is developed to demonstrate the optimization method. The numerical physics models include body-fitted Euler computational fluid dynamics; linear, panel aerodynamics; linear, finite-element computational structural mechanics; and reduced, modal structural bases. A central element of the multifidelity, multidisciplinary framework is a shared parametric, attributed geometric representation that ensures the analysis inputs are consistent between disciplines and fidelities. The attributed geometry also enables the transfer of data between disciplines. The new optimization algorithm, a standard trust region approach, and a single-fidelity quasi-Newton method are compared for a series of analytic test functions, using both polynomial chaos expansions and kriging to correct discrepancies between fidelity levels of data. In the aggregate, the new method requires fewer high-fidelity evaluations than the trust region approach in 51% of cases, and the same number of evaluations in 18%. The new approach also requires fewer low-fidelity evaluations, by up to an order of magnitude, in almost all cases. The efficacy of both multifidelity methods compared to single-fidelity optimization depends significantly on the behavior of the high-fidelity model and the quality of the low-fidelity approximation, though savings are realized in a large number of cases. The multifidelity algorithm is also compared to the single-fidelity quasi-Newton method for complex aeroelastic simulations. The vehicle design problem includes variables for planform shape, structural sizing, and cruise condition with constraints on trim and structural stresses. Considering the objective function reduction versus computational expenditure, the multifidelity process performs better in three of four cases in early iterations. However, the enforcement of a contracting trust region slows the multifidelity progress. Even so, leveraging the approximate inverse Hessian, the optimization can be seamlessly continued using high-fidelity data alone. Ultimately, the proposed new algorithm produced better designs in all four cases. Investigating the return on investment in terms of design improvement per computational hour confirms that the multifidelity advantage is greatest in early iterations, and managing the transition to high-fidelity optimization is critical.

Performance of Nonlinear Finite-Difference Poisson-Boltzmann Solvers

PubMed Central

Cai, Qin; Hsieh, Meng-Juei; Wang, Jun; Luo, Ray

2014-01-01

We implemented and optimized seven finite-difference solvers for the full nonlinear Poisson-Boltzmann equation in biomolecular applications, including four relaxation methods, one conjugate gradient method, and two inexact Newton methods. The performance of the seven solvers was extensively evaluated with a large number of nucleic acids and proteins. Worth noting is the inexact Newton method in our analysis. We investigated the role of linear solvers in its performance by incorporating the incomplete Cholesky conjugate gradient and the geometric multigrid into its inner linear loop. We tailored and optimized both linear solvers for faster convergence rate. In addition, we explored strategies to optimize the successive over-relaxation method to reduce its convergence failures without too much sacrifice in its convergence rate. Specifically we attempted to adaptively change the relaxation parameter and to utilize the damping strategy from the inexact Newton method to improve the successive over-relaxation method. Our analysis shows that the nonlinear methods accompanied with a functional-assisted strategy, such as the conjugate gradient method and the inexact Newton method, can guarantee convergence in the tested molecules. Especially the inexact Newton method exhibits impressive performance when it is combined with highly efficient linear solvers that are tailored for its special requirement. PMID:24723843

Fast Geostatistical Inversion using Randomized Matrix Decompositions and Sketchings for Heterogeneous Aquifer Characterization

NASA Astrophysics Data System (ADS)

O'Malley, D.; Le, E. B.; Vesselinov, V. V.

2015-12-01

We present a fast, scalable, and highly-implementable stochastic inverse method for characterization of aquifer heterogeneity. The method utilizes recent advances in randomized matrix algebra and exploits the structure of the Quasi-Linear Geostatistical Approach (QLGA), without requiring a structured grid like Fast-Fourier Transform (FFT) methods. The QLGA framework is a more stable version of Gauss-Newton iterates for a large number of unknown model parameters, but provides unbiased estimates. The methods are matrix-free and do not require derivatives or adjoints, and are thus ideal for complex models and black-box implementation. We also incorporate randomized least-square solvers and data-reduction methods, which speed up computation and simulate missing data points. The new inverse methodology is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). Julia is an advanced high-level scientific programing language that allows for efficient memory management and utilization of high-performance computational resources. Inversion results based on series of synthetic problems with steady-state and transient calibration data are presented.

Principal Component Geostatistical Approach for large-dimensional inverse problems

PubMed Central

Kitanidis, P K; Lee, J

2014-01-01

The quasi-linear geostatistical approach is for weakly nonlinear underdetermined inverse problems, such as Hydraulic Tomography and Electrical Resistivity Tomography. It provides best estimates as well as measures for uncertainty quantification. However, for its textbook implementation, the approach involves iterations, to reach an optimum, and requires the determination of the Jacobian matrix, i.e., the derivative of the observation function with respect to the unknown. Although there are elegant methods for the determination of the Jacobian, the cost is high when the number of unknowns, m, and the number of observations, n, is high. It is also wasteful to compute the Jacobian for points away from the optimum. Irrespective of the issue of computing derivatives, the computational cost of implementing the method is generally of the order of m2n, though there are methods to reduce the computational cost. In this work, we present an implementation that utilizes a matrix free in terms of the Jacobian matrix Gauss-Newton method and improves the scalability of the geostatistical inverse problem. For each iteration, it is required to perform K runs of the forward problem, where K is not just much smaller than m but can be smaller that n. The computational and storage cost of implementation of the inverse procedure scales roughly linearly with m instead of m2 as in the textbook approach. For problems of very large m, this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach. Results illustrate the validity of the approach and provide insight in the conditions under which this method perform best. PMID:25558113

Principal Component Geostatistical Approach for large-dimensional inverse problems.

PubMed

Kitanidis, P K; Lee, J

2014-07-01

The quasi-linear geostatistical approach is for weakly nonlinear underdetermined inverse problems, such as Hydraulic Tomography and Electrical Resistivity Tomography. It provides best estimates as well as measures for uncertainty quantification. However, for its textbook implementation, the approach involves iterations, to reach an optimum, and requires the determination of the Jacobian matrix, i.e., the derivative of the observation function with respect to the unknown. Although there are elegant methods for the determination of the Jacobian, the cost is high when the number of unknowns, m , and the number of observations, n , is high. It is also wasteful to compute the Jacobian for points away from the optimum. Irrespective of the issue of computing derivatives, the computational cost of implementing the method is generally of the order of m 2 n , though there are methods to reduce the computational cost. In this work, we present an implementation that utilizes a matrix free in terms of the Jacobian matrix Gauss-Newton method and improves the scalability of the geostatistical inverse problem. For each iteration, it is required to perform K runs of the forward problem, where K is not just much smaller than m but can be smaller that n . The computational and storage cost of implementation of the inverse procedure scales roughly linearly with m instead of m 2 as in the textbook approach. For problems of very large m , this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach. Results illustrate the validity of the approach and provide insight in the conditions under which this method perform best.

Quasi-static ensemble variational data assimilation: a theoretical and numerical study with the iterative ensemble Kalman smoother

NASA Astrophysics Data System (ADS)

Fillion, Anthony; Bocquet, Marc; Gratton, Serge

2018-04-01

The analysis in nonlinear variational data assimilation is the solution of a non-quadratic minimization. Thus, the analysis efficiency relies on its ability to locate a global minimum of the cost function. If this minimization uses a Gauss-Newton (GN) method, it is critical for the starting point to be in the attraction basin of a global minimum. Otherwise the method may converge to a local extremum, which degrades the analysis. With chaotic models, the number of local extrema often increases with the temporal extent of the data assimilation window, making the former condition harder to satisfy. This is unfortunate because the assimilation performance also increases with this temporal extent. However, a quasi-static (QS) minimization may overcome these local extrema. It accomplishes this by gradually injecting the observations in the cost function. This method was introduced by Pires et al. (1996) in a 4D-Var context. We generalize this approach to four-dimensional strong-constraint nonlinear ensemble variational (EnVar) methods, which are based on both a nonlinear variational analysis and the propagation of dynamical error statistics via an ensemble. This forces one to consider the cost function minimizations in the broader context of cycled data assimilation algorithms. We adapt this QS approach to the iterative ensemble Kalman smoother (IEnKS), an exemplar of nonlinear deterministic four-dimensional EnVar methods. Using low-order models, we quantify the positive impact of the QS approach on the IEnKS, especially for long data assimilation windows. We also examine the computational cost of QS implementations and suggest cheaper algorithms.

A Generic 1D Forward Modeling and Inversion Algorithm for TEM Sounding with an Arbitrary Horizontal Loop

NASA Astrophysics Data System (ADS)

Li, Zhanhui; Huang, Qinghua; Xie, Xingbing; Tang, Xingong; Chang, Liao

2016-08-01

We present a generic 1D forward modeling and inversion algorithm for transient electromagnetic (TEM) data with an arbitrary horizontal transmitting loop and receivers at any depth in a layered earth. Both the Hankel and sine transforms required in the forward algorithm are calculated using the filter method. The adjoint-equation method is used to derive the formulation of data sensitivity at any depth in non-permeable media. The inversion algorithm based on this forward modeling algorithm and sensitivity formulation is developed using the Gauss-Newton iteration method combined with the Tikhonov regularization. We propose a new data-weighting method to minimize the initial model dependence that enhances the convergence stability. On a laptop with a CPU of i7-5700HQ@3.5 GHz, the inversion iteration of a 200 layered input model with a single receiver takes only 0.34 s, while it increases to only 0.53 s for the data from four receivers at a same depth. For the case of four receivers at different depths, the inversion iteration runtime increases to 1.3 s. Modeling the data with an irregular loop and an equal-area square loop indicates that the effect of the loop geometry is significant at early times and vanishes gradually along the diffusion of TEM field. For a stratified earth, inversion of data from more than one receiver is useful in noise reducing to get a more credible layered earth. However, for a resistive layer shielded below a conductive layer, increasing the number of receivers on the ground does not have significant improvement in recovering the resistive layer. Even with a down-hole TEM sounding, the shielded resistive layer cannot be recovered if all receivers are above the shielded resistive layer. However, our modeling demonstrates remarkable improvement in detecting the resistive layer with receivers in or under this layer.

What can numerical computation do for the history of science? (a study of an orbit drawn by Newton in a letter to Hooke)

NASA Astrophysics Data System (ADS)

Cardozo Dias, Penha Maria; Stuchi, T. J.

2013-11-01

In a letter to Robert Hooke, Isaac Newton drew the orbit of a mass moving under a constant attracting central force. The drawing of the orbit may indicate how and when Newton developed dynamic categories. Some historians claim that Newton used a method contrived by Hooke; others that he used some method of curvature. We prove that Hooke’s method is a second-order symplectic area-preserving algorithm, and the method of curvature is a first-order algorithm without special features; then we integrate the Hamiltonian equations. Integration by the method of curvature can also be done, exploring the geometric properties of curves. We compare three methods: Hooke’s method, the method of curvature and a first-order method. A fourth-order algorithm sets a standard of comparison. We analyze which of these methods best explains Newton’s drawing.

Program for Calculating the Cubic and Fifth Roots of a Number of Newton's Method (610 IBM Electronic Computer); PROGRAMMA PER IL CALCOLO DELLE RADICI TERZA E QUINTA DI UN NUMERO, COL METODO DI NEWTON (CALCOLATORE ELETTRONICO IBM 610)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Giucci, D.

1963-01-01

A program was devised for calculating the cubic and fifth roots of a number of Newton's method using the 610 IBM electronic computer. For convenience a program was added for obtaining n/sup th/ roots by the logarithmic method. (auth)

Existence and amplitude bounds for irrotational water waves in finite depth

NASA Astrophysics Data System (ADS)

Kogelbauer, Florian

2017-12-01

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces. This article is part of the theme issue 'Nonlinear water waves'.

An efficient method for solving the steady Euler equations

NASA Technical Reports Server (NTRS)

Liou, M. S.

1986-01-01

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Combined genetic algorithm and multiple linear regression (GA-MLR) optimizer: Application to multi-exponential fluorescence decay surface.

PubMed

Fisz, Jacek J

2006-12-07

The optimization approach based on the genetic algorithm (GA) combined with multiple linear regression (MLR) method, is discussed. The GA-MLR optimizer is designed for the nonlinear least-squares problems in which the model functions are linear combinations of nonlinear functions. GA optimizes the nonlinear parameters, and the linear parameters are calculated from MLR. GA-MLR is an intuitive optimization approach and it exploits all advantages of the genetic algorithm technique. This optimization method results from an appropriate combination of two well-known optimization methods. The MLR method is embedded in the GA optimizer and linear and nonlinear model parameters are optimized in parallel. The MLR method is the only one strictly mathematical "tool" involved in GA-MLR. The GA-MLR approach simplifies and accelerates considerably the optimization process because the linear parameters are not the fitted ones. Its properties are exemplified by the analysis of the kinetic biexponential fluorescence decay surface corresponding to a two-excited-state interconversion process. A short discussion of the variable projection (VP) algorithm, designed for the same class of the optimization problems, is presented. VP is a very advanced mathematical formalism that involves the methods of nonlinear functionals, algebra of linear projectors, and the formalism of Fréchet derivatives and pseudo-inverses. Additional explanatory comments are added on the application of recently introduced the GA-NR optimizer to simultaneous recovery of linear and weakly nonlinear parameters occurring in the same optimization problem together with nonlinear parameters. The GA-NR optimizer combines the GA method with the NR method, in which the minimum-value condition for the quadratic approximation to chi(2), obtained from the Taylor series expansion of chi(2), is recovered by means of the Newton-Raphson algorithm. The application of the GA-NR optimizer to model functions which are multi-linear combinations of nonlinear functions, is indicated. The VP algorithm does not distinguish the weakly nonlinear parameters from the nonlinear ones and it does not apply to the model functions which are multi-linear combinations of nonlinear functions.

Rate of convergence of k-step Newton estimators to efficient likelihood estimators

Treesearch

Steve Verrill

2007-01-01

We make use of Cramer conditions together with the well-known local quadratic convergence of Newton?s method to establish the asymptotic closeness of k-step Newton estimators to efficient likelihood estimators. In Verrill and Johnson [2007. Confidence bounds and hypothesis tests for normal distribution coefficients of variation. USDA Forest Products Laboratory Research...

Cantera Integration with the Toolbox for Modeling and Analysis of Thermodynamic Systems (T-MATS)

NASA Technical Reports Server (NTRS)

Lavelle, Thomas M.; Chapman, Jeffryes W.; May, Ryan D.; Litt, Jonathan S.; Guo, Ten-Huei

2014-01-01

NASA Glenn Research Center (GRC) has recently developed a software package for modeling generic thermodynamic systems called the Toolbox for the Modeling and Analysis of Thermodynamic Systems (T-MATS). T-MATS is a library of building blocks that can be assembled to represent any thermodynamic system in the Simulink(Registered TradeMark) (The MathWorks, Inc.) environment. These elements, along with a Newton Raphson solver (also provided as part of the T-MATS package), enable users to create models of a wide variety of systems. The current version of T-MATS (v1.0.1) uses tabular data for providing information about a specific mixture of air, water (humidity), and hydrocarbon fuel in calculations of thermodynamic properties. The capabilities of T-MATS can be expanded by integrating it with the Cantera thermodynamic package. Cantera is an object-oriented analysis package that calculates thermodynamic solutions for any mixture defined by the user. Integration of Cantera with T-MATS extends the range of systems that may be modeled using the toolbox. In addition, the library of elements released with Cantera were developed using MATLAB native M-files, allowing for quicker prototyping of elements. This paper discusses how the new Cantera-based elements are created and provides examples for using T-MATS integrated with Cantera.

Cantera Integration with the Toolbox for Modeling and Analysis of Thermodynamic Systems (T-MATS)

NASA Technical Reports Server (NTRS)

Lavelle, Thomas M.; Chapman, Jeffryes W.; May, Ryan D.; Litt, Jonathan S.; Guo, Ten-Huei

2014-01-01

NASA Glenn Research Center (GRC) has recently developed a software package for modeling generic thermodynamic systems called the Toolbox for the Modeling and Analysis of Thermodynamic Systems (T-MATS). T-MATS is a library of building blocks that can be assembled to represent any thermodynamic system in the Simulink (The MathWorks, Inc.) environment. These elements, along with a Newton Raphson solver (also provided as part of the T-MATS package), enable users to create models of a wide variety of systems. The current version of T-MATS (v1.0.1) uses tabular data for providing information about a specific mixture of air, water (humidity), and hydrocarbon fuel in calculations of thermodynamic properties. The capabilities of T-MATS can be expanded by integrating it with the Cantera thermodynamic package. Cantera is an object-oriented analysis package that calculates thermodynamic solutions for any mixture defined by the user. Integration of Cantera with T-MATS extends the range of systems that may be modeled using the toolbox. In addition, the library of elements released with Cantera were developed using MATLAB native M-files, allowing for quicker prototyping of elements. This paper discusses how the new Cantera-based elements are created and provides examples for using T-MATS integrated with Cantera.

Voltage Analysis Improvement of 150 kV Transmission Subsystem Using Static Synchronous Compensator (STATCOM)

NASA Astrophysics Data System (ADS)

Akbar, P. A.; Hakim, D. L.; Sucita, T.

2018-02-01

In this research, testing improvements to the distribution voltage electricity at 150 kV transmission subsystem Bandung Selatan and New Ujungberung using Flexible AC Transmission System (FACTS) technology. One of them is by doing the control of active and reactive power through the power electronics equipment Static Synchronous Compensator (STATCOM). The subsystem is tested because it has a voltage profile are relatively less well when based on the IEEE / ANSI C.84.1 (142.5 - 157.5 kV). This study was conducted by analyzing the Newton-Raphson power flow on the simulator DigSilent Power Factory 15 to determine the profile of the voltage (V) on the system. Bus which has the lowest voltage to be a reference in the installation of STATCOM. From this research is known that the voltage on the conditions of the existing bus 28, as many as 21-23 still below standard buses (142.5 kV), after the installation is done using STATCOM, voltage on the buses improved by increasing the number of tracks that follow the standard / is in the range 142.5 kV -157.5 kV as many as 23-27 buses or 78.6% - 96%, with the optimum mounting on a bus Rancaekek STATCOM II with a capacity of 300 MVA.

LOGISTIC FUNCTION PROFILE FIT: A least-squares program for fitting interface profiles to an extended logistic function

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kirchhoff, William H.

2012-09-15

The extended logistic function provides a physically reasonable description of interfaces such as depth profiles or line scans of surface topological or compositional features. It describes these interfaces with the minimum number of parameters, namely, position, width, and asymmetry. Logistic Function Profile Fit (LFPF) is a robust, least-squares fitting program in which the nonlinear extended logistic function is linearized by a Taylor series expansion (equivalent to a Newton-Raphson approach) with no apparent introduction of bias in the analysis. The program provides reliable confidence limits for the parameters when systematic errors are minimal and provides a display of the residuals frommore » the fit for the detection of systematic errors. The program will aid researchers in applying ASTM E1636-10, 'Standard practice for analytically describing sputter-depth-profile and linescan-profile data by an extended logistic function,' and may also prove useful in applying ISO 18516: 2006, 'Surface chemical analysis-Auger electron spectroscopy and x-ray photoelectron spectroscopy-determination of lateral resolution.' Examples are given of LFPF fits to a secondary ion mass spectrometry depth profile, an Auger surface line scan, and synthetic data generated to exhibit known systematic errors for examining the significance of such errors to the extrapolation of partial profiles.« less

Flowing partially penetrating well: solution to a mixed-type boundary value problem

NASA Astrophysics Data System (ADS)

Cassiani, G.; Kabala, Z. J.; Medina, M. A.

A new semi-analytic solution to the mixed-type boundary value problem for a flowing partially penetrating well with infinitesimal skin situated in an anisotropic aquifer is developed. The solution is suited to aquifers having a semi-infinite vertical extent or to packer tests with aquifer horizontal boundaries far enough from the tested area. The problem reduces to a system of dual integral equations (DE) and further to a deconvolution problem. Unlike the analogous Dagan's steady-state solution [Water Resour. Res. 1978; 14:929-34], our DE solution does not suffer from numerical oscillations. The new solution is validated by matching the corresponding finite-difference solution and is computationally much more efficient. An automated (Newton-Raphson) parameter identification algorithm is proposed for field test inversion, utilizing the DE solution for the forward model. The procedure is computationally efficient and converges to correct parameter values. A solution for the partially penetrating flowing well with no skin and a drawdown-drawdown discontinuous boundary condition, analogous to that by Novakowski [Can. Geotech. J. 1993; 30:600-6], is compared to the DE solution. The D-D solution leads to physically inconsistent infinite total flow rate to the well, when no skin effect is considered. The DE solution, on the other hand, produces accurate results.

Jointly reconstructing ground motion and resistivity for ERT-based slope stability monitoring

NASA Astrophysics Data System (ADS)

Boyle, Alistair; Wilkinson, Paul B.; Chambers, Jonathan E.; Meldrum, Philip I.; Uhlemann, Sebastian; Adler, Andy

2018-02-01

Electrical resistivity tomography (ERT) is increasingly being used to investigate unstable slopes and monitor the hydrogeological processes within. But movement of electrodes or incorrect placement of electrodes with respect to an assumed model can introduce significant resistivity artefacts into the reconstruction. In this work, we demonstrate a joint resistivity and electrode movement reconstruction algorithm within an iterative Gauss-Newton framework. We apply this to ERT monitoring data from an active slow-moving landslide in the UK. Results show fewer resistivity artefacts and suggest that electrode movement and resistivity can be reconstructed at the same time under certain conditions. A new 2.5-D formulation for the electrode position Jacobian is developed and is shown to give accurate numerical solutions when compared to the adjoint method on 3-D models. On large finite element meshes, the calculation time of the newly developed approach was also proven to be orders of magnitude faster than the 3-D adjoint method and addressed modelling errors in the 2-D perturbation and adjoint electrode position Jacobian.

Biomolecular Interaction Analysis Using an Optical Surface Plasmon Resonance Biosensor: The Marquardt Algorithm vs Newton Iteration Algorithm

PubMed Central

Hu, Jiandong; Ma, Liuzheng; Wang, Shun; Yang, Jianming; Chang, Keke; Hu, Xinran; Sun, Xiaohui; Chen, Ruipeng; Jiang, Min; Zhu, Juanhua; Zhao, Yuanyuan

2015-01-01

Kinetic analysis of biomolecular interactions are powerfully used to quantify the binding kinetic constants for the determination of a complex formed or dissociated within a given time span. Surface plasmon resonance biosensors provide an essential approach in the analysis of the biomolecular interactions including the interaction process of antigen-antibody and receptors-ligand. The binding affinity of the antibody to the antigen (or the receptor to the ligand) reflects the biological activities of the control antibodies (or receptors) and the corresponding immune signal responses in the pathologic process. Moreover, both the association rate and dissociation rate of the receptor to ligand are the substantial parameters for the study of signal transmission between cells. A number of experimental data may lead to complicated real-time curves that do not fit well to the kinetic model. This paper presented an analysis approach of biomolecular interactions established by utilizing the Marquardt algorithm. This algorithm was intensively considered to implement in the homemade bioanalyzer to perform the nonlinear curve-fitting of the association and disassociation process of the receptor to ligand. Compared with the results from the Newton iteration algorithm, it shows that the Marquardt algorithm does not only reduce the dependence of the initial value to avoid the divergence but also can greatly reduce the iterative regression times. The association and dissociation rate constants, ka, kd and the affinity parameters for the biomolecular interaction, KA, KD, were experimentally obtained 6.969×105 mL·g-1·s-1, 0.00073 s-1, 9.5466×108 mL·g-1 and 1.0475×10-9 g·mL-1, respectively from the injection of the HBsAg solution with the concentration of 16ng·mL-1. The kinetic constants were evaluated distinctly by using the obtained data from the curve-fitting results. PMID:26147997

Frequency-domain full-waveform inversion with non-linear descent directions

NASA Astrophysics Data System (ADS)

Geng, Yu; Pan, Wenyong; Innanen, Kristopher A.

2018-05-01

Full-waveform inversion (FWI) is a highly non-linear inverse problem, normally solved iteratively, with each iteration involving an update constructed through linear operations on the residuals. Incorporating a flexible degree of non-linearity within each update may have important consequences for convergence rates, determination of low model wavenumbers and discrimination of parameters. We examine one approach for doing so, wherein higher order scattering terms are included within the sensitivity kernel during the construction of the descent direction, adjusting it away from that of the standard Gauss-Newton approach. These scattering terms are naturally admitted when we construct the sensitivity kernel by varying not the current but the to-be-updated model at each iteration. Linear and/or non-linear inverse scattering methodologies allow these additional sensitivity contributions to be computed from the current data residuals within any given update. We show that in the presence of pre-critical reflection data, the error in a second-order non-linear update to a background of s0 is, in our scheme, proportional to at most (Δs/s0)3 in the actual parameter jump Δs causing the reflection. In contrast, the error in a standard Gauss-Newton FWI update is proportional to (Δs/s0)2. For numerical implementation of more complex cases, we introduce a non-linear frequency-domain scheme, with an inner and an outer loop. A perturbation is determined from the data residuals within the inner loop, and a descent direction based on the resulting non-linear sensitivity kernel is computed in the outer loop. We examine the response of this non-linear FWI using acoustic single-parameter synthetics derived from the Marmousi model. The inverted results vary depending on data frequency ranges and initial models, but we conclude that the non-linear FWI has the capability to generate high-resolution model estimates in both shallow and deep regions, and to converge rapidly, relative to a benchmark FWI approach involving the standard gradient.

River velocities from sequential multispectral remote sensing images

NASA Astrophysics Data System (ADS)

Chen, Wei; Mied, Richard P.

2013-06-01

We address the problem of extracting surface velocities from a pair of multispectral remote sensing images over rivers using a new nonlinear multiple-tracer form of the global optimal solution (GOS). The derived velocity field is a valid solution across the image domain to the nonlinear system of equations obtained by minimizing a cost function inferred from the conservation constraint equations for multiple tracers. This is done by deriving an iteration equation for the velocity, based on the multiple-tracer displaced frame difference equations, and a local approximation to the velocity field. The number of velocity equations is greater than the number of velocity components, and thus overly constrain the solution. The iterative technique uses Gauss-Newton and Levenberg-Marquardt methods and our own algorithm of the progressive relaxation of the over-constraint. We demonstrate the nonlinear multiple-tracer GOS technique with sequential multispectral Landsat and ASTER images over a portion of the Potomac River in MD/VA, and derive a dense field of accurate velocity vectors. We compare the GOS river velocities with those from over 12 years of data at four NOAA reference stations, and find good agreement. We discuss how to find the appropriate spatial and temporal resolutions to allow optimization of the technique for specific rivers.

The trust-region self-consistent field method in Kohn-Sham density-functional theory.

PubMed

Thøgersen, Lea; Olsen, Jeppe; Köhn, Andreas; Jørgensen, Poul; Sałek, Paweł; Helgaker, Trygve

2005-08-15

The trust-region self-consistent field (TRSCF) method is extended to the optimization of the Kohn-Sham energy. In the TRSCF method, both the Roothaan-Hall step and the density-subspace minimization step are replaced by trust-region optimizations of local approximations to the Kohn-Sham energy, leading to a controlled, monotonic convergence towards the optimized energy. Previously the TRSCF method has been developed for optimization of the Hartree-Fock energy, which is a simple quadratic function in the density matrix. However, since the Kohn-Sham energy is a nonquadratic function of the density matrix, the local energy functions must be generalized for use with the Kohn-Sham model. Such a generalization, which contains the Hartree-Fock model as a special case, is presented here. For comparison, a rederivation of the popular direct inversion in the iterative subspace (DIIS) algorithm is performed, demonstrating that the DIIS method may be viewed as a quasi-Newton method, explaining its fast local convergence. In the global region the convergence behavior of DIIS is less predictable. The related energy DIIS technique is also discussed and shown to be inappropriate for the optimization of the Kohn-Sham energy.

"To Improve upon Hints of Things": Illustrating Isaac Newton.

PubMed

Schilt, Cornelis J

2016-01-01

When Isaac Newton died in 1727 he left a rich legacy in terms of draft manuscripts, encompassing a variety of topics: natural philosophy, mathematics, alchemy, theology, and chronology, as well as papers relating to his career at the Mint. One thing that immediately strikes us is the textuality of Newton's legacy: images are sparse. Regarding his scholarly endeavours we witness the same practice. Newton's extensive drafts on theology and chronology do not contain a single illustration or map. Today we have all of Newton's draft manuscripts as witnesses of his working methods, as well as access to a significant number of books from his own library. Drawing parallels between Newton's reading practices and his natural philosophical and scholarly work, this paper seeks to understand Newton's recondite writing and publishing politics.

The Linearized Bregman Method for Frugal Full-waveform Inversion with Compressive Sensing and Sparsity-promoting

NASA Astrophysics Data System (ADS)

Chai, Xintao; Tang, Genyang; Peng, Ronghua; Liu, Shaoyong

2018-03-01

Full-waveform inversion (FWI) reconstructs the subsurface properties from acquired seismic data via minimization of the misfit between observed and simulated data. However, FWI suffers from considerable computational costs resulting from the numerical solution of the wave equation for each source at each iteration. To reduce the computational burden, constructing supershots by combining several sources (aka source encoding) allows mitigation of the number of simulations at each iteration, but it gives rise to crosstalk artifacts because of interference between the individual sources of the supershot. A modified Gauss-Newton FWI (MGNFWI) approach showed that as long as the difference between the initial and true models permits a sparse representation, the ℓ _1-norm constrained model updates suppress subsampling-related artifacts. However, the spectral-projected gradient ℓ _1 (SPGℓ _1) algorithm employed by MGNFWI is rather complicated that makes its implementation difficult. To facilitate realistic applications, we adapt a linearized Bregman (LB) method to sparsity-promoting FWI (SPFWI) because of the efficiency and simplicity of LB in the framework of ℓ _1-norm constrained optimization problem and compressive sensing. Numerical experiments performed with the BP Salt model, the Marmousi model and the BG Compass model verify the following points. The FWI result with LB solving ℓ _1-norm sparsity-promoting problem for the model update outperforms that generated by solving ℓ _2-norm problem in terms of crosstalk elimination and high-fidelity results. The simpler LB method performs comparably and even superiorly to the complicated SPGℓ _1 method in terms of computational efficiency and model quality, making the LB method a viable alternative for realistic implementations of SPFWI.

Evaluation of kinetic constants of biomolecular interaction on optical surface plasmon resonance sensor with Newton Iteration Method

NASA Astrophysics Data System (ADS)

Zhao, Yuanyuan; Jiang, Guoliang; Hu, Jiandong; Hu, Fengjiang; Wei, Jianguang; Shi, Liang

2010-10-01

In the immunology, there are two important types of biomolecular interaction: antigens-antibodies and receptors-ligands. Monitoring the response rate and affinity of biomolecular interaction can help analyze the protein function, drug discover, genomics and proteomics research. Moreover the association rate constant and dissociation rate constant of receptors-ligands are the important parameters for the study of signal transmission between cells. Recent advances in bioanalyzer instruments have greatly simplified the measurement of the kinetics of molecular interactions. Non-destructive and real-time monitoring the response to evaluate the parameters between antigens and antibodies can be performed by using optical surface plasmon resonance (SPR) biosensor technology. This technology provides a quantitative analysis that is carried out rapidly with label-free high-throughput detection using the binding curves of antigens-antibodies. Consequently, the kinetic parameters of interaction between antigens and antibodies can be obtained. This article presents a low cost integrated SPR-based bioanalyzer (HPSPR-6000) designed by ourselves. This bioanalyzer is mainly composed of a biosensor TSPR1K23, a touch-screen monitor, a microprocessor PIC24F128, a microflow cell with three channels, a clamp and a photoelectric conversion device. To obtain the kinetic parameters, sensorgrams may be modeled using one of several binding models provided with BIAevaluation software 3.0, SensiQ or Autolab. This allows calculation of the association rate constant (ka) and the dissociation rate constant (kd). The ratio of ka to kd can be used to estimate the equilibrium constant. Another kind is the analysis software OriginPro, which can process the obtained data by nonlinear fitting and then get some correlative parameters, but it can't be embedded into the bioanalyzer, so the bioanalyzer don't support the use of OriginPro. This paper proposes a novel method to evaluate the kinetic parameters of biomolecular interaction by using Newton Iteration Method and Least Squares Method. First, the pseudo first order kinetic model of biomolecular interaction was established. Then the data of molecular interaction of HBsAg and HBsAb was obtained by bioanalyzer. Finally, we used the optical SPR bioanalyzer software which was written by ourselves to make nonlinear fit about the association and dissociation curves. The correlation coefficient R-squared is 0.99229 and 0.99593, respectively. Furthermore, the kinetic parameters and affinity constants were evaluated using the obtained data from the fitting results.