Numerical methods for engine-airframe integration
Murthy, S.N.B.; Paynter, G.C.
1986-01-01
Various papers on numerical methods for engine-airframe integration are presented. The individual topics considered include: scientific computing environment for the 1980s, overview of prediction of complex turbulent flows, numerical solutions of the compressible Navier-Stokes equations, elements of computational engine/airframe integrations, computational requirements for efficient engine installation, application of CAE and CFD techniques to complete tactical missile design, CFD applications to engine/airframe integration, and application of a second-generation low-order panel methods to powerplant installation studies. Also addressed are: three-dimensional flow analysis of turboprop inlet and nacelle configurations, application of computational methods to the design of large turbofan engine nacelles, comparison of full potential and Euler solution algorithms for aeropropulsive flow field computations, subsonic/transonic, supersonic nozzle flows and nozzle integration, subsonic/transonic prediction capabilities for nozzle/afterbody configurations, three-dimensional viscous design methodology of supersonic inlet systems for advanced technology aircraft, and a user's technology assessment.
Automatic numerical integration methods for Feynman integrals through 3-loop
NASA Astrophysics Data System (ADS)
de Doncker, E.; Yuasa, F.; Kato, K.; Ishikawa, T.; Olagbemi, O.
2015-05-01
We give numerical integration results for Feynman loop diagrams through 3-loop such as those covered by Laporta [1]. The methods are based on automatic adaptive integration, using iterated integration and extrapolation with programs from the QUADPACK package, or multivariate techniques from the ParInt package. The Dqags algorithm from QuadPack accommodates boundary singularities of fairly general types. PARINT is a package for multivariate integration layered over MPI (Message Passing Interface), which runs on clusters and incorporates advanced parallel/distributed techniques such as load balancing among processes that may be distributed over a network of nodes. Results are included for 3-loop self-energy diagrams without IR (infra-red) or UV (ultra-violet) singularities. A procedure based on iterated integration and extrapolation yields a novel method of numerical regularization for integrals with UV terms, and is applied to a set of 2-loop self-energy diagrams with UV singularities.
ERIC Educational Resources Information Center
Sozio, Gerry
2009-01-01
Senior secondary students cover numerical integration techniques in their mathematics courses. In particular, students would be familiar with the "midpoint rule," the elementary "trapezoidal rule" and "Simpson's rule." This article derives these techniques by methods which secondary students may not be familiar with and an approach that…
Singularity Preserving Numerical Methods for Boundary Integral Equations
NASA Technical Reports Server (NTRS)
Kaneko, Hideaki (Principal Investigator)
1996-01-01
In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.
Comparison of four stable numerical methods for Abel's integral equation
NASA Technical Reports Server (NTRS)
Murio, Diego A.; Mejia, Carlos E.
1991-01-01
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.
Trigonometrically fitted two step hybrid method for the numerical integration of second order IVPs
NASA Astrophysics Data System (ADS)
Monovasilis, Th.; Kalogiratou, Z.; Simos, T. E.
2016-06-01
In this work we consider the numerical integration of second order ODEs where the first derivative is missing. We construct trigonometrically fitted two step hybrid methods. We apply the new methods on the numerical integration of several test problems.
Path Integrals and Exotic Options:. Methods and Numerical Results
NASA Astrophysics Data System (ADS)
Bormetti, G.; Montagna, G.; Moreni, N.; Nicrosini, O.
2005-09-01
In the framework of Black-Scholes-Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price path dependent options on multidimensional and correlated underlying assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the case of Asian call options. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at the money (ATM) and out of the money (OTM) options, path integral exhibits competitive performances.
An Improved Numerical Integration Method for Springback Predictions
NASA Astrophysics Data System (ADS)
Ibrahim, R.; Smith, L. M.; Golovashchenko, Sergey F.
2011-08-01
In this investigation, the focus is on the springback of steel sheets in V-die air bending. A full replication to a numerical integration algorithm presented rigorously in [1] to predict the springback in air bending was performed and confirmed successfully. Algorithm alteration and extensions were proposed here. The altered approach used in solving the moment equation numerically resulted in springback values much closer to the trend presented by the experimental data, Although investigation here extended to use a more realistic work-hardening model, the differences in the springback values obtained by both hardening models were almost negligible. The algorithm was extended to be applied on thin sheets down to 0.8 mm. Results show that this extension is possible as verified by FEA and other published experiments on TRIP steel sheets.
Integrated numerical methods for hypersonic aircraft cooling systems analysis
NASA Technical Reports Server (NTRS)
Petley, Dennis H.; Jones, Stuart C.; Dziedzic, William M.
1992-01-01
Numerical methods have been developed for the analysis of hypersonic aircraft cooling systems. A general purpose finite difference thermal analysis code is used to determine areas which must be cooled. Complex cooling networks of series and parallel flow can be analyzed using a finite difference computer program. Both internal fluid flow and heat transfer are analyzed, because increased heat flow causes a decrease in the flow of the coolant. The steady state solution is a successive point iterative method. The transient analysis uses implicit forward-backward differencing. Several examples of the use of the program in studies of hypersonic aircraft and rockets are provided.
Numerical integration of population models satisfying conservation laws: NSFD methods.
Mickens, Ronald E
2007-10-01
Population models arising in ecology, epidemiology and mathematical biology may involve a conservation law, i.e. the total population is constant. In addition to these cases, other situations may occur for which the total population, asymptotically in time, approach a constant value. Since it is rarely the situation that the equations of motion can be analytically solved to obtain exact solutions, it follows that numerical techniques are needed to provide solutions. However, numerical procedures are only valid if they can reproduce fundamental properties of the differential equations modeling the phenomena of interest. We show that for population models, involving a dynamical conservation law the use of nonstandard finite difference (NSFD) methods allows the construction of discretization schemes such that they are dynamically consistent (DC) with the original differential equations. The paper will briefly discuss the NSFD methodology, the concept of DC, and illustrate their application to specific problems for population models. PMID:22876826
NASA Astrophysics Data System (ADS)
Min, Xiaoyi
This thesis first presents the study of the interaction of electromagnetic waves with three-dimensional heterogeneous, dielectric, magnetic, and lossy bodies by surface integral equation modeling. Based on the equivalence principle, a set of coupled surface integral equations is formulated and then solved numerically by the method of moments. Triangular elements are used to model the interfaces of the heterogeneous body, and vector basis functions are defined to expand the unknown current in the formulation. The validity of this formulation is verified by applying it to concentric spheres for which an exact solution exists. The potential applications of this formulation to a partially coated sphere and a homogeneous human body are discussed. Next, this thesis also introduces an efficient new set of integral equations for treating the scattering problem of a perfectly conducting body coated with a thin magnetically lossy layer. These electric field integral equations and magnetic field integral equations are numerically solved by the method of moments (MoM). To validate the derived integral equations, an alternative method to solve the scattering problem of an infinite circular cylinder coated with a thin magnetic lossy layer has also been developed, based on the eigenmode expansion. Results for the radar cross section and current densities via the MoM and the eigenmode expansion method are compared. The agreement is excellent. The finite difference time domain method is subsequently implemented to solve a metallic object coated with a magnetic thin layer and numerical results are compared with that by the MoM. Finally, this thesis presents an application of the finite-difference time-domain approach to the problem of electromagnetic receiving and scattering by a cavity -backed antenna situated on an infinite conducting plane. This application involves modifications of Yee's model, which applies the difference approximations of field derivatives to differential
An efficient step-size control method in numerical integration for astrodynamical equations
NASA Astrophysics Data System (ADS)
Liu, C. Z.; Cui, D. X.
2002-11-01
Using the curvature of the integral curve, a step-size control method is introduced in this paper. This method will prove to be the efficient scheme in the sense that it saves computation time and improve accuracy of numerical integration.
NASA Astrophysics Data System (ADS)
Plakhov, Iu. V.; Mytsenko, A. V.; Shel'Pov, V. A.
A numerical integration method is developed that is more accurate than Everhart's (1974) implicit single-sequence approach for integrating orbits. This method can be used to solve problems of space geodesy based on the use of highly precise laser observations.
Some numerical methods for integrating systems of first-order ordinary differential equations
NASA Technical Reports Server (NTRS)
Clark, N. W.
1969-01-01
Report on numerical methods of integration includes the extrapolation methods of Bulirsch-Stoer and Neville. A comparison is made nith the Runge-Kutta and Adams-Moulton methods, and circumstances are discussed under which the extrapolation method may be preferred.
NASA Technical Reports Server (NTRS)
Sidi, A.; Israeli, M.
1986-01-01
High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.
NASA Technical Reports Server (NTRS)
Rosenbaum, J. S.
1976-01-01
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
NASA Astrophysics Data System (ADS)
Tang, Xiaojun
2016-04-01
The main purpose of this work is to provide multiple-interval integral Gegenbauer pseudospectral methods for solving optimal control problems. The latest developed single-interval integral Gauss/(flipped Radau) pseudospectral methods can be viewed as special cases of the proposed methods. We present an exact and efficient approach to compute the mesh pseudospectral integration matrices for the Gegenbauer-Gauss and flipped Gegenbauer-Gauss-Radau points. Numerical results on benchmark optimal control problems confirm the ability of the proposed methods to obtain highly accurate solutions.
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.
NASA Astrophysics Data System (ADS)
Liu, Peigui; Elshall, Ahmed S.; Ye, Ming; Beerli, Peter; Zeng, Xiankui; Lu, Dan; Tao, Yuezan
2016-02-01
Evaluating marginal likelihood is the most critical and computationally expensive task, when conducting Bayesian model averaging to quantify parametric and model uncertainties. The evaluation is commonly done by using Laplace approximations to evaluate semianalytical expressions of the marginal likelihood or by using Monte Carlo (MC) methods to evaluate arithmetic or harmonic mean of a joint likelihood function. This study introduces a new MC method, i.e., thermodynamic integration, which has not been attempted in environmental modeling. Instead of using samples only from prior parameter space (as in arithmetic mean evaluation) or posterior parameter space (as in harmonic mean evaluation), the thermodynamic integration method uses samples generated gradually from the prior to posterior parameter space. This is done through a path sampling that conducts Markov chain Monte Carlo simulation with different power coefficient values applied to the joint likelihood function. The thermodynamic integration method is evaluated using three analytical functions by comparing the method with two variants of the Laplace approximation method and three MC methods, including the nested sampling method that is recently introduced into environmental modeling. The thermodynamic integration method outperforms the other methods in terms of their accuracy, convergence, and consistency. The thermodynamic integration method is also applied to a synthetic case of groundwater modeling with four alternative models. The application shows that model probabilities obtained using the thermodynamic integration method improves predictive performance of Bayesian model averaging. The thermodynamic integration method is mathematically rigorous, and its MC implementation is computationally general for a wide range of environmental problems.
Peskin, Michael E
2003-02-13
In upper-division undergraduate physics courses, it is desirable to give numerical problem-solving exercises integrated naturally into weekly problem sets. I explain a method for doing this that makes use of the built-in class structure of the Java programming language. I also supply a Java class library that can assist instructors in writing programs of this type.
Extended RKN-type methods for numerical integration of perturbed oscillators
NASA Astrophysics Data System (ADS)
Yang, Hongli; Wu, Xinyuan; You, Xiong; Fang, Yonglei
2009-10-01
In this paper, extended Runge-Kutta-Nyström-type methods for the numerical integration of perturbed oscillators with low frequencies are presented, which inherit the framework of RKN methods and make full use of the special feature of the true flows for both the internal stages and the updates. Following the approach of J. Butcher, E. Hairer and G. Wanner, we develop a new kind of tree set to derive order conditions for the extended Runge-Kutta-Nyström-type methods. The numerical stability and phase properties of the new methods are analyzed. Numerical experiments are accompanied to show the applicability and efficiency of our new methods in comparison with some well-known high quality methods proposed in the scientific literature.
NASA Technical Reports Server (NTRS)
Gottlieb, D.; Turkel, E.
1980-01-01
New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
A comparison of the efficiency of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations is presented. The methods examined include two general-purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient than evaluating the temperature by integrating its time-derivative.
NASA Astrophysics Data System (ADS)
Ding, Ye; Zhu, Limin; Zhang, Xiaojian; Ding, Han
2012-09-01
As one of the bases of gradient-based optimization algorithms, sensitivity analysis is usually required to calculate the derivatives of the system response with respect to the machining parameters. The most widely used approaches for sensitivity analysis are based on time-consuming numerical methods, such as finite difference methods. This paper presents a semi-analytical method for calculation of the sensitivity of the stability boundary in milling. After transforming the delay-differential equation with time-periodic coefficients governing the dynamic milling process into the integral form, the Floquet transition matrix is constructed by using the numerical integration method. Then, the analytical expressions of derivatives of the Floquet transition matrix with respect to the machining parameters are obtained. Thereafter, the classical analytical expression of the sensitivity of matrix eigenvalues is employed to calculate the sensitivity of the stability lobe diagram. The two-degree-of-freedom milling example illustrates the accuracy and efficiency of the proposed method. Compared with the existing methods, the unique merit of the proposed method is that it can be used for analytically computing the sensitivity of the stability boundary in milling, without employing any finite difference methods. Therefore, the high accuracy and high efficiency are both achieved. The proposed method can serve as an effective tool for machining parameter optimization and uncertainty analysis in high-speed milling.
Two step hybrid methods of 7th and 8th order for the numerical integration of second order IVPs
NASA Astrophysics Data System (ADS)
Kalogiratou, Z.; Monovasilis, Th.; Simos, T. E.
2016-06-01
In this work we consider the numerical integration of second order ODEs where the first derivative is missing. We construct two step hybrid methods with six and seven stages and seventh and eighth algebraic order. We apply the new methods on the numerical integration of several test problems.
Rythmos Numerical Integration Package
Coffey, Todd S.; Bartlett, Roscoe A.
2006-09-01
Rythmos numerically integrates transient differential equations. The differential equations can be explicit or implicit ordinary differential equations ofr formulated as fully implicit differential-algebraic equations. Methods include backward Euler, forward Euler, explicit Runge-Kutta, and implicit BDF at this time. Native support for operator split methods and strict modularity are strong design goals. Forward sensitivity computations will be included in the first release with adjoint sensitivities coming in the near future. Rythmos heavily relies on Thyra for linear algebra and nonlinear solver interfaces to AztecOO, Amesos, IFPack, and NOX in Tilinos. Rythmos is specially suited for stiff differential equations and thos applictions where operator split methods have a big advantage, e.g. Computational fluid dynamics, convection-diffusion equations, etc.
Rythmos Numerical Integration Package
Energy Science and Technology Software Center (ESTSC)
2006-09-01
Rythmos numerically integrates transient differential equations. The differential equations can be explicit or implicit ordinary differential equations ofr formulated as fully implicit differential-algebraic equations. Methods include backward Euler, forward Euler, explicit Runge-Kutta, and implicit BDF at this time. Native support for operator split methods and strict modularity are strong design goals. Forward sensitivity computations will be included in the first release with adjoint sensitivities coming in the near future. Rythmos heavily relies on Thyra formore » linear algebra and nonlinear solver interfaces to AztecOO, Amesos, IFPack, and NOX in Tilinos. Rythmos is specially suited for stiff differential equations and thos applictions where operator split methods have a big advantage, e.g. Computational fluid dynamics, convection-diffusion equations, etc.« less
Application of Numerical Integration and Data Fusion in Unit Vector Method
NASA Astrophysics Data System (ADS)
Zhang, J.
2012-01-01
The Unit Vector Method (UVM) is a series of orbit determination methods which are designed by Purple Mountain Observatory (PMO) and have been applied extensively. It gets the conditional equations for different kinds of data by projecting the basic equation to different unit vectors, and it suits for weighted process for different kinds of data. The high-precision data can play a major role in orbit determination, and accuracy of orbit determination is improved obviously. The improved UVM (PUVM2) promoted the UVM from initial orbit determination to orbit improvement, and unified the initial orbit determination and orbit improvement dynamically. The precision and efficiency are improved further. In this thesis, further research work has been done based on the UVM: Firstly, for the improvement of methods and techniques for observation, the types and decision of the observational data are improved substantially, it is also asked to improve the decision of orbit determination. The analytical perturbation can not meet the requirement. So, the numerical integration for calculating the perturbation has been introduced into the UVM. The accuracy of dynamical model suits for the accuracy of the real data, and the condition equations of UVM are modified accordingly. The accuracy of orbit determination is improved further. Secondly, data fusion method has been introduced into the UVM. The convergence mechanism and the defect of weighted strategy have been made clear in original UVM. The problem has been solved in this method, the calculation of approximate state transition matrix is simplified and the weighted strategy has been improved for the data with different dimension and different precision. Results of orbit determination of simulation and real data show that the work of this thesis is effective: (1) After the numerical integration has been introduced into the UVM, the accuracy of orbit determination is improved obviously, and it suits for the high-accuracy data of
NASA Astrophysics Data System (ADS)
Akamatsu, T.; Matsushita, M.; Murata, S.
1985-11-01
A two-parameter integral method is presented which is applicable even to separated boundary layers. The governing equation system, which consists of three moment equations of the boundary layer equation, is shown to be classifiable as a quasi-linear hyperbolic system under the assumed velocity profile function. The governing system is numerically solved by a dissipative finite difference scheme in order to capture a discontinuous solution associated with the singularity of unsteady separation. The spontaneous generation of singularity associated with unsteady separation is confirmed as the focusing of characteristics. The starting flows of a circular and an elliptic cylinder are considered as definite examples. This method is found to give excellent results in comparison with exact methods, not only for practically important boundary layer quantities such as displacement thickness or skin friction coefficient, but also for generation of separation singularity.
McCammon, R.B.; Finch, W.I.; Kork, J.O.; Bridges, N.J.
1994-01-01
An integrated data-directed numerical method has been developed to estimate the undiscovered mineral endowment within a given area. The method has been used to estimate the undiscovered uranium endowment in the San Juan Basin, New Mexico, U.S.A. The favorability of uranium concentration was evaluated in each of 2,068 cells defined within the Basin. Favorability was based on the correlated similarity of the geologic characteristics of each cell to the geologic characteristics of five area-related deposit models. Estimates of the undiscovered endowment for each cell were categorized according to deposit type, depth, and cutoff grade. The method can be applied to any mineral or energy commodity provided that the data collected reflect discovered endowment. ?? 1994 Oxford University Press.
NASA Technical Reports Server (NTRS)
Banyukevich, A.; Ziolkovski, K.
1975-01-01
A number of hybrid methods for solving Cauchy problems are described on the basis of an evaluation of advantages of single and multiple-point numerical integration methods. The selection criterion is the principle of minimizing computer time. The methods discussed include the Nordsieck method, the Bulirsch-Stoer extrapolation method, and the method of recursive Taylor-Steffensen power series.
Ianakiev, A.; Esat, I.I.
1995-09-01
Numerical solution of dynamical systems with widely varying motion characteristics, such as relatively slow motion coupled with high frequency as it would be in flexible mechanisms, likely to pose problems. In this paper the mathematical model of a flexible mechanism is solved by using a mixed integration method that attempts to deal with the complexity of the coupled differential equations of the rigid-body and elastic motion. The mixed integration method consists of two integration methods (Rossenbrock-Wanner and Hilber-Hughes-Taylor methods) that have been combined in order to minimize the computational complexity required for the approximation of the real system. The a Hilber-Hughes-Taylor methods incorporates numerical damping that selectively affects only the higher modes of vibration. The improvement of the stability and the accuracy of the solution due to the numerical damping has been demonstrated via a numerical example that represents a stiff system. The example system was selected to contain a physically important low frequency and spurious highly frequency oscillations. The solution method filtered the high numerical oscillations from the response results. The Rossenbrock-Wanner integration technique was also presented. In this case it is also shown that fine adjustment of integration parameters could effect the degree of numerical damping. A mixed integration method, combination of the two found to give the best performance and accuracy in the case of stiff problems.
NASA Astrophysics Data System (ADS)
Imada, Masatoshi; Kashima, Tsuyoshi
2000-09-01
A numerical algorithm for studying strongly correlated electron systems is proposed. The groundstate wavefunction is projected out after a numerical renormalization procedure in the path integral formalism. The wavefunction is expressed from the optimized linear combination of retained states in the truncated Hilbert space with a numerically chosen basis. This algorithm does not suffer from the negative sign problem and can be applied to any type of Hamiltonian in any dimension. The efficiency is tested in examples of the Hubbard model where the basis of Slater determinants is numerically optimized. We show results on fast convergence and accuracy achieved with a small number of retained states.
NASA Astrophysics Data System (ADS)
Calvisi, Michael; Manmi, Kawa; Wang, Qianxi
2014-11-01
Ultrasound contrast agents (UCAs) are microbubbles stabilized with a shell typically of lipid, polymer, or protein and are emerging as a unique tool for noninvasive therapies ranging from gene delivery to tumor ablation. The nonspherical dynamics of contrast agents are thought to play an important role in both diagnostic and therapeutic applications, for example, causing the emission of subharmonic frequency components and enhancing the uptake of therapeutic agents across cell membranes and tissue interfaces. A three-dimensional model for nonspherical contrast agent dynamics based on the boundary integral method is presented. The effects of the encapsulating shell are approximated by adapting Hoff's model for thin-shell, spherical contrast agents to the nonspherical case. A high-quality mesh of the bubble surface is maintained by implementing a hybrid approach of the Lagrangian method and elastic mesh technique. Numerical analyses for the dynamics of UCAs in an infinite liquid and near a rigid wall are performed in parameter regimes of clinical relevance. The results show that the presence of a coating significantly reduces the oscillation amplitude and period, increases the ultrasound pressure amplitude required to incite jetting, and reduces the jet width and velocity.
Cuba: Multidimensional numerical integration library
NASA Astrophysics Data System (ADS)
Hahn, Thomas
2016-08-01
The Cuba library offers four independent routines for multidimensional numerical integration: Vegas, Suave, Divonne, and Cuhre. The four algorithms work by very different methods, and can integrate vector integrands and have very similar Fortran, C/C++, and Mathematica interfaces. Their invocation is very similar, making it easy to cross-check by substituting one method by another. For further safeguarding, the output is supplemented by a chi-square probability which quantifies the reliability of the error estimate.
Comparison of symbolic and numerical integration methods for an assumed-stress hybrid shell element
NASA Technical Reports Server (NTRS)
Rengarajan, Govind; Knight, Norman F., Jr.; Aminpour, Mohammad A.
1993-01-01
Hybrid shell elements have long been regarded with reserve by the commercial finite element developers despite the high degree of reliability and accuracy associated with such formulations. The fundamental reason is the inherent higher computational cost of the hybrid approach as compared to the displacement-based formulations. However, a noteworthy factor in favor of hybrid elements is that numerical integration to generate element matrices can be entirely avoided by the use of symbolic integration. In this paper, the use of the symbolic computational approach is presented for an assumed-stress hybrid shell element with drilling degrees of freedom and the significant time savings achieved is demonstrated through an example.
NASA Technical Reports Server (NTRS)
Chan, William M.
1992-01-01
The following papers are presented: (1) numerical methods for the simulation of complex multi-body flows with applications for the Integrated Space Shuttle vehicle; (2) a generalized scheme for 3-D hyperbolic grid generation; (3) collar grids for intersecting geometric components within the Chimera overlapped grid scheme; and (4) application of the Chimera overlapped grid scheme to simulation of Space Shuttle ascent flows.
Time transformations and Cowell's method. [for numerical integration of satellite motion equations
NASA Technical Reports Server (NTRS)
Velez, C. E.; Hilinski, S.
1978-01-01
The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variable s defined by an equation of the form dt = cr to the n-th power ds, where t represents time, r the radial distance from the center of attraction, c is a constant, and n is a parameter. This has been primarily motivated by the 'uniformizing' effects of such a transformation resulting in desirable 'analytic' stepsize control for elliptical orbits. This report discusses the 'proper' choice of the parameter n defining the independent variable s for various types of orbits and perturbation models, and develops a criterion for its selection.
ICM: an Integrated Compartment Method for numerically solving partial differential equations
Yeh, G.T.
1981-05-01
An integrated compartment method (ICM) is proposed to construct a set of algebraic equations from a system of partial differential equations. The ICM combines the utility of integral formulation of finite element approach, the simplicity of interpolation of finite difference approximation, and the flexibility of compartment analyses. The integral formulation eases the treatment of boundary conditions, in particular, the Neumann-type boundary conditions. The simplicity of interpolation provides great economy in computation. The flexibility of discretization with irregular compartments of various shapes and sizes offers advantages in resolving complex boundaries enclosing compound regions of interest. The basic procedures of ICM are first to discretize the region of interest into compartments, then to apply three integral theorems of vectors to transform the volume integral to the surface integral, and finally to use interpolation to relate the interfacial values in terms of compartment values to close the system. The Navier-Stokes equations are used as an example of how to derive the corresponding ICM alogrithm for a given set of partial differential equations. Because of the structure of the algorithm, the basic computer program remains the same for cases in one-, two-, or three-dimensional problems.
High-performance Integrated numerical methods for Two-phase Flow in Heterogeneous Porous Media
NASA Astrophysics Data System (ADS)
Chueh, Chih-Che; Djilali, Ned; Bangerth, Wolfgang
2010-11-01
Modelling of two-phase flow in heterogeneous porous media has been playing a decisive role in a variety of areas. However, how to efficiently and accurately solve the governing equation in the flow in porous media remains a challenge. In order to ensure the accurate representative flow field and simultaneously increase the computational efficiency, we incorporate a number of state-of-the-art techniques into a numerical framework on which more complicated models in the field of multi-phase flow in porous media will be based. Such a numerical framework consists of a h-adaptive refinement method, an entropy-based artificial diffusive term, a new adaptive operator splitting method and efficient preconditioners. In particular, it is emphasized that we propose a new efficient adaptive operator splitting to avoid solving a time-consuming pressure-velocity part every saturation time step and, most importantly, we also provide a theoretically numerical analysis as well as proof. A few benchmarks will be demonstrated in the presentation.
NASA Astrophysics Data System (ADS)
Kalogiratou, Z.; Monovasilis, Th.; Psihoyios, G.; Simos, T. E.
2014-03-01
In this work we review single step methods of the Runge-Kutta type with special properties. Among them are methods specially tuned to integrate problems that exhibit a pronounced oscillatory character and such problems arise often in celestial mechanics and quantum mechanics. Symplectic methods, exponentially and trigonometrically fitted methods, minimum phase-lag and phase-fitted methods are presented. These are Runge-Kutta, Runge-Kutta-Nyström and Partitioned Runge-Kutta methods. The theory of constructing such methods is given as well as several specific methods. In order to present the performance of the methods we have tested 58 methods from all categories. We consider the two dimensional harmonic oscillator, the two body problem, the pendulum problem and the orbital problem studied by Stiefel and Bettis. Also we have tested the methods on the computation of the eigenvalues of the one dimensional time independent Schrödinger equation with the harmonic oscillator, the doubly anharmonic oscillator and the exponential potentials.
NASA Astrophysics Data System (ADS)
Mendoza-Suárez, A.; Pérez-Aguilar, H.
2016-09-01
We present several numerical integral methods for the study of a photonic crystal waveguide, formed by two parallel conducting plates and an array of circular inclusions involving a conducting material and a metamaterial. Band structures and reflectance were calculated, for infinite and finite photonic crystal waveguides, respectively. The numerical results obtained show that the numerical methods applied provide good accuracy and efficiency. An interesting detail that resulted from this study was the appearance of a propagating mode in a band gap due to defects in the middle of the photonic crystal waveguide. This is equivalent to dope a semiconductor to introduce allowed energy states within a band gap. Our main interest in this work is to model photonic crystal waveguides that involve left-handed materials (LHMs). For the specific LHM considered, a surface plasmon mode on the vacuum-LHM interface was found.
NASA Technical Reports Server (NTRS)
Yaros, S. F.; Carlson, J. R.; Chandrasekaran, B.
1986-01-01
An effort has been undertaken at the NASA Langley Research Center to assess the capabilities of available computational methods for use in propulsion integration design studies of transonic transport aircraft, particularly of pylon/nacelle combinations which exhibit essentially no interference drag. The three computer codes selected represent state-of-the-art computational methods for analyzing complex configurations at subsonic and transonic flight conditions. These are: EULER, a finitie volume solution of the Euler equation; VSAERO, a panel solution of the Laplace equation; and PPW, a finite difference solution of the small disturbance transonic equations. In general, all three codes have certain capabilities that allow them to be of some value in predicting the flows about transport configurations, but all have limitations. Until more accurate methods are available, careful application and interpretation of the results of these codes are needed.
NASA Astrophysics Data System (ADS)
Hellmich, S.; Mottola, S.; Hahn, G.; Kührt, E.; Hlawitschka, M.
2014-07-01
Simulations of dynamical processes in planetary systems represent an important tool for studying the orbital evolution of the systems [1--3]. Using modern numerical integration methods, it is possible to model systems containing many thousands of objects over timescales of several hundred million years. However, in general, supercomputers are needed to get reasonable simulation results in acceptable execution times [3]. To exploit the ever-growing computation power of Graphics Processing Units (GPUs) in modern desktop computers, we implemented cuSwift, a library of numerical integration methods for studying long-term dynamical processes in planetary systems. cuSwift can be seen as a re-implementation of the famous SWIFT integrator package written by Hal Levison and Martin Duncan. cuSwift is written in C/CUDA and contains different integration methods for various purposes. So far, we have implemented three algorithms: a 15th-order Radau integrator [4], the Wisdom-Holman Mapping (WHM) integrator [5], and the Regularized Mixed Variable Symplectic (RMVS) Method [6]. These algorithms treat only the planets as mutually gravitationally interacting bodies whereas asteroids and comets (or other minor bodies of interest) are treated as massless test particles which are gravitationally influenced by the massive bodies but do not affect each other or the massive bodies. The main focus of this work is on the symplectic methods (WHM and RMVS) which use a larger time step and thus are capable of integrating many particles over a large time span. As an additional feature, we implemented the non-gravitational Yarkovsky effect as described by M. Brož [7]. With cuSwift, we show that the use of modern GPUs makes it possible to speed up these methods by more than one order of magnitude compared to the single-core CPU implementation, thereby enabling modest workstation computers to perform long-term dynamical simulations. We use these methods to study the influence of the Yarkovsky
Numerical Integration: One Step at a Time
ERIC Educational Resources Information Center
Yang, Yajun; Gordon, Sheldon P.
2016-01-01
This article looks at the effects that adding a single extra subdivision has on the level of accuracy of some common numerical integration routines. Instead of automatically doubling the number of subdivisions for a numerical integration rule, we investigate what happens with a systematic method of judiciously selecting one extra subdivision for…
NASA Astrophysics Data System (ADS)
Kabuth, Alina; Dahmke, Andreas; Hagrey, Said Attia al; Berta, Márton; Dörr, Cordula; Koproch, Nicolas; Köber, Ralf; Köhn, Daniel; Nolde, Michael; Tilmann Pfeiffer, Wolf; Popp, Steffi; Schwanebeck, Malte; Bauer, Sebastian
2016-04-01
Within the framework of the transition to renewable energy sources ("Energiewende"), the German government defined the target of producing 60 % of the final energy consumption from renewable energy sources by the year 2050. However, renewable energies are subject to natural fluctuations. Energy storage can help to buffer the resulting time shifts between production and demand. Subsurface geological structures provide large potential capacities for energy stored in the form of heat or gas on daily to seasonal time scales. In order to explore this potential sustainably, the possible induced effects of energy storage operations have to be quantified for both specified normal operation and events of failure. The ANGUS+ project therefore integrates experimental laboratory studies with numerical approaches to assess subsurface energy storage scenarios and monitoring methods. Subsurface storage options for gas, i.e. hydrogen, synthetic methane and compressed air in salt caverns or porous structures, as well as subsurface heat storage are investigated with respect to site prerequisites, storage dimensions, induced effects, monitoring methods and integration into spatial planning schemes. The conceptual interdisciplinary approach of the ANGUS+ project towards the integration of subsurface energy storage into a sustainable subsurface planning scheme is presented here, and this approach is then demonstrated using the examples of two selected energy storage options: Firstly, the option of seasonal heat storage in a shallow aquifer is presented. Coupled thermal and hydraulic processes induced by periodic heat injection and extraction were simulated in the open-source numerical modelling package OpenGeoSys. Situations of specified normal operation as well as cases of failure in operational storage with leaking heat transfer fluid are considered. Bench-scale experiments provided parameterisations of temperature dependent changes in shallow groundwater hydrogeochemistry. As a
NASA Astrophysics Data System (ADS)
Kashirin, A. A.; Smagin, S. I.; Taltykina, M. Yu.
2016-04-01
Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure.
Numerical integration of subtraction terms
NASA Astrophysics Data System (ADS)
Seth, Satyajit; Weinzierl, Stefan
2016-06-01
Numerical approaches to higher-order calculations often employ subtraction terms, both for the real emission and the virtual corrections. These subtraction terms have to be added back. In this paper we show that at NLO the real subtraction terms, the virtual subtraction terms, the integral representations of the field renormalization constants and—in the case of initial-state partons—the integral representation for the collinear counterterm can be grouped together to give finite integrals, which can be evaluated numerically. This is useful for an extension towards next-to-next-to-leading order.
Giurgiutiu, V.; Ionita, A.; Dillard, D.A.; Graffeo, J.K.
1996-12-31
Fracture mechanics analysis of adhesively bonded joints has attracted considerable attention in recent years. A possible approach to the analysis of adhesive layer cracks is to study a brittle adhesive between 2 elastic half-planes representing the substrates. A 2-material 3-region elasticity problem is set up and has to be solved. A modeling technique based on the work of Fleck, Hutchinson, and Suo is used. Two complex potential problems using Muskelishvili`s formulation are set up for the 3-region, 2-material model: (a) a distribution of edge dislocations is employed to simulate the crack and its near field; and (b) a crack-free problem is used to simulate the effect of the external loading applied in the far field. Superposition of the two problems is followed by matching tractions and displacements at the bimaterial boundaries. The Cauchy principal value integral is used to treat the singularities. Imposing the traction-free boundary conditions over the entire crack length yielded a linear system of two integral equations. The parameters of the problem are Dundurs` elastic mismatch coefficients, {alpha} and {beta}, and the ratio c/H representing the geometric position of the crack in the adhesive layer.
Fast methods to numerically integrate the Reynolds equation for gas fluid films
NASA Technical Reports Server (NTRS)
Dimofte, Florin
1992-01-01
The alternating direction implicit (ADI) method is adopted, modified, and applied to the Reynolds equation for thin, gas fluid films. An efficient code is developed to predict both the steady-state and dynamic performance of an aerodynamic journal bearing. An alternative approach is shown for hybrid journal gas bearings by using Liebmann's iterative solution (LIS) for elliptic partial differential equations. The results are compared with known design criteria from experimental data. The developed methods show good accuracy and very short computer running time in comparison with methods based on an inverting of a matrix. The computer codes need a small amount of memory and can be run on either personal computers or on mainframe systems.
Numerical integration of a relativistic two-body problem via a multiple scales method
NASA Astrophysics Data System (ADS)
Abouelmagd, Elbaz I.; Elshaboury, S. M.; Selim, H. H.
2016-01-01
We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based on the separation of time scales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (Essential Relativistic Celestial Mechanics, Hilger, Bristol, 1991). The practical applicability of this model may be in studies of the long-term evolution of relativistic binaries (neutron stars or black holes).
Fresnel Integral Equations: Numerical Properties
Adams, R J; Champagne, N J II; Davis, B A
2003-07-22
A spatial-domain solution to the problem of electromagnetic scattering from a dielectric half-space is outlined. The resulting half-space operators are referred to as Fresnel surface integral operators. When used as preconditioners for nonplanar geometries, the Fresnel operators yield surface Fresnel integral equations (FIEs) which are stable with respect to dielectric constant, discretization, and frequency. Numerical properties of the formulations are discussed.
NASA Astrophysics Data System (ADS)
Wang, Qianxi; Manmi, Kawa; Calvisi, Michael L.
2015-02-01
Ultrasound contrast agents (UCAs) are microbubbles stabilized with a shell typically of lipid, polymer, or protein and are emerging as a unique tool for noninvasive therapies ranging from gene delivery to tumor ablation. While various models have been developed to describe the spherical oscillations of contrast agents, the treatment of nonspherical behavior has received less attention. However, the nonspherical dynamics of contrast agents are thought to play an important role in therapeutic applications, for example, enhancing the uptake of therapeutic agents across cell membranes and tissue interfaces, and causing tissue ablation. In this paper, a model for nonspherical contrast agent dynamics based on the boundary integral method is described. The effects of the encapsulating shell are approximated by adapting Hoff's model for thin-shell, spherical contrast agents. A high-quality mesh of the bubble surface is maintained by implementing a hybrid approach of the Lagrangian method and elastic mesh technique. The numerical model agrees well with a modified Rayleigh-Plesset equation for encapsulated spherical bubbles. Numerical analyses of the dynamics of UCAs in an infinite liquid and near a rigid wall are performed in parameter regimes of clinical relevance. The oscillation amplitude and period decrease significantly due to the coating. A bubble jet forms when the amplitude of ultrasound is sufficiently large, as occurs for bubbles without a coating; however, the threshold amplitude required to incite jetting increases due to the coating. When a UCA is near a rigid boundary subject to acoustic forcing, the jet is directed towards the wall if the acoustic wave propagates perpendicular to the boundary. When the acoustic wave propagates parallel to the rigid boundary, the jet direction has components both along the wave direction and towards the boundary that depend mainly on the dimensionless standoff distance of the bubble from the boundary. In all cases, the jet
Chern, I-Liang
1994-08-01
Two versions of a control volume method on a symmetrized icosahedral grid are proposed for solving the shallow-water equations on a sphere. One version expresses of the equations in the 3-D Cartersian coordinate system, while the other expresses the equations in the northern/southern polar sterographic coordinate systems. The pole problem is avoided because of these expressions in both versions and the quasi-homogenity of the icosahedral grid. Truncation errors and convergence tests of the numerical gradient and divergent operators associated with this method are studied. A convergence tests of the numerical gradient and divergent operators associated with this method are studied. A convergence test for a steady zonal flow is demonstrated. Several simulations of Rossby-Haurwitz waves with various numbers are also performed.
Effects of aliasing on numerical integration.
Edwards, Timothy S.
2005-02-01
During the course of processing acceleration data from mechanical systems it is often desirable to integrate the data to obtain velocity or displacement waveforms. However, those who have attempted these operations may be painfully aware that the integrated records often yield unrealistic residual values. This is true whether the data has been obtained experimentally or through numerical simulation such as Runge-Kutta integration or the explicit finite element method. In the case of experimentally obtained data, the integration errors are usually blamed on accelerometer zero shift or amplifier saturation. In the case of simulation data, incorrect integrations are often incorrectly blamed on the integration algorithm itself. This work demonstrates that seemingly small aliased content can cause appreciable errors in the integrated waveforms and explores the unavoidable source of aliasing in both experiment and simulation-the sampling operation. Numerical analysts are often puzzled as to why the integrated acceleration from their simulation does not match the displacement output from the same simulation. This work shows that these strange results can be caused by aliasing induced by interpolation of the model output during sampling regularization.
NASA Technical Reports Server (NTRS)
Chan, William M.
1992-01-01
This project forms part of the long term computational effort to simulate the time dependent flow over the integrated Space Shuttle vehicle (orbiter, solid rocket boosters (SRB's), external tank (ET), and attach hardware) during its ascent mode for various nominal and abort flight conditions. Due to the limitations of experimental data such as wind tunnel wall effects and the difficulty of safely obtaining valid flight data, numerical simulations are undertaken to supplement the existing data base. This data can then be used to predict the aerodynamic behavior over a wide range of flight conditions. Existing computational results show relatively good overall comparison with experiments but further refinement is required to reduce numerical errors and to obtain finer agreements over a larger parameter space. One of the important goals of this project is to obtain better comparisons between numerical simulations and experiments. In the simulations performed so far, the geometry has been simplified in various ways to reduce the complexity so that useful results can be obtained in a reasonable time frame due to limitations in computer resources. In this project, the finer details of the major components of the Space Shuttle are modeled better by including more complexity in the geometry definition. Smaller components not included in early Space Shuttle simulations will now be modeled and gridded.
Egorov, Alexander A
2011-07-31
We consider theoretical, experimental and numerical methods which make it possible to analyse the key characteristics of laser radiation scattered in the integrated-optical waveguide with three-dimensional irregularities. The main aspects of the three-dimensional vector electrodynamic problem of waveguide scattering are studied. The waveguide light scattering method is presented and its main advantages over the methods of single scattering of laser radiation are discussed. The experimental setup and results of measurements are described. Theoretical and experimental results confirming the validity of the vector theory of three-dimensional waveguide scattering of laser radiation developed by the author are compared for the first time. (fiber and integrated optics)
Numerical multi-loop integrals and applications
NASA Astrophysics Data System (ADS)
Freitas, A.
2016-09-01
Higher-order radiative corrections play an important role in precision studies of the electroweak and Higgs sector, as well as for the detailed understanding of large backgrounds to new physics searches. For corrections beyond the one-loop level and involving many independent mass and momentum scales, it is in general not possible to find analytic results, so that one needs to resort to numerical methods instead. This article presents an overview of a variety of numerical loop integration techniques, highlighting their range of applicability, suitability for automatization, and numerical precision and stability. In a second part of this article, the application of numerical loop integration methods in the area of electroweak precision tests is illustrated. Numerical methods were essential for obtaining full two-loop predictions for the most important precision observables within the Standard Model. The theoretical foundations for these corrections will be described in some detail, including aspects of the renormalization, resummation of leading log contributions, and the evaluation of the theory uncertainty from missing higher orders.
Numerical integration of asymptotic solutions of ordinary differential equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Safouhi, Hassan . E-mail: hassan.safouhi@ualberta.ca; Berlu, Lilian
2006-07-20
Molecular overlap-like quantum similarity measurements imply the evaluation of overlap integrals of two molecular electronic densities related by Dirac delta function. When the electronic densities are expanded over atomic orbitals using the usual LCAO-MO approach (linear combination of atomic orbitals), overlap-like quantum similarity integrals could be expressed in terms of four-center overlap integrals. It is shown that by introducing the Fourier transform of delta Dirac function in the integrals and using the Fourier transform approach combined with the so-called B functions, one can obtain analytic expressions of the integrals under consideration. These analytic expressions involve highly oscillatory semi-infinite spherical Bessel functions, which are the principal source of severe numerical and computational difficulties. In this work, we present a highly efficient algorithm for a fast and accurate numerical evaluation of these multicenter overlap-like quantum similarity integrals over Slater type functions. This algorithm is based on the SD-bar approach due to Safouhi. Recurrence formulae are used for a better control of the degree of accuracy and for a better stability of the algorithm. The numerical result section shows the efficiency of our algorithm, compared with the alternatives using the one-center two-range expansion method, which led to very complicated analytic expressions, the epsilon algorithm and the nonlinear D-bar transformation.
Numerical Analysis of the Symmetric Methods
NASA Astrophysics Data System (ADS)
Xu, Ji-Hong; Zhang, A.-Li
1995-03-01
Aimed at the initial value problem of the particular second-order ordinary differential equations,y ″=f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods.
Zhao, Peng; Wang, Qing-Hong; Tian, Cheng-Ming; Kakishima, Makoto
2015-01-01
The species in genus Melampsora are the causal agents of leaf rust diseases on willows in natural habitats and plantations. However, the classification and recognition of species diversity are challenging because morphological characteristics are scant and morphological variation in Melampsora on willows has not been thoroughly evaluated. Thus, the taxonomy of Melampsora species on willows remains confused, especially in China where 31 species were reported based on either European or Japanese taxonomic systems. To clarify the species boundaries of Melampsora species on willows in China, we tested two approaches for species delimitation inferred from morphological and molecular variations. Morphological species boundaries were determined based on numerical taxonomic analyses of morphological characteristics in the uredinial and telial stages by cluster analysis and one-way analysis of variance. Phylogenetic species boundaries were delineated based on the generalized mixed Yule-coalescent (GMYC) model analysis of the sequences of the internal transcribed spacer (ITS1 and ITS2) regions including the 5.8S and D1/D2 regions of the large nuclear subunit of the ribosomal RNA gene. Numerical taxonomic analyses of 14 morphological characteristics recognized in the uredinial-telial stages revealed 22 morphological species, whereas the GMYC results recovered 29 phylogenetic species. In total, 17 morphological species were in concordance with the phylogenetic species and 5 morphological species were in concordance with 12 phylogenetic species. Both the morphological and molecular data supported 14 morphological characteristics, including 5 newly recognized characteristics and 9 traditionally emphasized characteristics, as effective for the differentiation of Melampsora species on willows in China. Based on the concordance and discordance of the two species delimitation approaches, we concluded that integrative taxonomy by using both morphological and molecular variations was
NASA Astrophysics Data System (ADS)
Virgo, Simon; Ankit, Kumar; Nestler, Britta; Urai, Janos L.
2016-04-01
Crack-seal veins form in a complex interplay of coupled thermal, hydraulic, mechanical and chemical processes. Their formation and cyclic growth involves brittle fracturing and dilatancy, phases of increased fluid flow and the growth of crystals that fill the voids and reestablish the mechanical strength. Existing numerical models of vein formation focus on selected aspects of the coupled process. Until today, no model exists that is able to use a realistic representation of the fracturing AND sealing processes, simultaneously. To address this challenge, we propose the bidirectional coupling of two numerical methods that have proven themselves as very powerful to model the fundamental processes acting in crack-seal systems: Phase-field and the Discrete Element Method (DEM). The phase-field Method was recently successfully extended to model the precipitation of quartz crystals from an aqueous solution and applied to model the sealing of a vein over multiple opening events (Ankit et al., 2013; Ankit et al., 2015a; Ankit et al., 2015b). The advantage over former, purely kinematic approaches is that in phase-field, the crystal growth is modeled based on thermodynamic and kinetic principles. Different driving forces for microstructure evolution, such as chemical bulk free energy, interfacial energy, elastic strain energy and different transport processes, such as mass diffusion and advection, can be coupled and the effect on the evolution process can be studied in 3D. The Discrete Element Method was already used in several studies to model the fracturing of rocks and the incremental growth of veins by repeated fracturing (Virgo et al., 2013; Virgo et al., 2014). Materials in DEM are represented by volumes of packed spherical particles and the response to the material to stress is modeled by interaction of the particles with their nearest neighbours. For rocks, in 3D, the method provides a realistic brittle failure behaviour. Exchange Routines are being developed that
Efficient numerical evaluation of Feynman integrals
NASA Astrophysics Data System (ADS)
Li, Zhao; Wang, Jian; Yan, Qi-Shu; Zhao, Xiaoran
2016-03-01
Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector decomposition by implementing a quasi-Monte Carlo method associated with the CUDA/GPU technique. For demonstration we present the results of several Feynman integrals up to two loops in both Euclidean and physical kinematic regions in comparison with those obtained from FIESTA3. It is shown that both planar and non-planar two-loop master integrals in the physical kinematic region can be evaluated in less than half a minute with accuracy, which makes the direct numerical approach viable for precise investigation of higher order effects in multi-loop processes, e.g. the next-to-leading order QCD effect in Higgs pair production via gluon fusion with a finite top quark mass. Supported by the Natural Science Foundation of China (11305179 11475180), Youth Innovation Promotion Association, CAS, IHEP Innovation (Y4545170Y2), State Key Lab for Electronics and Particle Detectors, Open Project Program of State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, China (Y4KF061CJ1), Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA-EXC 1098)
Numerical integration of ordinary differential equations of various orders
NASA Technical Reports Server (NTRS)
Gear, C. W.
1969-01-01
Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.
NASA Astrophysics Data System (ADS)
Yolcu, Cem; Memiç, Muhammet; Şimşek, Kadir; Westin, Carl-Fredrik; Özarslan, Evren
2016-05-01
We study the influence of diffusion on NMR experiments when the molecules undergo random motion under the influence of a force field and place special emphasis on parabolic (Hookean) potentials. To this end, the problem is studied using path integral methods. Explicit relationships are derived for commonly employed gradient waveforms involving pulsed and oscillating gradients. The Bloch-Torrey equation, describing the temporal evolution of magnetization, is modified by incorporating potentials. A general solution to this equation is obtained for the case of parabolic potential by adopting the multiple correlation function (MCF) formalism, which has been used in the past to quantify the effects of restricted diffusion. Both analytical and MCF results were found to be in agreement with random walk simulations. A multidimensional formulation of the problem is introduced that leads to a new characterization of diffusion anisotropy. Unlike the case of traditional methods that employ a diffusion tensor, anisotropy originates from the tensorial force constant, and bulk diffusivity is retained in the formulation. Our findings suggest that some features of the NMR signal that have traditionally been attributed to restricted diffusion are accommodated by the Hookean model. Under certain conditions, the formalism can be envisioned to provide a viable approximation to the mathematically more challenging restricted diffusion problems.
NASA Technical Reports Server (NTRS)
Emukashvily, I. M.
1982-01-01
An extension of the method of moments is developed for the numerical integration of the kinetic equations of droplet spectra evolution by condensation/evaporation and by coalescence/breakup processes. The number density function n sub k (x,t) in each separate droplet packet between droplet mass grid points (x sub k, x sub k+1) is represented by an expansion in orthogonal polynomials with a given weighting function. In this way droplet number concentrations, liquid water contents and other moments in each droplet packet are conserved and the problem of solving the kinetic equations is replaced by one of solving a set of coupled differential equations for the number density function moments. The method is tested against analytic solutions of the corresponding kinetic equations. Numerical results are obtained for different coalescence/breakup and condensation/evaporation kernels and for different initial droplet spectra. Also droplet mass grid intervals, weighting functions, and time steps are varied.
Sun, Kerang
2015-09-01
A three-dimensional finite element model is constructed to simulate the experimental conditions presented in a paper published in this journal [Goltz et al., 2009. Validation of two innovative methods to measure contaminant mass flux in groundwater. Journal of Contaminant Hydrology 106 (2009) 51-61] where the modified integral pumping test (MIPT) method was found to significantly underestimate the specific discharge in an artificial aquifer. The numerical model closely replicates the experimental configuration with explicit representation of the pumping well column and skin, allowing for the model to simulate the wellbore flow in the pumping well as an integral part of the porous media flow in the aquifer using the equivalent hydraulic conductivity approach. The equivalent hydraulic conductivity is used to account for head losses due to friction within the wellbore of the pumping well. Applying the MIPT method on the model simulated piezometric heads resulted in a specific discharge that underestimates the true specific discharge in the experimental aquifer by 18.8%, compared with the 57% underestimation of mass flux by the experiment reported by Goltz et al. (2009). Alternative simulation shows that the numerical model is capable of approximately replicating the experiment results when the equivalent hydraulic conductivity is reduced by an order of magnitude, suggesting that the accuracy of the MIPT estimation could be improved by expanding the physical meaning of the equivalent hydraulic conductivity to account for other factors such as orifice losses in addition to frictional losses within the wellbore. Numerical experiments also show that when applying the MIPT method to estimate hydraulic parameters, use of depth-integrated piezometric head instead of the head near the pump intake can reduce the estimation error resulting from well losses, but not the error associated with the well not being fully screened. PMID:26210034
NASA Astrophysics Data System (ADS)
Sun, Kerang
2015-09-01
A three-dimensional finite element model is constructed to simulate the experimental conditions presented in a paper published in this journal [Goltz et al., 2009. Validation of two innovative methods to measure contaminant mass flux in groundwater. Journal of Contaminant Hydrology 106 (2009) 51-61] where the modified integral pumping test (MIPT) method was found to significantly underestimate the specific discharge in an artificial aquifer. The numerical model closely replicates the experimental configuration with explicit representation of the pumping well column and skin, allowing for the model to simulate the wellbore flow in the pumping well as an integral part of the porous media flow in the aquifer using the equivalent hydraulic conductivity approach. The equivalent hydraulic conductivity is used to account for head losses due to friction within the wellbore of the pumping well. Applying the MIPT method on the model simulated piezometric heads resulted in a specific discharge that underestimates the true specific discharge in the experimental aquifer by 18.8%, compared with the 57% underestimation of mass flux by the experiment reported by Goltz et al. (2009). Alternative simulation shows that the numerical model is capable of approximately replicating the experiment results when the equivalent hydraulic conductivity is reduced by an order of magnitude, suggesting that the accuracy of the MIPT estimation could be improved by expanding the physical meaning of the equivalent hydraulic conductivity to account for other factors such as orifice losses in addition to frictional losses within the wellbore. Numerical experiments also show that when applying the MIPT method to estimate hydraulic parameters, use of depth-integrated piezometric head instead of the head near the pump intake can reduce the estimation error resulting from well losses, but not the error associated with the well not being fully screened.
NASA Astrophysics Data System (ADS)
Kashima, Tsuyoshi; Imada, Masatoshi
2001-08-01
A new efficient numerical algorithm for interacting fermion systems is proposed and examined in detail. The ground state is expressed approximately by a linear combination of numerically chosen basis states in a truncated Hilbert space. Two procedures lead to a better approximation. The first is a numerical renormalization, which optimizes the chosen basis and projects onto the ground state within the fixed dimension, L, of the Hilbert space. The second is an increase of the dimension of the truncated Hilbert space, which enables the linear combination to converge to a better approximation. The extrapolation L→∞ after the convergence removes the approximation error systematically. This algorithm does not suffer from the negative sign problem and can be applied to systems in any spatial dimension and arbitrary lattice structure. The efficiency is tested and the implementation explained for two-dimensional Hubbard models where Slater determinants are employed as chosen basis. Our results with less than 400 chosen basis indicate good accuracy within the errorbar of the best available results as those of the quantum Monte Carlo for energy and other physical quantities.
An Integrative Theory of Numerical Development
ERIC Educational Resources Information Center
Siegler, Robert; Lortie-Forgues, Hugues
2014-01-01
Understanding of numerical development is growing rapidly, but the volume and diversity of findings can make it difficult to perceive any coherence in the process. The integrative theory of numerical development posits that a coherent theme is present, however--progressive broadening of the set of numbers whose magnitudes can be accurately…
Numerical methods in structural mechanics
NASA Astrophysics Data System (ADS)
Obraztsov, I. F.
The papers contained in this volume focus on numerical, numerical-analytical, and theoretical methods for dealing with strength, stability, and dynamics problems in the design of the structural elements of flight vehicles. Topics discussed include the solution of homogeneous boundary value problems for systems of ordinary differential equations modified by a difference factorization method, a study of the rupture strength of a welded joint between plates, singular solutions in mixed problems for a wedge and a half-strip, and a thermoelasticity problem for an open-profile cylindrical shell with a localized temperature field.
Numerical methods for turbulent flow
NASA Astrophysics Data System (ADS)
Turner, James C., Jr.
1988-09-01
It has generally become accepted that the Navier-Strokes equations predict the dynamic behavior of turbulent as well as laminar flows of a fluid at a point in space away form a discontinuity such as a shock wave. Turbulence is also closely related to the phenomena of non-uniqueness of solutions of the Navier-Strokes equations. These second order, nonlinear partial differential equations can be solved analytically for only a few simple flows. Turbulent flow fields are much to complex to lend themselves to these few analytical methods. Numerical methods, therefore, offer the only possibility of achieving a solution of turbulent flow equations. In spite of recent advances in computer technology, the direct solution, by discrete methods, of the Navier-Strokes equations for turbulent flow fields is today, and in the foreseeable future, impossible. Thus the only economically feasible way to solve practical turbulent flow problems numerically is to use statistically averaged equations governing mean-flow quantities. The objective is to study some recent developments relating to the use of numerical methods to study turbulent flow.
Numerical methods for molecular dynamics
Skeel, R.D.
1991-01-01
This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods which are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. We first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.
Translation and integration of numerical atomic orbitals in linear molecules.
Heinäsmäki, Sami
2014-02-14
We present algorithms for translation and integration of atomic orbitals for LCAO calculations in linear molecules. The method applies to arbitrary radial functions given on a numerical mesh. The algorithms are based on pseudospectral differentiation matrices in two dimensions and the corresponding two-dimensional Gaussian quadratures. As a result, multicenter overlap and Coulomb integrals can be evaluated effectively. PMID:24527905
Translation and integration of numerical atomic orbitals in linear molecules
NASA Astrophysics Data System (ADS)
Heinäsmäki, Sami
2014-02-01
We present algorithms for translation and integration of atomic orbitals for LCAO calculations in linear molecules. The method applies to arbitrary radial functions given on a numerical mesh. The algorithms are based on pseudospectral differentiation matrices in two dimensions and the corresponding two-dimensional Gaussian quadratures. As a result, multicenter overlap and Coulomb integrals can be evaluated effectively.
On Numerical Methods For Hypersonic Turbulent Flows
NASA Astrophysics Data System (ADS)
Yee, H. C.; Sjogreen, B.; Shu, C. W.; Wang, W.; Magin, T.; Hadjadj, A.
2011-05-01
Proper control of numerical dissipation in numerical methods beyond the standard shock-capturing dissipation at discontinuities is an essential element for accurate and stable simulation of hypersonic turbulent flows, including combustion, and thermal and chemical nonequilibrium flows. Unlike rapidly developing shock interaction flows, turbulence computations involve long time integrations. Improper control of numerical dissipation from one time step to another would be compounded over time, resulting in the smearing of turbulent fluctuations to an unrecognizable form. Hypersonic turbulent flows around re- entry space vehicles involve mixed steady strong shocks and turbulence with unsteady shocklets that pose added computational challenges. Stiffness of the source terms and material mixing in combustion pose yet other types of numerical challenges. A low dissipative high order well- balanced scheme, which can preserve certain non-trivial steady solutions of the governing equations exactly, may help minimize some of these difficulties. For stiff reactions it is well known that the wrong propagation speed of discontinuities occurs due to the under-resolved numerical solutions in both space and time. Schemes to improve the wrong propagation speed of discontinuities for systems of stiff reacting flows remain a challenge for algorithm development. Some of the recent algorithm developments for direct numerical simulations (DNS) and large eddy simulations (LES) for the subject physics, including the aforementioned numerical challenges, will be discussed.
Highly Parallel, High-Precision Numerical Integration
Bailey, David H.; Borwein, Jonathan M.
2005-04-22
This paper describes a scheme for rapidly computing numerical values of definite integrals to very high accuracy, ranging from ordinary machine precision to hundreds or thousands of digits, even for functions with singularities or infinite derivatives at endpoints. Such a scheme is of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where high-precision numerical values of definite integrals can be used to numerically discover new identities. This paper discusses techniques for a parallel implementation of this scheme, then presents performance results for 1-D and 2-D test suites. Results are also given for a certain problem from mathematical physics, which features a difficult singularity, confirming a conjecture to 20,000 digit accuracy. The performance rate for this latter calculation on 1024 CPUs is 690 Gflop/s. We believe that this and one other 20,000-digit integral evaluation that we report are the highest-precision non-trivial numerical integrations performed to date.
Numerical methods for multibody systems
NASA Technical Reports Server (NTRS)
Glowinski, Roland; Nasser, Mahmoud G.
1994-01-01
This article gives a brief summary of some results obtained by Nasser on modeling and simulation of inequality problems in multibody dynamics. In particular, the augmented Lagrangian method discussed here is applied to a constrained motion problem with impulsive inequality constraints. A fundamental characteristic of the multibody dynamics problem is the lack of global convexity of its Lagrangian. The problem is transformed into a convex analysis problem by localization (piecewise linearization), where the augmented Lagrangian has been successfully used. A model test problem is considered and a set of numerical experiments is presented.
Numerical solution of boundary-integral equations for molecular electrostatics.
Bardhan, Jaydeep P
2009-03-01
Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived. PMID:19275391
Numerical Methods for Stochastic Partial Differential Equations
Sharp, D.H.; Habib, S.; Mineev, M.B.
1999-07-08
This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
Numerical methods used in fusion science numerical modeling
NASA Astrophysics Data System (ADS)
Yagi, M.
2015-04-01
The dynamics of burning plasma is very complicated physics, which is dominated by multi-scale and multi-physics phenomena. To understand such phenomena, numerical simulations are indispensable. Fundamentals of numerical methods used in fusion science numerical modeling are briefly discussed in this paper. In addition, the parallelization technique such as open multi processing (OpenMP) and message passing interface (MPI) parallel programing are introduced and the loop-level parallelization is shown as an example.
Fast Quantum Algorithms for Numerical Integrals and Stochastic Processes
NASA Technical Reports Server (NTRS)
Abrams, D.; Williams, C.
1999-01-01
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known classical deterministic algotithms and a quadratic speed increase incomparison to classical Monte Carlo methods.
RELAP-7 Numerical Stabilization: Entropy Viscosity Method
R. A. Berry; M. O. Delchini; J. Ragusa
2014-06-01
The RELAP-7 code is the next generation nuclear reactor system safety analysis code being developed at the Idaho National Laboratory (INL). The code is based on the INL's modern scientific software development framework, MOOSE (Multi-Physics Object Oriented Simulation Environment). The overall design goal of RELAP-7 is to take advantage of the previous thirty years of advancements in computer architecture, software design, numerical integration methods, and physical models. The end result will be a reactor systems analysis capability that retains and improves upon RELAP5's capability and extends the analysis capability for all reactor system simulation scenarios. RELAP-7 utilizes a single phase and a novel seven-equation two-phase flow models as described in the RELAP-7 Theory Manual (INL/EXT-14-31366). The basic equation systems are hyperbolic, which generally require some type of stabilization (or artificial viscosity) to capture nonlinear discontinuities and to suppress advection-caused oscillations. This report documents one of the available options for this stabilization in RELAP-7 -- a new and novel approach known as the entropy viscosity method. Because the code is an ongoing development effort in which the physical sub models, numerics, and coding are evolving, so too must the specific details of the entropy viscosity stabilization method. Here the fundamentals of the method in their current state are presented.
Integrated optical circuits for numerical computation
NASA Technical Reports Server (NTRS)
Verber, C. M.; Kenan, R. P.
1983-01-01
The development of integrated optical circuits (IOC) for numerical-computation applications is reviewed, with a focus on the use of systolic architectures. The basic architecture criteria for optical processors are shown to be the same as those proposed by Kung (1982) for VLSI design, and the advantages of IOCs over bulk techniques are indicated. The operation and fabrication of electrooptic grating structures are outlined, and the application of IOCs of this type to an existing 32-bit, 32-Mbit/sec digital correlator, a proposed matrix multiplier, and a proposed pipeline processor for polynomial evaluation is discussed. The problems arising from the inherent nonlinearity of electrooptic gratings are considered. Diagrams and drawings of the application concepts are provided.
Numerical evaluation of Feynman path integrals
NASA Astrophysics Data System (ADS)
Baird, William Hugh
1999-11-01
The notion of path integration developed by Feynman, while an incredibly successful method of solving quantum mechanical problems, leads to frequently intractable integrations over an infinite number of paths. Two methods now exist which sidestep this difficulty by defining "densities" of actions which give the relative number of paths found at different values of the action. These densities are sampled by computer generation of paths and the propagators are found to a high degree of accuracy for the case of a particle on the infinite half line and in a finite square well in one dimension. The problem of propagation within a two dimensional radial well is also addressed as the precursor to the problem of a particle in a stadium (quantum billiard).
The core of the research effort in the Regional Vulnerability Assessment Program (ReVA) is a set of data integration methods ranging from simple overlays to complex multivariate statistics. These methods are described in the EPA publication titled, "Regional Vulnerability Assess...
Numerical integration of ordinary differential equations on manifolds
NASA Astrophysics Data System (ADS)
Crouch, P. E.; Grossman, R.
1993-12-01
This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.
Stability of numerical integration techniques for transient rotor dynamics
NASA Technical Reports Server (NTRS)
Kascak, A. F.
1977-01-01
A finite element model of a rotor bearing system was analyzed to determine the stability limits of the forward, backward, and centered Euler; Runge-Kutta; Milne; and Adams numerical integration techniques. The analysis concludes that the highest frequency mode determines the maximum time step for a stable solution. Thus, the number of mass elements should be minimized. Increasing the damping can sometimes cause numerical instability. For a uniform shaft, with 10 mass elements, operating at approximately the first critical speed, the maximum time step for the Runge-Kutta, Milne, and Adams methods is that which corresponds to approximately 1 degree of shaft movement. This is independent of rotor dimensions.
A Collocation Method for Volterra Integral Equations
NASA Astrophysics Data System (ADS)
Kolk, Marek
2010-09-01
We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with logarithmic kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration. An attainable order of the convergence of the method is studied. We illustrate our results with a numerical example.
NASA Astrophysics Data System (ADS)
Xu, Peiliang
2015-04-01
Satellite orbits have been routinely used to produce models of the Earth's gravity field. The numerical integration method is most widely used by almost all major institutions to determine standard gravity models from space geodetic measurements. As a basic component of the method, the partial derivatives of a satellite orbit with respect to the force parameters to be determined, namely, the unknown harmonic coefficients of the gravitational model, have been first computed by setting the initial values of partial derivatives to zero. In this talk, we first design some simple mathematical examples to show that setting the initial values of partial derivatives to zero is generally erroneous mathematically. We then prove that it is prohibited physically. In other words, setting the initial values of partial derivatives to zero violates the physics of motion of celestial bodies. To conclude, the numerical integration method, as is widely used today by major institutions to produce standard satellite gravity models, is simply incorrect mathematically. As a direct consequence, further work is required to confirm whether the numerical integration method can still be used as a mathematical foundation to produce standard satellite gravity models. More details can be found in Xu (2009, Sci China Ser D-Earth Sci, 52, 562-566).
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Jezewski, D. J.
1979-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
NASA Astrophysics Data System (ADS)
Mengistu, Haile; Tessema, Abera; Abiye, Tamiru; Demlie, Molla; Lin, Haili
2015-05-01
Improved groundwater flow conceptualization was achieved using environmental stable isotope (ESI) and hydrochemical information to complete a numerical groundwater flow model with reasonable certainty. The study aimed to assess the source of excess water at a pumping shaft located near the town of Stilfontein, North West Province, South Africa. The results indicate that the water intercepted at Margaret Shaft comes largely from seepage of a nearby mine tailings dam (Dam 5) and from the upper dolomite aquifer. If pumping at the shaft continues at the current rate and Dam 5 is decommissioned, neighbouring shallow farm boreholes would dry up within approximately 10 years. Stable isotope data of shaft water indicate that up to 50 % of the pumped water from Margaret Shaft is recirculated, mainly from Dam 5. The results are supplemented by tritium data, demonstrating that recent recharge is taking place through open fractures as well as man-made underground workings, whereas hydrochemical data of fissure water samples from roughly 950 m below ground level exhibit mine-water signatures. Pumping at the shaft, which captures shallow groundwater as well as seepage from surface dams, is a highly recommended option for preventing flooding of downstream mines. The results of this research highlight the importance of additional methods (ESI and hydrochemical analyses) to improve flow conceptualization and numerical modelling.
Simple and Efficient Numerical Evaluation of Near-Hypersingular Integrals
NASA Technical Reports Server (NTRS)
Fink, Patrick W.; Wilton, Donald R.; Khayat, Michael A.
2007-01-01
Recently, significant progress has been made in the handling of singular and nearly-singular potential integrals that commonly arise in the Boundary Element Method (BEM). To facilitate object-oriented programming and handling of higher order basis functions, cancellation techniques are favored over techniques involving singularity subtraction. However, gradients of the Newton-type potentials, which produce hypersingular kernels, are also frequently required in BEM formulations. As is the case with the potentials, treatment of the near-hypersingular integrals has proven more challenging than treating the limiting case in which the observation point approaches the surface. Historically, numerical evaluation of these near-hypersingularities has often involved a two-step procedure: a singularity subtraction to reduce the order of the singularity, followed by a boundary contour integral evaluation of the extracted part. Since this evaluation necessarily links basis function, Green s function, and the integration domain (element shape), the approach ill fits object-oriented programming concepts. Thus, there is a need for cancellation-type techniques for efficient numerical evaluation of the gradient of the potential. Progress in the development of efficient cancellation-type procedures for the gradient potentials was recently presented. To the extent possible, a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. However, since the gradient kernel involves singularities of different orders, we also require that the transformation leaves remaining terms that are analytic. The terms "normal" and "tangential" are used herein with reference to the source element. Also, since computational formulations often involve the numerical evaluation of both potentials and their gradients, it is highly desirable that a single integration procedure efficiently handles both.
Numerical Methods for Radiation Magnetohydrodynamics in Astrophysics
Klein, R I; Stone, J M
2007-11-20
We describe numerical methods for solving the equations of radiation magnetohydrodynamics (MHD) for astrophysical fluid flow. Such methods are essential for the investigation of the time-dependent and multidimensional dynamics of a variety of astrophysical systems, although our particular interest is motivated by problems in star formation. Over the past few years, the authors have been members of two parallel code development efforts, and this review reflects that organization. In particular, we discuss numerical methods for MHD as implemented in the Athena code, and numerical methods for radiation hydrodynamics as implemented in the Orion code. We discuss the challenges introduced by the use of adaptive mesh refinement in both codes, as well as the most promising directions for future developments.
Error Estimates for Numerical Integration Rules
ERIC Educational Resources Information Center
Mercer, Peter R.
2005-01-01
The starting point for this discussion of error estimates is the fact that integrals that arise in Fourier series have properties that can be used to get improved bounds. This idea is extended to more general situations.
The development of accurate and efficient methods of numerical quadrature
NASA Technical Reports Server (NTRS)
Feagin, T.
1973-01-01
Some new methods for performing numerical quadrature of an integrable function over a finite interval are described. Each method provides a sequence of approximations of increasing order to the value of the integral. Each approximation makes use of all previously computed values of the integrand. The points at which new values of the integrand are computed are selected in such a way that the order of the approximation is maximized. The methods are compared with the quadrature methods of Clenshaw and Curtis, Gauss, Patterson, and Romberg using several examples.
Research on the Evolutionary Strategy Based on AIS and Its Application on Numerical Integration
NASA Astrophysics Data System (ADS)
Bei, Li
Based on the features of artificial immune system, a new evolutionary strategy is proposed in order to calculate the numerical integration of functions. This evolutionary strategy includes the mechanisms of swarm searching and constructing the fitness function. Finally, numerical examples are given for verifying the effectiveness of evolutionary strategy. The results show that the performance of evolutionary strategy is satisfactory and more accurate than traditional methods of numerical integration, such as trapezoid formula and Simpson formula.
Accelerated adaptive integration method.
Kaus, Joseph W; Arrar, Mehrnoosh; McCammon, J Andrew
2014-05-15
Conformational changes that occur upon ligand binding may be too slow to observe on the time scales routinely accessible using molecular dynamics simulations. The adaptive integration method (AIM) leverages the notion that when a ligand is either fully coupled or decoupled, according to λ, barrier heights may change, making some conformational transitions more accessible at certain λ values. AIM adaptively changes the value of λ in a single simulation so that conformations sampled at one value of λ seed the conformational space sampled at another λ value. Adapting the value of λ throughout a simulation, however, does not resolve issues in sampling when barriers remain high regardless of the λ value. In this work, we introduce a new method, called Accelerated AIM (AcclAIM), in which the potential energy function is flattened at intermediate values of λ, promoting the exploration of conformational space as the ligand is decoupled from its receptor. We show, with both a simple model system (Bromocyclohexane) and the more complex biomolecule Thrombin, that AcclAIM is a promising approach to overcome high barriers in the calculation of free energies, without the need for any statistical reweighting or additional processors. PMID:24780083
Accelerated Adaptive Integration Method
2015-01-01
Conformational changes that occur upon ligand binding may be too slow to observe on the time scales routinely accessible using molecular dynamics simulations. The adaptive integration method (AIM) leverages the notion that when a ligand is either fully coupled or decoupled, according to λ, barrier heights may change, making some conformational transitions more accessible at certain λ values. AIM adaptively changes the value of λ in a single simulation so that conformations sampled at one value of λ seed the conformational space sampled at another λ value. Adapting the value of λ throughout a simulation, however, does not resolve issues in sampling when barriers remain high regardless of the λ value. In this work, we introduce a new method, called Accelerated AIM (AcclAIM), in which the potential energy function is flattened at intermediate values of λ, promoting the exploration of conformational space as the ligand is decoupled from its receptor. We show, with both a simple model system (Bromocyclohexane) and the more complex biomolecule Thrombin, that AcclAIM is a promising approach to overcome high barriers in the calculation of free energies, without the need for any statistical reweighting or additional processors. PMID:24780083
A numerical method of detecting singularity
NASA Technical Reports Server (NTRS)
Laporte, M.; Vignes, J.
1978-01-01
A numerical method is reported which determines a value C for the degree of conditioning of a matrix. This value is C = 0 for a singular matrix and has progressively larger values for matrices which are increasingly well-conditioned. This value is C sub = C max sub max (C defined by the precision of the computer) when the matrix is perfectly well conditioned.
A numerical method for predicting hypersonic flowfields
NASA Technical Reports Server (NTRS)
Maccormack, Robert W.; Candler, Graham V.
1989-01-01
The flow about a body traveling at hypersonic speed is energetic enough to cause the atmospheric gases to chemically react and reach states in thermal nonequilibrium. The prediction of hypersonic flowfields requires a numerical method capable of solving the conservation equations of fluid flow, the chemical rate equations for specie formation and dissociation, and the transfer of energy relations between translational and vibrational temperature states. Because the number of equations to be solved is large, the numerical method should also be as efficient as possible. The proposed paper presents a fully implicit method that fully couples the solution of the fluid flow equations with the gas physics and chemistry relations. The method flux splits the inviscid flow terms, central differences of the viscous terms, preserves element conservation in the strong chemistry source terms, and solves the resulting block matrix equation by Gauss Seidel line relaxation.
Daeva, S.G.; Setukha, A.V.
2015-03-10
A numerical method for solving a problem of diffraction of acoustic waves by system of solid and thin objects based on the reduction the problem to a boundary integral equation in which the integral is understood in the sense of finite Hadamard value is proposed. To solve this equation we applied piecewise constant approximations and collocation methods numerical scheme. The difference between the constructed scheme and earlier known is in obtaining approximate analytical expressions to appearing system of linear equations coefficients by separating the main part of the kernel integral operator. The proposed numerical scheme is tested on the solution of the model problem of diffraction of an acoustic wave by inelastic sphere.
Integral Deferred Correction methods for scientific computing
NASA Astrophysics Data System (ADS)
Morton, Maureen Marilla
Since high order numerical methods frequently can attain accurate solutions more efficiently than low order methods, we develop and analyze new high order numerical integrators for the time discretization of ordinary and partial differential equations. Our novel methods address some of the issues surrounding high order numerical time integration, such as the difficulty of many popular methods' construction and handling the effects of disparate behaviors produce by different terms in the equations to be solved. We are motivated by the simplicity of how Deferred Correction (DC) methods achieve high order accuracy [72, 27]. DC methods are numerical time integrators that, rather than calculating tedious coefficients for order conditions, instead construct high order accurate solutions by iteratively improving a low order preliminary numerical solution. With each iteration, an error equation is solved, the error decreases, and the order of accuracy increases. Later, DC methods were adjusted to include an integral formulation of the residual, which stabilizes the method. These Spectral Deferred Correction (SDC) methods [25] motivated Integral Deferred Corrections (IDC) methods. Typically, SDC methods are limited to increasing the order of accuracy by one with each iteration due to smoothness properties imposed by the gridspacing. However, under mild assumptions, explicit IDC methods allow for any explicit rth order Runge-Kutta (RK) method to be used within each iteration, and then an order of accuracy increase of r is attained after each iteration [18]. We extend these results to the construction of implicit IDC methods that use implicit RK methods, and we prove analogous results for order of convergence. One means of solving equations with disparate parts is by semi-implicit integrators, handling a "fast" part implicitly and a "slow" part explicitly. We incorporate additive RK (ARK) integrators into the iterations of IDC methods in order to construct new arbitrary order
A Numerical Study of Hypersonic Forebody/Inlet Integration Problem
NASA Technical Reports Server (NTRS)
Kumar, Ajay
1991-01-01
A numerical study of hypersonic forebody/inlet integration problem is presented in the form of the view-graphs. The following topics are covered: physical/chemical modeling; solution procedure; flow conditions; mass flow rate at inlet face; heating and skin friction loads; 3-D forebogy/inlet integration model; and sensitivity studies.
Numerical methods for supersonic astrophysical jets
NASA Astrophysics Data System (ADS)
Ha, Youngsoo
2003-09-01
The Euler equations of gas dynamics are used for the simulation of general astrophysical fluid flows including high Mach number astrophysical jets with radiative cooling. To accurately compute supersonic jet solutions with sharp resolution of shock waves, three modern numerical methods for gas dynamics were used: (1)a second-order Godunov method in LeVeque's software package CLAWPACK, (2)the Nessyahu-Tadmor-Kurganov (NTK) central hyperbolic scheme, and (3)the WENO-LF (Weighted Essentially Non-Oscillatory Lax-Friedrichs) scheme. Then simulations of supersonic astrophysical jets were compared, first without and then with radiative cooling. CLAWPACK consists of routines for solving time-dependent nonlinear hyperbolic conservation laws based on higher order Godunov methods and approximate Riemann problem solutions; the NTK scheme solves conservation laws using a modified Lax-Friedrichs central difference method without appealing to Riemann problem solutions; and the WENO-LF finite difference scheme is based on the Essentially Non-Oscillatory (ENO) idea by using Lax- Friedrichs flux splitting. The ENO method constructs a solution using the smoothness of the interpolating polynomial on given stencils; on the other hand, the WENO scheme uses a convex combination of the interpolate functions on all candidate stencils. The third-order and fifth-order WENO-LF methods were used to simulate the high Mach number jets. Appropriate numerical methods for incorporating radiative cooling in these numerical methods are also discussed. Interactions of supersonic jets with their environments (jet-“blob” interactions) are shown after modifying the codes to handle high Mach numbers and radiative cooling.
Hyperbolic conservation laws and numerical methods
NASA Technical Reports Server (NTRS)
Leveque, Randall J.
1990-01-01
The mathematical structure of hyperbolic systems and the scalar equation case of conservation laws are discussed. Linear, nonlinear systems and the Riemann problem for the Euler equations are also studied. The numerical methods for conservation laws are presented in a nonstandard manner which leads to large time steps generalizations and computations on irregular grids. The solution of conservation laws with stiff source terms is examined.
Numerical methods for molecular dynamics. Progress report
Skeel, R.D.
1991-12-31
This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods which are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. We first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.
Construction of the two-electron contribution to the Fock matrix by numerical integration
NASA Astrophysics Data System (ADS)
Losilla, Sergio A.; Mehine, Mooses M.; Sundholm, Dage
2012-10-01
A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10-12 E h using the numerical approach.
Numerical methods for finding stationary gravitational solutions
NASA Astrophysics Data System (ADS)
Dias, Óscar J. C.; Santos, Jorge E.; Way, Benson
2016-07-01
The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory–Laflamme zero modes of small rotating black holes in AdS{}5× {S}5. We also include several tools and tricks that have been useful throughout the literature.
Numerical analysis method for linear induction machines.
NASA Technical Reports Server (NTRS)
Elliott, D. G.
1972-01-01
A numerical analysis method has been developed for linear induction machines such as liquid metal MHD pumps and generators and linear motors. Arbitrary phase currents or voltages can be specified and the moving conductor can have arbitrary velocity and conductivity variations from point to point. The moving conductor is divided into a mesh and coefficients are calculated for the voltage induced at each mesh point by unit current at every other mesh point. Combining the coefficients with the mesh resistances yields a set of simultaneous equations which are solved for the unknown currents.
Numerical solution of a class of integral equations arising in two-dimensional aerodynamics
NASA Technical Reports Server (NTRS)
Fromme, J.; Golberg, M. A.
1978-01-01
We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.
Application of numerical methods to elasticity imaging.
Castaneda, Benjamin; Ormachea, Juvenal; Rodríguez, Paul; Parker, Kevin J
2013-03-01
Elasticity imaging can be understood as the intersection of the study of biomechanical properties, imaging sciences, and physics. It was mainly motivated by the fact that pathological tissue presents an increased stiffness when compared to surrounding normal tissue. In the last two decades, research on elasticity imaging has been an international and interdisciplinary pursuit aiming to map the viscoelastic properties of tissue in order to provide clinically useful information. As a result, several modalities of elasticity imaging, mostly based on ultrasound but also on magnetic resonance imaging and optical coherence tomography, have been proposed and applied to a number of clinical applications: cancer diagnosis (prostate, breast, liver), hepatic cirrhosis, renal disease, thyroiditis, arterial plaque evaluation, wall stiffness in arteries, evaluation of thrombosis in veins, and many others. In this context, numerical methods are applied to solve forward and inverse problems implicit in the algorithms in order to estimate viscoelastic linear and nonlinear parameters, especially for quantitative elasticity imaging modalities. In this work, an introduction to elasticity imaging modalities is presented. The working principle of qualitative modalities (sonoelasticity, strain elastography, acoustic radiation force impulse) and quantitative modalities (Crawling Waves Sonoelastography, Spatially Modulated Ultrasound Radiation Force (SMURF), Supersonic Imaging) will be explained. Subsequently, the areas in which numerical methods can be applied to elasticity imaging are highlighted and discussed. Finally, we present a detailed example of applying total variation and AM-FM techniques to the estimation of elasticity. PMID:24010245
Mathematica with a Numerical Methods Course
NASA Astrophysics Data System (ADS)
Varley, Rodney
2003-04-01
An interdisciplinary "Numerical Methods" course has been shared between physics, mathematics and computer science since 1992 at Hunter C. Recently, the lectures and workshops for this course have become formalized and placed on the internet at http://www.ph.hunter.cuny.edu (follow the links "Course Listings and Websites" >> "PHYS385 (Numerical Methods)". Mathematica notebooks for the lectures are available for automatic download (by "double clicking" the lecture icon) for student use in the classroom or at home. AOL (or Netscape/Explorer) can be used provided Mathematica (or the "free" MathReader) has been made a "helper application". Using Mathematica has the virtue that mathematical equations (no LaTex required) can easily be included with the text and Mathematica's graphing is easy to use. Computational cells can be included within the notebook and students may easily modify the calculation to see the result of "what if..." questions. Homework is sent as Mathematica notebooks to the instructor via the internet and the corrected workshops are returned in the same manner. Most exam questions require computational solutions.
A numerical method for interface problems in elastodynamics
NASA Technical Reports Server (NTRS)
Mcghee, D. S.
1984-01-01
The numerical implementation of a formulation for a class of interface problems in elastodynamics is discussed. This formulation combines the use of the finite element and boundary integral methods to represent the interior and the exteriro regions, respectively. In particular, the response of a semicylindrical alluvial valley in a homogeneous halfspace to incident antiplane SH waves is considered to determine the accuracy and convergence of the numerical procedure. Numerical results are obtained from several combinations of the incidence angle, frequency of excitation, and relative stiffness between the inclusion and the surrounding halfspace. The results tend to confirm the theoretical estimates that the convergence is of the order H(2) for the piecewise linear elements used. It was also observed that the accuracy descreases as the frequency of excitation increases or as the relative stiffness of the inclusion decreases.
Numerical Methods of Computational Electromagnetics for Complex Inhomogeneous Systems
Cai, Wei
2014-05-15
Understanding electromagnetic phenomena is the key in many scientific investigation and engineering designs such as solar cell designs, studying biological ion channels for diseases, and creating clean fusion energies, among other things. The objectives of the project are to develop high order numerical methods to simulate evanescent electromagnetic waves occurring in plasmon solar cells and biological ion-channels, where local field enhancement within random media in the former and long range electrostatic interactions in the latter are of major challenges for accurate and efficient numerical computations. We have accomplished these objectives by developing high order numerical methods for solving Maxwell equations such as high order finite element basis for discontinuous Galerkin methods, well-conditioned Nedelec edge element method, divergence free finite element basis for MHD, and fast integral equation methods for layered media. These methods can be used to model the complex local field enhancement in plasmon solar cells. On the other hand, to treat long range electrostatic interaction in ion channels, we have developed image charge based method for a hybrid model in combining atomistic electrostatics and continuum Poisson-Boltzmann electrostatics. Such a hybrid model will speed up the molecular dynamics simulation of transport in biological ion-channels.
Monograph - The Numerical Integration of Ordinary Differential Equations.
ERIC Educational Resources Information Center
Hull, T. E.
The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…
Integrated product definition representation for agile numerical control applications
Simons, W.R. Jr.; Brooks, S.L.; Kirk, W.J. III; Brown, C.W.
1994-11-01
Realization of agile manufacturing capabilities for a virtual enterprise requires the integration of technology, management, and work force into a coordinated, interdependent system. This paper is focused on technology enabling tools for agile manufacturing within a virtual enterprise specifically relating to Numerical Control (N/C) manufacturing activities and product definition requirements for these activities.
Numerical methods for problems in computational aeroacoustics
NASA Astrophysics Data System (ADS)
Mead, Jodi Lorraine
1998-12-01
A goal of computational aeroacoustics is the accurate calculation of noise from a jet in the far field. This work concerns the numerical aspects of accurately calculating acoustic waves over large distances and long time. More specifically, the stability, efficiency, accuracy, dispersion and dissipation in spatial discretizations, time stepping schemes, and absorbing boundaries for the direct solution of wave propagation problems are determined. Efficient finite difference methods developed by Tam and Webb, which minimize dispersion and dissipation, are commonly used for the spatial and temporal discretization. Alternatively, high order pseudospectral methods can be made more efficient by using the grid transformation introduced by Kosloff and Tal-Ezer. Work in this dissertation confirms that the grid transformation introduced by Kosloff and Tal-Ezer is not spectrally accurate because, in the limit, the grid transformation forces zero derivatives at the boundaries. If a small number of grid points are used, it is shown that approximations with the Chebyshev pseudospectral method with the Kosloff and Tal-Ezer grid transformation are as accurate as with the Chebyshev pseudospectral method. This result is based on the analysis of the phase and amplitude errors of these methods, and their use for the solution of a benchmark problem in computational aeroacoustics. For the grid transformed Chebyshev method with a small number of grid points it is, however, more appropriate to compare its accuracy with that of high- order finite difference methods. This comparison, for an order of accuracy 10-3 for a benchmark problem in computational aeroacoustics, is performed for the grid transformed Chebyshev method and the fourth order finite difference method of Tam. Solutions with the finite difference method are as accurate. and the finite difference method is more efficient than, the Chebyshev pseudospectral method with the grid transformation. The efficiency of the Chebyshev
Numerical methods for scattering from electrically large objects
NASA Astrophysics Data System (ADS)
Enguist, Bjorn; Murphy, W. D.; Rokhlin, Vladimir; Vassiliou, Marius S.
1991-05-01
A new and computationally very efficient integral equation numerical method for computing electromagnetic scattering and radar cross section (RCS) was developed. A theory of higher order impedance boundary conditions was derived to handle single and multiple dielectric coatings around conductors. The method was tested in two dimensions using a 14,000-line FORTRAN program and was found to be very promising for electrically large objects. Initial ideas for extensions to three dimensions were explored. Treatments of trailing edge and corner singularities were developed.
Technical Report: Scalable Parallel Algorithms for High Dimensional Numerical Integration
Masalma, Yahya; Jiao, Yu
2010-10-01
We implemented a scalable parallel quasi-Monte Carlo numerical high-dimensional integration for tera-scale data points. The implemented algorithm uses the Sobol s quasi-sequences to generate random samples. Sobol s sequence was used to avoid clustering effects in the generated random samples and to produce low-discrepancy random samples which cover the entire integration domain. The performance of the algorithm was tested. Obtained results prove the scalability and accuracy of the implemented algorithms. The implemented algorithm could be used in different applications where a huge data volume is generated and numerical integration is required. We suggest using the hyprid MPI and OpenMP programming model to improve the performance of the algorithms. If the mixed model is used, attention should be paid to the scalability and accuracy.
Controlled time integration for the numerical simulation of meteor radar reflections
NASA Astrophysics Data System (ADS)
Räbinä, Jukka; Mönkölä, Sanna; Rossi, Tuomo; Markkanen, Johannes; Gritsevich, Maria; Muinonen, Karri
2016-07-01
We model meteoroids entering the Earth's atmosphere as objects surrounded by non-magnetized plasma, and consider efficient numerical simulation of radar reflections from meteors in the time domain. Instead of the widely used finite difference time domain method (FDTD), we use more generalized finite differences by applying the discrete exterior calculus (DEC) and non-uniform leapfrog-style time discretization. The computational domain is presented by convex polyhedral elements. The convergence of the time integration is accelerated by the exact controllability method. The numerical experiments show that our code is efficiently parallelized. The DEC approach is compared to the volume integral equation (VIE) method by numerical experiments. The result is that both methods are competitive in modelling non-magnetized plasma scattering. For demonstrating the simulation capabilities of the DEC approach, we present numerical experiments of radar reflections and vary parameters in a wide range.
Numerical Method for the Astronomical Almanac and Orbit Calculations
NASA Astrophysics Data System (ADS)
Kim, Kap-Sung
1993-12-01
We have calculated the astronomical almanac 1994 and simulated the trajectory of a satellite orbit considering all perturbative forces with various initial conditions. In this work, Gauss Jackson multistep integration method has been used to calculate our basic equation of motion with high numerical accuracy. It has been found that our results agree well with the Astronomical Almanac Data distributed by JPL of NASA and the orbit simulations have been carried out with fast speed, stability and excellent round-off error accumulation, comparing with other numerical methods. In order to be carried out our works on almanac and orbit calculations easily by anyone who uses a personal computer, we have made a computer program on graphical user interface to provide various menus for detail works selected by a mouse.
Toyoda, Masayuki; Ozaki, Taisuke
2009-03-28
A numerical method to calculate the four-center electron-repulsion integrals for strictly localized pseudoatomic orbital basis sets has been developed. Compared to the conventional Gaussian expansion method, this method has an advantage in the ease of combination with O(N) density functional calculations. Additional mathematical derivations are also presented including the analytic derivatives of the integrals with respect to atomic positions and spatial damping of the Coulomb interaction due to the screening effect. In the numerical test for a simple molecule, the convergence up to 10(-5) hartree in energy is successfully obtained with a feasible cost of computation. PMID:19334815
Ensemble-type numerical uncertainty information from single model integrations
Rauser, Florian Marotzke, Jochem; Korn, Peter
2015-07-01
We suggest an algorithm that quantifies the discretization error of time-dependent physical quantities of interest (goals) for numerical models of geophysical fluid dynamics. The goal discretization error is estimated using a sum of weighted local discretization errors. The key feature of our algorithm is that these local discretization errors are interpreted as realizations of a random process. The random process is determined by the model and the flow state. From a class of local error random processes we select a suitable specific random process by integrating the model over a short time interval at different resolutions. The weights of the influences of the local discretization errors on the goal are modeled as goal sensitivities, which are calculated via automatic differentiation. The integration of the weighted realizations of local error random processes yields a posterior ensemble of goal approximations from a single run of the numerical model. From the posterior ensemble we derive the uncertainty information of the goal discretization error. This algorithm bypasses the requirement of detailed knowledge about the models discretization to generate numerical error estimates. The algorithm is evaluated for the spherical shallow-water equations. For two standard test cases we successfully estimate the error of regional potential energy, track its evolution, and compare it to standard ensemble techniques. The posterior ensemble shares linear-error-growth properties with ensembles of multiple model integrations when comparably perturbed. The posterior ensemble numerical error estimates are of comparable size as those of a stochastic physics ensemble.
A Hybrid Numerical Analysis Method for Structural Health Monitoring
NASA Technical Reports Server (NTRS)
Forth, Scott C.; Staroselsky, Alexander
2001-01-01
A new hybrid surface-integral-finite-element numerical scheme has been developed to model a three-dimensional crack propagating through a thin, multi-layered coating. The finite element method was used to model the physical state of the coating (far field), and the surface integral method was used to model the fatigue crack growth. The two formulations are coupled through the need to satisfy boundary conditions on the crack surface and the external boundary. The coupling is sufficiently weak that the surface integral mesh of the crack surface and the finite element mesh of the uncracked volume can be set up independently. Thus when modeling crack growth, the finite element mesh can remain fixed for the duration of the simulation as the crack mesh is advanced. This method was implemented to evaluate the feasibility of fabricating a structural health monitoring system for real-time detection of surface cracks propagating in engine components. In this work, the authors formulate the hybrid surface-integral-finite-element method and discuss the mechanical issues of implementing a structural health monitoring system in an aircraft engine environment.
COMPARING NUMERICAL METHODS FOR ISOTHERMAL MAGNETIZED SUPERSONIC TURBULENCE
Kritsuk, Alexei G.; Collins, David; Norman, Michael L.; Xu Hao E-mail: dccollins@lanl.gov
2011-08-10
Many astrophysical applications involve magnetized turbulent flows with shock waves. Ab initio star formation simulations require a robust representation of supersonic turbulence in molecular clouds on a wide range of scales imposing stringent demands on the quality of numerical algorithms. We employ simulations of supersonic super-Alfvenic turbulence decay as a benchmark test problem to assess and compare the performance of nine popular astrophysical MHD methods actively used to model star formation. The set of nine codes includes: ENZO, FLASH, KT-MHD, LL-MHD, PLUTO, PPML, RAMSES, STAGGER, and ZEUS. These applications employ a variety of numerical approaches, including both split and unsplit, finite difference and finite volume, divergence preserving and divergence cleaning, a variety of Riemann solvers, and a range of spatial reconstruction and time integration techniques. We present a comprehensive set of statistical measures designed to quantify the effects of numerical dissipation in these MHD solvers. We compare power spectra for basic fields to determine the effective spectral bandwidth of the methods and rank them based on their relative effective Reynolds numbers. We also compare numerical dissipation for solenoidal and dilatational velocity components to check for possible impacts of the numerics on small-scale density statistics. Finally, we discuss the convergence of various characteristics for the turbulence decay test and the impact of various components of numerical schemes on the accuracy of solutions. The nine codes gave qualitatively the same results, implying that they are all performing reasonably well and are useful for scientific applications. We show that the best performing codes employ a consistently high order of accuracy for spatial reconstruction of the evolved fields, transverse gradient interpolation, conservation law update step, and Lorentz force computation. The best results are achieved with divergence-free evolution of the
Fast integral methods for integrated optical systems simulations: a review
NASA Astrophysics Data System (ADS)
Kleemann, Bernd H.
2015-09-01
Boundary integral equation methods (BIM) or simply integral methods (IM) in the context of optical design and simulation are rigorous electromagnetic methods solving Helmholtz or Maxwell equations on the boundary (surface or interface of the structures between two materials) for scattering or/and diffraction purposes. This work is mainly restricted to integral methods for diffracting structures such as gratings, kinoforms, diffractive optical elements (DOEs), micro Fresnel lenses, computer generated holograms (CGHs), holographic or digital phase holograms, periodic lithographic structures, and the like. In most cases all of the mentioned structures have dimensions of thousands of wavelengths in diameter. Therefore, the basic methods necessary for the numerical treatment are locally applied electromagnetic grating diffraction algorithms. Interestingly, integral methods belong to the first electromagnetic methods investigated for grating diffraction. The development started in the mid 1960ies for gratings with infinite conductivity and it was mainly due to the good convergence of the integral methods especially for TM polarization. The first integral equation methods (IEM) for finite conductivity were the methods by D. Maystre at Fresnel Institute in Marseille: in 1972/74 for dielectric, and metallic gratings, and later for multiprofile, and other types of gratings and for photonic crystals. Other methods such as differential and modal methods suffered from unstable behaviour and slow convergence compared to BIMs for metallic gratings in TM polarization from the beginning to the mid 1990ies. The first BIM for gratings using a parametrization of the profile was developed at Karl-Weierstrass Institute in Berlin under a contract with Carl Zeiss Jena works in 1984-1986 by A. Pomp, J. Creutziger, and the author. Due to the parametrization, this method was able to deal with any kind of surface grating from the beginning: whether profiles with edges, overhanging non
On the stability of numerical integration routines for ordinary differential equations.
NASA Technical Reports Server (NTRS)
Glover, K.; Willems, J. C.
1973-01-01
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Integrated numeric and symbolic signal processing using a heterogeneous design environment
NASA Astrophysics Data System (ADS)
Mani, Ramamurthy; Nawab, S. Hamid; Winograd, Joseph M.; Evans, Brian L.
1996-10-01
We present a solution to a complex multi-tone transient detection problem to illustrate the integrated use of symbolic and numeric processing techniques which are supported by well-established underlying models. Examples of such models include synchronous dataflow for numeric processing and the blackboard paradigm for symbolic heuristic search. Our transient detection solution serves to emphasize the importance of developing system design methods and tools which can support the integrated use of well- established symbolic and numerical models of computation. Recently, we incorporated a blackboard-based model of computation underlying the Integrated Processing and Understanding of Signals (IPUS) paradigm into a system-level design environment for numeric processing called Ptolemy. Using the IPUS/Ptolemy environment, we are implementing our solution to the multi-tone transient detection problem.
Numerical computation of 2D Sommerfeld integrals - Decomposition of the angular integral
NASA Astrophysics Data System (ADS)
Dvorak, Steven L.; Kuester, Edward F.
1992-02-01
The computational efficiency of the 2D Sommerfeld integrals is shown to undergo improvement through the discovery of novel ways to compute the inner angular integral in polar representations. It is shown that the angular integral can be decomposed into a finite number of incomplete Lipschitz-Hankel integrals; these can in turn be calculated through a series of expansions, so that the angular integral can be computed by summing a series rather than applying a standard numerical integration algorithm. The technique is most efficient and accurate when piecewise-sinusoidal basis functions are employed to analyze a printed strip-dipole antenna in a layered medium.
Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow
NASA Astrophysics Data System (ADS)
Grooms, Ian G.
This thesis documents three investigations carried out in pursuance of a doctoral degree in applied mathematics at the University of Colorado (Boulder). The first investigation concerns the properties of rotating Rayleigh-Benard convection -- thermal convection in a rotating infinite plane layer between two constant-temperature boundaries. It is noted that in certain parameter regimes convective Taylor columns appear which dominate the dynamics, and a semi-analytical model of these is presented. Investigation of the columns and of various other properties of the flow is ongoing. The second investigation concerns the interactions between planetary-scale and mesoscale dynamics in the oceans. Using multiple-scale asymptotics the possible connections between planetary geostrophic and quasigeostrophic dynamics are investigated, and three different systems of coupled equations are derived. Possible use of these equations in conjunction with the method of superparameterization, and extension of the asymptotic methods to the interactions between mesoscale and submesoscale dynamics is ongoing. The third investigation concerns the linear stability properties of semi-implicit methods for the numerical integration of ordinary differential equations, focusing in particular on the linear stability of IMEX (Implicit-Explicit) methods and exponential integrators applied to systems of ordinary differential equations arising in the numerical solution of spatially discretized nonlinear partial differential equations containing both dispersive and dissipative linear terms. While these investigations may seem unrelated at first glance, some reflection shows that they are in fact closely linked. The investigation of rotating convection makes use of single-space, multiple-time-scale asymptotics to deal with dynamics strongly constrained by rotation. Although the context of thermal convection in an infinite layer seems somewhat removed from large-scale ocean dynamics, the asymptotic
Numerical methods for analyzing electromagnetic scattering
NASA Technical Reports Server (NTRS)
Lee, S. W.; Lo, Y. T.; Chuang, S. L.; Lee, C. S.
1985-01-01
Numerical methods to analyze electromagnetic scattering are presented. The dispersions and attenuations of the normal modes in a circular waveguide coated with lossy material were completely analyzed. The radar cross section (RCS) from a circular waveguide coated with lossy material was calculated. The following is observed: (1) the interior irradiation contributes to the RCS much more than does the rim diffraction; (2) at low frequency, the RCS from the circular waveguide terminated by a perfect electric conductor (PEC) can be reduced more than 13 dB down with a coating thickness less than 1% of the radius using the best lossy material available in a 6 radius-long cylinder; (3) at high frequency, a modal separation between the highly attenuated and the lowly attenuated modes is evident if the coating material is too lossy, however, a large RCS reduction can be achieved for a small incident angle with a thin layer of coating. It is found that the waveguide coated with a lossy magnetic material can be used as a substitute for a corrugated waveguide to produce a circularly polarized radiation yield.
Efficient integration method for fictitious domain approaches
NASA Astrophysics Data System (ADS)
Duczek, Sascha; Gabbert, Ulrich
2015-10-01
In the current article, we present an efficient and accurate numerical method for the integration of the system matrices in fictitious domain approaches such as the finite cell method (FCM). In the framework of the FCM, the physical domain is embedded in a geometrically larger domain of simple shape which is discretized using a regular Cartesian grid of cells. Therefore, a spacetree-based adaptive quadrature technique is normally deployed to resolve the geometry of the structure. Depending on the complexity of the structure under investigation this method accounts for most of the computational effort. To reduce the computational costs for computing the system matrices an efficient quadrature scheme based on the divergence theorem (Gauß-Ostrogradsky theorem) is proposed. Using this theorem the dimension of the integral is reduced by one, i.e. instead of solving the integral for the whole domain only its contour needs to be considered. In the current paper, we present the general principles of the integration method and its implementation. The results to several two-dimensional benchmark problems highlight its properties. The efficiency of the proposed method is compared to conventional spacetree-based integration techniques.
A numerical method for cardiac mechanoelectric simulations.
Pathmanathan, Pras; Whiteley, Jonathan P
2009-05-01
Much effort has been devoted to developing numerical techniques for solving the equations that describe cardiac electrophysiology, namely the monodomain equations and bidomain equations. Only a limited selection of publications, however, address the development of numerical techniques for mechanoelectric simulations where cardiac electrophysiology is coupled with deformation of cardiac tissue. One problem commonly encountered in mechanoelectric simulations is instability of the coupled numerical scheme. In this study, we develop a stable numerical scheme for mechanoelectric simulations. A number of convergence tests are carried out using this stable technique for simulations where deformations are of the magnitude typically observed in a beating heart. These convergence tests demonstrate that accurate computation of tissue deformation requires a nodal spacing of around 1 mm in the mesh used to calculate tissue deformation. This is a much finer computational grid than has previously been acknowledged, and has implications for the computational efficiency of the resulting numerical scheme. PMID:19263223
Romá, Federico; Cugliandolo, Leticia F; Lozano, Gustavo S
2014-08-01
We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy. PMID:25215839
Time-dependent corona models - A numerical method
NASA Astrophysics Data System (ADS)
Korevaar, P.; van Leer, B.
1988-07-01
A time-dependent numerical method for calculating gas flows is described. The method is implicit and especially suitable for finding stationary flow solutions. Although the method is general in its application to ideal compressible fluids, this paper applies it to a stellar atmosphere, heated to coronal temperatures by dissipation of mechanical energy. The integration scheme is based on conservative upwind spatial differencing. The upwind switching is provided by Van Leer's method of differentiable flux-splitting. It is shown that the code can handle large differences in density: up to 14 orders of magnitude. Special attention is paid to the boundary conditions, which are made completely transparent to disturbances. Besides some test-results, converged solutions for various values of the initial mechanical flux are presented which are in good agreement with previous time-independent calculations.
Integration methods for molecular dynamics
Leimkuhler, B.J.; Reich, S.; Skeel, R.D.
1996-12-31
Classical molecular dynamics simulation of a macromolecule requires the use of an efficient time-stepping scheme that can faithfully approximate the dynamics over many thousands of timesteps. Because these problems are highly nonlinear, accurate approximation of a particular solution trajectory on meaningful time intervals is neither obtainable nor desired, but some restrictions, such as symplecticness, can be imposed on the discretization which tend to imply good long term behavior. The presence of a variety of types and strengths of interatom potentials in standard molecular models places severe restrictions on the timestep for numerical integration used in explicit integration schemes, so much recent research has concentrated on the search for alternatives that possess (1) proper dynamical properties, and (2) a relative insensitivity to the fastest components of the dynamics. We survey several recent approaches. 48 refs., 2 figs.
Numerical integration of discontinuities on arbitrary domains based on moment fitting
NASA Astrophysics Data System (ADS)
Joulaian, Meysam; Hubrich, Simeon; Düster, Alexander
2016-03-01
Discretization methods based on meshes that do not conform to the geometry of the problem under consideration require special treatment when it comes to the integration of finite elements that are broken by the boundary or internal interfaces. To this end, we propose a numerical approach suitable for integrating broken elements with a low number of integration points. In this method, which is based on the moment fitting approach, an individual quadrature rule is set up for each cut element. The approach requires a B-rep representation of the broken element, which can be either achieved by processing a triangulated surface obtained from a CAD software or by taking advantage of a voxel model resulting from computed tomography. The numerical examples presented in this paper reveal that the proposed method delivers for a wide variety of geometrical situations very accurate results and requires a rather low number of integration points.
Numerical integration of discontinuities on arbitrary domains based on moment fitting
NASA Astrophysics Data System (ADS)
Joulaian, Meysam; Hubrich, Simeon; Düster, Alexander
2016-06-01
Discretization methods based on meshes that do not conform to the geometry of the problem under consideration require special treatment when it comes to the integration of finite elements that are broken by the boundary or internal interfaces. To this end, we propose a numerical approach suitable for integrating broken elements with a low number of integration points. In this method, which is based on the moment fitting approach, an individual quadrature rule is set up for each cut element. The approach requires a B-rep representation of the broken element, which can be either achieved by processing a triangulated surface obtained from a CAD software or by taking advantage of a voxel model resulting from computed tomography. The numerical examples presented in this paper reveal that the proposed method delivers for a wide variety of geometrical situations very accurate results and requires a rather low number of integration points.
[Numerical methods for multi-fluid flows]. Final progress report
Pozrikidis, C.
1998-07-21
The central objective of this research has been to develop efficient numerical methods for computing multi-fluid flows with large interfacial deformations, and apply these methods to study the rheology of suspensions of deformable particles with viscous and non-Newtonian interfacial behavior. The mathematical formulation employs boundary-integral, immersed-boundary, and related numerical methods. Particles of interest include liquid drops with constant surface tension and capsules whose interfaces exhibit viscoelastic and incompressible characteristics. In one family of problems, the author has considered the shear-driven and pressure-driven flow of a suspension of two-dimensional liquid drops with ordered and random structure. In a second series of investigations, the author carried out dynamic simulations of two-dimensional, unbounded, doubly-periodic shear flows with random structure. Another family of problems addresses the deformation of three-dimensional capsules whose interfaces exhibit isotropic surface tension, viscous, elastic, or incompressible behavior, in simple shear flow. The numerical results extend previous asymptotic theories for small deformations and illuminate the mechanism of membrane rupture.
The instanton method and its numerical implementation in fluid mechanics
NASA Astrophysics Data System (ADS)
Grafke, Tobias; Grauer, Rainer; Schäfer, Tobias
2015-08-01
A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin-Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler-Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional (2D or 3D) fluid dynamical problems. We illustrate these ideas using the 2D Burgers equation and the 3D Navier-Stokes equations.
Integrating Numerical Groundwater Modeling Results With Geographic Information Systems
NASA Astrophysics Data System (ADS)
Witkowski, M. S.; Robinson, B. A.; Linger, S. P.
2001-12-01
Many different types of data are used to create numerical models of flow and transport of groundwater in the vadose zone. Results from water balance studies, infiltration models, hydrologic properties, and digital elevation models (DEMs) are examples of such data. Because input data comes in a variety of formats, for consistency the data need to be assembled in a coherent fashion on a single platform. Through the use of a geographic information system (GIS), all data sources can effectively be integrated on one platform to store, retrieve, query, and display data. In our vadoze zone modeling studies in support of Los Alamos National Laboratory's Environmental Restoration Project, we employ a GIS comprised of a Raid storage device, an Oracle database, ESRI's spatial database engine (SDE), ArcView GIS, and custom GIS tools for three-dimensional (3D) analysis. We store traditional GIS data, such as, contours, historical building footprints, and study area locations, as points, lines, and polygons with attributes. Numerical flow and transport model results from the Finite Element Heat and Mass Transfer Code (FEHM) are stored as points with attributes, such as fluid saturation, or pressure, or contaminant concentration at a given location. We overlay traditional types of GIS data with numerical model results, thereby allowing us to better build conceptual models and perform spatial analyses. We have also developed specialized analysis tools to assist in the data and model analysis process. This approach provides an integrated framework for performing tasks such as comparing the model to data and understanding the relationship of model predictions to existing contaminant source locations and water supply wells. Our process of integrating GIS and numerical modeling results allows us to answer a wide variety of questions about our conceptual model design: - Which set of locations should be identified as contaminant sources based on known historical building operations
Comparison of integrated numerical experiments with accelerator and FEL experiments
Thode, L.E.; Carlsten, B.E.; Chan, K.C.D.; Cooper, R.K.; Elliott, J.C.; Gitomer, S.J.; Goldstein, J.C.; Jones, M.E.; McVey, B.D.; Schmitt, M.J.; Takeda, H.; Tokar, R.L.; Wang, T.S.; Young, L.M.
1991-01-01
Even at the conceptual level the strong coupling between the laser subsystem elements, such as the accelerator, wiggler, optics, and control, greatly complicates the understanding and design of an FEL. Given the requirements for a high-performance FEL, the coupling between the laser subsystems must be included in the design approach. To address the subsystem coupling the concept of an integrated numerical experiment (INEX) has been implemented. Unique features of the INEX approach are consistency and numerical equivalence of experimental diagnostic. The equivalent numerical diagnostics mitigates the major problem of misinterpretation that often occurs when theoretical and experimental data are compared. A complete INEX model has been applied to the 10{mu}m high-extraction-efficiency experiment at Los Alamos and the 0.6-{mu}m Burst Mode experiment at Boeing Aerospace. In addition, various subsets of the INEX model have been compared with a number of other experiments. Overall, the agreement between INEX and the experiments is very good. With the INEX approach, it now appears possible to design high-performance FELS for numerous applications. The first full-scale test of the INEX approach is the Los Alamos HIBAF experiment. The INEX concept, implementation, and validation with experiments are discussed. 28 refs., 13 figs., 1 tab.
INEX (integrated numerical experiment) simulations of the Boeing FEL system
Tokar, R.L.; Young, L.M.; Lumpkin, A.H.; McVey, B.D.; Thode, L.E.; Bender, S.C.; Chan, K.C.D. ); Yeremian, A.D.; Dowell, D.H.; Lowrey, A.R. )
1989-01-01
The INEX (integrated numerical experiment) numerical model is applied to the 0.6 {mu}m FEL oscillator at Boeing Aerospace and Electronics Company in Seattle, WA. This system consists of a 110 MeV L-band rf linac, a beam transport line from the accelerator to the entrance of the wiggler, the 5.0 meter THUNDER variable taper wiggler, and a near concentric two mirror optical oscillator. Many aspects of the model for the electron beam accelerator and transport line agree with experimental measurements. Predictions for lasing performance are compared with data obtained in May and June 1989 using a mild tapered wiggler. We obtain good agreement with the achieved extraction efficiency, while 1D pulse simulations reproduce the observed sideband instability. 15 refs., 11 figs.
Numerical methods for determining interstitial oxygen in silicon
Stevenson, J.O.; Medernach, J.W.
1995-01-01
The interstitial oxygen (O{sub i}) concentration in Czochralski silicon and the subsequent SiO{sub x} precipitation are important parameters for integrated circuit fabrication. Uncontrolled SiO{sub x} precipitation during processing can create detrimental mechanical and electrical effects that contribute to poor performance. An inability to consistently and accurately measure the initial O{sub i} concentration in heavily doped silicon has led to contradictory results regarding the effects of dopant type and concentration on SiO{sub x} precipitation. The authors have developed a software package for reliably determining and comparing O{sub i} in heavily doped silicon. The SiFTIR{copyright} code implements three independent oxygen analysis methods in a single integrated package. Routine oxygen measurements are desirable over a wide range of silicon resistivities, but there has been confusion concerning which of the three numerical methods is most suitable for the low resistivity portion of the continuum. A major strength of the software is an ability to rapidly produce results for all three methods using only a single Fourier Transform Infrared Spectroscopy (FTIR) spectrum as input. This ability to perform three analyses on a single data set allows a detailed comparison of the three methods across the entire range of resistivities in question. Integrated circuit manufacturers could use the enabling technology provided by SiFTIR{copyright} to monitor O{sub i} content. Early detection of O{sub i} using this diagnostic could be beneficial in controlling SiO{sub x} precipitation during integrated circuit processing.
The Use of Phase-Lag Derivatives in the Numerical Integration of ODEs with Oscillating Solutions
Anastassi, Z. A.; Vlachos, D. S.; Simos, T. E.
2008-09-01
In this paper we consider the fitting of the coefficients of a numerical method, not only due to the nullification of the phase-lag, but also to its derivatives. We show that the method gains efficiency with each derivative of the phase-lag nullified for various problems with oscillating solutions. The analysis of the local truncation error analysis and the stability of the methods show the importance of zero phase-lag derivatives when integrating oscillatory differential equations.
Numerical solutions to ill-posed and well-posed impedance boundary condition integral equations
NASA Astrophysics Data System (ADS)
Rogers, J. R.
1983-11-01
Exterior scattering from a three-dimensional impedance body can be formulated in terms of various integral equations derived from the Leontovich impedance boundary condition (IBC). The electric and magnetic field integral equations are ill-posed because they theoretically admit spurious solutions at the frequencies of interior perfect conductor cavity resonances. A combined field formulation is well-posed because it does not allow the spurious solutions. This report outlines the derivation of IBC integral equations and describes a procedure for constructing moment-method solutions for bodies of revolution. Numerical results for scattering from impedance spheres are presented which contrast the stability and accuracy of solutions to the ill-posed equations with those of the well-posed equation. The results show that numerical solutions for exterior scattering to the electric and magnetic field integral equations can be severely contaminated by spurious resonant solutions regardless of whether the surface impedance of the body is lossy or lossless.
NLOS UV channel modeling using numerical integration and an approximate closed-form path loss model
NASA Astrophysics Data System (ADS)
Gupta, Ankit; Noshad, Mohammad; Brandt-Pearce, Maïté
2012-10-01
In this paper we propose a simulation method using numerical integration, and develop a closed-form link loss model for physical layer channel characterization for non-line of sight (NLOS) ultraviolet (UV) communication systems. The impulse response of the channel is calculated by assuming both uniform and Gaussian profiles for transmitted beams and different geometries. The results are compared with previously published results. The accuracy of the integration approach is compared to the Monte Carlo simulation. Then the path loss using the simulation method and the suggested closed-form expression are presented for different link geometries. The accuracies are evaluated and compared to the results obtained using other methods.
Numerical performance of projection methods in finite element consolidation models
NASA Astrophysics Data System (ADS)
Gambolati, Giuseppe; Pini, Giorgio; Ferronato, Massimiliano
2001-12-01
Projection, or conjugate gradient like, methods are becoming increasingly popular for the efficient solution of large sparse sets of unsymmetric indefinite equations arising from the numerical integration of (initial) boundary value problems. One such problem is soil consolidation coupling a flow and a structural model, typically solved by finite elements (FE) in space and a marching scheme in time (e.g. the Crank-Nicolson scheme). The attraction of a projection method stems from a number of factors, including the ease of implementation, the requirement of limited core memory and the low computational cost if a cheap and effective matrix preconditioner is available. In the present paper, biconjugate gradient stabilized (Bi- CGSTAB) is used to solve FE consolidation equations in 2-D and 3-D settings with variable time integration steps. Three different nodal orderings are selected along with the preconditioner ILUT based on incomplete triangular factorization and variable fill-in. The overall cost of the solver is made up of the preconditioning cost plus the cost to converge which is in turn related to the number of iterations and the elementary operations required by each iteration. The results show that nodal ordering affects the perfor mance of Bi-CGSTAB. For normally conditioned consolidation problems Bi-CGSTAB with the best ILUT preconditioner may converge in a number of iterations up to two order of magnitude smaller than the size of the FE model and proves an accurate, cost-effective and robust alternative to direct methods.
Finite element methods in numerical relativity.
NASA Astrophysics Data System (ADS)
Mann, P. J.
The finite element method is very successful in Newtonian fluid simulations, and can be extended to relativitstic fluid flows. This paper describes the general method, and then outlines some preliminary results for spherically symmetric geometries. The mixed finite element - finite difference scheme is introduced, and used for the description of spherically symmetric collapse. Baker's (Newtonian) shock modelling method and Miller's moving finite element method are also mentioned. Collapse in double-null coordinates requires non-constant time slicing, so the full finite element method in space and time is described.
Numerical implications of stabilization by the use of integrals
NASA Technical Reports Server (NTRS)
Beaudet, P. R.
1975-01-01
Liapunov or energy restraint methods for dynamic stabilization in two body motion perturbation problems are considered. Results of computerized orbital stabilization estimates show that the application of energy restraint prevents the occurrence of consistent timing errors in the stepwise integration of equations of motion for a nearly circular orbit.
Teaching Thermal Hydraulics & Numerical Methods: An Introductory Control Volume Primer
D. S. Lucas
2004-10-01
A graduate level course for Thermal Hydraulics (T/H) was taught through Idaho State University in the spring of 2004. A numerical approach was taken for the content of this course since the students were employed at the Idaho National Laboratory and had been users of T/H codes. The majority of the students had expressed an interest in learning about the Courant Limit, mass error, semi-implicit and implicit numerical integration schemes in the context of a computer code. Since no introductory text was found the author developed notes taught from his own research and courses taught for Westinghouse on the subject. The course started with a primer on control volume methods and the construction of a Homogeneous Equilibrium Model (HEM) (T/H) code. The primer was valuable for giving the students the basics behind such codes and their evolution to more complex codes for Thermal Hydraulics and Computational Fluid Dynamics (CFD). The course covered additional material including the Finite Element Method and non-equilibrium (T/H). The control volume primer and the construction of a three-equation (mass, momentum and energy) HEM code are the subject of this paper . The Fortran version of the code covered in this paper is elementary compared to its descendants. The steam tables used are less accurate than the available commercial version written in C Coupled to a Graphical User Interface (GUI). The Fortran version and input files can be downloaded at www.microfusionlab.com.
Numerical matrix method for quantum periodic potentials
NASA Astrophysics Data System (ADS)
Le Vot, Felipe; Meléndez, Juan J.; Yuste, Santos B.
2016-06-01
A numerical matrix methodology is applied to quantum problems with periodic potentials. The procedure consists essentially in replacing the true potential by an alternative one, restricted by an infinite square well, and in expressing the wave functions as finite superpositions of eigenfunctions of the infinite well. A matrix eigenvalue equation then yields the energy levels of the periodic potential within an acceptable accuracy. The methodology has been successfully used to deal with problems based on the well-known Kronig-Penney (KP) model. Besides the original model, these problems are a dimerized KP solid, a KP solid containing a surface, and a KP solid under an external field. A short list of additional problems that can be solved with this procedure is presented.
Method for numerical simulations of metastable states
Heller, U.M.; Seiberg, N.
1983-06-15
We present a numerical simulation of metastable states near a first-order phase transition in the example of a U(1) lattice gauge theory with a generalized action. In order to make measurements in these states possible their decay has to be prevented. We achieve this by using a microcanonical simulation for a finite system. We then obtain the coupling constant (inverse temperature) as a function of the action density. It turns out to be nonmonotonic and hence not uniquely invertible. From it we derive the effective potential for the action density. This effective potential is not always convex, a property that seems to be in contradiction with the standard lore about its convexity. This apparent ''paradox'' is resolved in a discussion about different definitions of the effective potential.
Numerical methods in Markov chain modeling
NASA Technical Reports Server (NTRS)
Philippe, Bernard; Saad, Youcef; Stewart, William J.
1989-01-01
Several methods for computing stationary probability distributions of Markov chains are described and compared. The main linear algebra problem consists of computing an eigenvector of a sparse, usually nonsymmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous singular linear system. Several methods based on combinations of Krylov subspace techniques are presented. The performance of these methods on some realistic problems are compared.
Interpolation Method Needed for Numerical Uncertainty
NASA Technical Reports Server (NTRS)
Groves, Curtis E.; Ilie, Marcel; Schallhorn, Paul A.
2014-01-01
Using Computational Fluid Dynamics (CFD) to predict a flow field is an approximation to the exact problem and uncertainties exist. There is a method to approximate the errors in CFD via Richardson's Extrapolation. This method is based off of progressive grid refinement. To estimate the errors, the analyst must interpolate between at least three grids. This paper describes a study to find an appropriate interpolation scheme that can be used in Richardson's extrapolation or other uncertainty method to approximate errors.
Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations
Weinstein, Marvin; /SLAC
2009-02-12
It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will
A selective integrated tempering method.
Yang, Lijiang; Qin Gao, Yi
2009-12-01
In this paper, based on the integrated tempering sampling we introduce a selective integrated tempering sampling (SITS) method for the efficient conformation sampling and thermodynamics calculations for a subsystem in a large one, such as biomolecules solvated in aqueous solutions. By introducing a potential surface scaled with temperature, the sampling over the configuration space of interest (e.g., the solvated biomolecule) is selectively enhanced but the rest of the system (e.g., the solvent) stays largely unperturbed. The applications of this method to biomolecular systems allow highly efficient sampling over both energy and configuration spaces of interest. Comparing to the popular and powerful replica exchange molecular dynamics (REMD), the method presented in this paper is significantly more efficient in yielding relevant thermodynamics quantities (such as the potential of mean force for biomolecular conformational changes in aqueous solutions). It is more important that SITS but not REMD yielded results that are consistent with the traditional umbrella sampling free energy calculations when explicit solvent model is used since SITS avoids the sampling of the irrelevant phase space (such as the boiling water at high temperatures). PMID:19968339
A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation
NASA Astrophysics Data System (ADS)
Lu, Xinhua; Dong, Bingjiang; Mao, Bing; Zhang, Xiaofeng
2015-12-01
In this study, we develop a shallow-water depth-integrated non-hydrostatic numerical model (SNH model) using a hybrid finite-volume and finite-difference method. Numerical discretization is performed using the non-incremental pressure-correction method on a collocated grid. We demonstrate that an extension can easily be made from an existing finite-volume method and collocated-grid based hydrostatic shallow-water equations (SWE) model to a non-hydrostatic model. A series of benchmark tests are used to validate the proposed numerical model. Our results demonstrate that the proposed model is robust and well-balanced, and it captures the wet-dry fronts accurately. A comparison between the SNH and SWE models indicates the importance of considering the wave dispersion effect in simulations when the wave amplitude to water depth ratio is large.
Space-time adaptive numerical methods for geophysical applications.
Castro, C E; Käser, M; Toro, E F
2009-11-28
In this paper we present high-order formulations of the finite volume and discontinuous Galerkin finite-element methods for wave propagation problems with a space-time adaptation technique using unstructured meshes in order to reduce computational cost without reducing accuracy. Both methods can be derived in a similar mathematical framework and are identical in their first-order version. In their extension to higher order accuracy in space and time, both methods use spatial polynomials of higher degree inside each element, a high-order solution of the generalized Riemann problem and a high-order time integration method based on the Taylor series expansion. The static adaptation strategy uses locally refined high-resolution meshes in areas with low wave speeds to improve the approximation quality. Furthermore, the time step length is chosen locally adaptive such that the solution is evolved explicitly in time by an optimal time step determined by a local stability criterion. After validating the numerical approach, both schemes are applied to geophysical wave propagation problems such as tsunami waves and seismic waves comparing the new approach with the classical global time-stepping technique. The problem of mesh partitioning for large-scale applications on multi-processor architectures is discussed and a new mesh partition approach is proposed and tested to further reduce computational cost. PMID:19840984
Modelling asteroid brightness variations. I - Numerical methods
NASA Technical Reports Server (NTRS)
Karttunen, H.
1989-01-01
A method for generating lightcurves of asteroid models is presented. The effects of the shape of the asteroid and the scattering law of a surface element are distinctly separable, being described by chosen functions that can easily be changed. The shape is specified by means of two functions that yield the length of the radius vector and the normal vector of the surface at a given point. The general shape must be convex, but spherical concavities producing macroscopic shadowing can also be modeled.
A numerical method for power plant simulations
Carcasci, C.; Facchini, B.
1996-03-01
This paper describes a highly flexible computerized method of calculating operating data in a power cycle. The computerized method presented here permits the study of steam, gas and combined plants. Its flexibility is not restricted by any defined cycle scheme. A power plant consists of simple elements (turbine, compressor, combustor chamber, pump, etc.). Each power plant component is represented by its typical equations relating to fundamental mechanical and thermodynamic laws, so a power plant system is represented by algebraic equations, which are the typical equations of components, continuity equations, and data concerning plant conditions. This equation system is not linear, but can be reduced to a linear equation system with variable coefficients. The solution is simultaneous for each component and it is determined by an iterative process. An example of a simple gas turbine cycle demonstrates the applied technique. This paper also presents the user interface based on MS-Windows. The input data, the results, and any characteristic parameters of a complex cycle scheme are also shown.
Numerical evaluation of two-center integrals over Slater type orbitals
NASA Astrophysics Data System (ADS)
Kurt, S. A.; Yükçü, N.
2016-03-01
Slater Type Orbitals (STOs) which one of the types of exponential type orbitals (ETOs) are used usually as basis functions in the multicenter molecular integrals to better understand physical and chemical properties of matter. In this work, we develop algorithms for two-center overlap and two-center two-electron hybrid and Coulomb integrals which are calculated with help of translation method for STOs and some auxiliary functions by V. Magnasco's group. We use Mathematica programming language to produce algorithms for these calculations. Numerical results for some quantum numbers are presented in the tables. Consequently, we compare our obtained numerical results with the other known literature results and other details of evaluation method are discussed.
Numerical methods for analyzing electromagnetic scattering
NASA Technical Reports Server (NTRS)
Lee, S. W.; Lo, Y. T.; Chuang, S. L.; Lee, C. S.
1985-01-01
Attenuation properties of the normal modes in an overmoded waveguide coated with a lossy material were analyzed. It is found that the low-order modes, can be significantly attenuated even with a thin layer of coating if the coating material is not too lossy. A thinner layer of coating is required for large attenuation of the low-order modes if the coating material is magnetic rather than dielectric. The Radar Cross Section (RCS) from an uncoated circular guide terminated by a perfect electric conductor was calculated and compared with available experimental data. It is confirmed that the interior irradiation contributes to the RCS. The equivalent-current method based on the geometrical theory of diffraction (GTD) was chosen for the calculation of the contribution from the rim diffraction. The RCS reduction from a coated circular guide terminated by a PEC are planned schemes for the experiments are included. The waveguide coated with a lossy magnetic material is suggested as a substitute for the corrugated waveguide.
Hybrid Numerical Methods for Multiscale Simulations of Subsurface Biogeochemical Processes
Scheibe, Timothy D.; Tartakovsky, Alexandre M.; Tartakovsky, Daniel M.; Redden, George D.; Meakin, Paul
2007-08-01
Many subsurface flow and transport problems of importance today involve coupled non-linear flow, transport, and reaction in media exhibiting complex heterogeneity. In particular, problems involving biological mediation of reactions fall into this class of problems. Recent experimental research has revealed important details about the physical, chemical, and biological mechanisms involved in these processes at a variety of scales ranging from molecular to laboratory scales. However, it has not been practical or possible to translate detailed knowledge at small scales into reliable predictions of field-scale phenomena important for environmental management applications. A large assortment of numerical simulation tools have been developed, each with its own characteristic scale including molecular (e.g., molecular dynamics), microbial (e.g., cellular automata or particle individual-based models), pore (e.g., lattice-Boltzmann, pore network models, and discrete particle methods such as smoothed particle hydrodynamics) and continuum scales (e.g., traditional partial differential equations solved by finite difference or finite element methods). While many problems can be effectively addressed by one of these models at a single scale, some problems may require explicit integration of models across multiple scales. We are developing a hybrid multi-scale subsurface reactive transport modeling framework that integrates models with diverse representations of physics, chemistry and biology at different scales (sub-pore, pore and continuum). The modeling framework is being designed to take advantage of advanced computational technologies including parallel code components using the Common Component Architecture, parallel solvers, gridding, data and workflow management, and visualization. This paper describes the specific methods/codes being used at each scale, techniques used to directly and adaptively couple across model scales, and preliminary results of application to a
Development of numerical methods to problems of micromechanics
NASA Astrophysics Data System (ADS)
Garcia-Martinez, Jose Ramon
In this dissertation we utilize the finite element method to investigate three micromechanical problems. In Chapter 2, we study the compliance contribution tensor H of multiple branched cracks. The cracks grow from a deltoid pore at their center into a triple crack. For plain strain conditions, two-dimensional models of the branched crack are modeled and solved in ABAQUS. The displacement field over the surface of the branched crack and the deltoid is curve fitted to carry out the integral surface of the compliance contribution tensor H. The predicted values are in good agreement with analytical solution. In Chapter 3 a three-dimensional finite element program using an unaligned mesh with an eight-node isoparametric element is developed to study the compliance contribution tensor H of cavities with superellipsoid shapes. A mesh clustering algorithm to increase the number of elements inside and near the superellipsoid surface to obtain a mesh independent solution is used. The numerical results are compared with the analytical solution of a sphere; the error of the numerical approximation varied from 8 to 11%. It is found that the number of elements inside the superellipsoid are insufficient. An algorithm to mesh independently the volumes inside and outside the cube is proposed to increase the accuracy in the calculation of H. As n1 and n2 increase, the numerical solutions show that, H1111 → 0 and H2211 → 0. Although, for these concave shapes no analytical solution exists a bound of 0 for the terms H 1111 and H2211 is suggested. Finally, in Chapter 4 a numerical verification of the cross-property connection between the effective fluid permeability and the effective electrical conductivity is study. A molecular dynamics algorithm is used to generate a set of different microstructural patterns. The volumetric average over a cubic volume is used to obtain the effective electrical conductivity and the effective fluid permeability. The tortuosity of the porous phase
Exponential Methods for the Time Integration of Schroedinger Equation
Cano, B.; Gonzalez-Pachon, A.
2010-09-30
We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.
ERIC Educational Resources Information Center
Johnson, K. J.
1979-01-01
This workshop was for participants who were interested in developing a numerical methods course. The contents of a numerical methods text were covered, with special emphasis on nonlinear least squares analysis, and the Runge-Kutta method of integrating systems of first-order differential equations. (BB)
A Novel Numerical Method for Fuzzy Boundary Value Problems
NASA Astrophysics Data System (ADS)
Can, E.; Bayrak, M. A.; Hicdurmaz
2016-05-01
In the present paper, a new numerical method is proposed for solving fuzzy differential equations which are utilized for the modeling problems in science and engineering. Fuzzy approach is selected due to its important applications on processing uncertainty or subjective information for mathematical models of physical problems. A second-order fuzzy linear boundary value problem is considered in particular due to its important applications in physics. Moreover, numerical experiments are presented to show the effectiveness of the proposed numerical method on specific physical problems such as heat conduction in an infinite plate and a fin.
ERIC Educational Resources Information Center
Motter, Wendell L.
It is noted that there are some integrals which cannot be evaluated by determining an antiderivative, and these integrals must be subjected to other techniques. Numerical integration is one such method; it provides a sum that is an approximate value for some integral types. This module's purpose is to introduce methods of numerical integration and…
Integrated navigation method based on inertial navigation system and Lidar
NASA Astrophysics Data System (ADS)
Zhang, Xiaoyue; Shi, Haitao; Pan, Jianye; Zhang, Chunxi
2016-04-01
An integrated navigation method based on the inertial navigational system (INS) and Lidar was proposed for land navigation. Compared with the traditional integrated navigational method and dead reckoning (DR) method, the influence of the inertial measurement unit (IMU) scale factor and misalignment was considered in the new method. First, the influence of the IMU scale factor and misalignment on navigation accuracy was analyzed. Based on the analysis, the integrated system error model of INS and Lidar was established, in which the IMU scale factor and misalignment error states were included. Then the observability of IMU error states was analyzed. According to the results of the observability analysis, the integrated system was optimized. Finally, numerical simulation and a vehicle test were carried out to validate the availability and utility of the proposed INS/Lidar integrated navigational method. Compared with the test result of a traditional integrated navigation method and DR method, the proposed integrated navigational method could result in a higher navigation precision. Consequently, the IMU scale factor and misalignment error were effectively compensated by the proposed method and the new integrated navigational method is valid.
Carbon Dioxide Dispersion in the Combustion Integrated Rack Simulated Numerically
NASA Technical Reports Server (NTRS)
Wu, Ming-Shin; Ruff, Gary A.
2004-01-01
When discharged into an International Space Station (ISS) payload rack, a carbon dioxide (CO2) portable fire extinguisher (PFE) must extinguish a fire by decreasing the oxygen in the rack by 50 percent within 60 sec. The length of time needed for this oxygen reduction throughout the rack and the length of time that the CO2 concentration remains high enough to prevent the fire from reigniting is important when determining the effectiveness of the response and postfire procedures. Furthermore, in the absence of gravity, the local flow velocity can make the difference between a fire that spreads rapidly and one that self-extinguishes after ignition. A numerical simulation of the discharge of CO2 from PFE into the Combustion Integrated Rack (CIR) in microgravity was performed to obtain the local velocity and CO2 concentration. The complicated flow field around the PFE nozzle exits was modeled by sources of equivalent mass and momentum flux at a location downstream of the nozzle. The time for the concentration of CO2 to reach a level that would extinguish a fire anywhere in the rack was determined using the Fire Dynamics Simulator (FDS), a computational fluid dynamics code developed by the National Institute of Standards and Technology specifically to evaluate the development of a fire and smoke transport. The simulation shows that CO2, as well as any smoke and combustion gases produced by a fire, would be discharged into the ISS cabin through the resource utility panel at the bottom of the rack. These simulations will be validated by comparing the results with velocity and CO2 concentration measurements obtained during the fire suppression system verification tests conducted on the CIR in March 2003. Once these numerical simulations are validated, portions of the ISS labs and living areas will be modeled to determine the local flow conditions before, during, and after a fire event. These simulations can yield specific information about how long it takes for smoke and
Black shale weathering: An integrated field and numerical modeling study
NASA Astrophysics Data System (ADS)
Bolton, E. W.; Wildman, R. A., Jr.; Berner, R. A.; Eckert, J. O., Jr.; Petsch, S. T.; Mok, U.; Evans, B.
2003-04-01
We present an integrated study of black shale weathering in a near surface environment. Implications of this study contribute to our understanding of organic matter oxidation in uplifted sediments, along with erosion and reburial of ancient unoxidized organic matter, as major controls on atmospheric oxygen levels over geologic time. The field study used to launch the modeling effort is based on core samples from central-eastern Kentucky near Clay City (Late Devonian New Albany/Ohio Shale), where the strata are essentially horizontal. Samples from various depth intervals (up to 12 m depth) were analyzed for texture (SEM images), porosity fraction (0.02 to 0.1), and horizontal and vertical permeability (water and air permeabilities differ due to the fine-grained nature of the sediments, but are on the order of 0.01 to 1. millidarcies, respectively). Chemical analyses were also performed for per cent C, N, S, and basic mineralogy was determined (clays, quartz, pyrite, in addition to organic matter). The samples contained from 2 to 15 per cent ancient (non-modern soil) organic matter. These results were used in the creation of a numerical model for kinetically controlled oxidation of the organic matter within the shale (based on kinetics from Chang and Berner, 1999). The one-dimensional model includes erosion, oxygen diffusion in the partially saturated vadose zone as well as water percolation and solute transport. This study extends the studies of Petsch (2000) and the weathering component of Lasaga and Ohmoto (2002) to include more reactions (e.g., pyrite oxidation to sulfuric acid and weathering of silicates due to low pH) and to resolve the near-surface boundary layer. The model provides a convenient means of exploring the influence of variable rates of erosion, oxygen level, rainfall, as well as physical and chemical characteristics of the shale on organic matter oxidation.
A numerical method for the time coarsening of transport processes at the atomistic scale
NASA Astrophysics Data System (ADS)
Gonzalez-Ferreiro, B.; Romero, I.; Ortiz, M.
2016-05-01
We propose a novel numerical scheme for the simulation of slow transport processes at the atomistic scale. The scheme is based on a model for non-equilibrium statistical thermodynamics recently proposed by the authors, and extends it by formulating a variational integrator, i.e. a discrete functional whose optimality conditions provide all the governing equations of the problem. The method is employed to study surface segregation of AuAg alloys and its convergence is confirmed numerically.
Liu, Fang; Lin, Lin; Vigil-Fowler, Derek; Lischner, Johannes; Kemper, Alexander F.; Sharifzadeh, Sahar; Jornada, Felipe H. da; Deslippe, Jack; Yang, Chao; and others
2015-04-01
We present a numerical integration scheme for evaluating the convolution of a Green's function with a screened Coulomb potential on the real axis in the GW approximation of the self energy. Our scheme takes the zero broadening limit in Green's function first, replaces the numerator of the integrand with a piecewise polynomial approximation, and performs principal value integration on subintervals analytically. We give the error bound of our numerical integration scheme and show by numerical examples that it is more reliable and accurate than the standard quadrature rules such as the composite trapezoidal rule. We also discuss the benefit of using different self energy expressions to perform the numerical convolution at different frequencies.
NASA Astrophysics Data System (ADS)
Macomber, B.; Woollands, R. M.; Probe, A.; Younes, A.; Bai, X.; Junkins, J.
2013-09-01
Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for approximating solutions of linear or non-linear Ordinary Differential Equations (ODEs) to obtain time histories of system state trajectories. Unlike other step-by-step differential equation solvers, the Runge-Kutta family of numerical integrators for example, MCPI approximates long arcs of the state trajectory with an iterative path approximation approach, and is ideally suited to parallel computation. Orthogonal Chebyshev Polynomials are used as basis functions during each path iteration; the integrations of the Picard iteration are then done analytically. Due to the orthogonality of the Chebyshev basis functions, the least square approximations are computed without matrix inversion; the coefficients are computed robustly from discrete inner products. As a consequence of discrete sampling and weighting adopted for the inner product definition, Runge phenomena errors are minimized near the ends of the approximation intervals. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning they can be simultaneously computed in parallel for further decreased computational cost. Over an order of magnitude speedup from traditional methods is achieved in serial processing, and an additional order of magnitude is achievable in parallel architectures. This paper presents a new MCPI library, a modular toolset designed to allow MCPI to be easily applied to a wide variety of ODE systems. Library users will not have to concern themselves with the underlying mathematics behind the MCPI method. Inputs are the boundary conditions of the dynamical system, the integrand function governing system behavior, and the desired time interval of integration, and the output is a time history of the system states over the interval of interest. Examples from the field of astrodynamics are
Asymptotic-induced numerical methods for conservation laws
NASA Technical Reports Server (NTRS)
Garbey, Marc; Scroggs, Jeffrey S.
1990-01-01
Asymptotic-induced methods are presented for the numerical solution of hyperbolic conservation laws with or without viscosity. The methods consist of multiple stages. The first stage is to obtain a first approximation by using a first-order method, such as the Godunov scheme. Subsequent stages of the method involve solving internal-layer problems identified by using techniques derived via asymptotics. Finally, a residual correction increases the accuracy of the scheme. The method is derived and justified with singular perturbation techniques.
An iterative analytic—numerical method for scattering from a target buried beneath a rough surface
NASA Astrophysics Data System (ADS)
Xu, Run-Wen; Guo, Li-Xin; Wang, Rui
2014-11-01
An efficiently iterative analytical—numerical method is proposed for two-dimensional (2D) electromagnetic scattering from a perfectly electric conducting (PEC) target buried under a dielectric rough surface. The basic idea is to employ the Kirchhoff approximation (KA) to accelerate the boundary integral method (BIM). Below the rough surface, an iterative system is designed between the rough surface and the target. The KA is used to simulate the initial field on the rough surface based on the Fresnel theory, while the target is analyzed by the boundary integral method to obtain a precise result. The fields between the rough surface and the target can be linked by the boundary integral equations below the rough surface. The technique presented here is highly efficient in terms of computational memory, time, and versatility. Numerical simulations of two typical models are carried out to validate the method.
A numerical method for approximating antenna surfaces defined by discrete surface points
NASA Technical Reports Server (NTRS)
Lee, R. Q.; Acosta, R.
1985-01-01
A simple numerical method for the quadratic approximation of a discretely defined reflector surface is described. The numerical method was applied to interpolate the surface normal of a parabolic reflector surface from a grid of nine closest surface points to the point of incidence. After computing the surface normals, the geometrical optics and the aperture integration method using the discrete Fast Fourier Transform (FFT) were applied to compute the radiaton patterns for a symmetric and an offset antenna configurations. The computed patterns are compared to that of the analytic case and to the patterns generated from another numerical technique using the spline function approximation. In the paper, examples of computations are given. The accuracy of the numerical method is discussed.
Integrating Numerical Computation into the Modeling Instruction Curriculum
ERIC Educational Resources Information Center
Caballero, Marcos D.; Burk, John B.; Aiken, John M.; Thoms, Brian D.; Douglas, Scott S.; Scanlon, Erin M.; Schatz, Michael F.
2014-01-01
Numerical computation (the use of a computer to solve, simulate, or visualize a physical problem) has fundamentally changed the way scientific research is done. Systems that are too difficult to solve in closed form are probed using computation. Experiments that are impossible to perform in the laboratory are studied numerically. Consequently, in…
Numerical Integration with GeoGebra in High School
ERIC Educational Resources Information Center
Herceg, Dorde; Herceg, Dragoslav
2010-01-01
The concept of definite integral is almost always introduced as the Riemann integral, which is defined in terms of the Riemann sum, and its geometric interpretation. This definition is hard to understand for high school students. With the aid of mathematical software for visualisation and computation of approximate integrals, the notion of…
Numerical Asymptotic Solutions Of Differential Equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1992-01-01
Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.
Parallel processing numerical method for confined vortex dynamics and applications
NASA Astrophysics Data System (ADS)
Bistrian, Diana Alina
2013-10-01
This paper explores a combined analytical and numerical technique to investigate the hydrodynamic instability of confined swirling flows, with application to vortex rope dynamics in a Francis turbine diffuser, in condition of sophisticated boundary constraints. We present a new approach based on the method of orthogonal decomposition in the Hilbert space, implemented with a spectral descriptor scheme in discrete space. A parallel implementation of the numerical scheme is conducted reducing the computational time compared to other techniques.
Algorithms for the Fractional Calculus: A Selection of Numerical Methods
NASA Technical Reports Server (NTRS)
Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Yu.
2003-01-01
Many recently developed models in areas like viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of derivatives (and integrals) of fractional (non-integer) order. In this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context. We believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way.
Investigating Convergence Patterns for Numerical Methods Using Data Analysis
ERIC Educational Resources Information Center
Gordon, Sheldon P.
2013-01-01
The article investigates the patterns that arise in the convergence of numerical methods, particularly those in the errors involved in successive iterations, using data analysis and curve fitting methods. In particular, the results obtained are used to convey a deeper level of understanding of the concepts of linear, quadratic, and cubic…
Collocation Method for Numerical Solution of Coupled Nonlinear Schroedinger Equation
Ismail, M. S.
2010-09-30
The coupled nonlinear Schroedinger equation models several interesting physical phenomena presents a model equation for optical fiber with linear birefringence. In this paper we use collocation method to solve this equation, we test this method for stability and accuracy. Numerical tests using single soliton and interaction of three solitons are used to test the resulting scheme.
A numerical method for solving singular De`s
Mahaver, W.T.
1996-12-31
A numerical method is developed for solving singular differential equations using steepest descent based on weighted Sobolev gradients. The method is demonstrated on a variety of first and second order problems, including linear constrained, unconstrained, and partially constrained first order problems, a nonlinear first order problem with irregular singularity, and two second order variational problems.
A new numerical method of total solar eclipse photography processing
NASA Astrophysics Data System (ADS)
Druckmüller, M.; Rušin, V.; Minarovjech, M.
2006-10-01
A new numerical method of image processing suitable for visualization of corona images taken during total solar eclipses is presented. This method allows us to study both small- and large-scale coronal structures that remain invisible on original images because of their very high dynamic range of the coronal brightness. The method is based on the use of adaptive filters inspired by human vision and the sensitivity of resulting images is thus very close to that of the human eye during an eclipse. A high precision alignment method for white-light corona images is also discussed. The proposed method highly improves a widely used unsharp masking method employing a radially blurred mask. The results of these numerical image processing techniques are illustrated by a series of images taken during eclipses of the last decade. The method minimizes the risk of processing artifacts.
Numerical solution of a diffusion problem by exponentially fitted finite difference methods.
D'Ambrosio, Raffaele; Paternoster, Beatrice
2014-01-01
This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver. PMID:26034665
Numerical integration techniques for curved-element discretizations of molecule-solvent interfaces.
Bardhan, Jaydeep P; Altman, Michael D; Willis, David J; Lippow, Shaun M; Tidor, Bruce; White, Jacob K
2007-07-01
Surface formulations of biophysical modeling problems offer attractive theoretical and computational properties. Numerical simulations based on these formulations usually begin with discretization of the surface under consideration; often, the surface is curved, possessing complicated structure and possibly singularities. Numerical simulations commonly are based on approximate, rather than exact, discretizations of these surfaces. To assess the strength of the dependence of simulation accuracy on the fidelity of surface representation, here methods were developed to model several important surface formulations using exact surface discretizations. Following and refining Zauhar's work [J. Comput.-Aided Mol. Des. 9, 149 (1995)], two classes of curved elements were defined that can exactly discretize the van der Waals, solvent-accessible, and solvent-excluded (molecular) surfaces. Numerical integration techniques are presented that can accurately evaluate nonsingular and singular integrals over these curved surfaces. After validating the exactness of the surface discretizations and demonstrating the correctness of the presented integration methods, a set of calculations are presented that compare the accuracy of approximate, planar-triangle-based discretizations and exact, curved-element-based simulations of surface-generalized-Born (sGB), surface-continuum van der Waals (scvdW), and boundary-element method (BEM) electrostatics problems. Results demonstrate that continuum electrostatic calculations with BEM using curved elements, piecewise-constant basis functions, and centroid collocation are nearly ten times more accurate than planar-triangle BEM for basis sets of comparable size. The sGB and scvdW calculations give exceptional accuracy even for the coarsest obtainable discretized surfaces. The extra accuracy is attributed to the exact representation of the solute-solvent interface; in contrast, commonly used planar-triangle discretizations can only offer improved
25 Years of Self-organized Criticality: Numerical Detection Methods
NASA Astrophysics Data System (ADS)
McAteer, R. T. James; Aschwanden, Markus J.; Dimitropoulou, Michaila; Georgoulis, Manolis K.; Pruessner, Gunnar; Morales, Laura; Ireland, Jack; Abramenko, Valentyna
2016-01-01
The detection and characterization of self-organized criticality (SOC), in both real and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basic autocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numerical methods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines—the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expanding field of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.
Comparison of methods for numerical calculation of continuum damping
Bowden, G. W.; Hole, M. J.; Dennis, G. R.; Könies, A.; Gorelenkov, N. N.
2014-05-15
Continuum resonance damping is an important factor in determining the stability of certain global modes in fusion plasmas. A number of analytic and numerical approaches have been developed to compute this damping, particularly, in the case of the toroidicity-induced shear Alfvén eigenmode. This paper compares results obtained using an analytical perturbative approach with those found using resistive and complex contour numerical approaches. It is found that the perturbative method does not provide accurate agreement with reliable numerical methods for the range of parameters examined. This discrepancy exists even in the limit where damping approaches zero. When the perturbative technique is implemented using a standard finite element method, the damping estimate fails to converge with radial grid resolution. The finite elements used cannot accurately represent the eigenmode in the region of the continuum resonance, regardless of the number of radial grid points used.
SAMSAN- MODERN NUMERICAL METHODS FOR CLASSICAL SAMPLED SYSTEM ANALYSIS
NASA Technical Reports Server (NTRS)
Frisch, H. P.
1994-01-01
SAMSAN was developed to aid the control system analyst by providing a self consistent set of computer algorithms that support large order control system design and evaluation studies, with an emphasis placed on sampled system analysis. Control system analysts have access to a vast array of published algorithms to solve an equally large spectrum of controls related computational problems. The analyst usually spends considerable time and effort bringing these published algorithms to an integrated operational status and often finds them less general than desired. SAMSAN reduces the burden on the analyst by providing a set of algorithms that have been well tested and documented, and that can be readily integrated for solving control system problems. Algorithm selection for SAMSAN has been biased toward numerical accuracy for large order systems with computational speed and portability being considered important but not paramount. In addition to containing relevant subroutines from EISPAK for eigen-analysis and from LINPAK for the solution of linear systems and related problems, SAMSAN contains the following not so generally available capabilities: 1) Reduction of a real non-symmetric matrix to block diagonal form via a real similarity transformation matrix which is well conditioned with respect to inversion, 2) Solution of the generalized eigenvalue problem with balancing and grading, 3) Computation of all zeros of the determinant of a matrix of polynomials, 4) Matrix exponentiation and the evaluation of integrals involving the matrix exponential, with option to first block diagonalize, 5) Root locus and frequency response for single variable transfer functions in the S, Z, and W domains, 6) Several methods of computing zeros for linear systems, and 7) The ability to generate documentation "on demand". All matrix operations in the SAMSAN algorithms assume non-symmetric matrices with real double precision elements. There is no fixed size limit on any matrix in any
Implicit numerical integration for periodic solutions of autonomous nonlinear systems
NASA Technical Reports Server (NTRS)
Thurston, G. A.
1982-01-01
A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.
NASA Astrophysics Data System (ADS)
Jones, Marvin Quenten, Jr.
The motion and behavior of quantum processes can be described by the Schrodinger equation using the wave function, Psi(x,t). The use of the Schrodinger equation to study quantum phenomena is known as Quantum Mechanics, akin to classical mechanics being the tool to study classical physics. This research focuses on the emphasis of numerical techniques: Finite-Difference, Fast Fourier Transform (spectral method), finite difference schemes such as the Leapfrog method and the Crank-Nicolson scheme and second quantization to solve and analyze the Schrodinger equation for the infinite square well problem, the free particle with periodic boundary conditions, the barrier problem, tight-binding hamiltonians and a potential wall problem. We discuss these techniques and the problems created to test how these different techniques draw both physical and numerical conclusions in a tabular summary. We observed both numerical stability and quantum stability (conservation of energy, probability, momentum, etc.). We found in our results that the Crank-Nicolson scheme is an unconditionally stable scheme and conserves probability (unitary), and momentum, though dissipative with energy. The time-independent problems conserved energy, momentum and were unitary, which is of interest, but we found when time-dependence was introduced, quantum stability (i.e. conservation of mass, momentum, etc.) was not implied by numerical stability. Hence, we observed schemes that were numerically stable, but not quantum stable as well as schemes that were quantum stable, but not numerically stable for all of time, t. We also observed that second quantization removed the issues with stability as the problem was transformed into a discrete problem. Moreover, all quantum information is conserved in second quantization. This method, however, does not work universally for all problems.
Method of descent for integrable lattices
NASA Astrophysics Data System (ADS)
Bogoyavlensky, Oleg
2009-05-01
A method of descent for constructing integrable Hamiltonian systems is introduced. The derived periodic and nonperiodic lattices possess Lax representations with spectral parameter and have plenty of first integrals. Examples of Liouville-integrable four-dimensional Hamiltonian Lotka-Volterra systems are presented.
Evaluation of the Hubbell rectangular source integral using Haar wavelets method
NASA Astrophysics Data System (ADS)
Belkadhi, K.; Manai, K.
2016-05-01
Haar wavelets numerical integration method is exposed and used for evaluating the Hubbell rectangular source integral. The method convergence is studied to get the minimum iteration number for a desired precision. Haar wavelets results are finally compared to those obtained with other integration methods.
NASA Technical Reports Server (NTRS)
Lundberg, J. B.; Feulner, M. R.; Abusali, P. A. M.; Ho, C. S.
1991-01-01
The method of modified back differences, a technique that significantly reduces the numerical integration errors associated with crossing shadow boundaries using a fixed-mesh multistep integrator without a significant increase in computer run time, is presented. While Hubbard's integral approach can produce significant improvements to the trajectory solution, the interpolation method provides the best overall results. It is demonstrated that iterating on the point mass term correction is also important for achieving the best overall results. It is also shown that the method of modified back differences can be implemented with only a small increase in execution time.
Numerical modeling of magnetic induction tomography using the impedance method.
Ramos, Airton; Wolff, Julia G B
2011-02-01
This article discusses the impedance method in the forward calculation in magnetic induction tomography (MIT). Magnetic field and eddy current distributions were obtained numerically for a sphere in the field of a coil and were compared with an analytical model. Additionally, numerical and experimental results for phase sensitivity in MIT were obtained and compared for a cylindrical object in a planar array of sensors. The results showed that the impedance method provides results that agree very well with reality in the frequency range from 100 kHz to 20 MHz and for low conductivity objects (10 S/m or less). This opens the possibility of using this numerical approach in image reconstruction in MIT. PMID:21229327
Damping identification in frequency domain using integral method
NASA Astrophysics Data System (ADS)
Guo, Zhiwei; Sheng, Meiping; Ma, Jiangang; Zhang, Wulin
2015-03-01
A new method for damping identification of linear system in frequency domain is presented, by using frequency response function (FRF) with integral method. The FRF curve is firstly transformed to other type of frequency-related curve by changing the representations of horizontal and vertical axes. For the newly constructed frequency-related curve, integral is conducted and the area forming from the new curve is used to determine the damping. Three different methods based on integral are proposed in this paper, which are called FDI-1, FDI-2 and FDI-3 method, respectively. For a single degree of freedom (Sdof) system, the formulated relation of each method between integrated area and loss factor is derived theoretically. The numeral simulation and experiment results show that, the proposed integral methods have high precision, strong noise resistance and are very stable in repeated measurements. Among the three integral methods, FDI-3 method is the most recommended because of its higher accuracy and simpler algorithm. The new methods are limited to linear system in which modes are well separated, and for closely spaced mode system, mode decomposition process should be conducted firstly.
NASA Astrophysics Data System (ADS)
Hermand, Jean-Pierre; Berrada, Mohamed; Meyer, Matthias; Asch, Mark
2005-09-01
Recently, an analytic adjoint-based method of optimal nonlocal boundary control has been proposed for inversion of a waveguide acoustic field using the wide-angle parabolic equation [Meyer and Hermand, J. Acoust. Soc. Am. 117, 2937-2948 (2005)]. In this paper a numerical extension of this approach is presented that allows the direct inversion for the geoacoustic parameters which are embedded in a spectral integral representation of the nonlocal boundary condition. The adjoint model is generated numerically and the inversion is carried out jointly across multiple frequencies. The paper further discusses the application of the numerical adjoint PE method for ocean acoustic tomography. To show the effectiveness of the implemented numerical adjoint, preliminary inversion results of water sound-speed profile and bottom acoustic properties will be shown for the YELLOW SHARK '94 experimental conditions.
Willis, Catherine; Rubin, Jacob
1987-01-01
In this paper we consider examples of chemistry-affected transport processes in porous media. A moving boundary problem which arises during transport with precipitation-dissolution reactions is solved by three different numerical methods. Two of these methods (one explicit and one implicit) are based on an integral formulation of mass balance and lead to an approximation of a weak solution. These methods are compared to a front-tracking scheme. Although the two approaches are conceptually different, the numerical solutions showed good agreement. As the ratio of dispersion to convection decreases, the methods based on the integral formulation become computationally more efficient. Specific reactions were modeled to examine the dependence of the system on the physical and chemical parameters.
A numerical method of tracing a vortical axis along local topological axis line
NASA Astrophysics Data System (ADS)
Nakayama, Katsuyuki; Hasegawa, Hideki
2016-06-01
A new numerical method is presented to trace or identify a vortical axis in flow, which is based on Galilean invariant flow topology. We focus on the local flow topology specified by the eigenvalues and eigenvectors of the velocity gradient tensor, and extract the axis component from its flow trajectory. Eigen-vortical-axis line is defined from the eigenvector of the real eigenvalue of the velocity gradient tensor where the tensor has the conjugate complex eigenvalues. This numerical method integrates the eigen-vortical-axis line and traces a vortical axis in terms of the invariant flow topology, which enables to investigate the feature of the topology-based vortical axis.
NASA Technical Reports Server (NTRS)
Hickey, Michael Philip
1988-01-01
A proposed replacement scheme for the integration of the barometric and diffusion equations in the NASA Marshall Engineering Thermosphere (MET) model is presented. This proposed integration scheme is based on Gaussian Quadrature. Extensive numerical testing reveals it to be faster, more accurate and more reliable than the present integration scheme (a modified form of Simpson's Rule) used in the MET model. Numerous graphical examples are provided, along with a listing of a modified form of the MET model in which subroutine INTEGRATE (using Simpson's Rule) is replaced by subroutine GAUSS (which uses Gaussian Quadrature). It is recommended that the Gaussian Quadrature integration scheme, as used here, be used in the MET model.
Integrated control system and method
Wang, Paul Sai Keat; Baldwin, Darryl; Kim, Myoungjin
2013-10-29
An integrated control system for use with an engine connected to a generator providing electrical power to a switchgear is disclosed. The engine receives gas produced by a gasifier. The control system includes an electronic controller associated with the gasifier, engine, generator, and switchgear. A gas flow sensor monitors a gas flow from the gasifier to the engine through an engine gas control valve and provides a gas flow signal to the electronic controller. A gas oversupply sensor monitors a gas oversupply from the gasifier and provides an oversupply signal indicative of gas not provided to the engine. A power output sensor monitors a power output of the switchgear and provide a power output signal. The electronic controller changes gas production of the gasifier and the power output rating of the switchgear based on the gas flow signal, the oversupply signal, and the power output signal.
A novel gas-droplet numerical method for spray combustion
NASA Technical Reports Server (NTRS)
Chen, C. P.; Shang, H. M.; Jiang, Y.
1991-01-01
This paper presents a non-iterative numerical technique for computing time-dependent gas-droplet flows. The method is a fully-interacting combination of Eulerian fluid and Lagrangian particle calculation. The interaction calculations between the two phases are formulated on a pressure-velocity coupling procedure based on the operator-splitting technique. This procedure eliminates the global iterations required in the conventional particle-source-in-cell (PSIC) procedure. Turbulent dispersion calculations are treated by a stochastic procedure. Numerical calculations and comparisons with available experimental data, as well as efficiency assessments are given for some sprays typical of spray combustion applications.
Simple numerical method for predicting steady compressible flows
NASA Technical Reports Server (NTRS)
Vonlavante, Ernst; Nelson, N. Duane
1986-01-01
A numerical method for solving the isenthalpic form of the governing equations for compressible viscous and inviscid flows was developed. The method was based on the concept of flux vector splitting in its implicit form. The method was tested on several demanding inviscid and viscous configurations. Two different forms of the implicit operator were investigated. The time marching to steady state was accelerated by the implementation of the multigrid procedure. Its various forms very effectively increased the rate of convergence of the present scheme. High quality steady state results were obtained in most of the test cases; these required only short computational times due to the relative efficiency of the basic method.
The TAB method for numerical calculation of spray droplet breakup
NASA Astrophysics Data System (ADS)
Orourke, P. J.; Amsden, A. A.
A short history is given of the major milestones in the development of the stochastic particle method for calculating liquid fuel sprays. The most recent advance has been the discovery of the importance of drop breakup in engine sprays. A new method, called TAB, for calculating drop breakup is presented. Some theoretical properties of the method are derived; its numerical implementation in the computer program KIVA is described; and comparisons are presented between TAB-method calculations and experiments and calculations using another breakup model.
Numerical simulation of dynamic processes in biomechanics using the grid-characteristic method
NASA Astrophysics Data System (ADS)
Beklemysheva, K. A.; Vasyukov, A. V.; Petrov, I. B.
2015-08-01
Results of the numerical simulation of mechanical processes occurring in biological tissues under dynamic actions are presented. The grid-characteristic method on unstructured grids is used to solve the system of equations of mechanics of deformable solids; this method takes into account the characteristic properties of the constitutive system of partial differential equations and produces adequate algorithms on interfaces between media and on the boundaries of integration domains.
Robust rotational-velocity-Verlet integration methods
NASA Astrophysics Data System (ADS)
Rozmanov, Dmitri; Kusalik, Peter G.
2010-05-01
Two rotational integration algorithms for rigid-body dynamics are proposed in velocity-Verlet formulation. The first method uses quaternion dynamics and was derived from the original rotational leap-frog method by Svanberg [Mol. Phys. 92, 1085 (1997)]; it produces time consistent positions and momenta. The second method is also formulated in terms of quaternions but it is not quaternion specific and can be easily adapted for any other orientational representation. Both the methods are tested extensively and compared to existing rotational integrators. The proposed integrators demonstrated performance at least at the level of previously reported rotational algorithms. The choice of simulation parameters is also discussed.
NASA Astrophysics Data System (ADS)
Roig, Jaume; Stefanov, Evgueniy; Morancho, Frédéric
2007-07-01
The use of super-junction (SJ) techniques in PIN photodiodes is proposed in this letter for the first time with the objective to assist the optoelectronic integrated circuits (OEICs) implementation in complementary metal oxide semiconductor (CMOS), bipolar CMOS (BiCMOS) and bipolar-CMOS-double diffused MOS (BCD) technologies. Its technological viability is also discussed to make it credible as an alternative to other OEICs approaches. Numerical simulation of realistic SJ-PIN devices, widely used in high power electronics, demonstrates the possibility to integrate high-performance CMOS-based OEICs in epitaxial layers with doping concentrations above 1× 1015 cm-3. The induced lateral depletion at low reverse biased voltage, assisted by the alternated N and P-doped pillars, allows high-speed transient response in SJ-PIN detecting wavelengths between 400 and 800 nm. Moreover, other important parameters as the responsivity and the dark current are not degraded in respect to the conventional PIN (C-PIN) structures.
NASA Astrophysics Data System (ADS)
Evans, W. A. B.; Torre, A.
2012-11-01
The paper focusses on the advantages of using high-order Gauss-Legendre quadratures for the precise evaluation of integrals with both smooth and rapidly changing integrands. Aspects of their precision are analysed in the light of Gauss' error formula. Some "test examples" are considered and evaluated in multiple precision to ≈ 200 significant decimal digits with David Bailey's multiprecision package to eliminate truncation/rounding errors. The increase of precision on doubling the number of subintervals is analysed, the relevant quadrature attribute being the precision increment. In order to exemplify the advantages that high-order quadrature afford, the technique is then used to evaluate several plots of the Rayleigh-Sommerfeld diffraction integral for axi-symmetric source fields defined on a planar aperture. A comparison of the high-order quadrature method against various FFT-based methods is finally given.
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.
Impact of numerical integration on gas curtain simulations
Rider, W.; Kamm, J.
2000-11-01
In recent years, we have presented a less than glowing experimental comparison of hydrodynamic codes with the gas curtain experiment (e.g., Kamm et al. 1999a). Here, we discuss the manner in which the details of the hydrodynamic integration techniques may conspire to produce poor results. This also includes some progress in improving the results and agreement with experimental results. Because our comparison was conducted on the details of the experimental images (i.e., their detailed structural information), our results do not conflict with previously published results of good agreement with Richtmyer-Meshkov instabilities based on the integral scale of mixing. New experimental and analysis techniques are also discussed.
Numerical simulation methods for the Rouse model in flow
NASA Astrophysics Data System (ADS)
Howard, Michael P.; Milner, Scott T.
2011-11-01
Simulation of the Rouse model in flow underlies a great variety of numerical investigations of polymer dynamics, in both entangled melts and solutions and in dilute solution. Typically a simple explicit stochastic Euler method is used to evolve the Rouse model. Here we compare this approach to an operator splitting method, which splits the evolution operator into stochastic linear and deterministic nonlinear parts and takes advantage of an analytical solution for the linear Rouse model in terms of the noise history. We show that this splitting method has second-order weak convergence, whereas the Euler method has only first-order weak convergence. Furthermore, the splitting method is unconditionally stable, in contrast to the limited stability range of the Euler method. Similar splitting methods are applicable to a broad class of problems in stochastic dynamics in which noise competes with ordering and flow to determine steady-state order parameter structures.
Numerical Polynomial Homotopy Continuation Method and String Vacua
Mehta, Dhagash
2011-01-01
Finding vmore » acua for the four-dimensional effective theories for supergravity which descend from flux compactifications and analyzing them according to their stability is one of the central problems in string phenomenology. Except for some simple toy models, it is, however, difficult to find all the vacua analytically. Recently developed algorithmic methods based on symbolic computer algebra can be of great help in the more realistic models. However, they suffer from serious algorithmic complexities and are limited to small system sizes. In this paper, we review a numerical method called the numerical polynomial homotopy continuation (NPHC) method, first used in the areas of lattice field theories, which by construction finds all of the vacua of a given potential that is known to have only isolated solutions. The NPHC method is known to suffer from no major algorithmic complexities and is embarrassingly parallelizable , and hence its applicability goes way beyond the existing symbolic methods. We first solve a simple toy model as a warm-up example to demonstrate the NPHC method at work. We then show that all the vacua of a more complicated model of a compactified M theory model, which has an S U ( 3 ) structure, can be obtained by using a desktop machine in just about an hour, a feat which was reported to be prohibitively difficult by the existing symbolic methods. Finally, we compare the various technicalities between the two methods.« less
Numerical methods for characterization of synchrotron radiation based on the Wigner function method
NASA Astrophysics Data System (ADS)
Tanaka, Takashi
2014-06-01
Numerical characterization of synchrotron radiation based on the Wigner function method is explored in order to accurately evaluate the light source performance. A number of numerical methods to compute the Wigner functions for typical synchrotron radiation sources such as bending magnets, undulators and wigglers, are presented, which significantly improve the computation efficiency and reduce the total computation time. As a practical example of the numerical characterization, optimization of betatron functions to maximize the brilliance of undulator radiation is discussed.
Projected discrete ordinates methods for numerical transport problems
Larsen, E.W.
1985-01-01
A class of Projected Discrete-Ordinates (PDO) methods is described for obtaining iterative solutions of discrete-ordinates problems with convergence rates comparable to those observed using Diffusion Synthetic Acceleration (DSA). The spatially discretized PDO solutions are generally not equal to the DSA solutions, but unlike DSA, which requires great care in the use of spatial discretizations to preserve stability, the PDO solutions remain stable and rapidly convergent with essentially arbitrary spatial discretizations. Numerical results are presented which illustrate the rapid convergence and the accuracy of solutions obtained using PDO methods with commonplace differencing methods.
Computational methods for aerodynamic design using numerical optimization
NASA Technical Reports Server (NTRS)
Peeters, M. F.
1983-01-01
Five methods to increase the computational efficiency of aerodynamic design using numerical optimization, by reducing the computer time required to perform gradient calculations, are examined. The most promising method consists of drastically reducing the size of the computational domain on which aerodynamic calculations are made during gradient calculations. Since a gradient calculation requires the solution of the flow about an airfoil whose geometry was slightly perturbed from a base airfoil, the flow about the base airfoil is used to determine boundary conditions on the reduced computational domain. This method worked well in subcritical flow.
On numerical methods in non-Newtonian flows
NASA Astrophysics Data System (ADS)
Fileas, G.
1982-12-01
The constitutive equations for non-Newtonian flows are presented and the various flow models derived from continuum mechanics and molecular theories are considered and evaluated. Detailed account is given of numerical simulation employing differential and integral models of different kinds of non-Newtonian flows using finite difference and finite element techniques. Procedures for computer set ups are described and references are given for finite difference, finite element and molecular theory based programs for several kinds of flow. Achievements and unreached goals in the field of numerical simulation of non-Newtonian flows are discussed and the lack of numerical work in the fields of suspension flows and heat transfer is pointed out. Finally, FFOCUS is presented as a newly built computer program which can simulate freezing flows of Newtonian fluids through various geometries and is aimed to be further developed to handle non-Newtonian freezing flows and certain types of suspension phenomena involved in corium flow after a hypothetical core melt down accident in a pressurized water reactor.
NASA Astrophysics Data System (ADS)
Xie, Guizhong; Zhang, Dehai; Zhang, Jianming; Meng, Fannian; Du, Wenliao; Wen, Xiaoyu
2016-07-01
As a widely used numerical method, boundary element method (BEM) is efficient for computer aided engineering (CAE). However, boundary integrals with near singularity need to be calculated accurately and efficiently to implement BEM for CAE analysis on thin bodies successfully. In this paper, the distance in the denominator of the fundamental solution is first designed as an equivalent form using approximate expansion and the original sinh method can be revised into a new form considering the minimum distance and the approximate expansion. Second, the acquisition of the projection point by Newton-Raphson method is introduced. We acquire the nearest point between the source point and element edge by solving a cubic equation if the location of the projection point is outside the element, where boundary integrals with near singularity appear. Finally, the subtriangles of the local coordinate space are mapped into the integration space and the sinh method is applied in the integration space. The revised sinh method can be directly performed in the integration element. Averification test of our method is proposed. Results demonstrate that our method is effective for regularizing the boundary integrals with near singularity.
Tensor numerical methods in quantum chemistry: from Hartree-Fock to excitation energies.
Khoromskaia, Venera; Khoromskij, Boris N
2015-12-21
We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, first appeared as an accurate tensor calculus for the 3D Hartree potential using 1D complexity operations, and have evolved to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in O(n log n) complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D n × n × n Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D "density fitting" scheme, which yield an almost irreducible number of product basis functions involved in the 3D convolution integrals, depending on a threshold ε > 0. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excitation energies, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is towards the tensor-based Hartree-Fock numerical scheme for finite lattices, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a L × L × L lattice manifests the linear in L computational work, O(L), instead of the usual O(L(3) log L) scaling by the Ewald-type approaches. PMID:26016539
Fast and stable numerical method for neuronal modelling
NASA Astrophysics Data System (ADS)
Hashemi, Soheil; Abdolali, Ali
2016-11-01
Excitable cell modelling is of a prime interest in predicting and targeting neural activity. Two main limits in solving related equations are speed and stability of numerical method. Since there is a tradeoff between accuracy and speed, most previously presented methods for solving partial differential equations (PDE) are focused on one side. More speed means more accurate simulations and therefore better device designing. By considering the variables in finite differenced equation in proper time and calculating the unknowns in the specific sequence, a fast, stable and accurate method is introduced in this paper for solving neural partial differential equations. Propagation of action potential in giant axon is studied by proposed method and traditional methods. Speed, consistency and stability of the methods are compared and discussed. The proposed method is as fast as forward methods and as stable as backward methods. Forward methods are known as fastest methods and backward methods are stable in any circumstances. Complex structures can be simulated by proposed method due to speed and stability of the method.
NASA Astrophysics Data System (ADS)
Bao, WeiZhu; Cai, YongYong; Jia, XiaoWei; Yin, Jia
2016-08-01
We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\\varepsilon\\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\\tau$ as well as the small parameter $0<\\varepsilon\\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\\varepsilon\\ll 1$, the CNFD method requests the $\\varepsilon$-scalability: $\\tau=O(\\varepsilon^3)$ and $h=O(\\sqrt{\\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\\varepsilon$-scalability is improved to $\\tau=O(\\varepsilon^2)$ and $h=O(1)$ when $0<\\varepsilon\\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.
Numerical method for shear bands in ductile metal with inclusions
Plohr, Jee Yeon N; Plohr, Bradley J
2010-01-01
A numerical method for mesoscale simulation of high strain-rate loading of ductile metal containing inclusions is described. Because of small-scale inhomogeneities, such a composite material is prone to localized shear deformation (adiabatic shear bands). The modeling framework is the Generalized Method of Cells of Paley and Aboudi [Mech. Materials, vol. 14, pp. /27-139, 1992], which ensures that the micromechanical response of the material is reflected in the behavior of the composite at the mesoscale. To calculate the effective plastic strain rate when shear bands are present, the analytic and numerical analysis of shear bands by Glimm, Plohr, and Sharp [Mech. Materials, vol. 24, pp. 31-41, 1996] is adapted and extended.
An alternative numerical method for the stationary pulsar magnetosphere
NASA Astrophysics Data System (ADS)
Takamori, Yohsuke; Okawa, Hirotada; Takamoto, Makoto; Suwa, Yudai
2014-02-01
Stationary pulsar magnetospheres in the force-free system are governed by the pulsar equation. In 1999, Contopoulos, Kazanas, and Fendt (hereafter CKF) numerically solved the pulsar equation and obtained a pulsar magnetosphere model called the CKF solution that has both closed and open magnetic field lines. The CKF solution is a successful solution, but it contains a poloidal current sheet that flows along the last open field line. This current sheet is artificially added to make the current system closed. In this paper, we suggest an alternative method to solve the pulsar equation and construct pulsar magnetosphere models without a current sheet. In our method, the pulsar equation is decomposed into Ampère's law and the force-free condition. We numerically solve these equations simultaneously with a fixed poloidal current. As a result, we obtain a pulsar magnetosphere model without a current sheet, which is similar to the CKF solution near the neutron star and has a jet-like structure at a distance along the pole. In addition, we discuss physical properties of the model and find that the force-free condition breaks down in a vicinity of the light cylinder due to dissipation that is included implicitly in the numerical method.
Numerical method for wave forces acting on partially perforated caisson
NASA Astrophysics Data System (ADS)
Jiang, Feng; Tang, Xiao-cheng; Jin, Zhao; Zhang, Li; Chen, Hong-zhou
2015-04-01
The perforated caisson is widely applied to practical engineering because of its great advantages in effectively wave energy consumption and cost reduction. The attentions of many scientists were paid to the fluid-structure interaction between wave and perforated caisson studies, but until now, most concerns have been put on theoretical analysis and experimental model set up. In this paper, interaction between the wave and the partial perforated caisson in a 2D numerical wave flume is investigated by means of the renewed SPH algorithm, and the mathematical equations are in the form of SPH numerical approximation based on Navier-Stokes equations. The validity of the SPH mathematical method is examined and the simulated results are compared with the results of theoretical models, meanwhile the complex hydrodynamic characteristics when the water particles flow in or out of a wave absorbing chamber are analyzed and the wave pressure distribution of the perforated caisson is also addressed here. The relationship between the ratio of total horizontal force acting on caisson under regular waves and its influence factors is examined. The data show that the numerical calculation of the ratio of total horizontal force meets the empirical regression equation very well. The simulations of SPH about the wave nonlinearity and breaking are briefly depicted in the paper, suggesting that the advantages and great potentiality of the SPH method is significant compared with traditional methods.
EMERGY METHODS: VALUABLE INTEGRATED ASSESSMENT TOOLS
NHEERL's Atlantic Ecology Division is investigating emergy methods as tools for integrated assessment in several projects evaluating environmental impacts, policies, and alternatives for remediation and intervention. Emergy accounting is a methodology that provides a quantitative...
NASA Technical Reports Server (NTRS)
Thompson, J. F.; Mcwhorter, J. C.; Siddiqi, S. A.; Shanks, S. P.
1973-01-01
Numerical methods of integration of the equations of motion of a controlled satellite under the influence of gravity-gradient torque are considered. The results of computer experimentation using a number of Runge-Kutta, multi-step, and extrapolation methods for the numerical integration of this differential system are presented, and particularly efficient methods are noted. A large bibliography of numerical methods for initial value problems for ordinary differential equations is presented, and a compilation of Runge-Kutta and multistep formulas is given. Less common numerical integration techniques from the literature are noted for further consideration.
Dielectric Boundary Forces in Numerical Poisson-Boltzmann Methods: Theory and Numerical Strategies.
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-10-01
Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary forces based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems. PMID:22125339
Dielectric boundary force in numerical Poisson-Boltzmann methods: Theory and numerical strategies
NASA Astrophysics Data System (ADS)
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-10-01
Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary force based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.
Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus
NASA Astrophysics Data System (ADS)
Hirani, A. N.; Nakshatrala, K. B.; Chaudhry, J. H.
2015-05-01
We derive a numerical method for Darcy flow, and also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedral meshes. DEC is a coordinate invariant discretization, in that it does not depend on the embedding of the simplices or the whole mesh. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for a spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solutions in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. We also show numerical evidence of convergence of the flux and the pressure. A convergence experiment is included for Darcy flow on a surface. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this article. We also include a discussion of the boundary condition in terms of exterior calculus.
Teaching Thermal Hydraulics & Numerical Methods: An Introductory Control Volume Primer
Lucas, D.S.
2004-10-03
This paper covers the basics of the implementation of the control volume method in the context of the Homogeneous Equilibrium Model (HEM)(T/H) code using the conservation equations of mass, momentum, and energy. This primer uses the advection equation as a template. The discussion will cover the basic equations of the control volume portion of the course in the primer, which includes the advection equation, numerical methods, along with the implementation of the various equations via FORTRAN into computer programs and the final result for a three equation HEM code and its validation.
Explicit Integration of Extremely Stiff Reaction Networks: Partial Equilibrium Methods
Guidry, Mike W; Billings, J. J.; Hix, William Raphael
2013-01-01
In two preceding papers [1,2] we have shown that, when reaction networks are well removed from equilibrium, explicit asymptotic and quasi-steady-state approximations can give algebraically stabilized integration schemes that rival standard implicit methods in accuracy and speed for extremely stiff systems. However, we also showed that these explicit methods remain accurate but are no longer competitive in speed as the network approaches equilibrium. In this paper we analyze this failure and show that it is associated with the presence of fast equilibration timescales that neither asymptotic nor quasi-steady-state approximations are able to remove efficiently from the numerical integration. Based on this understanding, we develop a partial equilibrium method to deal effectively with the new partial equilibrium methods, give an integration scheme that plausibly can deal with the stiffest networks, even in the approach to equilibrium, with accuracy and speed competitive with that of implicit methods. Thus we demonstrate that algebraically stabilized explicit methods may offer alternatives to implicit integration of even extremely stiff systems, and that these methods may permit integration of much larger networks than have been feasible previously in a variety of fields.
On the accuracy of numerical integration over the unit sphere applied to full network models
NASA Astrophysics Data System (ADS)
Itskov, Mikhail
2016-05-01
This paper is motivated by a recent study by Verron (Mecha Mater 89:216-228, 2015) which revealed huge errors of the numerical integration over the unit sphere in application to large strain problems. For the verification of numerical integration schemes we apply here other analytical integrals over the unit sphere which demonstrate much more accurate results. Relative errors of these integrals with respect to corresponding analytical solutions are evaluated also for a full network model of rubber elasticity based on a Padé approximation of the inverse Langevin function as the chain force. According to the results of our study, the numerical integration over the unit sphere can still be considered as a reliable and accurate tool for full network models.
Optimization methods and silicon solar cell numerical models
NASA Technical Reports Server (NTRS)
Girardini, K.; Jacobsen, S. E.
1986-01-01
An optimization algorithm for use with numerical silicon solar cell models was developed. By coupling an optimization algorithm with a solar cell model, it is possible to simultaneously vary design variables such as impurity concentrations, front junction depth, back junction depth, and cell thickness to maximize the predicted cell efficiency. An optimization algorithm was developed and interfaced with the Solar Cell Analysis Program in 1 Dimension (SCAP1D). SCAP1D uses finite difference methods to solve the differential equations which, along with several relations from the physics of semiconductors, describe mathematically the performance of a solar cell. A major obstacle is that the numerical methods used in SCAP1D require a significant amount of computer time, and during an optimization the model is called iteratively until the design variables converge to the values associated with the maximum efficiency. This problem was alleviated by designing an optimization code specifically for use with numerically intensive simulations, to reduce the number of times the efficiency has to be calculated to achieve convergence to the optimal solution.
Numerical methods for the Poisson-Fermi equation in electrolytes
NASA Astrophysics Data System (ADS)
Liu, Jinn-Liang
2013-08-01
The Poisson-Fermi equation proposed by Bazant, Storey, and Kornyshev [Phys. Rev. Lett. 106 (2011) 046102] for ionic liquids is applied to and numerically studied for electrolytes and biological ion channels in three-dimensional space. This is a fourth-order nonlinear PDE that deals with both steric and correlation effects of all ions and solvent molecules involved in a model system. The Fermi distribution follows from classical lattice models of configurational entropy of finite size ions and solvent molecules and hence prevents the long and outstanding problem of unphysical divergence predicted by the Gouy-Chapman model at large potentials due to the Boltzmann distribution of point charges. The equation reduces to Poisson-Boltzmann if the correlation length vanishes. A simplified matched interface and boundary method exhibiting optimal convergence is first developed for this equation by using a gramicidin A channel model that illustrates challenging issues associated with the geometric singularities of molecular surfaces of channel proteins in realistic 3D simulations. Various numerical methods then follow to tackle a range of numerical problems concerning the fourth-order term, nonlinearity, stability, efficiency, and effectiveness. The most significant feature of the Poisson-Fermi equation, namely, its inclusion of steric and correlation effects, is demonstrated by showing good agreement with Monte Carlo simulation data for a charged wall model and an L type calcium channel model.
AN EFFICIENT NUMERICAL METHOD FOR THE SOLUTION OF THE L2 OPTIMAL MASS TRANSFER PROBLEM*
Haber, Eldad; Rehman, Tauseef; Tannenbaum, Allen
2010-01-01
In this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal L2 mass transport mapping. In contrast to the integration of a time dependent partial differential equation proposed in [S. Angenent, S. Haker, and A. Tannenbaum, SIAM J. Math. Anal., 35 (2003), pp. 61–97], we employ in the present work a direct variational method. The efficacy of the approach is demonstrated on both real and synthetic data. PMID:21278828
Analysis of free turbulent shear flows by numerical methods
NASA Technical Reports Server (NTRS)
Korst, H. H.; Chow, W. L.; Hurt, R. F.; White, R. A.; Addy, A. L.
1973-01-01
Studies are described in which the effort was essentially directed to classes of problems where the phenomenologically interpreted effective transport coefficients could be absorbed by, and subsequently extracted from (by comparison with experimental data), appropriate coordinate transformations. The transformed system of differential equations could then be solved without further specifications or assumptions by numerical integration procedures. An attempt was made to delineate different regimes for which specific eddy viscosity models could be formulated. In particular, this would account for the carryover of turbulence from attached boundary layers, the transitory adjustment, and the asymptotic behavior of initially disturbed mixing regions. Such models were subsequently used in seeking solutions for the prescribed two-dimensional test cases, yielding a better insight into overall aspects of the exchange mechanisms.
A Numerical Method for Solving Elasticity Equations with Interfaces
Li, Zhilin; Wang, Liqun; Wang, Wei
2012-01-01
Solving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L∞ norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner. PMID:22707984
Advanced numerical methods in mesh generation and mesh adaptation
Lipnikov, Konstantine; Danilov, A; Vassilevski, Y; Agonzal, A
2010-01-01
Numerical solution of partial differential equations requires appropriate meshes, efficient solvers and robust and reliable error estimates. Generation of high-quality meshes for complex engineering models is a non-trivial task. This task is made more difficult when the mesh has to be adapted to a problem solution. This article is focused on a synergistic approach to the mesh generation and mesh adaptation, where best properties of various mesh generation methods are combined to build efficiently simplicial meshes. First, the advancing front technique (AFT) is combined with the incremental Delaunay triangulation (DT) to build an initial mesh. Second, the metric-based mesh adaptation (MBA) method is employed to improve quality of the generated mesh and/or to adapt it to a problem solution. We demonstrate with numerical experiments that combination of all three methods is required for robust meshing of complex engineering models. The key to successful mesh generation is the high-quality of the triangles in the initial front. We use a black-box technique to improve surface meshes exported from an unattainable CAD system. The initial surface mesh is refined into a shape-regular triangulation which approximates the boundary with the same accuracy as the CAD mesh. The DT method adds robustness to the AFT. The resulting mesh is topologically correct but may contain a few slivers. The MBA uses seven local operations to modify the mesh topology. It improves significantly the mesh quality. The MBA method is also used to adapt the mesh to a problem solution to minimize computational resources required for solving the problem. The MBA has a solid theoretical background. In the first two experiments, we consider the convection-diffusion and elasticity problems. We demonstrate the optimal reduction rate of the discretization error on a sequence of adaptive strongly anisotropic meshes. The key element of the MBA method is construction of a tensor metric from hierarchical edge
Novel Parallel Numerical Methods for Radiation& Neutron Transport
Brown, P N
2001-03-06
In many of the multiphysics simulations performed at LLNL, transport calculations can take up 30 to 50% of the total run time. If Monte Carlo methods are used, the percentage can be as high as 80%. Thus, a significant core competence in the formulation, software implementation, and solution of the numerical problems arising in transport modeling is essential to Laboratory and DOE research. In this project, we worked on developing scalable solution methods for the equations that model the transport of photons and neutrons through materials. Our goal was to reduce the transport solve time in these simulations by means of more advanced numerical methods and their parallel implementations. These methods must be scalable, that is, the time to solution must remain constant as the problem size grows and additional computer resources are used. For iterative methods, scalability requires that (1) the number of iterations to reach convergence is independent of problem size, and (2) that the computational cost grows linearly with problem size. We focused on deterministic approaches to transport, building on our earlier work in which we performed a new, detailed analysis of some existing transport methods and developed new approaches. The Boltzmann equation (the underlying equation to be solved) and various solution methods have been developed over many years. Consequently, many laboratory codes are based on these methods, which are in some cases decades old. For the transport of x-rays through partially ionized plasmas in local thermodynamic equilibrium, the transport equation is coupled to nonlinear diffusion equations for the electron and ion temperatures via the highly nonlinear Planck function. We investigated the suitability of traditional-solution approaches to transport on terascale architectures and also designed new scalable algorithms; in some cases, we investigated hybrid approaches that combined both.
Mixed time integration methods for transient thermal analysis of structures
NASA Technical Reports Server (NTRS)
Liu, W. K.
1983-01-01
The computational methods used to predict and optimize the thermal-structural behavior of aerospace vehicle structures are reviewed. In general, two classes of algorithms, implicit and explicit, are used in transient thermal analysis of structures. Each of these two methods has its own merits. Due to the different time scales of the mechanical and thermal responses, the selection of a time integration method can be a difficult yet critical factor in the efficient solution of such problems. Therefore mixed time integration methods for transient thermal analysis of structures are being developed. The computer implementation aspects and numerical evaluation of these mixed time implicit-explicit algorithms in thermal analysis of structures are presented. A computationally-useful method of estimating the critical time step for linear quadrilateral element is also given. Numerical tests confirm the stability criterion and accuracy characteristics of the methods. The superiority of these mixed time methods to the fully implicit method or the fully explicit method is also demonstrated.
Numerical Simulation of Parachute Inflation Process Using AN Overset Deforming Grids Method
NASA Astrophysics Data System (ADS)
Xia, Jian; Tian, Shuling; Wu, Yizhao
A numerical method for the simulation of parachute inflation process is presented in this paper. The unsteady compressible N-S equations are fully coupled with MSD (Mass Spring Damper) structure model and integrated forward in time. The CFD solver is based on an unstructured finite volume algorithm and the preconditioning technique is applied to alleviate the stiffness caused by low Mach number. The Spalart-Allmaras one-equation turbulence model is implemented to evaluate the turbulent viscosity. The whole system (fluid equations and structural model equations) is marched implicitly in time using a dual time stepping method. An overset deforming grids method is adopted in this paper to deal with the very large domain deformation during the parachute inflation process. Finally numerical test is performed to validate the robustness of this method.
A survey of payload integration methods
NASA Technical Reports Server (NTRS)
Engels, R. C.; Harcrow, H. W.
1981-01-01
The most prominent payload integration methods are presented and evaluated. The paper outlines the problem and some of the difficulties encountered when analyzing a coupled booster/payload system. Descriptions of both full-scale and short-cut methods are given together with an assessment of their strengths and weaknesses. Finally, an extensive list of references is included.
Numerical Analysis of a Finite Element/Volume Penalty Method
NASA Astrophysics Data System (ADS)
Maury, Bertrand
The penalty method makes it possible to incorporate a large class of constraints in general purpose Finite Element solvers like freeFEM++. We present here some contributions to the numerical analysis of this method. We propose an abstract framework for this approach, together with some general error estimates based on the discretization parameter ɛ and the space discretization parameter h. As this work is motivated by the possibility to handle constraints like rigid motion for fluid-particle flows, we shall pay a special attention to a model problem of this kind, where the constraint is prescribed over a subdomain. We show how the abstract estimate can be applied to this situation, in the case where a non-body-fitted mesh is used. In addition, we describe how this method provides an approximation of the Lagrange multiplier associated to the constraint.
Numerical methods for high-dimensional probability density function equations
NASA Astrophysics Data System (ADS)
Cho, H.; Venturi, D.; Karniadakis, G. E.
2016-01-01
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker-Planck and Dostupov-Pugachev equations), random wave theory (Malakhov-Saichev equations) and coarse-grained stochastic systems (Mori-Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
Improved numerical method for subchannel cross-flow calculations
Kaya, S.; Anghaie, S.
1986-01-01
COBRA-OSU is a fast running computer code for coupled kinetic and thermal-hydraulic analysis of nuclear reactor core subchannels, currently under development at Oregon State University. This code is a modified version of COBRA-IV with two major improved features. First, COBRA-OSU uses the Gaussian elimination method instead of Gauss-Seidel iteration for subchannel cross-flow calculation. Second, COBRA-OSU has an additional model for regionwise point reactor kinetics which includes all major feedback reactivity effects on calculation of the axial power profile during the course of a transient. This paper summarizes the improved numerical features of the COBRA-OSU code.