From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method

PubMed Central

Bolis, A; Cantwell, C D; Kirby, R M; Sherwin, S J

2014-01-01

We investigate the relative performance of a second-order Adams–Bashforth scheme and second-order and fourth-order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd. PMID:25892840

A parallelization method for time periodic steady state in simulation of radio frequency sheath dynamics

NASA Astrophysics Data System (ADS)

Kwon, Deuk-Chul; Shin, Sung-Sik; Yu, Dong-Hun

2017-10-01

In order to reduce the computing time in simulation of radio frequency (rf) plasma sources, various numerical schemes were developed. It is well known that the upwind, exponential, and power-law schemes can efficiently overcome the limitation on the grid size for fluid transport simulations of high density plasma discharges. Also, the semi-implicit method is a well-known numerical scheme to overcome on the simulation time step. However, despite remarkable advances in numerical techniques and computing power over the last few decades, efficient multi-dimensional modeling of low temperature plasma discharges has remained a considerable challenge. In particular, there was a difficulty on parallelization in time for the time periodic steady state problems such as capacitively coupled plasma discharges and rf sheath dynamics because values of plasma parameters in previous time step are used to calculate new values each time step. Therefore, we present a parallelization method for the time periodic steady state problems by using period-slices. In order to evaluate the efficiency of the developed method, one-dimensional fluid simulations are conducted for describing rf sheath dynamics. The result shows that speedup can be achieved by using a multithreading method.

Asynchronous machine rotor speed estimation using a tabulated numerical approach

NASA Astrophysics Data System (ADS)

Nguyen, Huu Phuc; De Miras, Jérôme; Charara, Ali; Eltabach, Mario; Bonnet, Stéphane

2017-12-01

This paper proposes a new method to estimate the rotor speed of the asynchronous machine by looking at the estimation problem as a nonlinear optimal control problem. The behavior of the nonlinear plant model is approximated off-line as a prediction map using a numerical one-step time discretization obtained from simulations. At each time-step, the speed of the induction machine is selected satisfying the dynamic fitting problem between the plant output and the predicted output, leading the system to adopt its dynamical behavior. Thanks to the limitation of the prediction horizon to a single time-step, the execution time of the algorithm can be completely bounded. It can thus easily be implemented and embedded into a real-time system to observe the speed of the real induction motor. Simulation results show the performance and robustness of the proposed estimator.

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.

Numerical simulation of the kinetic effects in the solar wind

NASA Astrophysics Data System (ADS)

Sokolov, I.; Toth, G.; Gombosi, T. I.

2017-12-01

Global numerical simulations of the solar wind are usually based on the ideal or resistive MagnetoHydroDynamics (MHD) equations. Within a framework of MHD the electric field is assumed to vanish in the co-moving frame of reference (ideal MHD) or to obey a simple and non-physical scalar Ohm's law (resistive MHD). The Maxwellian distribution functions are assumed, the electron and ion temperatures may be different. Non-disversive MHD waves can be present in this numerical model. The averaged equations for MHD turbulence may be included as well as the energy and momentum exchange between the turbulent and regular motion. With the use of explicit numerical scheme, the time step is controlled by the MHD wave propagtion time across the numerical cell (the CFL condition) More refined approach includes the Hall effect vie the generalized Ohm's law. The Lorentz force acting on light electrons is assumed to vanish, which gives the expression for local electric field in terms of the total electric current, the ion current as well as the electron pressure gradient and magnetic field. The waves (whistlers, ion-cyclotron waves etc) aquire dispersion and the short-wavelength perturbations propagate with elevated speed thus strengthening the CFL condition. If the grid size is sufficiently small to resolve ion skindepth scale, then the timestep is much shorter than the ion gyration period. The next natural step is to use hybrid code to resolve the ion kinetic effects. The hybrid numerical scheme employs the same generalized Ohm's law as Hall MHD and suffers from the same constraint on the time step while solving evolution of the electromagnetic field. The important distiction, however, is that by sloving particle motion for ions we can achieve more detailed description of the kinetic effect without significant degrade in the computational efficiency, because the time-step is sufficient to resolve the particle gyration. We present the fisrt numerical results from coupled BATS-R-US+ALTOR code as applied to kinetic simulations of the solar wind.

Improved diffusion Monte Carlo propagators for bosonic systems using Itô calculus

NASA Astrophysics Data System (ADS)

Hâkansson, P.; Mella, M.; Bressanini, Dario; Morosi, Gabriele; Patrone, Marta

2006-11-01

The construction of importance sampled diffusion Monte Carlo (DMC) schemes accurate to second order in the time step is discussed. A central aspect in obtaining efficient second order schemes is the numerical solution of the stochastic differential equation (SDE) associated with the Fokker-Plank equation responsible for the importance sampling procedure. In this work, stochastic predictor-corrector schemes solving the SDE and consistent with Itô calculus are used in DMC simulations of helium clusters. These schemes are numerically compared with alternative algorithms obtained by splitting the Fokker-Plank operator, an approach that we analyze using the analytical tools provided by Itô calculus. The numerical results show that predictor-corrector methods are indeed accurate to second order in the time step and that they present a smaller time step bias and a better efficiency than second order split-operator derived schemes when computing ensemble averages for bosonic systems. The possible extension of the predictor-corrector methods to higher orders is also discussed.

Efficient computation of the Grünwald-Letnikov fractional diffusion derivative using adaptive time step memory

NASA Astrophysics Data System (ADS)

MacDonald, Christopher L.; Bhattacharya, Nirupama; Sprouse, Brian P.; Silva, Gabriel A.

2015-09-01

Computing numerical solutions to fractional differential equations can be computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In general, numerical approaches that depend on truncating part of the system history while efficient, can suffer from high degrees of error and inaccuracy. Here we present an adaptive time step memory method for smooth functions applied to the Grünwald-Letnikov fractional diffusion derivative. This method is computationally efficient and results in smaller errors during numerical simulations. Sampled points along the system's history at progressively longer intervals are assumed to reflect the values of neighboring time points. By including progressively fewer points backward in time, a temporally 'weighted' history is computed that includes contributions from the entire past of the system, maintaining accuracy, but with fewer points actually calculated, greatly improving computational efficiency.

Numerical solution of the incompressible Navier-Stokes equations. Ph.D. Thesis - Stanford Univ., Mar. 1989

NASA Technical Reports Server (NTRS)

Rogers, Stuart E.

1990-01-01

The current work is initiated in an effort to obtain an efficient, accurate, and robust algorithm for the numerical solution of the incompressible Navier-Stokes equations in two- and three-dimensional generalized curvilinear coordinates for both steady-state and time-dependent flow problems. This is accomplished with the use of the method of artificial compressibility and a high-order flux-difference splitting technique for the differencing of the convective terms. Time accuracy is obtained in the numerical solutions by subiterating the equations in psuedo-time for each physical time step. The system of equations is solved with a line-relaxation scheme which allows the use of very large pseudo-time steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. Numerous laminar test flow problems are computed and presented with a comparison against analytically known solutions or experimental results. These include the flow in a driven cavity, the flow over a backward-facing step, the steady and unsteady flow over a circular cylinder, flow over an oscillating plate, flow through a one-dimensional inviscid channel with oscillating back pressure, the steady-state flow through a square duct with a 90 degree bend, and the flow through an artificial heart configuration with moving boundaries. An adequate comparison with the analytical or experimental results is obtained in all cases. Numerical comparisons of the upwind differencing with central differencing plus artificial dissipation indicates that the upwind differencing provides a much more robust algorithm, which requires significantly less computing time. The time-dependent problems require on the order of 10 to 20 subiterations, indicating that the elliptical nature of the problem does require a substantial amount of computing effort.

An asymptotic-preserving Lagrangian algorithm for the time-dependent anisotropic heat transport equation

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

Chacon, Luis; del-Castillo-Negrete, Diego; Hauck, Cory D.

2014-09-01

We propose a Lagrangian numerical algorithm for a time-dependent, anisotropic temperature transport equation in magnetized plasmas in the large guide field regime. The approach is based on an analytical integral formal solution of the parallel (i.e., along the magnetic field) transport equation with sources, and it is able to accommodate both local and non-local parallel heat flux closures. The numerical implementation is based on an operator-split formulation, with two straightforward steps: a perpendicular transport step (including sources), and a Lagrangian (field-line integral) parallel transport step. Algorithmically, the first step is amenable to the use of modern iterative methods, while themore » second step has a fixed cost per degree of freedom (and is therefore scalable). Accuracy-wise, the approach is free from the numerical pollution introduced by the discrete parallel transport term when the perpendicular to parallel transport coefficient ratio X ⊥ /X ∥ becomes arbitrarily small, and is shown to capture the correct limiting solution when ε = X⊥L 2 ∥/X1L 2 ⊥ → 0 (with L∥∙ L⊥ , the parallel and perpendicular diffusion length scales, respectively). Therefore, the approach is asymptotic-preserving. We demonstrate the capabilities of the scheme with several numerical experiments with varying magnetic field complexity in two dimensions, including the case of transport across a magnetic island.« less

On improving the iterative convergence properties of an implicit approximate-factorization finite difference algorithm. [considering transonic flow

NASA Technical Reports Server (NTRS)

Desideri, J. A.; Steger, J. L.; Tannehill, J. C.

1978-01-01

The iterative convergence properties of an approximate-factorization implicit finite-difference algorithm are analyzed both theoretically and numerically. Modifications to the base algorithm were made to remove the inconsistency in the original implementation of artificial dissipation. In this way, the steady-state solution became independent of the time-step, and much larger time-steps can be used stably. To accelerate the iterative convergence, large time-steps and a cyclic sequence of time-steps were used. For a model transonic flow problem governed by the Euler equations, convergence was achieved with 10 times fewer time-steps using the modified differencing scheme. A particular form of instability due to variable coefficients is also analyzed.

Least-squares finite element methods for compressible Euler equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Carey, G. F.

1990-01-01

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

Counterrotating prop-fan simulations which feature a relative-motion multiblock grid decomposition enabling arbitrary time-steps

NASA Technical Reports Server (NTRS)

Janus, J. Mark; Whitfield, David L.

1990-01-01

Improvements are presented of a computer algorithm developed for the time-accurate flow analysis of rotating machines. The flow model is a finite volume method utilizing a high-resolution approximate Riemann solver for interface flux definitions. The numerical scheme is a block LU implicit iterative-refinement method which possesses apparent unconditional stability. Multiblock composite gridding is used to orderly partition the field into a specified arrangement of blocks exhibiting varying degrees of similarity. Block-block relative motion is achieved using local grid distortion to reduce grid skewness and accommodate arbitrary time step selection. A general high-order numerical scheme is applied to satisfy the geometric conservation law. An even-blade-count counterrotating unducted fan configuration is chosen for a computational study comparing solutions resulting from altering parameters such as time step size and iteration count. The solutions are compared with measured data.

Comprehensive Numerical Analysis of Finite Difference Time Domain Methods for Improving Optical Waveguide Sensor Accuracy

PubMed Central

Samak, M. Mosleh E. Abu; Bakar, A. Ashrif A.; Kashif, Muhammad; Zan, Mohd Saiful Dzulkifly

2016-01-01

This paper discusses numerical analysis methods for different geometrical features that have limited interval values for typically used sensor wavelengths. Compared with existing Finite Difference Time Domain (FDTD) methods, the alternating direction implicit (ADI)-FDTD method reduces the number of sub-steps by a factor of two to three, which represents a 33% time savings in each single run. The local one-dimensional (LOD)-FDTD method has similar numerical equation properties, which should be calculated as in the previous method. Generally, a small number of arithmetic processes, which result in a shorter simulation time, are desired. The alternating direction implicit technique can be considered a significant step forward for improving the efficiency of unconditionally stable FDTD schemes. This comparative study shows that the local one-dimensional method had minimum relative error ranges of less than 40% for analytical frequencies above 42.85 GHz, and the same accuracy was generated by both methods.

Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Cooke, C. H.; Blanchard, D. K.

1975-01-01

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem.

Eulerian Lagrangian Adaptive Fup Collocation Method for solving the conservative solute transport in heterogeneous porous media

NASA Astrophysics Data System (ADS)

Gotovac, Hrvoje; Srzic, Veljko

2014-05-01

Contaminant transport in natural aquifers is a complex, multiscale process that is frequently studied using different Eulerian, Lagrangian and hybrid numerical methods. Conservative solute transport is typically modeled using the advection-dispersion equation (ADE). Despite the large number of available numerical methods that have been developed to solve it, the accurate numerical solution of the ADE still presents formidable challenges. In particular, current numerical solutions of multidimensional advection-dominated transport in non-uniform velocity fields are affected by one or all of the following problems: numerical dispersion that introduces artificial mixing and dilution, grid orientation effects, unresolved spatial and temporal scales and unphysical numerical oscillations (e.g., Herrera et al, 2009; Bosso et al., 2012). In this work we will present Eulerian Lagrangian Adaptive Fup Collocation Method (ELAFCM) based on Fup basis functions and collocation approach for spatial approximation and explicit stabilized Runge-Kutta-Chebyshev temporal integration (public domain routine SERK2) which is especially well suited for stiff parabolic problems. Spatial adaptive strategy is based on Fup basis functions which are closely related to the wavelets and splines so that they are also compactly supported basis functions; they exactly describe algebraic polynomials and enable a multiresolution adaptive analysis (MRA). MRA is here performed via Fup Collocation Transform (FCT) so that at each time step concentration solution is decomposed using only a few significant Fup basis functions on adaptive collocation grid with appropriate scales (frequencies) and locations, a desired level of accuracy and a near minimum computational cost. FCT adds more collocations points and higher resolution levels only in sensitive zones with sharp concentration gradients, fronts and/or narrow transition zones. According to the our recent achievements there is no need for solving the large linear system on adaptive grid because each Fup coefficient is obtained by predefined formulas equalizing Fup expansion around corresponding collocation point and particular collocation operator based on few surrounding solution values. Furthermore, each Fup coefficient can be obtained independently which is perfectly suited for parallel processing. Adaptive grid in each time step is obtained from solution of the last time step or initial conditions and advective Lagrangian step in the current time step according to the velocity field and continuous streamlines. On the other side, we implement explicit stabilized routine SERK2 for dispersive Eulerian part of solution in the current time step on obtained spatial adaptive grid. Overall adaptive concept does not require the solving of large linear systems for the spatial and temporal approximation of conservative transport. Also, this new Eulerian-Lagrangian-Collocation scheme resolves all mentioned numerical problems due to its adaptive nature and ability to control numerical errors in space and time. Proposed method solves advection in Lagrangian way eliminating problems in Eulerian methods, while optimal collocation grid efficiently describes solution and boundary conditions eliminating usage of large number of particles and other problems in Lagrangian methods. Finally, numerical tests show that this approach enables not only accurate velocity field, but also conservative transport even in highly heterogeneous porous media resolving all spatial and temporal scales of concentration field.

Numerical characterisation of one-step and three-step solar air heating collectors used for cocoa bean solar drying.

PubMed

Orbegoso, Elder Mendoza; Saavedra, Rafael; Marcelo, Daniel; La Madrid, Raúl

2017-12-01

In the northern coastal and jungle areas of Peru, cocoa beans are dried using artisan methods, such as direct exposure to sunlight. This traditional process is time intensive, leading to a reduction in productivity and, therefore, delays in delivery times. The present study was intended to numerically characterise the thermal behaviour of three configurations of solar air heating collectors in order to determine which demonstrated the best thermal performance under several controlled operating conditions. For this purpose, a computational fluid dynamics model was developed to describe the simultaneous convective and radiative heat transfer phenomena under several operation conditions. The constructed computational fluid dynamics model was firstly validated through comparison with the data measurements of a one-step solar air heating collector. We then simulated two further three-step solar air heating collectors in order to identify which demonstrated the best thermal performance in terms of outlet air temperature and thermal efficiency. The numerical results show that under the same solar irradiation area of exposition and operating conditions, the three-step solar air heating collector with the collector plate mounted between the second and third channels was 67% more thermally efficient compared to the one-step solar air heating collector. This is because the air exposition with the surface of the collector plate for the three-step solar air heating collector former device was twice than the one-step solar air heating collector. Copyright © 2017 Elsevier Ltd. All rights reserved.

An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm

NASA Astrophysics Data System (ADS)

Chen, G.; Chacón, L.; Barnes, D. C.

2011-08-01

This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov-Poisson formulation), ours is based on a nonlinearly converged Vlasov-Ampére (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant-Friedrichs-Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier "energy-conserving" explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton-Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.

Exact charge and energy conservation in implicit PIC with mapped computational meshes

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

Chen, Guangye; Barnes, D. C.

This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov Poisson formulation), ours is based on a nonlinearly converged Vlasov Amp re (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant Friedrichs Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicitmore » time steps (unlike the earlier energy-conserving explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible.« less

Development and Implementation of a Transport Method for the Transport and Reaction Simulation Engine (TaRSE) based on the Godunov-Mixed Finite Element Method

USGS Publications Warehouse

James, Andrew I.; Jawitz, James W.; Munoz-Carpena, Rafael

2009-01-01

A model to simulate transport of materials in surface water and ground water has been developed to numerically approximate solutions to the advection-dispersion equation. This model, known as the Transport and Reaction Simulation Engine (TaRSE), uses an algorithm that incorporates a time-splitting technique where the advective part of the equation is solved separately from the dispersive part. An explicit finite-volume Godunov method is used to approximate the advective part, while a mixed-finite element technique is used to approximate the dispersive part. The dispersive part uses an implicit discretization, which allows it to run stably with a larger time step than the explicit advective step. The potential exists to develop algorithms that run several advective steps, and then one dispersive step that encompasses the time interval of the advective steps. Because the dispersive step is computationally most expensive, schemes can be implemented that are more computationally efficient than non-time-split algorithms. This technique enables scientists to solve problems with high grid Peclet numbers, such as transport problems with sharp solute fronts, without spurious oscillations in the numerical approximation to the solution and with virtually no artificial diffusion.

A fast quadrature-based numerical method for the continuous spectrum biphasic poroviscoelastic model of articular cartilage.

PubMed

Stuebner, Michael; Haider, Mansoor A

2010-06-18

A new and efficient method for numerical solution of the continuous spectrum biphasic poroviscoelastic (BPVE) model of articular cartilage is presented. Development of the method is based on a composite Gauss-Legendre quadrature approximation of the continuous spectrum relaxation function that leads to an exponential series representation. The separability property of the exponential terms in the series is exploited to develop a numerical scheme that can be reduced to an update rule requiring retention of the strain history at only the previous time step. The cost of the resulting temporal discretization scheme is O(N) for N time steps. Application and calibration of the method is illustrated in the context of a finite difference solution of the one-dimensional confined compression BPVE stress-relaxation problem. Accuracy of the numerical method is demonstrated by comparison to a theoretical Laplace transform solution for a range of viscoelastic relaxation times that are representative of articular cartilage. Copyright (c) 2010 Elsevier Ltd. All rights reserved.

Arbitrarily high-order time-stepping schemes based on the operator spectrum theory for high-dimensional nonlinear Klein-Gordon equations

NASA Astrophysics Data System (ADS)

Liu, Changying; Wu, Xinyuan

2017-07-01

In this paper we explore arbitrarily high-order Lagrange collocation-type time-stepping schemes for effectively solving high-dimensional nonlinear Klein-Gordon equations with different boundary conditions. We begin with one-dimensional periodic boundary problems and first formulate an abstract ordinary differential equation (ODE) on a suitable infinity-dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula which is essential for the derivation of our arbitrarily high-order Lagrange collocation-type time-stepping schemes for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix under some suitable smoothness assumptions. With regard to the two dimensional Dirichlet or Neumann boundary problems, our new time-stepping schemes coupled with discrete Fast Sine / Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein-Gordon equations effectively. All essential features of the methodology are present in one-dimensional and two-dimensional cases, although the schemes to be analysed lend themselves with equal to higher-dimensional case. The numerical simulation is implemented and the numerical results clearly demonstrate the advantage and effectiveness of our new schemes in comparison with the existing numerical methods for solving nonlinear Klein-Gordon equations in the literature.

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

NASA Technical Reports Server (NTRS)

Nielsen, Tanner B.; Carpenter, Mark H.; Fisher, Travis C.; Frankel, Steven H.

2014-01-01

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sweby, P. K.; Griffiths, D. F.

1991-01-01

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Short‐term time step convergence in a climate model

PubMed Central

Rasch, Philip J.; Taylor, Mark A.; Jablonowski, Christiane

2015-01-01

Abstract This paper evaluates the numerical convergence of very short (1 h) simulations carried out with a spectral‐element (SE) configuration of the Community Atmosphere Model version 5 (CAM5). While the horizontal grid spacing is fixed at approximately 110 km, the process‐coupling time step is varied between 1800 and 1 s to reveal the convergence rate with respect to the temporal resolution. Special attention is paid to the behavior of the parameterized subgrid‐scale physics. First, a dynamical core test with reduced dynamics time steps is presented. The results demonstrate that the experimental setup is able to correctly assess the convergence rate of the discrete solutions to the adiabatic equations of atmospheric motion. Second, results from full‐physics CAM5 simulations with reduced physics and dynamics time steps are discussed. It is shown that the convergence rate is 0.4—considerably slower than the expected rate of 1.0. Sensitivity experiments indicate that, among the various subgrid‐scale physical parameterizations, the stratiform cloud schemes are associated with the largest time‐stepping errors, and are the primary cause of slow time step convergence. While the details of our findings are model specific, the general test procedure is applicable to any atmospheric general circulation model. The need for more accurate numerical treatments of physical parameterizations, especially the representation of stratiform clouds, is likely common in many models. The suggested test technique can help quantify the time‐stepping errors and identify the related model sensitivities. PMID:27660669

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

NASA Technical Reports Server (NTRS)

Yee, H. C.; Sweby, P. K.; Griffiths, D. F.

1990-01-01

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Direct Numerical Simulation of Turbulent Flow Over Complex Bathymetry

NASA Astrophysics Data System (ADS)

Yue, L.; Hsu, T. J.

2017-12-01

Direct numerical simulation (DNS) is regarded as a powerful tool in the investigation of turbulent flow featured with a wide range of time and spatial scales. With the application of coordinate transformation in a pseudo-spectral scheme, a parallelized numerical modeling system was created aiming at simulating flow over complex bathymetry with high numerical accuracy and efficiency. The transformed governing equations were integrated in time using a third-order low-storage Runge-Kutta method. For spatial discretization, the discrete Fourier expansion was adopted in the streamwise and spanwise direction, enforcing the periodic boundary condition in both directions. The Chebyshev expansion on Chebyshev-Gauss-Lobatto points was used in the wall-normal direction, assuming there is no-slip on top and bottom walls. The diffusion terms were discretized with a Crank-Nicolson scheme, while the advection terms dealiased with the 2/3 rule were discretized with an Adams-Bashforth scheme. In the prediction step, the velocity was calculated in physical domain by solving the resulting linear equation directly. However, the extra terms introduced by coordinate transformation impose a strict limitation to time step and an iteration method was applied to overcome this restriction in the correction step for pressure by solving the Helmholtz equation. The numerical solver is written in object-oriented C++ programing language utilizing Armadillo linear algebra library for matrix computation. Several benchmarking cases in laminar and turbulent flow were carried out to verify/validate the numerical model and very good agreements are achieved. Ongoing work focuses on implementing sediment transport capability for multiple sediment classes and parameterizations for flocculation processes.

A family of compact high order coupled time-space unconditionally stable vertical advection schemes

NASA Astrophysics Data System (ADS)

Lemarié, Florian; Debreu, Laurent

2016-04-01

Recent papers by Shchepetkin (2015) and Lemarié et al. (2015) have emphasized that the time-step of an oceanic model with an Eulerian vertical coordinate and an explicit time-stepping scheme is very often restricted by vertical advection in a few hot spots (i.e. most of the grid points are integrated with small Courant numbers, compared to the Courant-Friedrichs-Lewy (CFL) condition, except just few spots where numerical instability of the explicit scheme occurs first). The consequence is that the numerics for vertical advection must have good stability properties while being robust to changes in Courant number in terms of accuracy. An other constraint for oceanic models is the strict control of numerical mixing imposed by the highly adiabatic nature of the oceanic interior (i.e. mixing must be very small in the vertical direction below the boundary layer). We examine in this talk the possibility of mitigating vertical Courant-Friedrichs-Lewy (CFL) restriction, while avoiding numerical inaccuracies associated with standard implicit advection schemes (i.e. large sensitivity of the solution on Courant number, large phase delay, and possibly excess of numerical damping with unphysical orientation). Most regional oceanic models have been successfully using fourth order compact schemes for vertical advection. In this talk we present a new general framework to derive generic expressions for (one-step) coupled time and space high order compact schemes (see Daru & Tenaud (2004) for a thorough description of coupled time and space schemes). Among other properties, we show that those schemes are unconditionally stable and have very good accuracy properties even for large Courant numbers while having a very reasonable computational cost.

Numerical modeling and optimization of the Iguassu gas centrifuge

NASA Astrophysics Data System (ADS)

Bogovalov, S. V.; Borman, V. D.; Borisevich, V. D.; Tronin, V. N.; Tronin, I. V.

2017-07-01

The full procedure of the numerical calculation of the optimized parameters of the Iguassu gas centrifuge (GC) is under discussion. The procedure consists of a few steps. On the first step the problem of a hydrodynamical flow of the gas in the rotating rotor of the GC is solved numerically. On the second step the problem of diffusion of the binary mixture of isotopes is solved. The separation power of the gas centrifuge is calculated after that. On the last step the time consuming procedure of optimization of the GC is performed providing us the maximum of the separation power. The optimization is based on the BOBYQA method exploring the results of numerical simulations of the hydrodynamics and diffusion of the mixture of isotopes. Fast convergence of calculations is achieved due to exploring of a direct solver at the solution of the hydrodynamical and diffusion parts of the problem. Optimized separative power and optimal internal parameters of the Iguassu GC with 1 m rotor were calculated using the developed approach. Optimization procedure converges in 45 iterations taking 811 minutes.

Analysis of High Order Difference Methods for Multiscale Complex Compressible Flows

NASA Technical Reports Server (NTRS)

Sjoegreen, Bjoern; Yee, H. C.; Tang, Harry (Technical Monitor)

2002-01-01

Accurate numerical simulations of complex multiscale compressible viscous flows, especially high speed turbulence combustion and acoustics, demand high order schemes with adaptive numerical dissipation controls. Standard high resolution shock-capturing methods are too dissipative to capture the small scales and/or long-time wave propagations without extreme grid refinements and small time steps. An integrated approach for the control of numerical dissipation in high order schemes with incremental studies was initiated. Here we further refine the analysis on, and improve the understanding of the adaptive numerical dissipation control strategy. Basically, the development of these schemes focuses on high order nondissipative schemes and takes advantage of the progress that has been made for the last 30 years in numerical methods for conservation laws, such as techniques for imposing boundary conditions, techniques for stability at shock waves, and techniques for stable and accurate long-time integration. We concentrate on high order centered spatial discretizations and a fourth-order Runge-Kutta temporal discretizations as the base scheme. Near the bound-aries, the base scheme has stable boundary difference operators. To further enhance stability, the split form of the inviscid flux derivatives is frequently used for smooth flow problems. To enhance nonlinear stability, linear high order numerical dissipations are employed away from discontinuities, and nonlinear filters are employed after each time step in order to suppress spurious oscillations near discontinuities to minimize the smearing of turbulent fluctuations. Although these schemes are built from many components, each of which is well-known, it is not entirely obvious how the different components be best connected. For example, the nonlinear filter could instead have been built into the spatial discretization, so that it would have been activated at each stage in the Runge-Kutta time stepping. We could think of a mechanism that activates the split form of the equations only at some parts of the domain. Another issue is how to define good sensors for determining in which parts of the computational domain a certain feature should be filtered by the appropriate numerical dissipation. For the present study we employ a wavelet technique introduced in as sensors. Here, the method is briefly described with selected numerical experiments.

An algorithm for fast elastic wave simulation using a vectorized finite difference operator

NASA Astrophysics Data System (ADS)

Malkoti, Ajay; Vedanti, Nimisha; Tiwari, Ram Krishna

2018-07-01

Modern geophysical imaging techniques exploit the full wavefield information which can be simulated numerically. These numerical simulations are computationally expensive due to several factors, such as a large number of time steps and nodes, big size of the derivative stencil and huge model size. Besides these constraints, it is also important to reformulate the numerical derivative operator for improved efficiency. In this paper, we have introduced a vectorized derivative operator over the staggered grid with shifted coordinate systems. The operator increases the efficiency of simulation by exploiting the fact that each variable can be represented in the form of a matrix. This operator allows updating all nodes of a variable defined on the staggered grid, in a manner similar to the collocated grid scheme and thereby reducing the computational run-time considerably. Here we demonstrate an application of this operator to simulate the seismic wave propagation in elastic media (Marmousi model), by discretizing the equations on a staggered grid. We have compared the performance of this operator on three programming languages, which reveals that it can increase the execution speed by a factor of at least 2-3 times for FORTRAN and MATLAB; and nearly 100 times for Python. We have further carried out various tests in MATLAB to analyze the effect of model size and the number of time steps on total simulation run-time. We find that there is an additional, though small, computational overhead for each step and it depends on total number of time steps used in the simulation. A MATLAB code package, 'FDwave', for the proposed simulation scheme is available upon request.

Unconditionally energy stable time stepping scheme for Cahn–Morral equation: Application to multi-component spinodal decomposition and optimal space tiling

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

Tavakoli, Rouhollah, E-mail: rtavakoli@sharif.ir

An unconditionally energy stable time stepping scheme is introduced to solve Cahn–Morral-like equations in the present study. It is constructed based on the combination of David Eyre's time stepping scheme and Schur complement approach. Although the presented method is general and independent of the choice of homogeneous free energy density function term, logarithmic and polynomial energy functions are specifically considered in this paper. The method is applied to study the spinodal decomposition in multi-component systems and optimal space tiling problems. A penalization strategy is developed, in the case of later problem, to avoid trivial solutions. Extensive numerical experiments demonstrate themore » success and performance of the presented method. According to the numerical results, the method is convergent and energy stable, independent of the choice of time stepsize. Its MATLAB implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results. -- Highlights: •Extension of Eyre's convex–concave splitting scheme to multiphase systems. •Efficient solution of spinodal decomposition in multi-component systems. •Efficient solution of least perimeter periodic space partitioning problem. •Developing a penalization strategy to avoid trivial solutions. •Presentation of MATLAB implementation of the introduced algorithm.« less

Stability analysis of implicit time discretizations for the Compton-scattering Fokker-Planck equation

NASA Astrophysics Data System (ADS)

Densmore, Jeffery D.; Warsa, James S.; Lowrie, Robert B.; Morel, Jim E.

2009-09-01

The Fokker-Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker-Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

NASA Astrophysics Data System (ADS)

Rößler, Thomas; Stein, Olaf; Heng, Yi; Baumeister, Paul; Hoffmann, Lars

2018-02-01

The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge-Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge-Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge-Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge-Kutta method, respectively.

Numerical simulation of an elastic structure behavior under transient fluid flow excitation

NASA Astrophysics Data System (ADS)

Afanasyeva, Irina N.; Lantsova, Irina Yu.

2017-01-01

This paper deals with the verification of a numerical technique of modeling fluid-structure interaction (FSI) problems. The configuration consists of incompressible viscous fluid around an elastic structure in the channel. External flow is laminar. Multivariate calculations are performed using special software ANSYS CFX and ANSYS Mechanical. Different types of parameters of mesh deformation and solver controls (time step, under relaxation factor, number of iterations at coupling step) were tested. The results are presented in tables and plots in comparison with reference data.

Quadratic adaptive algorithm for solving cardiac action potential models.

PubMed

Chen, Min-Hung; Chen, Po-Yuan; Luo, Ching-Hsing

2016-10-01

An adaptive integration method is proposed for computing cardiac action potential models accurately and efficiently. Time steps are adaptively chosen by solving a quadratic formula involving the first and second derivatives of the membrane action potential. To improve the numerical accuracy, we devise an extremum-locator (el) function to predict the local extremum when approaching the peak amplitude of the action potential. In addition, the time step restriction (tsr) technique is designed to limit the increase in time steps, and thus prevent the membrane potential from changing abruptly. The performance of the proposed method is tested using the Luo-Rudy phase 1 (LR1), dynamic (LR2), and human O'Hara-Rudy dynamic (ORd) ventricular action potential models, and the Courtemanche atrial model incorporating a Markov sodium channel model. Numerical experiments demonstrate that the action potential generated using the proposed method is more accurate than that using the traditional Hybrid method, especially near the peak region. The traditional Hybrid method may choose large time steps near to the peak region, and sometimes causes the action potential to become distorted. In contrast, the proposed new method chooses very fine time steps in the peak region, but large time steps in the smooth region, and the profiles are smoother and closer to the reference solution. In the test on the stiff Markov ionic channel model, the Hybrid blows up if the allowable time step is set to be greater than 0.1ms. In contrast, our method can adjust the time step size automatically, and is stable. Overall, the proposed method is more accurate than and as efficient as the traditional Hybrid method, especially for the human ORd model. The proposed method shows improvement for action potentials with a non-smooth morphology, and it needs further investigation to determine whether the method is helpful during propagation of the action potential. Copyright © 2016 Elsevier Ltd. All rights reserved.

Time optimal control of a jet engine using a quasi-Hermite interpolation model. M.S. Thesis

NASA Technical Reports Server (NTRS)

Comiskey, J. G.

1979-01-01

This work made preliminary efforts to generate nonlinear numerical models of a two-spooled turbofan jet engine, and subject these models to a known method of generating global, nonlinear, time optimal control laws. The models were derived numerically, directly from empirical data, as a first step in developing an automatic modelling procedure.

Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

NASA Technical Reports Server (NTRS)

Rosenfeld, Moshe

1990-01-01

The main goals are the development, validation, and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems. A solution method that combines a finite volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.

Time-Accurate Solutions of Incompressible Navier-Stokes Equations for Potential Turbopump Applications

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Kwak, Dochan

2001-01-01

Two numerical procedures, one based on artificial compressibility method and the other pressure projection method, are outlined for obtaining time-accurate solutions of the incompressible Navier-Stokes equations. The performance of the two method are compared by obtaining unsteady solutions for the evolution of twin vortices behind a at plate. Calculated results are compared with experimental and other numerical results. For an un- steady ow which requires small physical time step, pressure projection method was found to be computationally efficient since it does not require any subiterations procedure. It was observed that the artificial compressibility method requires a fast convergence scheme at each physical time step in order to satisfy incompressibility condition. This was obtained by using a GMRES-ILU(0) solver in our computations. When a line-relaxation scheme was used, the time accuracy was degraded and time-accurate computations became very expensive.

Efficient adaptive pseudo-symplectic numerical integration techniques for Landau-Lifshitz dynamics

NASA Astrophysics Data System (ADS)

d'Aquino, M.; Capuano, F.; Coppola, G.; Serpico, C.; Mayergoyz, I. D.

2018-05-01

Numerical time integration schemes for Landau-Lifshitz magnetization dynamics are considered. Such dynamics preserves the magnetization amplitude and, in the absence of dissipation, also implies the conservation of the free energy. This property is generally lost when time discretization is performed for the numerical solution. In this work, explicit numerical schemes based on Runge-Kutta methods are introduced. The schemes are termed pseudo-symplectic in that they are accurate to order p, but preserve magnetization amplitude and free energy to order q > p. An effective strategy for adaptive time-stepping control is discussed for schemes of this class. Numerical tests against analytical solutions for the simulation of fast precessional dynamics are performed in order to point out the effectiveness of the proposed methods.

A New Family of Compact High Order Coupled Time-Space Unconditionally Stable Vertical Advection Schemes

NASA Astrophysics Data System (ADS)

Lemarié, F.; Debreu, L.

2016-02-01

Recent papers by Shchepetkin (2015) and Lemarié et al. (2015) have emphasized that the time-step of an oceanic model with an Eulerian vertical coordinate and an explicit time-stepping scheme is very often restricted by vertical advection in a few hot spots (i.e. most of the grid points are integrated with small Courant numbers, compared to the Courant-Friedrichs-Lewy (CFL) condition, except just few spots where numerical instability of the explicit scheme occurs first). The consequence is that the numerics for vertical advection must have good stability properties while being robust to changes in Courant number in terms of accuracy. An other constraint for oceanic models is the strict control of numerical mixing imposed by the highly adiabatic nature of the oceanic interior (i.e. mixing must be very small in the vertical direction below the boundary layer). We examine in this talk the possibility of mitigating vertical Courant-Friedrichs-Lewy (CFL) restriction, while avoiding numerical inaccuracies associated with standard implicit advection schemes (i.e. large sensitivity of the solution on Courant number, large phase delay, and possibly excess of numerical damping with unphysical orientation). Most regional oceanic models have been successfully using fourth order compact schemes for vertical advection. In this talk we present a new general framework to derive generic expressions for (one-step) coupled time and space high order compact schemes (see Daru & Tenaud (2004) for a thorough description of coupled time and space schemes). Among other properties, we show that those schemes are unconditionally stable and have very good accuracy properties even for large Courant numbers while having a very reasonable computational cost. To our knowledge no unconditionally stable scheme with such high order accuracy in time and space have been presented so far in the literature. Furthermore, we show how those schemes can be made monotonic without compromising their stability properties.

Enabling fast, stable and accurate peridynamic computations using multi-time-step integration

DOE PAGES

Lindsay, P.; Parks, M. L.; Prakash, A.

2016-04-13

Peridynamics is a nonlocal extension of classical continuum mechanics that is well-suited for solving problems with discontinuities such as cracks. This paper extends the peridynamic formulation to decompose a problem domain into a number of smaller overlapping subdomains and to enable the use of different time steps in different subdomains. This approach allows regions of interest to be isolated and solved at a small time step for increased accuracy while the rest of the problem domain can be solved at a larger time step for greater computational efficiency. Lastly, performance of the proposed method in terms of stability, accuracy, andmore » computational cost is examined and several numerical examples are presented to corroborate the findings.« less

Structured Overlapping Grid Simulations of Contra-rotating Open Rotor Noise

NASA Technical Reports Server (NTRS)

Housman, Jeffrey A.; Kiris, Cetin C.

2015-01-01

Computational simulations using structured overlapping grids with the Launch Ascent and Vehicle Aerodynamics (LAVA) solver framework are presented for predicting tonal noise generated by a contra-rotating open rotor (CROR) propulsion system. A coupled Computational Fluid Dynamics (CFD) and Computational AeroAcoustics (CAA) numerical approach is applied. Three-dimensional time-accurate hybrid Reynolds Averaged Navier-Stokes/Large Eddy Simulation (RANS/LES) CFD simulations are performed in the inertial frame, including dynamic moving grids, using a higher-order accurate finite difference discretization on structured overlapping grids. A higher-order accurate free-stream preserving metric discretization with discrete enforcement of the Geometric Conservation Law (GCL) on moving curvilinear grids is used to create an accurate, efficient, and stable numerical scheme. The aeroacoustic analysis is based on a permeable surface Ffowcs Williams-Hawkings (FW-H) approach, evaluated in the frequency domain. A time-step sensitivity study was performed using only the forward row of blades to determine an adequate time-step. The numerical approach is validated against existing wind tunnel measurements.

Developing Teaching Material Software Assisted for Numerical Methods

NASA Astrophysics Data System (ADS)

Handayani, A. D.; Herman, T.; Fatimah, S.

2017-09-01

The NCTM vision shows the importance of two things in school mathematics, which is knowing the mathematics of the 21st century and the need to continue to improve mathematics education to answer the challenges of a changing world. One of the competencies associated with the great challenges of the 21st century is the use of help and tools (including IT), such as: knowing the existence of various tools for mathematical activity. One of the significant challenges in mathematical learning is how to teach students about abstract concepts. In this case, technology in the form of mathematics learning software can be used more widely to embed the abstract concept in mathematics. In mathematics learning, the use of mathematical software can make high level math activity become easier accepted by student. Technology can strengthen student learning by delivering numerical, graphic, and symbolic content without spending the time to calculate complex computing problems manually. The purpose of this research is to design and develop teaching materials software assisted for numerical method. The process of developing the teaching material starts from the defining step, the process of designing the learning material developed based on information obtained from the step of early analysis, learners, materials, tasks that support then done the design step or design, then the last step is the development step. The development of teaching materials software assisted for numerical methods is valid in content. While validator assessment for teaching material in numerical methods is good and can be used with little revision.

Nonlinearly preconditioned semismooth Newton methods for variational inequality solution of two-phase flow in porous media

NASA Astrophysics Data System (ADS)

Yang, Haijian; Sun, Shuyu; Yang, Chao

2017-03-01

Most existing methods for solving two-phase flow problems in porous media do not take the physically feasible saturation fractions between 0 and 1 into account, which often destroys the numerical accuracy and physical interpretability of the simulation. To calculate the solution without the loss of this basic requirement, we introduce a variational inequality formulation of the saturation equilibrium with a box inequality constraint, and use a conservative finite element method for the spatial discretization and a backward differentiation formula with adaptive time stepping for the temporal integration. The resulting variational inequality system at each time step is solved by using a semismooth Newton algorithm. To accelerate the Newton convergence and improve the robustness, we employ a family of adaptive nonlinear elimination methods as a nonlinear preconditioner. Some numerical results are presented to demonstrate the robustness and efficiency of the proposed algorithm. A comparison is also included to show the superiority of the proposed fully implicit approach over the classical IMplicit Pressure-Explicit Saturation (IMPES) method in terms of the time step size and the total execution time measured on a parallel computer.

Runge-Kutta methods combined with compact difference schemes for the unsteady Euler equations

NASA Technical Reports Server (NTRS)

Yu, Sheng-Tao

1992-01-01

Recent development using compact difference schemes to solve the Navier-Stokes equations show spectral-like accuracy. A study was made of the numerical characteristics of various combinations of the Runge-Kutta (RK) methods and compact difference schemes to calculate the unsteady Euler equations. The accuracy of finite difference schemes is assessed based on the evaluations of dissipative error. The objectives are reducing the numerical damping and, at the same time, preserving numerical stability. While this approach has tremendous success solving steady flows, numerical characteristics of unsteady calculations remain largely unclear. For unsteady flows, in addition to the dissipative errors, phase velocity and harmonic content of the numerical results are of concern. As a result of the discretization procedure, the simulated unsteady flow motions actually propagate in a dispersive numerical medium. Consequently, the dispersion characteristics of the numerical schemes which relate the phase velocity and wave number may greatly impact the numerical accuracy. The aim is to assess the numerical accuracy of the simulated results. To this end, the Fourier analysis is to provide the dispersive correlations of various numerical schemes. First, a detailed investigation of the existing RK methods is carried out. A generalized form of an N-step RK method is derived. With this generalized form, the criteria are derived for the three and four-step RK methods to be third and fourth-order time accurate for the non-linear equations, e.g., flow equations. These criteria are then applied to commonly used RK methods such as Jameson's 3-step and 4-step schemes and Wray's algorithm to identify the accuracy of the methods. For the spatial discretization, compact difference schemes are presented. The schemes are formulated in the operator-type to render themselves suitable for the Fourier analyses. The performance of the numerical methods is shown by numerical examples. These examples are detailed. described. The third case is a two-dimensional simulation of a Lamb vortex in an uniform flow. This calculation provides a realistic assessment of various finite difference schemes in terms of the conservation of the vortex strength and the harmonic content after travelling a substantial distance. The numerical implementation of Giles' non-refelctive equations coupled with the characteristic equations as the boundary condition is discussed in detail. Finally, the single vortex calculation is extended to simulate vortex pairing. For the distance between two vortices less than a threshold value, numerical results show crisp resolution of the vortex merging.

On the error propagation of semi-Lagrange and Fourier methods for advection problems☆

PubMed Central

Einkemmer, Lukas; Ostermann, Alexander

2015-01-01

In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley–Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme. PMID:25844018

Mesh and Time-Step Independent Computational Fluid Dynamics (CFD) Solutions

ERIC Educational Resources Information Center

Nijdam, Justin J.

2013-01-01

A homework assignment is outlined in which students learn Computational Fluid Dynamics (CFD) concepts of discretization, numerical stability and accuracy, and verification in a hands-on manner by solving physically realistic problems of practical interest to engineers. The students solve a transient-diffusion problem numerically using the common…

Experimental localization of an acoustic sound source in a wind-tunnel flow by using a numerical time-reversal technique.

PubMed

Padois, Thomas; Prax, Christian; Valeau, Vincent; Marx, David

2012-10-01

The possibility of using the time-reversal technique to localize acoustic sources in a wind-tunnel flow is investigated. While the technique is widespread, it has scarcely been used in aeroacoustics up to now. The proposed method consists of two steps: in a first experimental step, the acoustic pressure fluctuations are recorded over a linear array of microphones; in a second numerical step, the experimental data are time-reversed and used as input data for a numerical code solving the linearized Euler equations. The simulation achieves the back-propagation of the waves from the array to the source and takes into account the effect of the mean flow on sound propagation. The ability of the method to localize a sound source in a typical wind-tunnel flow is first demonstrated using simulated data. A generic experiment is then set up in an anechoic wind tunnel to validate the proposed method with a flow at Mach number 0.11. Monopolar sources are first considered that are either monochromatic or have a narrow or wide-band frequency content. The source position estimation is well-achieved with an error inferior to the wavelength. An application to a dipolar sound source shows that this type of source is also very satisfactorily characterized.

Unsteady Computations of a Jet in a Crossflow with Ground Effect

NASA Technical Reports Server (NTRS)

Pandya, Shishir; Murman, Scott; Venkateswaran, Sankaran; Kwak, Dochan (Technical Monitor)

2002-01-01

A numerical study of a jet in crossflow with ground effect is conducted using OVERFLOW with dual time-stepping and low Mach number preconditioning. The results of the numerical study are compared to an experiment to show that the numerical methods are capable of capturing the dominant features of the flow field as well as the unsteadiness associated with the ground vortex.

Choice of Variables and Preconditioning for Time Dependent Problems

NASA Technical Reports Server (NTRS)

Turkel, Eli; Vatsa, Verr N.

2003-01-01

We consider the use of low speed preconditioning for time dependent problems. These are solved using a dual time step approach. We consider the effect of this dual time step on the parameter of the low speed preconditioning. In addition, we compare the use of two sets of variables, conservation and primitive variables, to solve the system. We show the effect of these choices on both the convergence to a steady state and the accuracy of the numerical solutions for low Mach number steady state and time dependent flows.

Efficiency and Accuracy of Time-Accurate Turbulent Navier-Stokes Computations

NASA Technical Reports Server (NTRS)

Rumsey, Christopher L.; Sanetrik, Mark D.; Biedron, Robert T.; Melson, N. Duane; Parlette, Edward B.

1995-01-01

The accuracy and efficiency of two types of subiterations in both explicit and implicit Navier-Stokes codes are explored for unsteady laminar circular-cylinder flow and unsteady turbulent flow over an 18-percent-thick circular-arc (biconvex) airfoil. Grid and time-step studies are used to assess the numerical accuracy of the methods. Nonsubiterative time-stepping schemes and schemes with physical time subiterations are subject to time-step limitations in practice that are removed by pseudo time sub-iterations. Computations for the circular-arc airfoil indicate that a one-equation turbulence model predicts the unsteady separated flow better than an algebraic turbulence model; also, the hysteresis with Mach number of the self-excited unsteadiness due to shock and boundary-layer separation is well predicted.

Spurious Behavior of Shock-Capturing Methods: Problems Containing Stiff Source Terms and Discontinuities

NASA Technical Reports Server (NTRS)

Yee, Helen M. C.; Kotov, D. V.; Wang, Wei; Shu, Chi-Wang

2013-01-01

The goal of this paper is to relate numerical dissipations that are inherited in high order shock-capturing schemes with the onset of wrong propagation speed of discontinuities. For pointwise evaluation of the source term, previous studies indicated that the phenomenon of wrong propagation speed of discontinuities is connected with the smearing of the discontinuity caused by the discretization of the advection term. The smearing introduces a nonequilibrium state into the calculation. Thus as soon as a nonequilibrium value is introduced in this manner, the source term turns on and immediately restores equilibrium, while at the same time shifting the discontinuity to a cell boundary. The present study is to show that the degree of wrong propagation speed of discontinuities is highly dependent on the accuracy of the numerical method. The manner in which the smearing of discontinuities is contained by the numerical method and the overall amount of numerical dissipation being employed play major roles. Moreover, employing finite time steps and grid spacings that are below the standard Courant-Friedrich-Levy (CFL) limit on shockcapturing methods for compressible Euler and Navier-Stokes equations containing stiff reacting source terms and discontinuities reveals surprising counter-intuitive results. Unlike non-reacting flows, for stiff reactions with discontinuities, employing a time step and grid spacing that are below the CFL limit (based on the homogeneous part or non-reacting part of the governing equations) does not guarantee a correct solution of the chosen governing equations. Instead, depending on the numerical method, time step and grid spacing, the numerical simulation may lead to (a) the correct solution (within the truncation error of the scheme), (b) a divergent solution, (c) a wrong propagation speed of discontinuities solution or (d) other spurious solutions that are solutions of the discretized counterparts but are not solutions of the governing equations. The present investigation for three very different stiff system cases confirms some of the findings of Lafon & Yee (1996) and LeVeque & Yee (1990) for a model scalar PDE. The findings might shed some light on the reported difficulties in numerical combustion and problems with stiff nonlinear (homogeneous) source terms and discontinuities in general.

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

DOE PAGES

Steyer, Andrew J.; Van Vleck, Erik S.

2018-04-13

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly,more » exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.« less

A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

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

Steyer, Andrew J.; Van Vleck, Erik S.

Approximation theory for Lyapunov and Sacker–Sell spectra based upon QR techniques is used to analyze the stability of a one-step method solving a time-dependent (nonautonomous) linear ordinary differential equation (ODE) initial value problem in terms of the local error. Integral separation is used to characterize the conditioning of stability spectra calculations. The stability of the numerical solution by a one-step method of a nonautonomous linear ODE using real-valued, scalar, nonautonomous linear test equations is justified. This analysis is used to approximate exponential growth/decay rates on finite and infinite time intervals and establish global error bounds for one-step methods approximating uniformly,more » exponentially stable trajectories of nonautonomous and nonlinear ODEs. A time-dependent stiffness indicator and a one-step method that switches between explicit and implicit Runge–Kutta methods based upon time-dependent stiffness are developed based upon the theoretical results.« less

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

NASA Astrophysics Data System (ADS)

Lafitte, Pauline; Melis, Ward; Samaey, Giovanni

2017-07-01

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.

Constant pressure and temperature discrete-time Langevin molecular dynamics

NASA Astrophysics Data System (ADS)

Grønbech-Jensen, Niels; Farago, Oded

2014-11-01

We present a new and improved method for simultaneous control of temperature and pressure in molecular dynamics simulations with periodic boundary conditions. The thermostat-barostat equations are built on our previously developed stochastic thermostat, which has been shown to provide correct statistical configurational sampling for any time step that yields stable trajectories. Here, we extend the method and develop a set of discrete-time equations of motion for both particle dynamics and system volume in order to seek pressure control that is insensitive to the choice of the numerical time step. The resulting method is simple, practical, and efficient. The method is demonstrated through direct numerical simulations of two characteristic model systems—a one-dimensional particle chain for which exact statistical results can be obtained and used as benchmarks, and a three-dimensional system of Lennard-Jones interacting particles simulated in both solid and liquid phases. The results, which are compared against the method of Kolb and Dünweg [J. Chem. Phys. 111, 4453 (1999)], show that the new method behaves according to the objective, namely that acquired statistical averages and fluctuations of configurational measures are accurate and robust against the chosen time step applied to the simulation.

Re-evaluation of an Optimized Second Order Backward Difference (BDF2OPT) Scheme for Unsteady Flow Applications

NASA Technical Reports Server (NTRS)

Vatsa, Veer N.; Carpenter, Mark H.; Lockard, David P.

2009-01-01

Recent experience in the application of an optimized, second-order, backward-difference (BDF2OPT) temporal scheme is reported. The primary focus of the work is on obtaining accurate solutions of the unsteady Reynolds-averaged Navier-Stokes equations over long periods of time for aerodynamic problems of interest. The baseline flow solver under consideration uses a particular BDF2OPT temporal scheme with a dual-time-stepping algorithm for advancing the flow solutions in time. Numerical difficulties are encountered with this scheme when the flow code is run for a large number of time steps, a behavior not seen with the standard second-order, backward-difference, temporal scheme. Based on a stability analysis, slight modifications to the BDF2OPT scheme are suggested. The performance and accuracy of this modified scheme is assessed by comparing the computational results with other numerical schemes and experimental data.

Asynchronous variational integration using continuous assumed gradient elements.

PubMed

Wolff, Sebastian; Bucher, Christian

2013-03-01

Asynchronous variational integration (AVI) is a tool which improves the numerical efficiency of explicit time stepping schemes when applied to finite element meshes with local spatial refinement. This is achieved by associating an individual time step length to each spatial domain. Furthermore, long-term stability is ensured by its variational structure. This article presents AVI in the context of finite elements based on a weakened weak form (W2) Liu (2009) [1], exemplified by continuous assumed gradient elements Wolff and Bucher (2011) [2]. The article presents the main ideas of the modified AVI, gives implementation notes and a recipe for estimating the critical time step.

A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation

USGS Publications Warehouse

Smith, Peter E.

2006-01-01

A semi-implicit, finite-difference method for the numerical solution of the three-dimensional equations for circulation in estuaries is presented and tested. The method uses a three-time-level, leapfrog-trapezoidal scheme that is essentially second-order accurate in the spatial and temporal numerical approximations. The three-time-level scheme is shown to be preferred over a two-time-level scheme, especially for problems with strong nonlinearities. The stability of the semi-implicit scheme is free from any time-step limitation related to the terms describing vertical diffusion and the propagation of the surface gravity waves. The scheme does not rely on any form of vertical/horizontal mode-splitting to treat the vertical diffusion implicitly. At each time step, the numerical method uses a double-sweep method to transform a large number of small tridiagonal equation systems and then uses the preconditioned conjugate-gradient method to solve a single, large, five-diagonal equation system for the water surface elevation. The governing equations for the multi-level scheme are prepared in a conservative form by integrating them over the height of each horizontal layer. The layer-integrated volumetric transports replace velocities as the dependent variables so that the depth-integrated continuity equation that is used in the solution for the water surface elevation is linear. Volumetric transports are computed explicitly from the momentum equations. The resulting method is mass conservative, efficient, and numerically accurate.

Teaching BASIC. A Step by Step Guide.

ERIC Educational Resources Information Center

Allen, M. F.

This three-chapter guide provides simple explanations about BASIC programming for a teacher to use in a classroom situation, and suggests procedures for a "hands-on" course. Numerous examples are presented of the questions, problems, and level of understanding to expect from first-time, adult users (ages 13 and up). The course materials…

Analysis of 3D poroelastodynamics using BEM based on modified time-step scheme

NASA Astrophysics Data System (ADS)

Igumnov, L. A.; Petrov, A. N.; Vorobtsov, I. V.

2017-10-01

The development of 3d boundary elements modeling of dynamic partially saturated poroelastic media using a stepping scheme is presented in this paper. Boundary Element Method (BEM) in Laplace domain and the time-stepping scheme for numerical inversion of the Laplace transform are used to solve the boundary value problem. The modified stepping scheme with a varied integration step for quadrature coefficients calculation using the symmetry of the integrand function and integral formulas of Strongly Oscillating Functions was applied. The problem with force acting on a poroelastic prismatic console end was solved using the developed method. A comparison of the results obtained by the traditional stepping scheme with the solutions obtained by this modified scheme shows that the computational efficiency is better with usage of combined formulas.

Hyperbolic heat conduction problems involving non-Fourier effects - Numerical simulations via explicit Lax-Wendroff/Taylor-Galerkin finite element formulations

NASA Technical Reports Server (NTRS)

Tamma, Kumar K.; Namburu, Raju R.

1989-01-01

Numerical simulations are presented for hyperbolic heat-conduction problems that involve non-Fourier effects, using explicit, Lax-Wendroff/Taylor-Galerkin FEM formulations as the principal computational tool. Also employed are smoothing techniques which stabilize the numerical noise and accurately predict the propagating thermal disturbances. The accurate capture of propagating thermal disturbances at characteristic time-step values is achieved; numerical test cases are presented which validate the proposed hyperbolic heat-conduction problem concepts.

An efficient unstructured WENO method for supersonic reactive flows

NASA Astrophysics Data System (ADS)

Zhao, Wen-Geng; Zheng, Hong-Wei; Liu, Feng-Jun; Shi, Xiao-Tian; Gao, Jun; Hu, Ning; Lv, Meng; Chen, Si-Cong; Zhao, Hong-Da

2018-03-01

An efficient high-order numerical method for supersonic reactive flows is proposed in this article. The reactive source term and convection term are solved separately by splitting scheme. In the reaction step, an adaptive time-step method is presented, which can improve the efficiency greatly. In the convection step, a third-order accurate weighted essentially non-oscillatory (WENO) method is adopted to reconstruct the solution in the unstructured grids. Numerical results show that our new method can capture the correct propagation speed of the detonation wave exactly even in coarse grids, while high order accuracy can be achieved in the smooth region. In addition, the proposed adaptive splitting method can reduce the computational cost greatly compared with the traditional splitting method.

The classification of two-loop integrand basis in pure four-dimension

NASA Astrophysics Data System (ADS)

Feng, Bo; Huang, Rijun

2013-02-01

In this paper, we have made the attempt to classify the integrand basis of all two-loop diagrams in pure four-dimensional space-time. The first step of our classification is to determine all different topologies of two-loop diagrams, i.e., the structure of denominators. The second step is to determine the set of independent numerators for each topology using Gröbner basis method. For the second step, varieties defined by putting all propagators on-shell has played an important role. We discuss the structures of varieties and how they split to various irreducible branches under specific kinematic configurations of external momenta. The structures of varieties are crucial to determine coefficients of integrand basis in reduction both numerically or analytically.

On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

NASA Astrophysics Data System (ADS)

Fehn, Niklas; Wall, Wolfgang A.; Kronbichler, Martin

2017-12-01

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for small time step sizes. Since the critical time step size depends on the viscosity and the spatial resolution, these instabilities limit the robustness of the Navier-Stokes solver in case of complex engineering applications characterized by coarse spatial resolutions and small viscosities. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.

Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical (PEC) solar cells

NASA Astrophysics Data System (ADS)

Harmon, Michael; Gamba, Irene M.; Ren, Kui

2016-12-01

This work concerns the numerical solution of a coupled system of self-consistent reaction-drift-diffusion-Poisson equations that describes the macroscopic dynamics of charge transport in photoelectrochemical (PEC) solar cells with reactive semiconductor and electrolyte interfaces. We present three numerical algorithms, mainly based on a mixed finite element and a local discontinuous Galerkin method for spatial discretization, with carefully chosen numerical fluxes, and implicit-explicit time stepping techniques, for solving the time-dependent nonlinear systems of partial differential equations. We perform computational simulations under various model parameters to demonstrate the performance of the proposed numerical algorithms as well as the impact of these parameters on the solution to the model.

Efficiency analysis of numerical integrations for finite element substructure in real-time hybrid simulation

NASA Astrophysics Data System (ADS)

Wang, Jinting; Lu, Liqiao; Zhu, Fei

2018-01-01

Finite element (FE) is a powerful tool and has been applied by investigators to real-time hybrid simulations (RTHSs). This study focuses on the computational efficiency, including the computational time and accuracy, of numerical integrations in solving FE numerical substructure in RTHSs. First, sparse matrix storage schemes are adopted to decrease the computational time of FE numerical substructure. In this way, the task execution time (TET) decreases such that the scale of the numerical substructure model increases. Subsequently, several commonly used explicit numerical integration algorithms, including the central difference method (CDM), the Newmark explicit method, the Chang method and the Gui-λ method, are comprehensively compared to evaluate their computational time in solving FE numerical substructure. CDM is better than the other explicit integration algorithms when the damping matrix is diagonal, while the Gui-λ (λ = 4) method is advantageous when the damping matrix is non-diagonal. Finally, the effect of time delay on the computational accuracy of RTHSs is investigated by simulating structure-foundation systems. Simulation results show that the influences of time delay on the displacement response become obvious with the mass ratio increasing, and delay compensation methods may reduce the relative error of the displacement peak value to less than 5% even under the large time-step and large time delay.

Short-term Time Step Convergence in a Climate Model

DOE PAGES

Wan, Hui; Rasch, Philip J.; Taylor, Mark; ...

2015-02-11

A testing procedure is designed to assess the convergence property of a global climate model with respect to time step size, based on evaluation of the root-mean-square temperature difference at the end of very short (1 h) simulations with time step sizes ranging from 1 s to 1800 s. A set of validation tests conducted without sub-grid scale parameterizations confirmed that the method was able to correctly assess the convergence rate of the dynamical core under various configurations. The testing procedure was then applied to the full model, and revealed a slow convergence of order 0.4 in contrast to themore » expected first-order convergence. Sensitivity experiments showed without ambiguity that the time stepping errors in the model were dominated by those from the stratiform cloud parameterizations, in particular the cloud microphysics. This provides a clear guidance for future work on the design of more accurate numerical methods for time stepping and process coupling in the model.« less

A local time stepping algorithm for GPU-accelerated 2D shallow water models

NASA Astrophysics Data System (ADS)

Dazzi, Susanna; Vacondio, Renato; Dal Palù, Alessandro; Mignosa, Paolo

2018-01-01

In the simulation of flooding events, mesh refinement is often required to capture local bathymetric features and/or to detail areas of interest; however, if an explicit finite volume scheme is adopted, the presence of small cells in the domain can restrict the allowable time step due to the stability condition, thus reducing the computational efficiency. With the aim of overcoming this problem, the paper proposes the application of a Local Time Stepping (LTS) strategy to a GPU-accelerated 2D shallow water numerical model able to handle non-uniform structured meshes. The algorithm is specifically designed to exploit the computational capability of GPUs, minimizing the overheads associated with the LTS implementation. The results of theoretical and field-scale test cases show that the LTS model guarantees appreciable reductions in the execution time compared to the traditional Global Time Stepping strategy, without compromising the solution accuracy.

Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

NASA Technical Reports Server (NTRS)

Rosenfeld, Moshe

1990-01-01

The development, validation and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems are discussed. A solution method that combines a finite-volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries was previously developed for fixed-grids. In the present research effort, this solution method is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.

A time-spectral approach to numerical weather prediction

NASA Astrophysics Data System (ADS)

Scheffel, Jan; Lindvall, Kristoffer; Yik, Hiu Fai

2018-05-01

Finite difference methods are traditionally used for modelling the time domain in numerical weather prediction (NWP). Time-spectral solution is an attractive alternative for reasons of accuracy and efficiency and because time step limitations associated with causal CFL-like criteria, typical for explicit finite difference methods, are avoided. In this work, the Lorenz 1984 chaotic equations are solved using the time-spectral algorithm GWRM (Generalized Weighted Residual Method). Comparisons of accuracy and efficiency are carried out for both explicit and implicit time-stepping algorithms. It is found that the efficiency of the GWRM compares well with these methods, in particular at high accuracy. For perturbative scenarios, the GWRM was found to be as much as four times faster than the finite difference methods. A primary reason is that the GWRM time intervals typically are two orders of magnitude larger than those of the finite difference methods. The GWRM has the additional advantage to produce analytical solutions in the form of Chebyshev series expansions. The results are encouraging for pursuing further studies, including spatial dependence, of the relevance of time-spectral methods for NWP modelling.

A time step criterion for the stable numerical simulation of hydraulic fracturing

NASA Astrophysics Data System (ADS)

Juan-Lien Ramirez, Alina; Löhnert, Stefan; Neuweiler, Insa

2017-04-01

The process of propagating or widening cracks in rock formations by means of fluid flow, known as hydraulic fracturing, has been gaining attention in the last couple of decades. There is growing interest in its numerical simulation to make predictions. Due to the complexity of the processes taking place, e.g. solid deformation, fluid flow in an open channel, fluid flow in a porous medium and crack propagation, this is a challenging task. Hydraulic fracturing has been numerically simulated for some years now [1] and new methods to take more of its processes into account (increasing accuracy) while modeling in an efficient way (lower computational effort) have been developed in recent years. An example is the use of the Extended Finite Element Method (XFEM), whose application originated within the framework of solid mechanics, but is now seen as an effective method for the simulation of discontinuities with no need for re-meshing [2]. While more focus has been put to the correct coupling of the processes mentioned above, less attention has been paid to the stability of the model. When using a quasi-static approach for the simulation of hydraulic fracturing, choosing an adequate time step is not trivial. This is in particular true if the equations are solved in a staggered way. The difficulty lies within the inconsistency between the static behavior of the solid and the dynamic behavior of the fluid. It has been shown that too small time steps may lead to instabilities early into the simulation time [3]. While the solid reaches a stationary state instantly, the fluid is not able to achieve equilibrium with its new surrounding immediately. This is why a time step criterion has been developed to quantify the instability of the model concerning the time step. The presented results were created with a 2D poroelastic model, using the XFEM for both the solid and the fluid phases. An embedded crack propagates following the energy release rate criteria when the fluid pressure within the crack rises. The fluid flow within the crack and in the porous medium are simulated using the mass balance for water and Darcy's law for flow. The equations for flow and deformation in the rock and that for flow in the fracture are solved in a staggered manner. The two sets of equations are coupled via Lagrange multipliers. We present a time step criterion for the stability of the scheme and illustrate this criterion with test examples of crack propagation. [1] T. Boone and A. Ingraffea. A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media. Int. J. Numer. Anal. Met. 14, 27-47, (1990) [2] T. Mohammadnejad and A. Khoei. An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model. Finite Elements in Analysis and Design. 73, 77-95, (2013) [3] E.W. Remij, J.J.C. Remmers, J.M. Huyghe, D.M.J. Smeulders. The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Comput. Methods Appl. Mech. Engrg. 286, 293-312, (2015)

A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

NASA Astrophysics Data System (ADS)

Witte, J. H.; Reisinger, C.

2010-09-01

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desireable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to an order of O(1/ρ), where ρ>0 is the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.

Numerical solution methods for viscoelastic orthotropic materials

NASA Technical Reports Server (NTRS)

Gramoll, K. C.; Dillard, D. A.; Brinson, H. F.

1988-01-01

Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time.

A numerical method for computing unsteady 2-D boundary layer flows

NASA Technical Reports Server (NTRS)

Krainer, Andreas

1988-01-01

A numerical method for computing unsteady two-dimensional boundary layers in incompressible laminar and turbulent flows is described and applied to a single airfoil changing its incidence angle in time. The solution procedure adopts a first order panel method with a simple wake model to solve for the inviscid part of the flow, and an implicit finite difference method for the viscous part of the flow. Both procedures integrate in time in a step-by-step fashion, in the course of which each step involves the solution of the elliptic Laplace equation and the solution of the parabolic boundary layer equations. The Reynolds shear stress term of the boundary layer equations is modeled by an algebraic eddy viscosity closure. The location of transition is predicted by an empirical data correlation originating from Michel. Since transition and turbulence modeling are key factors in the prediction of viscous flows, their accuracy will be of dominant influence to the overall results.

Experimental calibration procedures for rotating Lorentz-force flowmeters

DOE PAGES

Hvasta, M. G.; Slighton, N. T.; Kolemen, E.; ...

2017-07-14

Rotating Lorentz-force flowmeters are a novel and useful technology with a range of applications in a variety of different industries. However, calibrating these flowmeters can be challenging, time-consuming, and expensive. In this paper, simple calibration procedures for rotating Lorentz-force flowmeters are presented. These procedures eliminate the need for expensive equipment, numerical modeling, redundant flowmeters, and system down-time. Finally, the calibration processes are explained in a step-by-step manner and compared to experimental results.

Experimental calibration procedures for rotating Lorentz-force flowmeters

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

Hvasta, M. G.; Slighton, N. T.; Kolemen, E.

Rotating Lorentz-force flowmeters are a novel and useful technology with a range of applications in a variety of different industries. However, calibrating these flowmeters can be challenging, time-consuming, and expensive. In this paper, simple calibration procedures for rotating Lorentz-force flowmeters are presented. These procedures eliminate the need for expensive equipment, numerical modeling, redundant flowmeters, and system down-time. Finally, the calibration processes are explained in a step-by-step manner and compared to experimental results.

Time-symmetric integration in astrophysics

NASA Astrophysics Data System (ADS)

Hernandez, David M.; Bertschinger, Edmund

2018-04-01

Calculating the long-term solution of ordinary differential equations, such as those of the N-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods that are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty in accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time-stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show that it gives secular energy error increase for a pendulum problem and for a Hénon-Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favoured. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator.

Solution procedure of dynamical contact problems with friction

NASA Astrophysics Data System (ADS)

Abdelhakim, Lotfi

2017-07-01

Dynamical contact is one of the common research topics because of its wide applications in the engineering field. The main goal of this work is to develop a time-stepping algorithm for dynamic contact problems. We propose a finite element approach for elastodynamics contact problems [1]. Sticking, sliding and frictional contact can be taken into account. Lagrange multipliers are used to enforce non-penetration condition. For the time discretization, we propose a scheme equivalent to the explicit Newmark scheme. Each time step requires solving a nonlinear problem similar to a static friction problem. The nonlinearity of the system of equation needs an iterative solution procedure based on Uzawa's algorithm [2][3]. The applicability of the algorithm is illustrated by selected sample numerical solutions to static and dynamic contact problems. Results obtained with the model have been compared and verified with results from an independent numerical method.

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

NASA Technical Reports Server (NTRS)

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

1977-01-01

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

Variational Algorithms for Test Particle Trajectories

NASA Astrophysics Data System (ADS)

Ellison, C. Leland; Finn, John M.; Qin, Hong; Tang, William M.

2015-11-01

The theory of variational integration provides a novel framework for constructing conservative numerical methods for magnetized test particle dynamics. The retention of conservation laws in the numerical time advance captures the correct qualitative behavior of the long time dynamics. For modeling the Lorentz force system, new variational integrators have been developed that are both symplectic and electromagnetically gauge invariant. For guiding center test particle dynamics, discretization of the phase-space action principle yields multistep variational algorithms, in general. Obtaining the desired long-term numerical fidelity requires mitigation of the multistep method's parasitic modes or applying a discretization scheme that possesses a discrete degeneracy to yield a one-step method. Dissipative effects may be modeled using Lagrange-D'Alembert variational principles. Numerical results will be presented using a new numerical platform that interfaces with popular equilibrium codes and utilizes parallel hardware to achieve reduced times to solution. This work was supported by DOE Contract DE-AC02-09CH11466.

On time discretizations for spectral methods. [numerical integration of Fourier and Chebyshev methods for dynamic partial differential equations

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.

Predictable 'meta-mechanisms' emerge from feedbacks between transpiration and plant growth and cannot be simply deduced from short-term mechanisms.

PubMed

Tardieu, François; Parent, Boris

2017-06-01

Growth under water deficit is controlled by short-term mechanisms but, because of numerous feedbacks, the combination of these mechanisms over time often results in outputs that cannot be deduced from the simple inspection of individual mechanisms. It can be analysed with dynamic models in which causal relationships between variables are considered at each time-step, allowing calculation of outputs that are routed back to inputs for the next time-step and that can change the system itself. We first review physiological mechanisms involved in seven feedbacks of transpiration on plant growth, involving changes in tissue hydraulic conductance, stomatal conductance, plant architecture and underlying factors such as hormones or aquaporins. The combination of these mechanisms over time can result in non-straightforward conclusions as shown by examples of simulation outputs: 'over production of abscisic acid (ABA) can cause a lower concentration of ABA in the xylem sap ', 'decreasing root hydraulic conductance when evaporative demand is maximum can improve plant performance' and 'rapid root growth can decrease yield'. Systems of equations simulating feedbacks over numerous time-steps result in logical and reproducible emergent properties that can be viewed as 'meta-mechanisms' at plant level, which have similar roles as mechanisms at cell level. © 2016 John Wiley & Sons Ltd.

Modelling to very high strains

NASA Astrophysics Data System (ADS)

Bons, P. D.; Jessell, M. W.; Griera, A.; Evans, L. A.; Wilson, C. J. L.

2009-04-01

Ductile strains in shear zones often reach extreme values, resulting in typical structures, such as winged porphyroclasts and several types of shear bands. The numerical simulation of the development of such structures has so far been inhibited by the low maximum strains that numerical models can normally achieve. Typical numerical models collapse at shear strains in the order of one to three. We have implemented a number of new functionalities in the numerical platform "Elle" (Jessell et al. 2001), which significantly increases the amount of strain that can be achieved and simultaneously reduces boundary effects that become increasingly disturbing at higher strain. Constant remeshing, while maintaining the polygonal phase regions, is the first step to avoid collapse of the finite-element grid required by finite-element solvers, such as Basil (Houseman et al. 2008). The second step is to apply a grain-growth routine to the boundaries of polygons that represent phase regions. This way, the development of sharp angles is avoided. A second advantage is that phase regions may merge or become separated (boudinage). Such topological changes are normally not possible in finite element deformation codes. The third step is the use of wrapping vertical model boundaries, with which optimal and unchanging model boundaries are maintained for the application of stress or velocity boundary conditions. The fourth step is to shift the model by a random amount in the vertical direction every time step. This way, the fixed horizontal boundary conditions are applied to different material points within the model every time step. Disturbing boundary effects are thus averaged out over the whole model and not localised to e.g. top and bottom of the model. Reduction of boundary effects has the additional advantage that model can be smaller and, therefore, numerically more efficient. Owing to the combination of these existing and new functionalities it is now possible to simulate the development of very high-strain structures. Jessell, M.W., Bons, P.D., Evans, L., Barr, T., Stüwe, K. 2001. Elle: a micro-process approach to the simulation of microstructures. Computers & Geosciences 27, 17-30. Houseman, G., Barr, T., Evans, L. 2008. Basil: stress and deformation in a viscous material. In: P.D. Bons, D. Koehn & M.W.Jessell (Eds.) Microdynamics Simulation. Lecture Notes in Earth Sciences 106, Springer, Berlin, 405p.

Numerical Issues Associated with Compensating and Competing Processes in Climate Models: an Example from ECHAM-HAM

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

Wan, Hui; Rasch, Philip J.; Zhang, Kai

2013-06-26

The purpose of this paper is to draw attention to the need for appropriate numerical techniques to represent process interactions in climate models. In two versions of the ECHAM-HAM model, different time integration methods are used to solve the sulfuric acid (H2SO4) gas evolution equation, which lead to substantially different results in the H2SO4 gas concentration and the aerosol nucleation rate. Using convergence tests and sensitivity simulations performed with various time stepping schemes, it is confirmed that numerical errors in the second model version are significantly smaller than those in version one. The use of sequential operator splitting in combinationmore » with long time step is identified as the main reason for the large systematic biases in the old model. The remaining errors in version two in the nucleation rate, related to the competition between condensation and nucleation, have a clear impact on the simulated concentration of cloud condensation nuclei in the lower troposphere. These errors can be significantly reduced by employing an implicit solver that handles production, condensation and nucleation at the same time. Lessons learned in this work underline the need for more caution when treating multi-time-scale problems involving compensating and competing processes, a common occurrence in current climate models.« less

Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks.

PubMed

Rangan, Aaditya V; Cai, David

2007-02-01

We discuss numerical methods for simulating large-scale, integrate-and-fire (I&F) neuronal networks. Important elements in our numerical methods are (i) a neurophysiologically inspired integrating factor which casts the solution as a numerically tractable integral equation, and allows us to obtain stable and accurate individual neuronal trajectories (i.e., voltage and conductance time-courses) even when the I&F neuronal equations are stiff, such as in strongly fluctuating, high-conductance states; (ii) an iterated process of spike-spike corrections within groups of strongly coupled neurons to account for spike-spike interactions within a single large numerical time-step; and (iii) a clustering procedure of firing events in the network to take advantage of localized architectures, such as spatial scales of strong local interactions, which are often present in large-scale computational models-for example, those of the primary visual cortex. (We note that the spike-spike corrections in our methods are more involved than the correction of single neuron spike-time via a polynomial interpolation as in the modified Runge-Kutta methods commonly used in simulations of I&F neuronal networks.) Our methods can evolve networks with relatively strong local interactions in an asymptotically optimal way such that each neuron fires approximately once in [Formula: see text] operations, where N is the number of neurons in the system. We note that quantifications used in computational modeling are often statistical, since measurements in a real experiment to characterize physiological systems are typically statistical, such as firing rate, interspike interval distributions, and spike-triggered voltage distributions. We emphasize that it takes much less computational effort to resolve statistical properties of certain I&F neuronal networks than to fully resolve trajectories of each and every neuron within the system. For networks operating in realistic dynamical regimes, such as strongly fluctuating, high-conductance states, our methods are designed to achieve statistical accuracy when very large time-steps are used. Moreover, our methods can also achieve trajectory-wise accuracy when small time-steps are used.

Numerical Study of a High Head Francis Turbine with Measurements from the Francis-99 Project

NASA Astrophysics Data System (ADS)

Wallimann, H.; Neubauer, R.

2015-01-01

For the Francis-99 project initiated by the Norwegian University of Science and Technology (NTNU, Norway) and the Luleå University of Technology (LTU, Sweden) numerical flow simulation has been performed and the results compared to experimentally obtained data. The full machine including spiral casing, stay vanes, guide vanes, runner and draft tube was simulated transient for three operating points defined by the Francis-99 organisers. Two sets of results were created with differing time steps. Additionally, a reduced domain was simulated in a stationary manner to create a complete cut along constant prototype head and constant prototype discharge. The efficiency values and shape of the curves have been investigated and compared to the experimental data. Special attention has been given to rotor stator interaction (RSI). Signals from several probes and their counterpart in the simulation have been processed to evaluate the pressure fluctuations occurring due to the RSI. The direct comparison of the hydraulic efficiency obtained by the full machine simulation compared to the experimental data showed no improvement when using a 1° time step compared to a coarser 2° time step. At the BEP the 2° time step even showed a slightly better result with an absolute deviation 1.08% compared with 1.24% for the 1° time step. At the other two operating points the simulation results were practically identical but fell short of predicting the measured values. The RSI evaluation was done using the results of the 2° time step simulation, which proved to be an adequate setting to reproduce pressure signals with peaks at the correct frequencies. The simulation results showed the highest amplitudes in the vaneless space at the BEP operating point at a location different from the probe measurements available. This implies that not only the radial distance, but the shape of the vaneless space influences the RSI.

Variable Step Integration Coupled with the Method of Characteristics Solution for Water-Hammer Analysis, A Case Study

NASA Technical Reports Server (NTRS)

Turpin, Jason B.

2004-01-01

One-dimensional water-hammer modeling involves the solution of two coupled non-linear hyperbolic partial differential equations (PDEs). These equations result from applying the principles of conservation of mass and momentum to flow through a pipe, and usually the assumption that the speed at which pressure waves propagate through the pipe is constant. In order to solve these equations for the interested quantities (i.e. pressures and flow rates), they must first be converted to a system of ordinary differential equations (ODEs) by either approximating the spatial derivative terms with numerical techniques or using the Method of Characteristics (MOC). The MOC approach is ideal in that no numerical approximation errors are introduced in converting the original system of PDEs into an equivalent system of ODEs. Unfortunately this resulting system of ODEs is bound by a time step constraint so that when integrating the equations the solution can only be obtained at fixed time intervals. If the fluid system to be modeled also contains dynamic components (i.e. components that are best modeled by a system of ODEs), it may be necessary to take extremely small time steps during certain points of the model simulation in order to achieve stability and/or accuracy in the solution. Coupled together, the fixed time step constraint invoked by the MOC, and the occasional need for extremely small time steps in order to obtain stability and/or accuracy, can greatly increase simulation run times. As one solution to this problem, a method for combining variable step integration (VSI) algorithms with the MOC was developed for modeling water-hammer in systems with highly dynamic components. A case study is presented in which reverse flow through a dual-flapper check valve introduces a water-hammer event. The predicted pressure responses upstream of the check-valve are compared with test data.

A splitting scheme based on the space-time CE/SE method for solving multi-dimensional hydrodynamical models of semiconductor devices

NASA Astrophysics Data System (ADS)

Nisar, Ubaid Ahmed; Ashraf, Waqas; Qamar, Shamsul

2016-08-01

Numerical solutions of the hydrodynamical model of semiconductor devices are presented in one and two-space dimension. The model describes the charge transport in semiconductor devices. Mathematically, the models can be written as a convection-diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the conservation element and solution element (CE/SE) method for hyperbolic step, and a semi-implicit scheme for the relaxation step. The numerical results of the suggested scheme are compared with the splitting scheme based on Nessyahu-Tadmor (NT) central scheme for convection step and the same semi-implicit scheme for the relaxation step. The effects of various parameters such as low field mobility, device length, lattice temperature and voltages for one-space dimensional hydrodynamic model are explored to further validate the generic applicability of the CE/SE method for the current model equations. A two dimensional simulation is also performed by CE/SE method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes

NASA Astrophysics Data System (ADS)

Zhu, Yajun; Zhong, Chengwen; Xu, Kun

2016-06-01

This paper presents an implicit unified gas-kinetic scheme (UGKS) for non-equilibrium steady state flow computation. The UGKS is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function. By solving the macroscopic equations implicitly, a predicted equilibrium state can be obtained first through iterations. With the newly predicted equilibrium state, the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well. The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization method is implemented to solve both macroscopic and microscopic equations, which improves the efficiency of the scheme. Since the UGKS is a direct modeling method and its physical solution depends on the mesh resolution and the local time step, a physical time step needs to be fixed before using an implicit iterative technique with a pseudo-time marching step. Therefore, the physical time step in the current implicit scheme is determined by the same way as that in the explicit UGKS for capturing the physical solution in all flow regimes, but the convergence to a steady state speeds up through the adoption of a numerical time step with large CFL number. Many numerical test cases in different flow regimes from low speed to hypersonic ones, such as the Couette flow, cavity flow, and the flow passing over a cylinder, are computed to validate the current implicit method. The overall efficiency of the implicit UGKS can be improved by one or two orders of magnitude in comparison with the explicit one.

Krylov Deferred Correction Accelerated Method of Lines Transpose for Parabolic Problems

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

Jia, Jun; Jingfang, Huang

2008-01-01

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

Multilevel ensemble Kalman filtering

DOE PAGES

Hoel, Hakon; Law, Kody J. H.; Tempone, Raul

2016-06-14

This study embeds a multilevel Monte Carlo sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. Finally, the resulting multilevel EnKF is proved to asymptotically outperform EnKF in terms of computational cost versus approximation accuracy. The theoretical results are illustrated numerically.

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.

Boundary particle method for Laplace transformed time fractional diffusion equations

NASA Astrophysics Data System (ADS)

Fu, Zhuo-Jia; Chen, Wen; Yang, Hai-Tian

2013-02-01

This paper develops a novel boundary meshless approach, Laplace transformed boundary particle method (LTBPM), for numerical modeling of time fractional diffusion equations. It implements Laplace transform technique to obtain the corresponding time-independent inhomogeneous equation in Laplace space and then employs a truly boundary-only meshless boundary particle method (BPM) to solve this Laplace-transformed problem. Unlike the other boundary discretization methods, the BPM does not require any inner nodes, since the recursive composite multiple reciprocity technique (RC-MRM) is used to convert the inhomogeneous problem into the higher-order homogeneous problem. Finally, the Stehfest numerical inverse Laplace transform (NILT) is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding BPM solutions. In comparison with finite difference discretization, the LTBPM introduces Laplace transform and Stehfest NILT algorithm to deal with time fractional derivative term, which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability. Consequently, it can effectively simulate long time-history fractional diffusion systems. Error analysis and numerical experiments demonstrate that the present LTBPM is highly accurate and computationally efficient for 2D and 3D time fractional diffusion equations.

Advances in numerical and applied mathematics

NASA Technical Reports Server (NTRS)

South, J. C., Jr. (Editor); Hussaini, M. Y. (Editor)

1986-01-01

This collection of papers covers some recent developments in numerical analysis and computational fluid dynamics. Some of these studies are of a fundamental nature. They address basic issues such as intermediate boundary conditions for approximate factorization schemes, existence and uniqueness of steady states for time dependent problems, and pitfalls of implicit time stepping. The other studies deal with modern numerical methods such as total variation diminishing schemes, higher order variants of vortex and particle methods, spectral multidomain techniques, and front tracking techniques. There is also a paper on adaptive grids. The fluid dynamics papers treat the classical problems of imcompressible flows in helically coiled pipes, vortex breakdown, and transonic flows.

Influence of the random walk finite step on the first-passage probability

NASA Astrophysics Data System (ADS)

Klimenkova, Olga; Menshutin, Anton; Shchur, Lev

2018-01-01

A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with fixed step length. We measure first-passage probability to touch the absorbing sphere of radius R in 2D. We found a regular deviation of the first-passage probability from the exact function, which we attribute to the finiteness of the random walk step.

Saddlepoint approximation to the distribution of the total distance of the continuous time random walk

NASA Astrophysics Data System (ADS)

Gatto, Riccardo

2017-12-01

This article considers the random walk over Rp, with p ≥ 2, where a given particle starts at the origin and moves stepwise with uniformly distributed step directions and step lengths following a common distribution. Step directions and step lengths are independent. The case where the number of steps of the particle is fixed and the more general case where it follows an independent continuous time inhomogeneous counting process are considered. Saddlepoint approximations to the distribution of the distance from the position of the particle to the origin are provided. Despite the p-dimensional nature of the random walk, the computations of the saddlepoint approximations are one-dimensional and thus simple. Explicit formulae are derived with dimension p = 3: for uniformly and exponentially distributed step lengths, for fixed and for Poisson distributed number of steps. In these situations, the high accuracy of the saddlepoint approximations is illustrated by numerical comparisons with Monte Carlo simulation. Contribution to the "Topical Issue: Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook", edited by Ryszard Kutner and Jaume Masoliver.

High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields

NASA Astrophysics Data System (ADS)

He, Yang; Sun, Yajuan; Zhang, Ruili; Wang, Yulei; Liu, Jian; Qin, Hong

2016-09-01

We construct high order symmetric volume-preserving methods for the relativistic dynamics of a charged particle by the splitting technique with processing. By expanding the phase space to include the time t, we give a more general construction of volume-preserving methods that can be applied to systems with time-dependent electromagnetic fields. The newly derived methods provide numerical solutions with good accuracy and conservative properties over long time of simulation. Furthermore, because of the use of an accuracy-enhancing processing technique, the explicit methods obtain high-order accuracy and are more efficient than the methods derived from standard compositions. The results are verified by the numerical experiments. Linear stability analysis of the methods shows that the high order processed method allows larger time step size in numerical integrations.

Melatonin: a universal time messenger.

PubMed

Erren, Thomas C; Reiter, Russel J

2015-01-01

Temporal organization plays a key role in humans, and presumably all species on Earth. A core building block of the chronobiological architecture is the master clock, located in the suprachi asmatic nuclei [SCN], which organizes "when" things happen in sub-cellular biochemistry, cells, organs and organisms, including humans. Conceptually, time messenging should follow a 5 step-cascade. While abundant evidence suggests how steps 1 through 4 work, step 5 of "how is central time information transmitted througout the body?" awaits elucidation. Step 1: Light provides information on environmental (external) time; Step 2: Ocular interfaces between light and biological (internal) time are intrinsically photosensitive retinal ganglion cells [ipRGS] and rods and cones; Step 3: Via the retinohypothalamic tract external time information reaches the light-dependent master clock in the brain, viz the SCN; Step 4: The SCN translate environmental time information into biological time and distribute this information to numerous brain structures via a melanopsin-based network. Step 5: Melatonin, we propose, transmits, or is a messenger of, internal time information to all parts of the body to allow temporal organization which is orchestrated by the SCN. Key reasons why we expect melatonin to have such role include: First, melatonin, as the chemical expression of darkness, is centrally involved in time- and timing-related processes such as encoding clock and calendar information in the brain; Second, melatonin travels throughout the body without limits and is thus a ubiquitous molecule. The chemial conservation of melatonin in all tested species could make this molecule a candidate for a universal time messenger, possibly constituting a legacy of an all-embracing evolutionary history.

An online-coupled NWP/ACT model with conserved Lagrangian levels

NASA Astrophysics Data System (ADS)

Sørensen, B.; Kaas, E.; Lauritzen, P. H.

2012-04-01

Numerical weather and climate modelling is under constant development. Semi-implicit semi-Lagrangian (SISL) models have proven to be numerically efficient in both short-range weather forecasts and climate models, due to the ability to use long time steps. Chemical/aerosol feedback mechanism are becoming more and more relevant in NWP as well as climate models, since the biogenic and anthropogenic emissions can have a direct effect on the dynamics and radiative properties of the atmosphere. To include chemical feedback mechanisms in the NWP models, on-line coupling is crucial. In 3D semi-Lagrangian schemes with quasi-Lagrangian vertical coordinates the Lagrangian levels are remapped to Eulerian model levels each time step. This remapping introduces an undesirable tendency to smooth sharp gradients and creates unphysical numerical diffusion in the vertical distribution. A semi-Lagrangian advection method is introduced, it combines an inherently mass conserving 2D semi-Lagrangian scheme, with a SISL scheme employing both hybrid vertical coordinates and a fully Lagrangian vertical coordinate. This minimizes the vertical diffusion and thus potentially improves the simulation of the vertical profiles of moisture, clouds, and chemical constituents. Since the Lagrangian levels suffer from traditional Lagrangian limitations caused by the convergence and divergence of the flow, remappings to the Eulerian model levels are generally still required - but this need only be applied after a number of time steps - unless dynamic remapping methods are used. For this several different remapping methods has been implemented. The combined scheme is mass conserving, consistent, and multi-tracer efficient.

Sedimentary Geothermal Feasibility Study: October 2016

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

Augustine, Chad; Zerpa, Luis

The objective of this project is to analyze the feasibility of commercial geothermal projects using numerical reservoir simulation, considering a sedimentary reservoir with low permeability that requires productivity enhancement. A commercial thermal reservoir simulator (STARS, from Computer Modeling Group, CMG) is used in this work for numerical modeling. In the first stage of this project (FY14), a hypothetical numerical reservoir model was developed, and validated against an analytical solution. The following model parameters were considered to obtain an acceptable match between the numerical and analytical solutions: grid block size, time step and reservoir areal dimensions; the latter related to boundarymore » effects on the numerical solution. Systematic model runs showed that insufficient grid sizing generates numerical dispersion that causes the numerical model to underestimate the thermal breakthrough time compared to the analytic model. As grid sizing is decreased, the model results converge on a solution. Likewise, insufficient reservoir model area introduces boundary effects in the numerical solution that cause the model results to differ from the analytical solution.« less

Controlling dental enamel-cavity ablation depth with optimized stepping parameters along the focal plane normal using a three axis, numerically controlled picosecond laser.

PubMed

Yuan, Fusong; Lv, Peijun; Wang, Dangxiao; Wang, Lei; Sun, Yuchun; Wang, Yong

2015-02-01

The purpose of this study was to establish a depth-control method in enamel-cavity ablation by optimizing the timing of the focal-plane-normal stepping and the single-step size of a three axis, numerically controlled picosecond laser. Although it has been proposed that picosecond lasers may be used to ablate dental hard tissue, the viability of such a depth-control method in enamel-cavity ablation remains uncertain. Forty-two enamel slices with approximately level surfaces were prepared and subjected to two-dimensional ablation by a picosecond laser. The additive-pulse layer, n, was set to 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70. A three-dimensional microscope was then used to measure the ablation depth, d, to obtain a quantitative function relating n and d. Six enamel slices were then subjected to three dimensional ablation to produce 10 cavities, respectively, with additive-pulse layer and single-step size set to corresponding values. The difference between the theoretical and measured values was calculated for both the cavity depth and the ablation depth of a single step. These were used to determine minimum-difference values for both the additive-pulse layer (n) and single-step size (d). When the additive-pulse layer and the single-step size were set 5 and 45, respectively, the depth error had a minimum of 2.25 μm, and 450 μm deep enamel cavities were produced. When performing three-dimensional ablating of enamel with a picosecond laser, adjusting the timing of the focal-plane-normal stepping and the single-step size allows for the control of ablation-depth error to the order of micrometers.

Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.

2005-09-01

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

Adaptive time stepping for fluid-structure interaction solvers

DOE PAGES

Mayr, M.; Wall, W. A.; Gee, M. W.

2017-12-22

In this work, a novel adaptive time stepping scheme for fluid-structure interaction (FSI) problems is proposed that allows for controlling the accuracy of the time-discrete solution. Furthermore, it eases practical computations by providing an efficient and very robust time step size selection. This has proven to be very useful, especially when addressing new physical problems, where no educated guess for an appropriate time step size is available. The fluid and the structure field, but also the fluid-structure interface are taken into account for the purpose of a posteriori error estimation, rendering it easy to implement and only adding negligible additionalmore » cost. The adaptive time stepping scheme is incorporated into a monolithic solution framework, but can straightforwardly be applied to partitioned solvers as well. The basic idea can be extended to the coupling of an arbitrary number of physical models. Accuracy and efficiency of the proposed method are studied in a variety of numerical examples ranging from academic benchmark tests to complex biomedical applications like the pulsatile blood flow through an abdominal aortic aneurysm. Finally, the demonstrated accuracy of the time-discrete solution in combination with reduced computational cost make this algorithm very appealing in all kinds of FSI applications.« less

Adaptive time stepping for fluid-structure interaction solvers

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

Mayr, M.; Wall, W. A.; Gee, M. W.

In this work, a novel adaptive time stepping scheme for fluid-structure interaction (FSI) problems is proposed that allows for controlling the accuracy of the time-discrete solution. Furthermore, it eases practical computations by providing an efficient and very robust time step size selection. This has proven to be very useful, especially when addressing new physical problems, where no educated guess for an appropriate time step size is available. The fluid and the structure field, but also the fluid-structure interface are taken into account for the purpose of a posteriori error estimation, rendering it easy to implement and only adding negligible additionalmore » cost. The adaptive time stepping scheme is incorporated into a monolithic solution framework, but can straightforwardly be applied to partitioned solvers as well. The basic idea can be extended to the coupling of an arbitrary number of physical models. Accuracy and efficiency of the proposed method are studied in a variety of numerical examples ranging from academic benchmark tests to complex biomedical applications like the pulsatile blood flow through an abdominal aortic aneurysm. Finally, the demonstrated accuracy of the time-discrete solution in combination with reduced computational cost make this algorithm very appealing in all kinds of FSI applications.« less

An explicit scheme for ohmic dissipation with smoothed particle magnetohydrodynamics

NASA Astrophysics Data System (ADS)

Tsukamoto, Yusuke; Iwasaki, Kazunari; Inutsuka, Shu-ichiro

2013-09-01

In this paper, we present an explicit scheme for Ohmic dissipation with smoothed particle magnetohydrodynamics (SPMHD). We propose an SPH discretization of Ohmic dissipation and solve Ohmic dissipation part of induction equation with the super-time-stepping method (STS) which allows us to take a longer time step than Courant-Friedrich-Levy stability condition. Our scheme is second-order accurate in space and first-order accurate in time. Our numerical experiments show that optimal choice of the parameters of STS for Ohmic dissipation of SPMHD is νsts ˜ 0.01 and Nsts ˜ 5.

Spectral-based propagation schemes for time-dependent quantum systems with application to carbon nanotubes

NASA Astrophysics Data System (ADS)

Chen, Zuojing; Polizzi, Eric

2010-11-01

Effective modeling and numerical spectral-based propagation schemes are proposed for addressing the challenges in time-dependent quantum simulations of systems ranging from atoms, molecules, and nanostructures to emerging nanoelectronic devices. While time-dependent Hamiltonian problems can be formally solved by propagating the solutions along tiny simulation time steps, a direct numerical treatment is often considered too computationally demanding. In this paper, however, we propose to go beyond these limitations by introducing high-performance numerical propagation schemes to compute the solution of the time-ordered evolution operator. In addition to the direct Hamiltonian diagonalizations that can be efficiently performed using the new eigenvalue solver FEAST, we have designed a Gaussian propagation scheme and a basis-transformed propagation scheme (BTPS) which allow to reduce considerably the simulation times needed by time intervals. It is outlined that BTPS offers the best computational efficiency allowing new perspectives in time-dependent simulations. Finally, these numerical schemes are applied to study the ac response of a (5,5) carbon nanotube within a three-dimensional real-space mesh framework.

Numerical Inverse Scattering for the Toda Lattice

NASA Astrophysics Data System (ADS)

Bilman, Deniz; Trogdon, Thomas

2017-06-01

We present a method to compute the inverse scattering transform (IST) for the famed Toda lattice by solving the associated Riemann-Hilbert (RH) problem numerically. Deformations for the RH problem are incorporated so that the IST can be evaluated in O(1) operations for arbitrary points in the ( n, t)-domain, including short- and long-time regimes. No time-stepping is required to compute the solution because ( n, t) appear as parameters in the associated RH problem. The solution of the Toda lattice is computed in long-time asymptotic regions where the asymptotics are not known rigorously.

Application of symbolic/numeric matrix solution techniques to the NASTRAN program

NASA Technical Reports Server (NTRS)

Buturla, E. M.; Burroughs, S. H.

1977-01-01

The matrix solving algorithm of any finite element algorithm is extremely important since solution of the matrix equations requires a large amount of elapse time due to null calculations and excessive input/output operations. An alternate method of solving the matrix equations is presented. A symbolic processing step followed by numeric solution yields the solution very rapidly and is especially useful for nonlinear problems.

A quick response four decade logarithmic high-voltage stepping supply

NASA Technical Reports Server (NTRS)

Doong, H.

1978-01-01

An improved high-voltage stepping supply, for space instrumentation is described where low power consumption and fast settling time between steps are required. The high-voltage stepping supply, utilizing an average power of 750 milliwatts, delivers a pair of mirror images with 64 level logarithmic outputs. It covers a four decade range of + or - 2500 to + or - 0.29 volts having an output stability of + or - 0.5 percent or + or - 20 millivolts for all line load and temperature variations. The supply provides a typical step setting time of 1 millisecond with 100 microseconds for the lower two decades. The versatile design features of the high-voltage stepping supply provides a quick response staircase generator as described or a fixed voltage with the option to change levels as required over large dynamic ranges without circuit modifications. The concept can be implemented up to + or - 5000 volts. With these design features, the high-voltage stepping supply should find numerous applications where charged particle detection, electro-optical systems, and high voltage scientific instruments are used.

A comparison of artificial compressibility and fractional step methods for incompressible flow computations

NASA Technical Reports Server (NTRS)

Chan, Daniel C.; Darian, Armen; Sindir, Munir

1992-01-01

We have applied and compared the efficiency and accuracy of two commonly used numerical methods for the solution of Navier-Stokes equations. The artificial compressibility method augments the continuity equation with a transient pressure term and allows one to solve the modified equations as a coupled system. Due to its implicit nature, one can have the luxury of taking a large temporal integration step at the expense of higher memory requirement and larger operation counts per step. Meanwhile, the fractional step method splits the Navier-Stokes equations into a sequence of differential operators and integrates them in multiple steps. The memory requirement and operation count per time step are low, however, the restriction on the size of time marching step is more severe. To explore the strengths and weaknesses of these two methods, we used them for the computation of a two-dimensional driven cavity flow with Reynolds number of 100 and 1000, respectively. Three grid sizes, 41 x 41, 81 x 81, and 161 x 161 were used. The computations were considered after the L2-norm of the change of the dependent variables in two consecutive time steps has fallen below 10(exp -5).

Unsteady Crystal Growth Due to Step-Bunch Cascading

NASA Technical Reports Server (NTRS)

Vekilov, Peter G.; Lin, Hong; Rosenberger, Franz

1997-01-01

Based on our experimental findings of growth rate fluctuations during the crystallization of the protein lysozym, we have developed a numerical model that combines diffusion in the bulk of a solution with diffusive transport to microscopic growth steps that propagate on a finite crystal facet. Nonlinearities in layer growth kinetics arising from step interaction by bulk and surface diffusion, and from step generation by surface nucleation, are taken into account. On evaluation of the model with properties characteristic for the solute transport, and the generation and propagation of steps in the lysozyme system, growth rate fluctuations of the same magnitude and characteristic time, as in the experiments, are obtained. The fluctuation time scale is large compared to that of step generation. Variations of the governing parameters of the model reveal that both the nonlinearity in step kinetics and mixed transport-kinetics control of the crystallization process are necessary conditions for the fluctuations. On a microscopic scale, the fluctuations are associated with a morphological instability of the vicinal face, in which a step bunch triggers a cascade of new step bunches through the microscopic interfacial supersaturation distribution.

Numerical simulation of pseudoelastic shape memory alloys using the large time increment method

NASA Astrophysics Data System (ADS)

Gu, Xiaojun; Zhang, Weihong; Zaki, Wael; Moumni, Ziad

2017-04-01

The paper presents a numerical implementation of the large time increment (LATIN) method for the simulation of shape memory alloys (SMAs) in the pseudoelastic range. The method was initially proposed as an alternative to the conventional incremental approach for the integration of nonlinear constitutive models. It is adapted here for the simulation of pseudoelastic SMA behavior using the Zaki-Moumni model and is shown to be especially useful in situations where the phase transformation process presents little or lack of hardening. In these situations, a slight stress variation in a load increment can result in large variations of strain and local state variables, which may lead to difficulties in numerical convergence. In contrast to the conventional incremental method, the LATIN method solve the global equilibrium and local consistency conditions sequentially for the entire loading path. The achieved solution must satisfy the conditions of static and kinematic admissibility and consistency simultaneously after several iterations. 3D numerical implementation is accomplished using an implicit algorithm and is then used for finite element simulation using the software Abaqus. Computational tests demonstrate the ability of this approach to simulate SMAs presenting flat phase transformation plateaus and subjected to complex loading cases, such as the quasi-static behavior of a stent structure. Some numerical results are contrasted to those obtained using step-by-step incremental integration.

Large-eddy simulation of a backward facing step flow using a least-squares spectral element method

NASA Technical Reports Server (NTRS)

Chan, Daniel C.; Mittal, Rajat

1996-01-01

We report preliminary results obtained from the large eddy simulation of a backward facing step at a Reynolds number of 5100. The numerical platform is based on a high order Legendre spectral element spatial discretization and a least squares time integration scheme. A non-reflective outflow boundary condition is in place to minimize the effect of downstream influence. Smagorinsky model with Van Driest near wall damping is used for sub-grid scale modeling. Comparisons of mean velocity profiles and wall pressure show good agreement with benchmark data. More studies are needed to evaluate the sensitivity of this method on numerical parameters before it is applied to complex engineering problems.

Accurate Monotonicity - Preserving Schemes With Runge-Kutta Time Stepping

NASA Technical Reports Server (NTRS)

Suresh, A.; Huynh, H. T.

1997-01-01

A new class of high-order monotonicity-preserving schemes for the numerical solution of conservation laws is presented. The interface value in these schemes is obtained by limiting a higher-order polynominal reconstruction. The limiting is designed to preserve accuracy near extrema and to work well with Runge-Kutta time stepping. Computational efficiency is enhanced by a simple test that determines whether the limiting procedure is needed. For linear advection in one dimension, these schemes are shown as well as the Euler equations also confirm their high accuracy, good shock resolution, and computational efficiency.

Navier-Stokes calculations for DFVLR F5-wing in wind tunnel using Runge-Kutta time-stepping scheme

NASA Technical Reports Server (NTRS)

Vatsa, V. N.; Wedan, B. W.

1988-01-01

A three-dimensional Navier-Stokes code using an explicit multistage Runge-Kutta type of time-stepping scheme is used for solving the transonic flow past a finite wing mounted inside a wind tunnel. Flow past the same wing in free air was also computed to assess the effect of wind-tunnel walls on such flows. Numerical efficiency is enhanced through vectorization of the computer code. A Cyber 205 computer with 32 million words of internal memory was used for these computations.

Robust numerical solution of the reservoir routing equation

NASA Astrophysics Data System (ADS)

Fiorentini, Marcello; Orlandini, Stefano

2013-09-01

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson-Pilgrim method, (2) the fourth-order Runge-Kutta method, and (3) the fixed order Cash-Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12 min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.

A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

NASA Astrophysics Data System (ADS)

Igumnov, Leonid; Ipatov, Aleksandr; Belov, Aleksandr; Petrov, Andrey

2015-09-01

The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on direct-approach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-by-element approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary) and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.

Non-Ideal Compressible-Fluid Dynamics of Fast-Response Pressure Probes for Unsteady Flow Measurements in Turbomachinery

NASA Astrophysics Data System (ADS)

Gori, G.; Molesini, P.; Persico, G.; Guardone, A.

2017-03-01

The dynamic response of pressure probes for unsteady flow measurements in turbomachinery is investigated numerically for fluids operating in non-ideal thermodynamic conditions, which are relevant for e.g. Organic Rankine Cycles (ORC) and super-critical CO2 applications. The step response of a fast-response pressure probe is investigated numerically in order to assess the expected time response when operating in the non-ideal fluid regime. Numerical simulations are carried out exploiting the Non-Ideal Compressible Fluid-Dynamics (NICFD) solver embedded in the open-source fluid dynamics code SU2. The computational framework is assessed against available experimental data for air in dilute conditions. Then, polytropic ideal gas (PIG), i.e. constant specific heats, and Peng-Robinson Stryjek-Vera (PRSV) models are applied to simulate the flow field within the probe operating with siloxane fluid octamethyltrisiloxane (MDM). The step responses are found to depend mainly on the speed of sound of the working fluid, indicating that molecular complexity plays a major role in determining the promptness of the measurement devices. According to the PRSV model, non-ideal effects can increase the step response time with respect to the acoustic theory predictions. The fundamental derivative of gas-dynamic is confirmed to be the driving parameter for evaluating non-ideal thermodynamic effects related to the dynamic calibration of fast-response aerodynamic pressure probes.

Optimal subinterval selection approach for power system transient stability simulation

DOE PAGES

Kim, Soobae; Overbye, Thomas J.

2015-10-21

Power system transient stability analysis requires an appropriate integration time step to avoid numerical instability as well as to reduce computational demands. For fast system dynamics, which vary more rapidly than what the time step covers, a fraction of the time step, called a subinterval, is used. However, the optimal value of this subinterval is not easily determined because the analysis of the system dynamics might be required. This selection is usually made from engineering experiences, and perhaps trial and error. This paper proposes an optimal subinterval selection approach for power system transient stability analysis, which is based on modalmore » analysis using a single machine infinite bus (SMIB) system. Fast system dynamics are identified with the modal analysis and the SMIB system is used focusing on fast local modes. An appropriate subinterval time step from the proposed approach can reduce computational burden and achieve accurate simulation responses as well. As a result, the performance of the proposed method is demonstrated with the GSO 37-bus system.« less

Modelling of current loads on aquaculture net cages

NASA Astrophysics Data System (ADS)

Kristiansen, Trygve; Faltinsen, Odd M.

2012-10-01

In this paper we propose and discuss a screen type of force model for the viscous hydrodynamic load on nets. The screen model assumes that the net is divided into a number of flat net panels, or screens. It may thus be applied to any kind of net geometry. In this paper we focus on circular net cages for fish farms. The net structure itself is modelled by an existing truss model. The net shape is solved for in a time-stepping procedure that involves solving a linear system of equations for the unknown tensions at each time step. We present comparisons to experiments with circular net cages in steady current, and discuss the sensitivity of the numerical results to a set of chosen parameters. Satisfactory agreement between experimental and numerical prediction of drag and lift as function of the solidity ratio of the net and the current velocity is documented.

A numerical solution of the supersonic flow over a rearward facing step with transverse non-reacting hydrogen injection

NASA Technical Reports Server (NTRS)

Berman, H. A.; Anderson, J. D., Jr.; Drummond, J. P.

1982-01-01

The present investigation represents an application of computational fluid dynamics to a problem associated with the flow in the combustor region of a supersonic combustion ramjet engine (scramjet). The governing equations are considered, taking into account the Navier-Stokes equations, a molecular viscosity calculation, the molecular thermal conductivity, molecular diffusion, and a turbulence model. The employed numerical solution is patterned after the explicit, time-dependent, unsplit, predictor-corrector, finite-difference method given by MacCormack (1969). The calculation is concerned with the supersonic flow over a rearward-facing step with transverse H2 injection at conditions germane to the combustor region of a scramjet engine. The H2 jet acts as an effective body which essentially shields the primary flow from the rearward-facing step, thus substantially changing the wave pattern in the primary flow.

Effects of Turbulence Model and Numerical Time Steps on Von Karman Flow Behavior and Drag Accuracy of Circular Cylinder

NASA Astrophysics Data System (ADS)

Amalia, E.; Moelyadi, M. A.; Ihsan, M.

2018-04-01

The flow of air passing around a circular cylinder on the Reynolds number of 250,000 is to show Von Karman Vortex Street Phenomenon. This phenomenon was captured well by using a right turbulence model. In this study, some turbulence models available in software ANSYS Fluent 16.0 was tested to simulate Von Karman vortex street phenomenon, namely k- epsilon, SST k-omega and Reynolds Stress, Detached Eddy Simulation (DES), and Large Eddy Simulation (LES). In addition, it was examined the effect of time step size on the accuracy of CFD simulation. The simulations are carried out by using two-dimensional and three- dimensional models and then compared with experimental data. For two-dimensional model, Von Karman Vortex Street phenomenon was captured successfully by using the SST k-omega turbulence model. As for the three-dimensional model, Von Karman Vortex Street phenomenon was captured by using Reynolds Stress Turbulence Model. The time step size value affects the smoothness quality of curves of drag coefficient over time, as well as affecting the running time of the simulation. The smaller time step size, the better inherent drag coefficient curves produced. Smaller time step size also gives faster computation time.

Computational issues in the simulation of two-dimensional discrete dislocation mechanics

NASA Astrophysics Data System (ADS)

Segurado, J.; LLorca, J.; Romero, I.

2007-06-01

The effect of the integration time step and the introduction of a cut-off velocity for the dislocation motion was analysed in discrete dislocation dynamics (DD) simulations of a single crystal microbeam. Two loading modes, bending and uniaxial tension, were examined. It was found that a longer integration time step led to a progressive increment of the oscillations in the numerical solution, which would eventually diverge. This problem could be corrected in the simulations carried out in bending by introducing a cut-off velocity for the dislocation motion. This strategy (long integration times and a cut-off velocity for the dislocation motion) did not recover, however, the solution computed with very short time steps in uniaxial tension: the dislocation density was overestimated and the dislocation patterns modified. The different response to the same numerical algorithm was explained in terms of the nature of the dislocations generated in each case: geometrically necessary in bending and statistically stored in tension. The evolution of the dislocation density in the former was controlled by the plastic curvature of the beam and was independent of the details of the simulations. On the contrary, the steady-state dislocation density in tension was determined by the balance between nucleation of dislocations and those which are annihilated or which exit the beam. Changes in the DD imposed by the cut-off velocity altered this equilibrium and the solution. These results point to the need for detailed analyses of the accuracy and stability of the dislocation dynamic simulations to ensure that the results obtained are not fundamentally affected by the numerical strategies used to solve this complex problem.

Exploiting Superconvergence in Discontinuous Galerkin Methods for Improved Time-Stepping and Visualization

DTIC Science & Technology

2016-09-08

Accuracy Conserving (SIAC) filter when applied to nonuniform meshes; 2) Theoretically and numerical demonstration of the 2k+1 order accuracy of the SIAC...Establishing a more theoretical and numerical understanding of a computationally efficient scaling for the SIAC filter for nonuniform meshes [7]; 2...Li, “SIAC Filtering of DG Methods – Boundary and Nonuniform Mesh”, International Conference on Spectral and Higher Order Methods (ICOSAHOM

Unsteady Flow Simulation: A Numerical Challenge

DTIC Science & Technology

2003-03-01

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

Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model

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

Baudron, Anne-Marie, E-mail: anne-marie.baudron@cea.fr; CEA-DRN/DMT/SERMA, CEN-Saclay, 91191 Gif sur Yvette Cedex; Lautard, Jean-Jacques, E-mail: jean-jacques.lautard@cea.fr

2014-12-15

In this paper we present a time-parallel algorithm for the 3D neutrons calculation of a transient model in a nuclear reactor core. The neutrons calculation consists in numerically solving the time dependent diffusion approximation equation, which is a simplified transport equation. The numerical resolution is done with finite elements method based on a tetrahedral meshing of the computational domain, representing the reactor core, and time discretization is achieved using a θ-scheme. The transient model presents moving control rods during the time of the reaction. Therefore, cross-sections (piecewise constants) are taken into account by interpolations with respect to the velocity ofmore » the control rods. The parallelism across the time is achieved by an adequate use of the parareal in time algorithm to the handled problem. This parallel method is a predictor corrector scheme that iteratively combines the use of two kinds of numerical propagators, one coarse and one fine. Our method is made efficient by means of a coarse solver defined with large time step and fixed position control rods model, while the fine propagator is assumed to be a high order numerical approximation of the full model. The parallel implementation of our method provides a good scalability of the algorithm. Numerical results show the efficiency of the parareal method on large light water reactor transient model corresponding to the Langenbuch–Maurer–Werner benchmark.« less

Four-level conservative finite-difference schemes for Boussinesq paradigm equation

NASA Astrophysics Data System (ADS)

Kolkovska, N.

2013-10-01

In this paper a two-parametric family of four level conservative finite difference schemes is constructed for the multidimensional Boussinesq paradigm equation. The schemes are explicit in the sense that no inner iterations are needed for evaluation of the numerical solution. The preservation of the discrete energy with this method is proved. The schemes have been numerically tested on one soliton propagation model and two solitons interaction model. The numerical experiments demonstrate that the proposed family of schemes has second order of convergence in space and time steps in the discrete maximal norm.

A Class of Prediction-Correction Methods for Time-Varying Convex Optimization

NASA Astrophysics Data System (ADS)

Simonetto, Andrea; Mokhtari, Aryan; Koppel, Alec; Leus, Geert; Ribeiro, Alejandro

2016-09-01

This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of $1/h$, where $h$ is the length of the sampling interval. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves as $O(h^2)$, and in some cases as $O(h^4)$, which outperforms the state-of-the-art error bound of $O(h)$ for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

NASA Astrophysics Data System (ADS)

Parand, K.; Latifi, S.; Moayeri, M. M.; Delkhosh, M.

2018-05-01

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective, reliable and does not require any restrictive assumptions for nonlinear terms.

A time-accurate finite volume method valid at all flow velocities

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1993-01-01

A finite volume method to solve the Navier-Stokes equations at all flow velocities (e.g., incompressible, subsonic, transonic, supersonic and hypersonic flows) is presented. The numerical method is based on a finite volume method that incorporates a pressure-staggered mesh and an incremental pressure equation for the conservation of mass. Comparison of three generally accepted time-advancing schemes, i.e., Simplified Marker-and-Cell (SMAC), Pressure-Implicit-Splitting of Operators (PISO), and Iterative-Time-Advancing (ITA) scheme, are made by solving a lid-driven polar cavity flow and self-sustained oscillatory flows over circular and square cylinders. Calculated results show that the ITA is the most stable numerically and yields the most accurate results. The SMAC is the most efficient computationally and is as stable as the ITA. It is shown that the PISO is the most weakly convergent and it exhibits an undesirable strong dependence on the time-step size. The degenerated numerical results obtained using the PISO are attributed to its second corrector step that cause the numerical results to deviate further from a divergence free velocity field. The accurate numerical results obtained using the ITA is attributed to its capability to resolve the nonlinearity of the Navier-Stokes equations. The present numerical method that incorporates the ITA is used to solve an unsteady transitional flow over an oscillating airfoil and a chemically reacting flow of hydrogen in a vitiated supersonic airstream. The turbulence fields in these flow cases are described using multiple-time-scale turbulence equations. For the unsteady transitional over an oscillating airfoil, the fluid flow is described using ensemble-averaged Navier-Stokes equations defined on the Lagrangian-Eulerian coordinates. It is shown that the numerical method successfully predicts the large dynamic stall vortex (DSV) and the trailing edge vortex (TEV) that are periodically generated by the oscillating airfoil. The calculated streaklines are in very good comparison with the experimentally obtained smoke picture. The calculated turbulent viscosity contours show that the transition from laminar to turbulent state and the relaminarization occur widely in space as well as in time. The ensemble-averaged velocity profiles are also in good agreement with the measured data and the good comparison indicates that the numerical method as well as the multipletime-scale turbulence equations successfully predict the unsteady transitional turbulence field. The chemical reactions for the hydrogen in the vitiated supersonic airstream are described using 9 chemical species and 48 reaction-steps. Consider that a fast chemistry can not be used to describe the fine details (such as the instability) of chemically reacting flows while a reduced chemical kinetics can not be used confidently due to the uncertainty contained in the reaction mechanisms. However, the use of a detailed finite rate chemistry may make it difficult to obtain a fully converged solution due to the coupling between the large number of flow, turbulence, and chemical equations. The numerical results obtained in the present study are in good agreement with the measured data. The good comparison is attributed to the numerical method that can yield strongly converged results for the reacting flow and to the use of the multiple-time-scale turbulence equations that can accurately describe the mixing of the fuel and the oxidant.

Solving transient acoustic boundary value problems with equivalent sources using a lumped parameter approach.

PubMed

Fahnline, John B

2016-12-01

An equivalent source method is developed for solving transient acoustic boundary value problems. The method assumes the boundary surface is discretized in terms of triangular or quadrilateral elements and that the solution is represented using the acoustic fields of discrete sources placed at the element centers. Also, the boundary condition is assumed to be specified for the normal component of the surface velocity as a function of time, and the source amplitudes are determined to match the known elemental volume velocity vector at a series of discrete time steps. Equations are given for marching-on-in-time schemes to solve for the source amplitudes at each time step for simple, dipole, and tripole source formulations. Several example problems are solved to illustrate the results and to validate the formulations, including problems with closed boundary surfaces where long-time numerical instabilities typically occur. A simple relationship between the simple and dipole source amplitudes in the tripole source formulation is derived so that the source radiates primarily in the direction of the outward surface normal. The tripole source formulation is shown to eliminate interior acoustic resonances and long-time numerical instabilities.

Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks.

PubMed

Shelley, M J; Tao, L

2001-01-01

To avoid the numerical errors associated with resetting the potential following a spike in simulations of integrate-and-fire neuronal networks, Hansel et al. and Shelley independently developed a modified time-stepping method. Their particular scheme consists of second-order Runge-Kutta time-stepping, a linear interpolant to find spike times, and a recalibration of postspike potential using the spike times. Here we show analytically that such a scheme is second order, discuss the conditions under which efficient, higher-order algorithms can be constructed to treat resets, and develop a modified fourth-order scheme. To support our analysis, we simulate a system of integrate-and-fire conductance-based point neurons with all-to-all coupling. For six-digit accuracy, our modified Runge-Kutta fourth-order scheme needs a time-step of Delta(t) = 0.5 x 10(-3) seconds, whereas to achieve comparable accuracy using a recalibrated second-order or a first-order algorithm requires time-steps of 10(-5) seconds or 10(-9) seconds, respectively. Furthermore, since the cortico-cortical conductances in standard integrate-and-fire neuronal networks do not depend on the value of the membrane potential, we can attain fourth-order accuracy with computational costs normally associated with second-order schemes.

Molecular dynamics with rigid bodies: Alternative formulation and assessment of its limitations when employed to simulate liquid water

NASA Astrophysics Data System (ADS)

Silveira, Ana J.; Abreu, Charlles R. A.

2017-09-01

Sets of atoms collectively behaving as rigid bodies are often used in molecular dynamics to model entire molecules or parts thereof. This is a coarse-graining strategy that eliminates degrees of freedom and supposedly admits larger time steps without abandoning the atomistic character of a model. In this paper, we rely on a particular factorization of the rotation matrix to simplify the mechanical formulation of systems containing rigid bodies. We then propose a new derivation for the exact solution of torque-free rotations, which are employed as part of a symplectic numerical integration scheme for rigid-body dynamics. We also review methods for calculating pressure in systems of rigid bodies with pairwise-additive potentials and periodic boundary conditions. Finally, simulations of liquid phases, with special focus on water, are employed to analyze the numerical aspects of the proposed methodology. Our results show that energy drift is avoided for time step sizes up to 5 fs, but only if a proper smoothing is applied to the interatomic potentials. Despite this, the effects of discretization errors are relevant, even for smaller time steps. These errors induce, for instance, a systematic failure of the expected equipartition of kinetic energy between translational and rotational degrees of freedom.

Implicit and semi-implicit schemes in the Versatile Advection Code: numerical tests

NASA Astrophysics Data System (ADS)

Toth, G.; Keppens, R.; Botchev, M. A.

1998-04-01

We describe and evaluate various implicit and semi-implicit time integration schemes applied to the numerical simulation of hydrodynamical and magnetohydrodynamical problems. The schemes were implemented recently in the software package Versatile Advection Code, which uses modern shock capturing methods to solve systems of conservation laws with optional source terms. The main advantage of implicit solution strategies over explicit time integration is that the restrictive constraint on the allowed time step can be (partially) eliminated, thus the computational cost is reduced. The test problems cover one and two dimensional, steady state and time accurate computations, and the solutions contain discontinuities. For each test, we confront explicit with implicit solution strategies.

Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data

NASA Astrophysics Data System (ADS)

Chang, Chueh-Hsin; Yu, Ching-Hao; Sheu, Tony Wen-Hann

2016-10-01

In this article, we numerically revisit the long-time solution behavior of the Camassa-Holm equation ut - uxxt + 2ux + 3uux = 2uxuxx + uuxxx. The finite difference solution of this integrable equation is sought subject to the newly derived initial condition with Delta-function potential. Our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers. Additionally, to achieve the goal of conserving Hamiltonians in the completely integrable equation of current interest, a symplecticity-preserving time-stepping scheme is developed. Based on the solutions computed from the temporally symplecticity-preserving and the spatially wavenumber-preserving schemes, the long-time asymptotic CH solution characters can be accurately depicted in distinct regions of the space-time domain featuring with their own quantitatively very different solution behaviors. We also aim to numerically confirm that in the two transition zones their long-time asymptotics can indeed be described in terms of the theoretically derived Painlevé transcendents. Another attempt of this study is to numerically exhibit a close connection between the presently predicted finite-difference solution and the solution of the Painlevé ordinary differential equation of type II in two different transition zones.

Non-hydrostatic semi-elastic hybrid-coordinate SISL extension of HIRLAM. Part II: numerical testing

NASA Astrophysics Data System (ADS)

Rõõm, Rein; Männik, Aarne; Luhamaa, Andres; Zirk, Marko

2007-10-01

The semi-implicit semi-Lagrangian (SISL), two-time-level, non-hydrostatic numerical scheme, based on the non-hydrostatic, semi-elastic pressure-coordinate equations, is tested in model experiments with flow over given orography (elliptical hill, mountain ridge, system of successive ridges) in a rectangular domain with emphasis on the numerical accuracy and non-hydrostatic effect presentation capability. Comparison demonstrates good (in strong primary wave generation) to satisfactory (in weak secondary wave reproduction in some cases) consistency of the numerical modelling results with known stationary linear test solutions. Numerical stability of the developed model is investigated with respect to the reference state choice, modelling dynamics of a stationary front. The horizontally area-mean reference temperature proves to be the optimal stability warrant. The numerical scheme with explicit residual in the vertical forcing term becomes unstable for cross-frontal temperature differences exceeding 30 K. Stability is restored, if the vertical forcing is treated implicitly, which enables to use time steps, comparable with the hydrostatic SISL.

A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

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

Zheng, Xiang; Yang, Chao; State Key Laboratory of Computer Science, Chinese Academy of Sciences, Beijing 100190

2015-03-15

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracymore » (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.« less

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

NASA Technical Reports Server (NTRS)

Mei, Chuh; Moorthy, Jayashree

1995-01-01

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

Symplectic molecular dynamics simulations on specially designed parallel computers.

PubMed

Borstnik, Urban; Janezic, Dusanka

2005-01-01

We have developed a computer program for molecular dynamics (MD) simulation that implements the Split Integration Symplectic Method (SISM) and is designed to run on specialized parallel computers. The MD integration is performed by the SISM, which analytically treats high-frequency vibrational motion and thus enables the use of longer simulation time steps. The low-frequency motion is treated numerically on specially designed parallel computers, which decreases the computational time of each simulation time step. The combination of these approaches means that less time is required and fewer steps are needed and so enables fast MD simulations. We study the computational performance of MD simulation of molecular systems on specialized computers and provide a comparison to standard personal computers. The combination of the SISM with two specialized parallel computers is an effective way to increase the speed of MD simulations up to 16-fold over a single PC processor.

hp-Adaptive time integration based on the BDF for viscous flows

NASA Astrophysics Data System (ADS)

Hay, A.; Etienne, S.; Pelletier, D.; Garon, A.

2015-06-01

This paper presents a procedure based on the Backward Differentiation Formulas of order 1 to 5 to obtain efficient time integration of the incompressible Navier-Stokes equations. The adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process. The stepsize selection (h-adaptivity) is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance. The order selection (p-adaptivity) relies on the idea that low-accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low. The selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize. Hence, it guarantees both that (1) the used method of integration operates inside of its stability region and (2) the time integration procedure is computationally efficient. The proposed time integration procedure also features a time-step rejection and quarantine mechanisms, a modified Newton method with a predictor and dense output techniques to compute solution at off-step points.

Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

NASA Technical Reports Server (NTRS)

Majda, George

1986-01-01

One-leg and multistep discretizations of variable-coefficient linear systems of ODEs having both slow and fast time scales are investigated analytically. The stability properties of these discretizations are obtained independent of ODE stiffness and compared. The results of numerical computations are presented in tables, and it is shown that for large step sizes the stability of one-leg methods is better than that of the corresponding linear multistep methods.

Fine Tuning Cell Migration by a Disintegrin and Metalloproteinases

PubMed Central

Theodorou, K.

2017-01-01

Cell migration is an instrumental process involved in organ development, tissue homeostasis, and various physiological processes and also in numerous pathologies. Both basic cell migration and migration towards chemotactic stimulus consist of changes in cell polarity and cytoskeletal rearrangement, cell detachment from, invasion through, and reattachment to their neighboring cells, and numerous interactions with the extracellular matrix. The different steps of immune cell, tissue cell, or cancer cell migration are tightly coordinated in time and place by growth factors, cytokines/chemokines, adhesion molecules, and receptors for these ligands. This review describes how a disintegrin and metalloproteinases interfere with several steps of cell migration, either by proteolytic cleavage of such molecules or by functions independent of proteolytic activity. PMID:28260841

Efficient C1-continuous phase-potential upwind (C1-PPU) schemes for coupled multiphase flow and transport with gravity

NASA Astrophysics Data System (ADS)

Jiang, Jiamin; Younis, Rami M.

2017-10-01

In the presence of counter-current flow, nonlinear convergence problems may arise in implicit time-stepping when the popular phase-potential upwinding (PPU) scheme is used. The PPU numerical flux is non-differentiable across the co-current/counter-current flow regimes. This may lead to cycles or divergence in the Newton iterations. Recently proposed methods address improved smoothness of the numerical flux. The objective of this work is to devise and analyze an alternative numerical flux scheme called C1-PPU that, in addition to improving smoothness with respect to saturations and phase potentials, also improves the level of scalar nonlinearity and accuracy. C1-PPU involves a novel use of the flux limiter concept from the context of high-resolution methods, and allows a smooth variation between the co-current/counter-current flow regimes. The scheme is general and applies to fully coupled flow and transport formulations with an arbitrary number of phases. We analyze the consistency property of the C1-PPU scheme, and derive saturation and pressure estimates, which are used to prove the solution existence. Several numerical examples for two- and three-phase flows in heterogeneous and multi-dimensional reservoirs are presented. The proposed scheme is compared to the conventional PPU and the recently proposed Hybrid Upwinding schemes. We investigate three properties of these numerical fluxes: smoothness, nonlinearity, and accuracy. The results indicate that in addition to smoothness, nonlinearity may also be critical for convergence behavior and thus needs to be considered in the design of an efficient numerical flux scheme. Moreover, the numerical examples show that the C1-PPU scheme exhibits superior convergence properties for large time steps compared to the other alternatives.

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

PubMed

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

NASA Astrophysics Data System (ADS)

Fang, Gang; Ba, Jing; Liu, Xin-xin; Zhu, Kun; Liu, Guo-Chang

2017-06-01

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.

Procedures for numerical analysis of circadian rhythms

PubMed Central

REFINETTI, ROBERTO; LISSEN, GERMAINE CORNÉ; HALBERG, FRANZ

2010-01-01

This article reviews various procedures used in the analysis of circadian rhythms at the populational, organismal, cellular and molecular levels. The procedures range from visual inspection of time plots and actograms to several mathematical methods of time series analysis. Computational steps are described in some detail, and additional bibliographic resources and computer programs are listed. PMID:23710111

Time-Accurate Numerical Simulations of Synthetic Jet Quiescent Air

NASA Technical Reports Server (NTRS)

Rupesh, K-A. B.; Ravi, B. R.; Mittal, R.; Raju, R.; Gallas, Q.; Cattafesta, L.

2007-01-01

The unsteady evolution of three-dimensional synthetic jet into quiescent air is studied by time-accurate numerical simulations using a second-order accurate mixed explicit-implicit fractional step scheme on Cartesian grids. Both two-dimensional and three-dimensional calculations of synthetic jet are carried out at a Reynolds number (based on average velocity during the discharge phase of the cycle V(sub j), and jet width d) of 750 and Stokes number of 17.02. The results obtained are assessed against PIV and hotwire measurements provided for the NASA LaRC workshop on CFD validation of synthetic jets.

Pull-out fibers from composite materials at high rate of loading

NASA Technical Reports Server (NTRS)

Amijima, S.; Fujii, T.

1981-01-01

Numerical and experimental results are presented on the pullout phenomenon in composite materials at a high rate of loading. The finite element method was used, taking into account the existence of a virtual shear deformation layer as the interface between fiber and matrix. Experimental results agree well with those obtained by the finite element method. Numerical results show that the interlaminar shear stress is time dependent, in addition, it is shown to depend on the applied load time history. Under step pulse loading, the interlaminar shear stress fluctuates, finally decaying to its value under static loading.

Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping

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

Finn, John M., E-mail: finn@lanl.gov

2015-03-15

Properties of integration schemes for solenoidal fields in three dimensions are studied, with a focus on integrating magnetic field lines in a plasma using adaptive time stepping. It is shown that implicit midpoint (IM) and a scheme we call three-dimensional leapfrog (LF) can do a good job (in the sense of preserving KAM tori) of integrating fields that are reversible, or (for LF) have a “special divergence-free” (SDF) property. We review the notion of a self-adjoint scheme, showing that such schemes are at least second order accurate and can always be formed by composing an arbitrary scheme with its adjoint.more » We also review the concept of reversibility, showing that a reversible but not exactly volume-preserving scheme can lead to a fractal invariant measure in a chaotic region, although this property may not often be observable. We also show numerical results indicating that the IM and LF schemes can fail to preserve KAM tori when the reversibility property (and the SDF property for LF) of the field is broken. We discuss extensions to measure preserving flows, the integration of magnetic field lines in a plasma and the integration of rays for several plasma waves. The main new result of this paper relates to non-uniform time stepping for volume-preserving flows. We investigate two potential schemes, both based on the general method of Feng and Shang [Numer. Math. 71, 451 (1995)], in which the flow is integrated in split time steps, each Hamiltonian in two dimensions. The first scheme is an extension of the method of extended phase space, a well-proven method of symplectic integration with non-uniform time steps. This method is found not to work, and an explanation is given. The second method investigated is a method based on transformation to canonical variables for the two split-step Hamiltonian systems. This method, which is related to the method of non-canonical generating functions of Richardson and Finn [Plasma Phys. Controlled Fusion 54, 014004 (2012)], appears to work very well.« less

Special Year on Numerical Linear Algebra

DTIC Science & Technology

1988-09-01

ORNL) Worley, Pat (ORNL) A special acknowledgement should go to Mary Drake (UT) and Mitzy Denson (ORNL) who carried the burden of making the innumerable...a time step appropriate for the regular cells with no stability restriction. Entrance to Y-12 requires a pass. Contact Mitzy Denson (615) 574-3125 to...requires a pass. Contact Mitzy Denson (615) 574-3125 to obtain one. ’This seminar is part of the Special Year on Numerical Linear Algebra sponsored by the

The Root Cause of the Overheating Problem

NASA Technical Reports Server (NTRS)

Liou, Meng-Sing

2017-01-01

Previously we identified the receding flow, where two fluid streams recede from each other, as an open numerical problem, because all well-known numerical fluxes give an anomalous temperature rise, thus called the overheating problem. This phenomenon, although presented in several textbooks, and many previous publications, has scarcely been satisfactorily addressed and the root cause of the overheating problem not well understood. We found that this temperature rise was solely connected to entropy rise and proposed to use the method of characteristics to eradicate the problem. However, the root cause of the entropy production was still unclear. In the present study, we identify the cause of this problem: the entropy rise is rooted in the pressure flux in a finite volume formulation and is implanted at the first time step. It is found theoretically inevitable for all existing numerical flux schemes used in the finite volume setting, as confirmed by numerical tests. This difficulty cannot be eliminated by manipulating time step, grid size, spatial accuracy, etc, although the rate of overheating depends on the flux scheme used. Finally, we incorporate the entropy transport equation, in place of the energy equation, to ensure preservation of entropy, thus correcting this temperature anomaly. Its applicability is demonstrated for some relevant 1D and 2D problems. Thus, the present study validates that the entropy generated ab initio is the genesis of the overheating problem.

Multigrid for hypersonic viscous two- and three-dimensional flows

NASA Technical Reports Server (NTRS)

Turkel, E.; Swanson, R. C.; Vatsa, V. N.; White, J. A.

1991-01-01

The use of a multigrid method with central differencing to solve the Navier-Stokes equations for hypersonic flows is considered. The time dependent form of the equations is integrated with an explicit Runge-Kutta scheme accelerated by local time stepping and implicit residual smoothing. Variable coefficients are developed for the implicit process that removes the diffusion limit on the time step, producing significant improvement in convergence. A numerical dissipation formulation that provides good shock capturing capability for hypersonic flows is presented. This formulation is shown to be a crucial aspect of the multigrid method. Solutions are given for two-dimensional viscous flow over a NACA 0012 airfoil and three-dimensional flow over a blunt biconic.

A far-field non-reflecting boundary condition for two-dimensional wake flows

NASA Technical Reports Server (NTRS)

Danowitz, Jeffrey S.; Abarbanel, Saul A.; Turkel, Eli

1995-01-01

Far-field boundary conditions for external flow problems have been developed based upon long-wave perturbations of linearized flow equations about a steady state far field solution. The boundary improves convergence to steady state in single-grid temporal integration schemes using both regular-time-stepping and local-time-stepping. The far-field boundary may be near the trailing edge of the body which significantly reduces the number of grid points, and therefore the computational time, in the numerical calculation. In addition the solution produced is smoother in the far-field than when using extrapolation conditions. The boundary condition maintains the convergence rate to steady state in schemes utilizing multigrid acceleration.

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

NASA Astrophysics Data System (ADS)

Semplice, Matteo; Loubère, Raphaël

2018-02-01

In this paper we propose a third order accurate finite volume scheme based on a posteriori limiting of polynomial reconstructions within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge-Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a non-oscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using a more dissipative scheme but only locally, close to these cells. By locally decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate but more stable scheme. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed with the refined mesh, but only locally, close to the new cells. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology can maintain not only optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is at least comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.

Step-and-Repeat Nanoimprint-, Photo- and Laser Lithography from One Customised CNC Machine.

PubMed

Greer, Andrew Im; Della-Rosa, Benoit; Khokhar, Ali Z; Gadegaard, Nikolaj

2016-12-01

The conversion of a computer numerical control machine into a nanoimprint step-and-repeat tool with additional laser- and photolithography capacity is documented here. All three processes, each demonstrated on a variety of photoresists, are performed successfully and analysed so as to enable the reader to relate their known lithography process(es) to the findings. Using the converted tool, 1 cm(2) of nanopattern may be exposed in 6 s, over 3300 times faster than the electron beam equivalent. Nanoimprint tools are commercially available, but these can cost around 1000 times more than this customised computer numerical control (CNC) machine. The converted equipment facilitates rapid production and large area micro- and nanoscale research on small grants, ultimately enabling faster and more diverse growth in this field of science. In comparison to commercial tools, this converted CNC also boasts capacity to handle larger substrates, temperature control and active force control, up to ten times more curing dose and compactness. Actual devices are fabricated using the machine including an expanded nanotopographic array and microfluidic PDMS Y-channel mixers.

Step-and-Repeat Nanoimprint-, Photo- and Laser Lithography from One Customised CNC Machine

NASA Astrophysics Data System (ADS)

Greer, Andrew IM; Della-Rosa, Benoit; Khokhar, Ali Z.; Gadegaard, Nikolaj

2016-03-01

The conversion of a computer numerical control machine into a nanoimprint step-and-repeat tool with additional laser- and photolithography capacity is documented here. All three processes, each demonstrated on a variety of photoresists, are performed successfully and analysed so as to enable the reader to relate their known lithography process(es) to the findings. Using the converted tool, 1 cm2 of nanopattern may be exposed in 6 s, over 3300 times faster than the electron beam equivalent. Nanoimprint tools are commercially available, but these can cost around 1000 times more than this customised computer numerical control (CNC) machine. The converted equipment facilitates rapid production and large area micro- and nanoscale research on small grants, ultimately enabling faster and more diverse growth in this field of science. In comparison to commercial tools, this converted CNC also boasts capacity to handle larger substrates, temperature control and active force control, up to ten times more curing dose and compactness. Actual devices are fabricated using the machine including an expanded nanotopographic array and microfluidic PDMS Y-channel mixers.

Direct numerical simulations of premixed autoignition in compressible uniformly-sheared turbulence

NASA Astrophysics Data System (ADS)

Towery, Colin; Darragh, Ryan; Poludnenko, Alexei; Hamlington, Peter

2017-11-01

High-speed combustion systems, such as scramjet engines, operate at high temperatures and pressures, extremely short combustor residence times, very high rates of shear stress, and intense turbulent mixing. As a result, the reacting flow can be premixed and have highly-compressible turbulence fluctuations. We investigate the effects of compressible turbulence on the ignition delay time, heat-release-rate (HRR) intermittency, and mode of autoignition of premixed Hydrogen-air fuel in uniformly-sheared turbulence using new three-dimensional direct numerical simulations with a multi-step chemistry mechanism. We analyze autoignition in both the Eulerian and Lagrangian reference frames at eight different turbulence Mach numbers, Mat , spanning the quasi-isentropic, linear thermodynamic, and nonlinear compressibility regimes, with eddy shocklets appearing in the nonlinear regime. Results are compared to our previous study of premixed autoignition in isotropic turbulence at the same Mat and with a single-step reaction mechanism. This previous study found large decreases in delay times and large increases in HRR intermittency between the linear and nonlinear compressibility regimes and that detonation waves could form in both regimes.

Theoretical stability in coefficient inverse problems for general hyperbolic equations with numerical reconstruction

NASA Astrophysics Data System (ADS)

Yu, Jie; Liu, Yikan; Yamamoto, Masahiro

2018-04-01

In this article, we investigate the determination of the spatial component in the time-dependent second order coefficient of a hyperbolic equation from both theoretical and numerical aspects. By the Carleman estimates for general hyperbolic operators and an auxiliary Carleman estimate, we establish local Hölder stability with either partial boundary or interior measurements under certain geometrical conditions. For numerical reconstruction, we minimize a Tikhonov functional which penalizes the gradient of the unknown function. Based on the resulting variational equation, we design an iteration method which is updated by solving a Poisson equation at each step. One-dimensional prototype examples illustrate the numerical performance of the proposed iteration.

Stability analysis of Eulerian-Lagrangian methods for the one-dimensional shallow-water equations

USGS Publications Warehouse

Casulli, V.; Cheng, R.T.

1990-01-01

In this paper stability and error analyses are discussed for some finite difference methods when applied to the one-dimensional shallow-water equations. Two finite difference formulations, which are based on a combined Eulerian-Lagrangian approach, are discussed. In the first part of this paper the results of numerical analyses for an explicit Eulerian-Lagrangian method (ELM) have shown that the method is unconditionally stable. This method, which is a generalized fixed grid method of characteristics, covers the Courant-Isaacson-Rees method as a special case. Some artificial viscosity is introduced by this scheme. However, because the method is unconditionally stable, the artificial viscosity can be brought under control either by reducing the spatial increment or by increasing the size of time step. The second part of the paper discusses a class of semi-implicit finite difference methods for the one-dimensional shallow-water equations. This method, when the Eulerian-Lagrangian approach is used for the convective terms, is also unconditionally stable and highly accurate for small space increments or large time steps. The semi-implicit methods seem to be more computationally efficient than the explicit ELM; at each time step a single tridiagonal system of linear equations is solved. The combined explicit and implicit ELM is best used in formulating a solution strategy for solving a network of interconnected channels. The explicit ELM is used at channel junctions for each time step. The semi-implicit method is then applied to the interior points in each channel segment. Following this solution strategy, the channel network problem can be reduced to a set of independent one-dimensional open-channel flow problems. Numerical results support properties given by the stability and error analyses. ?? 1990.

The Julia programming language: the future of scientific computing

NASA Astrophysics Data System (ADS)

Gibson, John

2017-11-01

Julia is an innovative new open-source programming language for high-level, high-performance numerical computing. Julia combines the general-purpose breadth and extensibility of Python, the ease-of-use and numeric focus of Matlab, the speed of C and Fortran, and the metaprogramming power of Lisp. Julia uses type inference and just-in-time compilation to compile high-level user code to machine code on the fly. A rich set of numeric types and extensive numerical libraries are built-in. As a result, Julia is competitive with Matlab for interactive graphical exploration and with C and Fortran for high-performance computing. This talk interactively demonstrates Julia's numerical features and benchmarks Julia against C, C++, Fortran, Matlab, and Python on a spectral time-stepping algorithm for a 1d nonlinear partial differential equation. The Julia code is nearly as compact as Matlab and nearly as fast as Fortran. This material is based upon work supported by the National Science Foundation under Grant No. 1554149.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

NASA Astrophysics Data System (ADS)

Liu, Xinxin; Yin, Xingyao; Wu, Guochen

2014-04-01

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However, the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap, combined with variable grid-size and time-step, this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

Polynomial elimination theory and non-linear stability analysis for the Euler equations

NASA Technical Reports Server (NTRS)

Kennon, S. R.; Dulikravich, G. S.; Jespersen, D. C.

1986-01-01

Numerical methods are presented that exploit the polynomial properties of discretizations of the Euler equations. It is noted that most finite difference or finite volume discretizations of the steady-state Euler equations produce a polynomial system of equations to be solved. These equations are solved using classical polynomial elimination theory, with some innovative modifications. This paper also presents some preliminary results of a new non-linear stability analysis technique. This technique is applicable to determining the stability of polynomial iterative schemes. Results are presented for applying the elimination technique to a one-dimensional test case. For this test case, the exact solution is computed in three iterations. The non-linear stability analysis is applied to determine the optimal time step for solving Burgers' equation using the MacCormack scheme. The estimated optimal time step is very close to the time step that arises from a linear stability analysis.

Optimal variable-grid finite-difference modeling for porous media

NASA Astrophysics Data System (ADS)

Liu, Xinxin; Yin, Xingyao; Li, Haishan

2014-12-01

Numerical modeling of poroelastic waves by the finite-difference (FD) method is more expensive than that of acoustic or elastic waves. To improve the accuracy and computational efficiency of seismic modeling, variable-grid FD methods have been developed. In this paper, we derived optimal staggered-grid finite difference schemes with variable grid-spacing and time-step for seismic modeling in porous media. FD operators with small grid-spacing and time-step are adopted for low-velocity or small-scale geological bodies, while FD operators with big grid-spacing and time-step are adopted for high-velocity or large-scale regions. The dispersion relations of FD schemes were derived based on the plane wave theory, then the FD coefficients were obtained using the Taylor expansion. Dispersion analysis and modeling results demonstrated that the proposed method has higher accuracy with lower computational cost for poroelastic wave simulation in heterogeneous reservoirs.

Advanced in Visualization of 3D Time-Dependent CFD Solutions

NASA Technical Reports Server (NTRS)

Lane, David A.; Lasinski, T. A. (Technical Monitor)

1995-01-01

Numerical simulations of complex 3D time-dependent (unsteady) flows are becoming increasingly feasible because of the progress in computing systems. Unfortunately, many existing flow visualization systems were developed for time-independent (steady) solutions and do not adequately depict solutions from unsteady flow simulations. Furthermore, most systems only handle one time step of the solutions individually and do not consider the time-dependent nature of the solutions. For example, instantaneous streamlines are computed by tracking the particles using one time step of the solution. However, for streaklines and timelines, particles need to be tracked through all time steps. Streaklines can reveal quite different information about the flow than those revealed by instantaneous streamlines. Comparisons of instantaneous streamlines with dynamic streaklines are shown. For a complex 3D flow simulation, it is common to generate a grid system with several millions of grid points and to have tens of thousands of time steps. The disk requirement for storing the flow data can easily be tens of gigabytes. Visualizing solutions of this magnitude is a challenging problem with today's computer hardware technology. Even interactive visualization of one time step of the flow data can be a problem for some existing flow visualization systems because of the size of the grid. Current approaches for visualizing complex 3D time-dependent CFD solutions are described. The flow visualization system developed at NASA Ames Research Center to compute time-dependent particle traces from unsteady CFD solutions is described. The system computes particle traces (streaklines) by integrating through the time steps. This system has been used by several NASA scientists to visualize their CFD time-dependent solutions. The flow visualization capabilities of this system are described, and visualization results are shown.

Multi-step-ahead Method for Wind Speed Prediction Correction Based on Numerical Weather Prediction and Historical Measurement Data

NASA Astrophysics Data System (ADS)

Wang, Han; Yan, Jie; Liu, Yongqian; Han, Shuang; Li, Li; Zhao, Jing

2017-11-01

Increasing the accuracy of wind speed prediction lays solid foundation to the reliability of wind power forecasting. Most traditional correction methods for wind speed prediction establish the mapping relationship between wind speed of the numerical weather prediction (NWP) and the historical measurement data (HMD) at the corresponding time slot, which is free of time-dependent impacts of wind speed time series. In this paper, a multi-step-ahead wind speed prediction correction method is proposed with consideration of the passing effects from wind speed at the previous time slot. To this end, the proposed method employs both NWP and HMD as model inputs and the training labels. First, the probabilistic analysis of the NWP deviation for different wind speed bins is calculated to illustrate the inadequacy of the traditional time-independent mapping strategy. Then, support vector machine (SVM) is utilized as example to implement the proposed mapping strategy and to establish the correction model for all the wind speed bins. One Chinese wind farm in northern part of China is taken as example to validate the proposed method. Three benchmark methods of wind speed prediction are used to compare the performance. The results show that the proposed model has the best performance under different time horizons.

Adaptive [theta]-methods for pricing American options

NASA Astrophysics Data System (ADS)

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

2008-12-01

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

An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems

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

Padgett, Jill M. A.; Ilie, Silvana, E-mail: silvana@ryerson.ca

2016-03-15

Stochastic modelling is critical for studying many biochemical processes in a cell, in particular when some reacting species have low population numbers. For many such cellular processes the spatial distribution of the molecular species plays a key role. The evolution of spatially heterogeneous biochemical systems with some species in low amounts is accurately described by the mesoscopic model of the Reaction-Diffusion Master Equation. The Inhomogeneous Stochastic Simulation Algorithm provides an exact strategy to numerically solve this model, but it is computationally very expensive on realistic applications. We propose a novel adaptive time-stepping scheme for the tau-leaping method for approximating themore » solution of the Reaction-Diffusion Master Equation. This technique combines effective strategies for variable time-stepping with path preservation to reduce the computational cost, while maintaining the desired accuracy. The numerical tests on various examples arising in applications show the improved efficiency achieved by the new adaptive method.« less

On the performance of voltage stepping for the simulation of adaptive, nonlinear integrate-and-fire neuronal networks.

PubMed

Kaabi, Mohamed Ghaith; Tonnelier, Arnaud; Martinez, Dominique

2011-05-01

In traditional event-driven strategies, spike timings are analytically given or calculated with arbitrary precision (up to machine precision). Exact computation is possible only for simplified neuron models, mainly the leaky integrate-and-fire model. In a recent paper, Zheng, Tonnelier, and Martinez (2009) introduced an approximate event-driven strategy, named voltage stepping, that allows the generic simulation of nonlinear spiking neurons. Promising results were achieved in the simulation of single quadratic integrate-and-fire neurons. Here, we assess the performance of voltage stepping in network simulations by considering more complex neurons (quadratic integrate-and-fire neurons with adaptation) coupled with multiple synapses. To handle the discrete nature of synaptic interactions, we recast voltage stepping in a general framework, the discrete event system specification. The efficiency of the method is assessed through simulations and comparisons with a modified time-stepping scheme of the Runge-Kutta type. We demonstrated numerically that the original order of voltage stepping is preserved when simulating connected spiking neurons, independent of the network activity and connectivity.

Efficient high-order structure-preserving methods for the generalized Rosenau-type equation with power law nonlinearity

NASA Astrophysics Data System (ADS)

Cai, Jiaxiang; Liang, Hua; Zhang, Chun

2018-06-01

Based on the multi-symplectic Hamiltonian formula of the generalized Rosenau-type equation, a multi-symplectic scheme and an energy-preserving scheme are proposed. To improve the accuracy of the solution, we apply the composition technique to the obtained schemes to develop high-order schemes which are also multi-symplectic and energy-preserving respectively. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results verify that all the proposed schemes have satisfactory performance in providing accurate solution and preserving the discrete mass and energy invariants. Numerical results also show that although each basic time step is divided into several composition steps, the computational efficiency of the composition schemes is much higher than that of the non-composite schemes.

A Semi-implicit Method for Resolution of Acoustic Waves in Low Mach Number Flows

NASA Astrophysics Data System (ADS)

Wall, Clifton; Pierce, Charles D.; Moin, Parviz

2002-09-01

A semi-implicit numerical method for time accurate simulation of compressible flow is presented. By extending the low Mach number pressure correction method, a Helmholtz equation for pressure is obtained in the case of compressible flow. The method avoids the acoustic CFL limitation, allowing a time step restricted only by the convective velocity, resulting in significant efficiency gains. Use of a discretization that is centered in both time and space results in zero artificial damping of acoustic waves. The method is attractive for problems in which Mach numbers are low, and the acoustic waves of most interest are those having low frequency, such as acoustic combustion instabilities. Both of these characteristics suggest the use of time steps larger than those allowable by an acoustic CFL limitation. In some cases it may be desirable to include a small amount of numerical dissipation to eliminate oscillations due to small-wavelength, high-frequency, acoustic modes, which are not of interest; therefore, a provision for doing this in a controlled manner is included in the method. Results of the method for several model problems are presented, and the performance of the method in a large eddy simulation is examined.

Meshless Method with Operator Splitting Technique for Transient Nonlinear Bioheat Transfer in Two-Dimensional Skin Tissues

PubMed Central

Zhang, Ze-Wei; Wang, Hui; Qin, Qing-Hua

2015-01-01

A meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue. PMID:25603180

Meshless method with operator splitting technique for transient nonlinear bioheat transfer in two-dimensional skin tissues.

PubMed

Zhang, Ze-Wei; Wang, Hui; Qin, Qing-Hua

2015-01-16

A meshless numerical scheme combining the operator splitting method (OSM), the radial basis function (RBF) interpolation, and the method of fundamental solutions (MFS) is developed for solving transient nonlinear bioheat problems in two-dimensional (2D) skin tissues. In the numerical scheme, the nonlinearity caused by linear and exponential relationships of temperature-dependent blood perfusion rate (TDBPR) is taken into consideration. In the analysis, the OSM is used first to separate the Laplacian operator and the nonlinear source term, and then the second-order time-stepping schemes are employed for approximating two splitting operators to convert the original governing equation into a linear nonhomogeneous Helmholtz-type governing equation (NHGE) at each time step. Subsequently, the RBF interpolation and the MFS involving the fundamental solution of the Laplace equation are respectively employed to obtain approximated particular and homogeneous solutions of the nonhomogeneous Helmholtz-type governing equation. Finally, the full fields consisting of the particular and homogeneous solutions are enforced to fit the NHGE at interpolation points and the boundary conditions at boundary collocations for determining unknowns at each time step. The proposed method is verified by comparison of other methods. Furthermore, the sensitivity of the coefficients in the cases of a linear and an exponential relationship of TDBPR is investigated to reveal their bioheat effect on the skin tissue.

A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows

NASA Astrophysics Data System (ADS)

Veerapaneni, Shravan K.; Gueyffier, Denis; Biros, George; Zorin, Denis

2009-10-01

We extend [Shravan K. Veerapaneni, Denis Gueyffier, Denis Zorin, George Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, Journal of Computational Physics 228(7) (2009) 2334-2353] to the case of three-dimensional axisymmetric vesicles of spherical or toroidal topology immersed in viscous flows. Although the main components of the algorithm are similar in spirit to the 2D case—spectral approximation in space, semi-implicit time-stepping scheme—the main differences are that the bending and viscous force require new analysis, the linearization for the semi-implicit schemes must be rederived, a fully implicit scheme must be used for the toroidal topology to eliminate a CFL-type restriction and a novel numerical scheme for the evaluation of the 3D Stokes single layer potential on an axisymmetric surface is necessary to speed up the calculations. By introducing these novel components, we obtain a time-scheme that experimentally is unconditionally stable, has low cost per time step, and is third-order accurate in time. We present numerical results to analyze the cost and convergence rates of the scheme. To verify the solver, we compare it to a constrained variational approach to compute equilibrium shapes that does not involve interactions with a viscous fluid. To illustrate the applicability of method, we consider a few vesicle-flow interaction problems: the sedimentation of a vesicle, interactions of one and three vesicles with a background Poiseuille flow.

An extended validation of the last generation of particle finite element method for free surface flows

NASA Astrophysics Data System (ADS)

Gimenez, Juan M.; González, Leo M.

2015-03-01

In this paper, a new generation of the particle method known as Particle Finite Element Method (PFEM), which combines convective particle movement and a fixed mesh resolution, is applied to free surface flows. This interesting variant, previously described in the literature as PFEM-2, is able to use larger time steps when compared to other similar numerical tools which implies shorter computational times while maintaining the accuracy of the computation. PFEM-2 has already been extended to free surface problems, being the main topic of this paper a deep validation of this methodology for a wider range of flows. To accomplish this task, different improved versions of discontinuous and continuous enriched basis functions for the pressure field have been developed to capture the free surface dynamics without artificial diffusion or undesired numerical effects when different density ratios are involved. A collection of problems has been carefully selected such that a wide variety of Froude numbers, density ratios and dominant dissipative cases are reported with the intention of presenting a general methodology, not restricted to a particular range of parameters, and capable of using large time-steps. The results of the different free-surface problems solved, which include: Rayleigh-Taylor instability, sloshing problems, viscous standing waves and the dam break problem, are compared to well validated numerical alternatives or experimental measurements obtaining accurate approximations for such complex flows.

Numerical analysis of transient laminar forced convection of nanofluids in circular ducts

NASA Astrophysics Data System (ADS)

Sert, İsmail Ozan; Sezer-Uzol, Nilay; Kakaç, Sadık

2013-10-01

In this study, forced convection heat transfer characteristics of nanofluids are investigated by numerical analysis of incompressible transient laminar flow in a circular duct under step change in wall temperature and wall heat flux. The thermal responses of the system are obtained by solving energy equation under both transient and steady-state conditions for hydro-dynamically fully-developed flow. In the analyses, temperature dependent thermo-physical properties are also considered. In the numerical analysis, Al2O3/water nanofluid is assumed as a homogenous single-phase fluid. For the effective thermal conductivity of nanofluids, Hamilton-Crosser model is used together with a model for Brownian motion in the analysis which takes the effects of temperature and the particle diameter into account. Temperature distributions across the tube for a step jump of wall temperature and also wall heat flux are obtained for various times during the transient calculations at a given location for a constant value of Peclet number and a particle diameter. Variations of thermal conductivity in turn, heat transfer enhancement is obtained at various times as a function of nanoparticle volume fractions, at a given nanoparticle diameter and Peclet number. The results are given under transient and steady-state conditions; steady-state conditions are obtained at larger times and enhancements are found by comparison to the base fluid heat transfer coefficient under the same conditions.

Finite Volume Method for Pricing European Call Option with Regime-switching Volatility

NASA Astrophysics Data System (ADS)

Lista Tauryawati, Mey; Imron, Chairul; Putri, Endah RM

2018-03-01

In this paper, we present a finite volume method for pricing European call option using Black-Scholes equation with regime-switching volatility. In the first step, we formulate the Black-Scholes equations with regime-switching volatility. we use a finite volume method based on fitted finite volume with spatial discretization and an implicit time stepping technique for the case. We show that the regime-switching scheme can revert to the non-switching Black Scholes equation, both in theoretical evidence and numerical simulations.

Branching Patterns and Stepped Leaders in an Electric-Circuit Model for Creeping Discharge

NASA Astrophysics Data System (ADS)

Hidetsugu Sakaguchi,; Sahim M. Kourkouss,

2010-06-01

We construct a two-dimensional electric circuit model for creeping discharge. Two types of discharge, surface corona and surface leader, are modeled by a two-step function of conductance. Branched patterns of surface leaders surrounded by the surface corona appear in numerical simulation. The fractal dimension of branched discharge patterns is calculated by changing voltage and capacitance. We find that surface leaders often grow stepwise in time, as is observed in lightning leaders of thunder.

Smoothing-based compressed state Kalman filter for joint state-parameter estimation: Applications in reservoir characterization and CO2 storage monitoring

NASA Astrophysics Data System (ADS)

Li, Y. J.; Kokkinaki, Amalia; Darve, Eric F.; Kitanidis, Peter K.

2017-08-01

The operation of most engineered hydrogeological systems relies on simulating physical processes using numerical models with uncertain parameters and initial conditions. Predictions by such uncertain models can be greatly improved by Kalman-filter techniques that sequentially assimilate monitoring data. Each assimilation constitutes a nonlinear optimization, which is solved by linearizing an objective function about the model prediction and applying a linear correction to this prediction. However, if model parameters and initial conditions are uncertain, the optimization problem becomes strongly nonlinear and a linear correction may yield unphysical results. In this paper, we investigate the utility of one-step ahead smoothing, a variant of the traditional filtering process, to eliminate nonphysical results and reduce estimation artifacts caused by nonlinearities. We present the smoothing-based compressed state Kalman filter (sCSKF), an algorithm that combines one step ahead smoothing, in which current observations are used to correct the state and parameters one step back in time, with a nonensemble covariance compression scheme, that reduces the computational cost by efficiently exploring the high-dimensional state and parameter space. Numerical experiments show that when model parameters are uncertain and the states exhibit hyperbolic behavior with sharp fronts, as in CO2 storage applications, one-step ahead smoothing reduces overshooting errors and, by design, gives physically consistent state and parameter estimates. We compared sCSKF with commonly used data assimilation methods and showed that for the same computational cost, combining one step ahead smoothing and nonensemble compression is advantageous for real-time characterization and monitoring of large-scale hydrogeological systems with sharp moving fronts.

Method of Lines Transpose an Implicit Vlasov Maxwell Solver for Plasmas

DTIC Science & Technology

2015-04-17

boundary crossings should be rare. Numerical results for the Bennett pinch are given in Figure 9. In order to resolve large gradients near the center of the...contributing to the large error at the center of the beam due to large gradients there) and with the finite beam cut-off radius and the outflow boundary...usable time step size can be limited by the numerical accuracy of the method when there are large gradients (high-frequency content) in the solution. We

Constrained Self-adaptive Solutions Procedures for Structure Subject to High Temperature Elastic-plastic Creep Effects

NASA Technical Reports Server (NTRS)

Padovan, J.; Tovichakchaikul, S.

1983-01-01

This paper will develop a new solution strategy which can handle elastic-plastic-creep problems in an inherently stable manner. This is achieved by introducing a new constrained time stepping algorithm which will enable the solution of creep initiated pre/postbuckling behavior where indefinite tangent stiffnesses are encountered. Due to the generality of the scheme, both monotone and cyclic loading histories can be handled. The presentation will give a thorough overview of current solution schemes and their short comings, the development of constrained time stepping algorithms as well as illustrate the results of several numerical experiments which benchmark the new procedure.

Dynamics of Numerics & Spurious Behaviors in CFD Computations. Revised

NASA Technical Reports Server (NTRS)

Yee, Helen C.; Sweby, Peter K.

1997-01-01

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD.

Numerical System Solver Developed for the National Cycle Program

NASA Technical Reports Server (NTRS)

Binder, Michael P.

1999-01-01

As part of the National Cycle Program (NCP), a powerful new numerical solver has been developed to support the simulation of aeropropulsion systems. This software uses a hierarchical object-oriented design. It can provide steady-state and time-dependent solutions to nonlinear and even discontinuous problems typically encountered when aircraft and spacecraft propulsion systems are simulated. It also can handle constrained solutions, in which one or more factors may limit the behavior of the engine system. Timedependent simulation capabilities include adaptive time-stepping and synchronization with digital control elements. The NCP solver is playing an important role in making the NCP a flexible, powerful, and reliable simulation package.

The dynamic behaviour of data-driven Δ-M and ΔΣ-M in sliding mode control

NASA Astrophysics Data System (ADS)

Almakhles, Dhafer; Swain, Akshya K.; Nasiri, Alireza

2017-11-01

In recent years, delta (Δ-M) and delta-sigma modulators (ΔΣ-M) are increasingly being used as efficient data converters due to numerous advantages they offer. This paper investigates various dynamical features of these modulators/systems (both in continuous and discrete time domain) and derives their stability conditions using the theory of sliding mode. The upper bound of the hitting time (step) has been estimated. The equivalent mode conditions, i.e. where the outputs of the modulators are equivalent to the inputs, are established. The results of the analysis are validated through simulations considering a numerical example.

On the efficient and reliable numerical solution of rate-and-state friction problems

NASA Astrophysics Data System (ADS)

Pipping, Elias; Kornhuber, Ralf; Rosenau, Matthias; Oncken, Onno

2016-03-01

We present a mathematically consistent numerical algorithm for the simulation of earthquake rupture with rate-and-state friction. Its main features are adaptive time stepping, a novel algebraic solution algorithm involving nonlinear multigrid and a fixed point iteration for the rate-and-state decoupling. The algorithm is applied to a laboratory scale subduction zone which allows us to compare our simulations with experimental results. Using physical parameters from the experiment, we find a good fit of recurrence time of slip events as well as their rupture width and peak slip. Computations in 3-D confirm efficiency and robustness of our algorithm.

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

NASA Technical Reports Server (NTRS)

Chang, S. C.

1986-01-01

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. 2; Global Asymptotic Behavior of Time Discretizations; 2. Global Asymptotic Behavior of time Discretizations

NASA Technical Reports Server (NTRS)

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

1995-01-01

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

A cellular automata approach for modeling surface water runoff

NASA Astrophysics Data System (ADS)

Jozefik, Zoltan; Nanu Frechen, Tobias; Hinz, Christoph; Schmidt, Heiko

2015-04-01

This abstract reports the development and application of a two-dimensional cellular automata based model, which couples the dynamics of overland flow, infiltration processes and surface evolution through sediment transport. The natural hill slopes are represented by their topographic elevation and spatially varying soil properties infiltration rates and surface roughness coefficients. This model allows modeling of Hortonian overland flow and infiltration during complex rainfall events. An advantage of the cellular automata approach over the kinematic wave equations is that wet/dry interfaces that often appear with rainfall overland flows can be accurately captured and are not a source of numerical instabilities. An adaptive explicit time stepping scheme allows for rainfall events to be adequately resolved in time, while large time steps are taken during dry periods to provide for simulation run time efficiency. The time step is constrained by the CFL condition and mass conservation considerations. The spatial discretization is shown to be first-order accurate. For validation purposes, hydrographs for non-infiltrating and infiltrating plates are compared to the kinematic wave analytic solutions and data taken from literature [1,2]. Results show that our cellular automata model quantitatively accurately reproduces hydrograph patterns. However, recent works have showed that even through the hydrograph is satisfyingly reproduced, the flow field within the plot might be inaccurate [3]. For a more stringent validation, we compare steady state velocity, water flux, and water depth fields to rainfall simulation experiments conducted in Thies, Senegal [3]. Comparisons show that our model is able to accurately capture these flow properties. Currently, a sediment transport and deposition module is being implemented and tested. [1] M. Rousseau, O. Cerdan, O. Delestre, F. Dupros, F. James, S. Cordier. Overland flow modeling with the Shallow Water Equation using a well balanced numerical scheme: Adding efficiency or sum more complexity?. 2012. [2] Fritz R. Fiedler, J. A. Ramirez. A numerical method for simulating discontinuous shallow flow over an infiltrating surface. In. J. Numer. Mech. Fluids 200: 32: 219-240. [3] C. Mügler, O. Planchon, J. Patin, S. Weill, N. Silvera, P. Richard, E. Mouche. Comparison of Roughness models to simulate overland flow and tracer transport experiments under simulated rainfall at plot scale. Journal of Hydrology. 402 (2011) 25-40.

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

NASA Technical Reports Server (NTRS)

Sankar, Lakshmi N.; Hixon, Duane

1992-01-01

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step.

Efficient numerical method for solving Cauchy problem for the Gamma equation

NASA Astrophysics Data System (ADS)

Koleva, Miglena N.

2011-12-01

In this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) Γ = VSS of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard-Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non-differentiable functions. Results of numerical simulations are given.

A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equation

NASA Astrophysics Data System (ADS)

Zabihi, F.; Saffarian, M.

2016-07-01

The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Numerical Grid Generation. Proceedings of a Symposium on the Numerical Generation of Curvilinear Coordinate Systems and their Use in the Numerical Solution of Partial Differential Equations, held April 1982, Nashville, Tennessee

DTIC Science & Technology

1982-04-01

the transformation 2 1 V I " |€ , (22) 121 I which transforms a unit circle in the Z plane to a unit upper half ...circle (and to the real axis from -1. to 1.) in the 4 plane as illustrated in Fig. 6. I Zpbns rpIUII A A , FIGURE 6. , CIRCLE TO HALF CIRCLE MAPPING The ...an airfoil, or inlet, etc. ... ) to a rectangle (or equivalently to a circle or half plane ). The cumputing time required for this step is

Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method

NASA Technical Reports Server (NTRS)

Whitaker, David L.

1993-01-01

A method has been developed to solve the Euler equations on a three-dimensional unstructured grid composed of tetrahedra. The method uses an upwind flow solver with a linearized, backward-Euler time integration scheme. Each time step results in a sparse linear system of equations which is solved by an iterative, sparse matrix solver. Local-time stepping, switched evolution relaxation (SER), preconditioning and reuse of the Jacobian are employed to accelerate the convergence rate. Implicit boundary conditions were found to be extremely important for fast convergence. Numerical experiments have shown that convergence rates comparable to that of a multigrid, central-difference scheme are achievable on the same mesh. Results are presented for several grids about an ONERA M6 wing.

Adaptive Numerical Algorithms in Space Weather Modeling

NASA Technical Reports Server (NTRS)

Toth, Gabor; vanderHolst, Bart; Sokolov, Igor V.; DeZeeuw, Darren; Gombosi, Tamas I.; Fang, Fang; Manchester, Ward B.; Meng, Xing; Nakib, Dalal; Powell, Kenneth G.;

Regionally Implicit Discontinuous Galerkin Methods for Solving the Relativistic Vlasov-Maxwell System Submitted to Iowa State University

NASA Astrophysics Data System (ADS)

Guthrey, Pierson Tyler

The relativistic Vlasov-Maxwell system (RVM) models the behavior of collisionless plasma, where electrons and ions interact via the electromagnetic fields they generate. In the RVM system, electrons could accelerate to significant fractions of the speed of light. An idea that is actively being pursued by several research groups around the globe is to accelerate electrons to relativistic speeds by hitting a plasma with an intense laser beam. As the laser beam passes through the plasma it creates plasma wakes, much like a ship passing through water, which can trap electrons and push them to relativistic speeds. Such setups are known as laser wakefield accelerators, and have the potential to yield particle accelerators that are significantly smaller than those currently in use. Ultimately, the goal of such research is to harness the resulting electron beams to generate electromagnetic waves that can be used in medical imaging applications. High-order accurate numerical discretizations of kinetic Vlasov plasma models are very effective at yielding low-noise plasma simulations, but are computationally expensive to solve because of the high dimensionality. In addition to the general difficulties inherent to numerically simulating Vlasov models, the relativistic Vlasov-Maxwell system has unique challenges not present in the non-relativistic case. One such issue is that operator splitting of the phase gradient leads to potential instabilities, thus we require an alternative to operator splitting of the phase. The goal of the current work is to develop a new class of high-order accurate numerical methods for solving kinetic Vlasov models of plasma. The main discretization in configuration space is handled via a high-order finite element method called the discontinuous Galerkin method (DG). One difficulty is that standard explicit time-stepping methods for DG suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. The maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work, we overcome this difficulty by introducing a novel time-stepping strategy: the regionally-implicit discontinuous Galerkin (RIDG) method. The RIDG is method is based on an extension of the Lax-Wendroff DG (LxW-DG) method, which previously had been shown to be equivalent (for linear constant coefficient problems) to a predictor-corrector approach, where the prediction is computed by a space-time DG method (STDG). The corrector is an explicit method that uses the space-time reconstructed solution from the predictor step. In this work, we modify the predictor to include not just local information, but also neighboring information. With this modification, we show that the stability is greatly enhanced; we show that we can remove the polynomial degree dependence of the maximum time-step and show vastly improved time-steps in multiple spatial dimensions. Upon the development of the general RIDG method, we apply it to the non-relativistic 1D1V Vlasov-Poisson equations and the relativistic 1D2V Vlasov-Maxwell equations. For each we validate the high-order method on several test cases. In the final test case, we demonstrate the ability of the method to simulate the acceleration of electrons to relativistic speeds in a simplified test case.

CFD analysis of a scramjet combustor with cavity based flame holders

NASA Astrophysics Data System (ADS)

Kummitha, Obula Reddy; Pandey, Krishna Murari; Gupta, Rajat

2018-03-01

Numerical analysis of a scramjet combustor with different cavity flame holders has been carried out using ANSYS 16 - FLUENT tool. In this research article the internal fluid flow behaviour of the scramjet combustor with different cavity based flame holders have been discussed in detail. Two dimensional Reynolds-Averaged Navier-Stokes governing(RANS) equations and shear stress turbulence (SST) k - ω model along with finite rate/eddy dissipation chemistry turbulence have been considered for modelling chemical reacting flows. Due to the advantage of less computational time, global one step reaction mechanism has been used for combustion modelling of hydrogen and air. The performance of the scramjet combustor with two different cavities namely spherical and step cavity has been compared with the standard DLR scramjet. From the comparison of numerical results, it is found that the development of recirculation regions and additional shock waves from the edge of cavity flame holder is increased. And also it is observed that with the cavity flame holder the residence time of air in the scramjet combustor is also increased and achieved stabilized combustion. From this research analysis, it has been found that the mixing and combustion efficiency of scramjet combustor with step cavity design is optimum as compared to other models.

Magnetization-induced dynamics of a Josephson junction coupled to a nanomagnet

NASA Astrophysics Data System (ADS)

Ghosh, Roopayan; Maiti, Moitri; Shukrinov, Yury M.; Sengupta, K.

2017-11-01

We study the superconducting current of a Josephson junction (JJ) coupled to an external nanomagnet driven by a time-dependent magnetic field both without and in the presence of an external ac drive. We provide an analytic, albeit perturbative, solution for the Landau-Lifshitz (LL) equations governing the coupled JJ-nanomagnet system in the presence of a magnetic field with arbitrary time dependence oriented along the easy axis of the nanomagnet's magnetization and in the limit of weak dimensionless coupling ɛ0 between the JJ and the nanomagnet. We show the existence of Shapiro-type steps in the I -V characteristics of the JJ subjected to a voltage bias for a constant or periodically varying magnetic field and explore the effect of rotation of the magnetic field and the presence of an external ac drive on these steps. We support our analytic results with exact numerical solution of the LL equations. We also extend our results to dissipative nanomagnets by providing a perturbative solution to the Landau-Lifshitz-Gilbert (LLG) equations for weak dissipation. We study the fate of magnetization-induced Shapiro steps in the presence of dissipation both from our analytical results and via numerical solution of the coupled LLG equations. We discuss experiments which can test our theory.

A higher order numerical method for time fractional partial differential equations with nonsmooth data

NASA Astrophysics Data System (ADS)

Xing, Yanyuan; Yan, Yubin

2018-03-01

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 < α < 1 for any fixed tn > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

Individualizing drug dosage with longitudinal data.

PubMed

Zhu, Xiaolu; Qu, Annie

2016-10-30

We propose a two-step procedure to personalize drug dosage over time under the framework of a log-linear mixed-effect model. We model patients' heterogeneity using subject-specific random effects, which are treated as the realizations of an unspecified stochastic process. We extend the conditional quadratic inference function to estimate both fixed-effect coefficients and individual random effects on a longitudinal training data sample in the first step and propose an adaptive procedure to estimate new patients' random effects and provide dosage recommendations for new patients in the second step. An advantage of our approach is that we do not impose any distribution assumption on estimating random effects. Moreover, the new approach can accommodate more general time-varying covariates corresponding to random effects. We show in theory and numerical studies that the proposed method is more efficient compared with existing approaches, especially when covariates are time varying. In addition, a real data example of a clozapine study confirms that our two-step procedure leads to more accurate drug dosage recommendations. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Artificial acoustic stiffness reduction in fully compressible, direct numerical simulation of combustion

NASA Astrophysics Data System (ADS)

Wang, Yi; Trouvé, Arnaud

2004-09-01

A pseudo-compressibility method is proposed to modify the acoustic time step restriction found in fully compressible, explicit flow solvers. The method manipulates terms in the governing equations of order Ma2, where Ma is a characteristic flow Mach number. A decrease in the speed of acoustic waves is obtained by adding an extra term in the balance equation for total energy. This term is proportional to flow dilatation and uses a decomposition of the dilatational field into an acoustic component and a component due to heat transfer. The present method is a variation of the pressure gradient scaling (PGS) method proposed in Ramshaw et al (1985 Pressure gradient scaling method for fluid flow with nearly uniform pressure J. Comput. Phys. 58 361-76). It achieves gains in computational efficiencies similar to PGS: at the cost of a slightly more involved right-hand-side computation, the numerical time step increases by a full order of magnitude. It also features the added benefit of preserving the hydrodynamic pressure field. The original and modified PGS methods are implemented into a parallel direct numerical simulation solver developed for applications to turbulent reacting flows with detailed chemical kinetics. The performance of the pseudo-compressibility methods is illustrated in a series of test problems ranging from isothermal sound propagation to laminar premixed flame problems.

The development and validation of a numerical integration method for non-linear viscoelastic modeling

PubMed Central

Ramo, Nicole L.; Puttlitz, Christian M.

2018-01-01

Compelling evidence that many biological soft tissues display both strain- and time-dependent behavior has led to the development of fully non-linear viscoelastic modeling techniques to represent the tissue’s mechanical response under dynamic conditions. Since the current stress state of a viscoelastic material is dependent on all previous loading events, numerical analyses are complicated by the requirement of computing and storing the stress at each step throughout the load history. This requirement quickly becomes computationally expensive, and in some cases intractable, for finite element models. Therefore, we have developed a strain-dependent numerical integration approach for capturing non-linear viscoelasticity that enables calculation of the current stress from a strain-dependent history state variable stored from the preceding time step only, which improves both fitting efficiency and computational tractability. This methodology was validated based on its ability to recover non-linear viscoelastic coefficients from simulated stress-relaxation (six strain levels) and dynamic cyclic (three frequencies) experimental stress-strain data. The model successfully fit each data set with average errors in recovered coefficients of 0.3% for stress-relaxation fits and 0.1% for cyclic. The results support the use of the presented methodology to develop linear or non-linear viscoelastic models from stress-relaxation or cyclic experimental data of biological soft tissues. PMID:29293558

A MULTIPLE GRID ALGORITHM FOR ONE-DIMENSIONAL TRANSIENT OPEN CHANNEL FLOWS. (R825200)

EPA Science Inventory

Numerical modeling of open channel flows with shocks using explicit finite difference schemes is constrained by the choice of time step, which is limited by the CFL stability criteria. To overcome this limitation, in this work we introduce the application of a multiple grid al...

Continuous-Time Bilinear System Identification

NASA Technical Reports Server (NTRS)

Juang, Jer-Nan

2003-01-01

The objective of this paper is to describe a new method for identification of a continuous-time multi-input and multi-output bilinear system. The approach is to make judicious use of the linear-model properties of the bilinear system when subjected to a constant input. Two steps are required in the identification process. The first step is to use a set of pulse responses resulting from a constant input of one sample period to identify the state matrix, the output matrix, and the direct transmission matrix. The second step is to use another set of pulse responses with the same constant input over multiple sample periods to identify the input matrix and the coefficient matrices associated with the coupling terms between the state and the inputs. Numerical examples are given to illustrate the concept and the computational algorithm for the identification method.

General methods for analysis of sequential "n-step" kinetic mechanisms: application to single turnover kinetics of helicase-catalyzed DNA unwinding.

PubMed

Lucius, Aaron L; Maluf, Nasib K; Fischer, Christopher J; Lohman, Timothy M

2003-10-01

Helicase-catalyzed DNA unwinding is often studied using "all or none" assays that detect only the final product of fully unwound DNA. Even using these assays, quantitative analysis of DNA unwinding time courses for DNA duplexes of different lengths, L, using "n-step" sequential mechanisms, can reveal information about the number of intermediates in the unwinding reaction and the "kinetic step size", m, defined as the average number of basepairs unwound between two successive rate limiting steps in the unwinding cycle. Simultaneous nonlinear least-squares analysis using "n-step" sequential mechanisms has previously been limited by an inability to float the number of "unwinding steps", n, and m, in the fitting algorithm. Here we discuss the behavior of single turnover DNA unwinding time courses and describe novel methods for nonlinear least-squares analysis that overcome these problems. Analytic expressions for the time courses, f(ss)(t), when obtainable, can be written using gamma and incomplete gamma functions. When analytic expressions are not obtainable, the numerical solution of the inverse Laplace transform can be used to obtain f(ss)(t). Both methods allow n and m to be continuous fitting parameters. These approaches are generally applicable to enzymes that translocate along a lattice or require repetition of a series of steps before product formation.

A split-step method to include electron–electron collisions via Monte Carlo in multiple rate equation simulations

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

Huthmacher, Klaus; Molberg, Andreas K.; Rethfeld, Bärbel

2016-10-01

A split-step numerical method for calculating ultrafast free-electron dynamics in dielectrics is introduced. The two split steps, independently programmed in C++11 and FORTRAN 2003, are interfaced via the presented open source wrapper. The first step solves a deterministic extended multi-rate equation for the ionization, electron–phonon collisions, and single photon absorption by free-carriers. The second step is stochastic and models electron–electron collisions using Monte-Carlo techniques. This combination of deterministic and stochastic approaches is a unique and efficient method of calculating the nonlinear dynamics of 3D materials exposed to high intensity ultrashort pulses. Results from simulations solving the proposed model demonstrate howmore » electron–electron scattering relaxes the non-equilibrium electron distribution on the femtosecond time scale.« less

A numerically efficient damping model for acoustic resonances in microfluidic cavities

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

Hahn, P., E-mail: hahnp@ethz.ch; Dual, J.

Bulk acoustic wave devices are typically operated in a resonant state to achieve enhanced acoustic amplitudes and high acoustofluidic forces for the manipulation of microparticles. Among other loss mechanisms related to the structural parts of acoustofluidic devices, damping in the fluidic cavity is a crucial factor that limits the attainable acoustic amplitudes. In the analytical part of this study, we quantify all relevant loss mechanisms related to the fluid inside acoustofluidic micro-devices. Subsequently, a numerical analysis of the time-harmonic visco-acoustic and thermo-visco-acoustic equations is carried out to verify the analytical results for 2D and 3D examples. The damping results aremore » fitted into the framework of classical linear acoustics to set up a numerically efficient device model. For this purpose, all damping effects are combined into an acoustofluidic loss factor. Since some components of the acoustofluidic loss factor depend on the acoustic mode shape in the fluid cavity, we propose a two-step simulation procedure. In the first step, the loss factors are deduced from the simulated mode shape. Subsequently, a second simulation is invoked, taking all losses into account. Owing to its computational efficiency, the presented numerical device model is of great relevance for the simulation of acoustofluidic particle manipulation by means of acoustic radiation forces or acoustic streaming. For the first time, accurate 3D simulations of realistic micro-devices for the quantitative prediction of pressure amplitudes and the related acoustofluidic forces become feasible.« less

An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - III. Viscoelastic attenuation

NASA Astrophysics Data System (ADS)

Käser, Martin; Dumbser, Michael; de la Puente, Josep; Igel, Heiner

2007-01-01

We present a new numerical method to solve the heterogeneous anelastic, seismic wave equations with arbitrary high order accuracy in space and time on 3-D unstructured tetrahedral meshes. Using the velocity-stress formulation provides a linear hyperbolic system of equations with source terms that is completed by additional equations for the anelastic functions including the strain history of the material. These additional equations result from the rheological model of the generalized Maxwell body and permit the incorporation of realistic attenuation properties of viscoelastic material accounting for the behaviour of elastic solids and viscous fluids. The proposed method combines the Discontinuous Galerkin (DG) finite element (FE) method with the ADER approach using Arbitrary high order DERivatives for flux calculations. The DG approach, in contrast to classical FE methods, uses a piecewise polynomial approximation of the numerical solution which allows for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann problems can be applied as in the finite volume framework. The main idea of the ADER time integration approach is a Taylor expansion in time in which all time derivatives are replaced by space derivatives using the so-called Cauchy-Kovalewski procedure which makes extensive use of the governing PDE. Due to the ADER time integration technique the same approximation order in space and time is achieved automatically and the method is a one-step scheme advancing the solution for one time step without intermediate stages. To this end, we introduce a new unrolled recursive algorithm for efficiently computing the Cauchy-Kovalewski procedure by making use of the sparsity of the system matrices. The numerical convergence analysis demonstrates that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes while computational cost and storage space for a desired accuracy can be reduced when applying higher degree approximation polynomials. In addition, we investigate the increase in computing time, when the number of relaxation mechanisms due to the generalized Maxwell body are increased. An application to a well-acknowledged test case and comparisons with analytic and reference solutions, obtained by different well-established numerical methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER-DG approach for tetrahedral meshes including viscoelastic material provides a novel, flexible and efficient numerical technique to approach 3-D wave propagation problems including realistic attenuation and complex geometry.

Numerical prediction of fire resistance of RC beams

NASA Astrophysics Data System (ADS)

Serega, Szymon; Wosatko, Adam

2018-01-01

Fire resistance of different structural members is an important issue of their strength and durability. A simple but effective tool to investigate multi-span reinforced concrete beams exposed to fire is discussed in the paper. Assumptions and simplifications of the theory as well as numerical aspects are briefly reviewed. Two steps of nonlinear finite element analysis and two levels of observation are distinguished. The first step is the solution of transient heat transfer problem in representative two-dimensional reinforced concrete cross-section of a beam. The second part is a nonlinear mechanical analysis of the whole beam. All spans are uniformly loaded, but an additional time-dependent thermal load due to fire acts on selected ones. Global changes of curvature and bending moment functions induce deterioration of the stiffness. Benchmarks are shown to confirm the correctness of the model.

Numerical Modeling of Turbulent Combustion

NASA Technical Reports Server (NTRS)

Ghoneim, A. F.; Chorin, A. J.; Oppenheim, A. K.

1983-01-01

The work in numerical modeling is focused on the use of the random vortex method to treat turbulent flow fields associated with combustion while flame fronts are considered as interfaces between reactants and products, propagating with the flow and at the same time advancing in the direction normal to themselves at a prescribed burning speed. The latter is associated with the generation of specific volume (the flame front acting, in effect, as the locus of volumetric sources) to account for the expansion of the flow field due to the exothermicity of the combustion process. The model was applied to the flow in a channel equipped with a rearward facing step. The results obtained revealed the mechanism of the formation of large scale turbulent structure in the wake of the step, while it showed the flame to stabilize on the outer edges of these eddies.

Towards numerical prediction of cavitation erosion.

PubMed

Fivel, Marc; Franc, Jean-Pierre; Chandra Roy, Samir

2015-10-06

This paper is intended to provide a potential basis for a numerical prediction of cavitation erosion damage. The proposed method can be divided into two steps. The first step consists in determining the loading conditions due to cavitation bubble collapses. It is shown that individual pits observed on highly polished metallic samples exposed to cavitation for a relatively small time can be considered as the signature of bubble collapse. By combining pitting tests with an inverse finite-element modelling (FEM) of the material response to a representative impact load, loading conditions can be derived for each individual bubble collapse in terms of stress amplitude (in gigapascals) and radial extent (in micrometres). This step requires characterizing as accurately as possible the properties of the material exposed to cavitation. This characterization should include the effect of strain rate, which is known to be high in cavitation erosion (typically of the order of several thousands s(-1)). Nanoindentation techniques as well as compressive tests at high strain rate using, for example, a split Hopkinson pressure bar test system may be used. The second step consists in developing an FEM approach to simulate the material response to the repetitive impact loads determined in step 1. This includes a detailed analysis of the hardening process (isotropic versus kinematic) in order to properly account for fatigue as well as the development of a suitable model of material damage and failure to account for mass loss. Although the whole method is not yet fully operational, promising results are presented that show that such a numerical method might be, in the long term, an alternative to correlative techniques used so far for cavitation erosion prediction.

Towards numerical prediction of cavitation erosion

PubMed Central

Fivel, Marc; Franc, Jean-Pierre; Chandra Roy, Samir

2015-01-01

This paper is intended to provide a potential basis for a numerical prediction of cavitation erosion damage. The proposed method can be divided into two steps. The first step consists in determining the loading conditions due to cavitation bubble collapses. It is shown that individual pits observed on highly polished metallic samples exposed to cavitation for a relatively small time can be considered as the signature of bubble collapse. By combining pitting tests with an inverse finite-element modelling (FEM) of the material response to a representative impact load, loading conditions can be derived for each individual bubble collapse in terms of stress amplitude (in gigapascals) and radial extent (in micrometres). This step requires characterizing as accurately as possible the properties of the material exposed to cavitation. This characterization should include the effect of strain rate, which is known to be high in cavitation erosion (typically of the order of several thousands s−1). Nanoindentation techniques as well as compressive tests at high strain rate using, for example, a split Hopkinson pressure bar test system may be used. The second step consists in developing an FEM approach to simulate the material response to the repetitive impact loads determined in step 1. This includes a detailed analysis of the hardening process (isotropic versus kinematic) in order to properly account for fatigue as well as the development of a suitable model of material damage and failure to account for mass loss. Although the whole method is not yet fully operational, promising results are presented that show that such a numerical method might be, in the long term, an alternative to correlative techniques used so far for cavitation erosion prediction. PMID:26442139

Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model

NASA Astrophysics Data System (ADS)

Cheng, Qing; Yang, Xiaofeng; Shen, Jie

2017-07-01

In this paper, we consider numerical approximations of a hydro-dynamically coupled phase field diblock copolymer model, in which the free energy contains a kinetic potential, a gradient entropy, a Ginzburg-Landau double well potential, and a long range nonlocal type potential. We develop a set of second order time marching schemes for this system using the "Invariant Energy Quadratization" approach for the double well potential, the projection method for the Navier-Stokes equation, and a subtle implicit-explicit treatment for the stress and convective term. The resulting schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable. Various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.

A computational method for sharp interface advection.

PubMed

Roenby, Johan; Bredmose, Henrik; Jasak, Hrvoje

2016-11-01

We devise a numerical method for passive advection of a surface, such as the interface between two incompressible fluids, across a computational mesh. The method is called isoAdvector, and is developed for general meshes consisting of arbitrary polyhedral cells. The algorithm is based on the volume of fluid (VOF) idea of calculating the volume of one of the fluids transported across the mesh faces during a time step. The novelty of the isoAdvector concept consists of two parts. First, we exploit an isosurface concept for modelling the interface inside cells in a geometric surface reconstruction step. Second, from the reconstructed surface, we model the motion of the face-interface intersection line for a general polygonal face to obtain the time evolution within a time step of the submerged face area. Integrating this submerged area over the time step leads to an accurate estimate for the total volume of fluid transported across the face. The method was tested on simple two-dimensional and three-dimensional interface advection problems on both structured and unstructured meshes. The results are very satisfactory in terms of volume conservation, boundedness, surface sharpness and efficiency. The isoAdvector method was implemented as an OpenFOAM ® extension and is published as open source.

Simplified Two-Time Step Method for Calculating Combustion and Emission Rates of Jet-A and Methane Fuel With and Without Water Injection

NASA Technical Reports Server (NTRS)

Molnar, Melissa; Marek, C. John

2005-01-01

A simplified kinetic scheme for Jet-A, and methane fuels with water injection was developed to be used in numerical combustion codes, such as the National Combustor Code (NCC) or even simple FORTRAN codes. The two time step method is either an initial time averaged value (step one) or an instantaneous value (step two). The switch is based on the water concentration in moles/cc of 1x10(exp -20). The results presented here results in a correlation that gives the chemical kinetic time as two separate functions. This two time step method is used as opposed to a one step time averaged method previously developed to determine the chemical kinetic time with increased accuracy. The first time averaged step is used at the initial times for smaller water concentrations. This gives the average chemical kinetic time as a function of initial overall fuel air ratio, initial water to fuel mass ratio, temperature, and pressure. The second instantaneous step, to be used with higher water concentrations, gives the chemical kinetic time as a function of instantaneous fuel and water mole concentration, pressure and temperature (T4). The simple correlations would then be compared to the turbulent mixing times to determine the limiting rates of the reaction. The NASA Glenn GLSENS kinetics code calculates the reaction rates and rate constants for each species in a kinetic scheme for finite kinetic rates. These reaction rates are used to calculate the necessary chemical kinetic times. Chemical kinetic time equations for fuel, carbon monoxide and NOx are obtained for Jet-A fuel and methane with and without water injection to water mass loadings of 2/1 water to fuel. A similar correlation was also developed using data from NASA's Chemical Equilibrium Applications (CEA) code to determine the equilibrium concentrations of carbon monoxide and nitrogen oxide as functions of overall equivalence ratio, water to fuel mass ratio, pressure and temperature (T3). The temperature of the gas entering the turbine (T4) was also correlated as a function of the initial combustor temperature (T3), equivalence ratio, water to fuel mass ratio, and pressure.

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

NASA Technical Reports Server (NTRS)

Yee, H. C.; Mansour, Nagi N. (Technical Monitor)

2002-01-01

This talk describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

NASA Technical Reports Server (NTRS)

Yee, H. C.

2005-01-01

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations.

Building Blocks for Reliable Complex Nonlinear Numerical Simulations. Chapter 2

NASA Technical Reports Server (NTRS)

Yee, H. C.; Mansour, Nagi N. (Technical Monitor)

2001-01-01

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

An in-depth stability analysis of nonuniform FDTD combined with novel local implicitization techniques

NASA Astrophysics Data System (ADS)

Van Londersele, Arne; De Zutter, Daniël; Vande Ginste, Dries

2017-08-01

This work focuses on efficient full-wave solutions of multiscale electromagnetic problems in the time domain. Three local implicitization techniques are proposed and carefully analyzed in order to relax the traditional time step limit of the Finite-Difference Time-Domain (FDTD) method on a nonuniform, staggered, tensor product grid: Newmark, Crank-Nicolson (CN) and Alternating-Direction-Implicit (ADI) implicitization. All of them are applied in preferable directions, alike Hybrid Implicit-Explicit (HIE) methods, as to limit the rank of the sparse linear systems. Both exponential and linear stability are rigorously investigated for arbitrary grid spacings and arbitrary inhomogeneous, possibly lossy, isotropic media. Numerical examples confirm the conservation of energy inside a cavity for a million iterations if the time step is chosen below the proposed, relaxed limit. Apart from the theoretical contributions, new accomplishments such as the development of the leapfrog Alternating-Direction-Hybrid-Implicit-Explicit (ADHIE) FDTD method and a less stringent Courant-like time step limit for the conventional, fully explicit FDTD method on a nonuniform grid, have immediate practical applications.

Fast and high-order numerical algorithms for the solution of multidimensional nonlinear fractional Ginzburg-Landau equation

NASA Astrophysics Data System (ADS)

Mohebbi, Akbar

2018-02-01

In this paper we propose two fast and accurate numerical methods for the solution of multidimensional space fractional Ginzburg-Landau equation (FGLE). In the presented methods, to avoid solving a nonlinear system of algebraic equations and to increase the accuracy and efficiency of method, we split the complex problem into simpler sub-problems using the split-step idea. For a homogeneous FGLE, we propose a method which has fourth-order of accuracy in time component and spectral accuracy in space variable and for nonhomogeneous one, we introduce another scheme based on the Crank-Nicolson approach which has second-order of accuracy in time variable. Due to using the Fourier spectral method for fractional Laplacian operator, the resulting schemes are fully diagonal and easy to code. Numerical results are reported in terms of accuracy, computational order and CPU time to demonstrate the accuracy and efficiency of the proposed methods and to compare the results with the analytical solutions. The results show that the present methods are accurate and require low CPU time. It is illustrated that the numerical results are in good agreement with the theoretical ones.

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations. Part 2; Global Asymptotic Behavior of Time Discretizations

NASA Technical Reports Server (NTRS)

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

1995-01-01

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Evaluation of integration methods for hybrid simulation of complex structural systems through collapse

NASA Astrophysics Data System (ADS)

Del Carpio R., Maikol; Hashemi, M. Javad; Mosqueda, Gilberto

2017-10-01

This study examines the performance of integration methods for hybrid simulation of large and complex structural systems in the context of structural collapse due to seismic excitations. The target application is not necessarily for real-time testing, but rather for models that involve large-scale physical sub-structures and highly nonlinear numerical models. Four case studies are presented and discussed. In the first case study, the accuracy of integration schemes including two widely used methods, namely, modified version of the implicit Newmark with fixed-number of iteration (iterative) and the operator-splitting (non-iterative) is examined through pure numerical simulations. The second case study presents the results of 10 hybrid simulations repeated with the two aforementioned integration methods considering various time steps and fixed-number of iterations for the iterative integration method. The physical sub-structure in these tests consists of a single-degree-of-freedom (SDOF) cantilever column with replaceable steel coupons that provides repeatable highlynonlinear behavior including fracture-type strength and stiffness degradations. In case study three, the implicit Newmark with fixed-number of iterations is applied for hybrid simulations of a 1:2 scale steel moment frame that includes a relatively complex nonlinear numerical substructure. Lastly, a more complex numerical substructure is considered by constructing a nonlinear computational model of a moment frame coupled to a hybrid model of a 1:2 scale steel gravity frame. The last two case studies are conducted on the same porotype structure and the selection of time steps and fixed number of iterations are closely examined in pre-test simulations. The generated unbalance forces is used as an index to track the equilibrium error and predict the accuracy and stability of the simulations.

Reconstruction of local perturbations in periodic surfaces

NASA Astrophysics Data System (ADS)

Lechleiter, Armin; Zhang, Ruming

2018-03-01

This paper concerns the inverse scattering problem to reconstruct a local perturbation in a periodic structure. Unlike the periodic problems, the periodicity for the scattered field no longer holds, thus classical methods, which reduce quasi-periodic fields in one periodic cell, are no longer available. Based on the Floquet-Bloch transform, a numerical method has been developed to solve the direct problem, that leads to a possibility to design an algorithm for the inverse problem. The numerical method introduced in this paper contains two steps. The first step is initialization, that is to locate the support of the perturbation by a simple method. This step reduces the inverse problem in an infinite domain into one periodic cell. The second step is to apply the Newton-CG method to solve the associated optimization problem. The perturbation is then approximated by a finite spline basis. Numerical examples are given at the end of this paper, showing the efficiency of the numerical method.

Four decades of implicit Monte Carlo

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

Wollaber, Allan B.

In 1971, Fleck and Cummings derived a system of equations to enable robust Monte Carlo simulations of time-dependent, thermal radiative transfer problems. Denoted the “Implicit Monte Carlo” (IMC) equations, their solution remains the de facto standard of high-fidelity radiative transfer simulations. Over the course of 44 years, their numerical properties have become better understood, and accuracy enhancements, novel acceleration methods, and variance reduction techniques have been suggested. In this review, we rederive the IMC equations—explicitly highlighting assumptions as they are made—and outfit the equations with a Monte Carlo interpretation. We put the IMC equations in context with other approximate formsmore » of the radiative transfer equations and present a new demonstration of their equivalence to another well-used linearization solved with deterministic transport methods for frequency-independent problems. We discuss physical and numerical limitations of the IMC equations for asymptotically small time steps, stability characteristics and the potential of maximum principle violations for large time steps, and solution behaviors in an asymptotically thick diffusive limit. We provide a new stability analysis for opacities with general monomial dependence on temperature. Here, we consider spatial accuracy limitations of the IMC equations and discussion acceleration and variance reduction techniques.« less

A three-dimensional numerical simulation of cell behavior in a flow chamber based on fluid-solid interaction.

PubMed

Bai, Long; Cui, Yuhong; Zhang, Yixia; Zhao, Na

2014-01-01

The mechanical behavior of blood cells in the vessels has a close relationship with the physical characteristics of the blood and the cells. In this paper, a numerical simulation method was proposed to understand a single-blood cell's behavior in the vessels based on fluid-solid interaction method, which was conducted under adaptive time step and fixed time step, respectively. The main programme was C++ codes, which called FLUENT and ANSYS software, and UDF and APDL acted as a messenger to connect FLUENT and ANSYS for exchanging data. The computing results show: (1) the blood cell moved towards the bottom of the flow chamber in the beginning due to the influence of gravity, then it began to jump up when reached a certain height rather than touching the bottom. It could move downwards again after jump up, the blood cell could keep this way of moving like dancing continuously in the vessels; (2) the blood cell was rolling and deforming all the time; the rotation had oscillatory changes and the deformation became conspicuously when the blood cell was dancing. This new simulation method and results can be widely used in the researches of cytology, blood, cells, etc.

Four decades of implicit Monte Carlo

DOE PAGES

Wollaber, Allan B.

2016-02-23

In 1971, Fleck and Cummings derived a system of equations to enable robust Monte Carlo simulations of time-dependent, thermal radiative transfer problems. Denoted the “Implicit Monte Carlo” (IMC) equations, their solution remains the de facto standard of high-fidelity radiative transfer simulations. Over the course of 44 years, their numerical properties have become better understood, and accuracy enhancements, novel acceleration methods, and variance reduction techniques have been suggested. In this review, we rederive the IMC equations—explicitly highlighting assumptions as they are made—and outfit the equations with a Monte Carlo interpretation. We put the IMC equations in context with other approximate formsmore » of the radiative transfer equations and present a new demonstration of their equivalence to another well-used linearization solved with deterministic transport methods for frequency-independent problems. We discuss physical and numerical limitations of the IMC equations for asymptotically small time steps, stability characteristics and the potential of maximum principle violations for large time steps, and solution behaviors in an asymptotically thick diffusive limit. We provide a new stability analysis for opacities with general monomial dependence on temperature. Here, we consider spatial accuracy limitations of the IMC equations and discussion acceleration and variance reduction techniques.« less

Numerical investigations of low-density nozzle flow by solving the Boltzmann equation

NASA Technical Reports Server (NTRS)

Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen

1995-01-01

A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.

Numerical simulations of the flow with the prescribed displacement of the airfoil and comparison with experiment

NASA Astrophysics Data System (ADS)

Řidký, V.; Šidlof, P.; Vlček, V.

2013-04-01

The work is devoted to comparing measured data with the results of numerical simulations. As mathematical model was used mathematical model whitout turbulence for incompressible flow In the experiment was observed the behavior of designed NACA0015 airfoil in airflow. For the numerical solution was used OpenFOAM computational package, this is open-source software based on finite volume method. In the numerical solution is prescribed displacement of the airfoil, which corresponds to the experiment. The velocity at a point close to the airfoil surface is compared with the experimental data obtained from interferographic measurements of the velocity field. Numerical solution is computed on a 3D mesh composed of about 1 million ortogonal hexahedron elements. The time step is limited by the Courant number. Parallel computations are run on supercomputers of the CIV at Technical University in Prague (HAL and FOX) and on a computer cluster of the Faculty of Mechatronics of Liberec (HYDRA). Run time is fixed at five periods, the results from the fifth periods and average value for all periods are then be compared with experiment.

Modeling the tides of Massachusetts and Cape Cod Bays

USGS Publications Warehouse

Jenter, H.L.; Signell, R.P.; Blumberg, A.F.; ,

1993-01-01

A time-dependent, three-dimensional numerical modeling study of the tides of Massachusetts and Cape Code Bays, motivated by construction of a new sewage treatment plant and ocean outfall for the city of Boston, has been undertaken by the authors. The numerical model being used is a hybrid version of the Blumberg and Mellor ECOM3D model, modified to include a semi-implicit time-stepping scheme and transport of a non-reactive dissolved constituent. Tides in the bays are dominated by the semi-diurnal frequencies, in particular by the M2 tide, due to the resonance of these frequencies in the Gulf of Maine. The numerical model reproduces, well, measured tidal ellipses in unstratified wintertime conditions. Stratified conditions present more of a problem because tidal-frequency internal wave generation and propagation significantly complicates the structure of the resulting tidal field. Nonetheless, the numerical model reproduces qualitative aspects of the stratified tidal flow that are consistent with observations in the bays.

A new unconditionally stable and consistent quasi-analytical in-stream water quality solution scheme for CSTR-based water quality simulators

NASA Astrophysics Data System (ADS)

Woldegiorgis, Befekadu Taddesse; van Griensven, Ann; Pereira, Fernando; Bauwens, Willy

2017-06-01

Most common numerical solutions used in CSTR-based in-stream water quality simulators are susceptible to instabilities and/or solution inconsistencies. Usually, they cope with instability problems by adopting computationally expensive small time steps. However, some simulators use fixed computation time steps and hence do not have the flexibility to do so. This paper presents a novel quasi-analytical solution for CSTR-based water quality simulators of an unsteady system. The robustness of the new method is compared with the commonly used fourth-order Runge-Kutta methods, the Euler method and three versions of the SWAT model (SWAT2012, SWAT-TCEQ, and ESWAT). The performance of each method is tested for different hypothetical experiments. Besides the hypothetical data, a real case study is used for comparison. The growth factors we derived as stability measures for the different methods and the R-factor—considered as a consistency measure—turned out to be very useful for determining the most robust method. The new method outperformed all the numerical methods used in the hypothetical comparisons. The application for the Zenne River (Belgium) shows that the new method provides stable and consistent BOD simulations whereas the SWAT2012 model is shown to be unstable for the standard daily computation time step. The new method unconditionally simulates robust solutions. Therefore, it is a reliable scheme for CSTR-based water quality simulators that use first-order reaction formulations.

Smoothing and the second law

NASA Technical Reports Server (NTRS)

Merriam, Marshal L.

1987-01-01

The technique of obtaining second-order oscillation-free total -variation-diminishing (TVD), scalar difference schemes by adding a limited diffusive flux ('smoothing') to a second-order centered scheme is explored. It is shown that such schemes do not always converge to the correct physical answer. The approach presented here is to construct schemes that numerically satisfy the second law of thermodynamics on a cell-by-cell basis. Such schemes can only converge to the correct physical solution and in some cases can be shown to be TVD. An explicit scheme with this property and second-order spatial accuracy was found to have extremely restrictive time-step limitation. Switching to an implicit scheme removed the time-step limitation.

A fast numerical scheme for causal relativistic hydrodynamics with dissipation

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

Takamoto, Makoto, E-mail: takamoto@tap.scphys.kyoto-u.ac.jp; Inutsuka, Shu-ichiro

2011-08-01

Highlights: {yields} We have developed a new multi-dimensional numerical scheme for causal relativistic hydrodynamics with dissipation. {yields} Our new scheme can calculate the evolution of dissipative relativistic hydrodynamics faster and more effectively than existing schemes. {yields} Since we use the Riemann solver for solving the advection steps, our method can capture shocks very accurately. - Abstract: In this paper, we develop a stable and fast numerical scheme for relativistic dissipative hydrodynamics based on Israel-Stewart theory. Israel-Stewart theory is a stable and causal description of dissipation in relativistic hydrodynamics although it includes relaxation process with the timescale for collision of constituentmore » particles, which introduces stiff equations and makes practical numerical calculation difficult. In our new scheme, we use Strang's splitting method, and use the piecewise exact solutions for solving the extremely short timescale problem. In addition, since we split the calculations into inviscid step and dissipative step, Riemann solver can be used for obtaining numerical flux for the inviscid step. The use of Riemann solver enables us to capture shocks very accurately. Simple numerical examples are shown. The present scheme can be applied to various high energy phenomena of astrophysics and nuclear physics.« less

Unexpected Reaction Pathway for butyrylcholinesterase-catalyzed inactivation of “hunger hormone” ghrelin

NASA Astrophysics Data System (ADS)

Yao, Jianzhuang; Yuan, Yaxia; Zheng, Fang; Zhan, Chang-Guo

2016-02-01

Extensive computational modeling and simulations have been carried out, in the present study, to uncover the fundamental reaction pathway for butyrylcholinesterase (BChE)-catalyzed hydrolysis of ghrelin, demonstrating that the acylation process of BChE-catalyzed hydrolysis of ghrelin follows an unprecedented single-step reaction pathway and the single-step acylation process is rate-determining. The free energy barrier (18.8 kcal/mol) calculated for the rate-determining step is reasonably close to the experimentally-derived free energy barrier (~19.4 kcal/mol), suggesting that the obtained mechanistic insights are reasonable. The single-step reaction pathway for the acylation is remarkably different from the well-known two-step acylation reaction pathway for numerous ester hydrolysis reactions catalyzed by a serine esterase. This is the first time demonstrating that a single-step reaction pathway is possible for an ester hydrolysis reaction catalyzed by a serine esterase and, therefore, one no longer can simply assume that the acylation process must follow the well-known two-step reaction pathway.

Progress in development of HEDP capabilities in FLASH's Unsplit Staggered Mesh MHD solver

NASA Astrophysics Data System (ADS)

Lee, D.; Xia, G.; Daley, C.; Dubey, A.; Gopal, S.; Graziani, C.; Lamb, D.; Weide, K.

2011-11-01

FLASH is a publicly available astrophysical community code designed to solve highly compressible multi-physics reactive flows. We are adding capabilities to FLASH that will make it an open science code for the academic HEDP community. Among many important numerical requirements, we consider the following features to be important components necessary to meet our goals for FLASH as an HEDP open toolset. First, we are developing computationally efficient time-stepping integration methods that overcome the stiffness that arises in the equations describing a physical problem when there are disparate time scales. To this end, we are adding two different time-stepping schemes to FLASH that relax the time step limit when diffusive effects are present: an explicit super-time-stepping algorithm (Alexiades et al. in Com. Num. Mech. Eng. 12:31-42, 1996) and a Jacobian-Free Newton-Krylov implicit formulation. These two methods will be integrated into a robust, efficient, and high-order accurate Unsplit Staggered Mesh MHD (USM) solver (Lee and Deane in J. Comput. Phys. 227, 2009). Second, we have implemented an anisotropic Spitzer-Braginskii conductivity model to treat thermal heat conduction along magnetic field lines. Finally, we are implementing the Biermann Battery term to account for spontaneous generation of magnetic fields in the presence of non-parallel temperature and density gradients.

On large time step TVD scheme for hyperbolic conservation laws and its efficiency evaluation

NASA Astrophysics Data System (ADS)

Qian, ZhanSen; Lee, Chun-Hian

2012-08-01

A large time step (LTS) TVD scheme originally proposed by Harten is modified and further developed in the present paper and applied to Euler equations in multidimensional problems. By firstly revealing the drawbacks of Harten's original LTS TVD scheme, and reasoning the occurrence of the spurious oscillations, a modified formulation of its characteristic transformation is proposed and a high resolution, strongly robust LTS TVD scheme is formulated. The modified scheme is proven to be capable of taking larger number of time steps than the original one. Following the modified strategy, the LTS TVD schemes for Yee's upwind TVD scheme and Yee-Roe-Davis's symmetric TVD scheme are constructed. The family of the LTS schemes is then extended to multidimensional by time splitting procedure, and the associated boundary condition treatment suitable for the LTS scheme is also imposed. The numerical experiments on Sod's shock tube problem, inviscid flows over NACA0012 airfoil and ONERA M6 wing are performed to validate the developed schemes. Computational efficiencies for the respective schemes under different CFL numbers are also evaluated and compared. The results reveal that the improvement is sizable as compared to the respective single time step schemes, especially for the CFL number ranging from 1.0 to 4.0.

Influence of numerical dissipation in computing supersonic vortex-dominated flows

NASA Technical Reports Server (NTRS)

Kandil, O. A.; Chuang, A.

1986-01-01

Steady supersonic vortex-dominated flows are solved using the unsteady Euler equations for conical and three-dimensional flows around sharp- and round-edged delta wings. The computational method is a finite-volume scheme which uses a four-stage Runge-Kutta time stepping with explicit second- and fourth-order dissipation terms. The grid is generated by a modified Joukowski transformation. The steady flow solution is obtained through time-stepping with initial conditions corresponding to the freestream conditions, and the bow shock is captured as a part of the solution. The scheme is applied to flat-plate and elliptic-section wings with a leading edge sweep of 70 deg at an angle of attack of 10 deg and a freestream Mach number of 2.0. Three grid sizes of 29 x 39, 65 x 65 and 100 x 100 have been used. The results for sharp-edged wings show that they are consistent with all grid sizes and variation of the artificial viscosity coefficients. The results for round-edged wings show that separated and attached flow solutions can be obtained by varying the artificial viscosity coefficients. They also show that the solutions are independent of the way time stepping is done. Local time-stepping and global minimum time-steeping produce same solutions.

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

NASA Technical Reports Server (NTRS)

Mazaheri, Alireza; Nishikawa, Hiroaki

2014-01-01

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

Alternating Direction Implicit (ADI) schemes for a PDE-based image osmosis model

NASA Astrophysics Data System (ADS)

Calatroni, L.; Estatico, C.; Garibaldi, N.; Parisotto, S.

2017-10-01

We consider Alternating Direction Implicit (ADI) splitting schemes to compute efficiently the numerical solution of the PDE osmosis model considered by Weickert et al. in [10] for several imaging applications. The discretised scheme is shown to preserve analogous properties to the continuous model. The dimensional splitting strategy traduces numerically into the solution of simple tridiagonal systems for which standard matrix factorisation techniques can be used to improve upon the performance of classical implicit methods, even for large time steps. Applications to the shadow removal problem are presented.

Numerical flow analysis of axial flow compressor for steady and unsteady flow cases

NASA Astrophysics Data System (ADS)

Prabhudev, B. M.; Satish kumar, S.; Rajanna, D.

2017-07-01

Performance of jet engine is dependent on the performance of compressor. This paper gives numerical study of performance characteristics for axial compressor. The test rig is present at CSIR LAB Bangalore. Flow domains are meshed and fluid dynamic equations are solved using ANSYS package. Analysis is done for six different speeds and for operating conditions like choke, maximum efficiency & before stall point. Different plots are compared and results are discussed. Shock displacement, vortex flows, leakage patterns are presented along with unsteady FFT plot and time step plot.

An Operator-Integration-Factor Splitting (OIFS) method for Incompressible Flows in Moving Domains

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

Patel, Saumil S.; Fischer, Paul F.; Min, Misun

In this paper, we present a characteristic-based numerical procedure for simulating incompressible flows in domains with moving boundaries. Our approach utilizes an operator-integration-factor splitting technique to help produce an effcient and stable numerical scheme. Using the spectral element method and an arbitrary Lagrangian-Eulerian formulation, we investigate flows where the convective acceleration effects are non-negligible. Several examples, ranging from laminar to turbulent flows, are considered. Comparisons with a standard, semi-implicit time-stepping procedure illustrate the improved performance of the scheme.

Length and time for development of laminar flow in tubes following a step increase of volume flux

NASA Astrophysics Data System (ADS)

Chaudhury, Rafeed A.; Herrmann, Marcus; Frakes, David H.; Adrian, Ronald J.

2015-01-01

Laminar flows starting up from rest in round tubes are relevant to numerous industrial and biomedical applications. The two most common types are flows driven by an abruptly imposed constant pressure gradient or by an abruptly imposed constant volume flux. Analytical solutions are available for transient, fully developed flows, wherein streamwise development over the entrance length is absent (Szymanski in J de Mathématiques Pures et Appliquées 11:67-107, 1932; Andersson and Tiseth in Chem Eng Commun 112(1):121-133, 1992, respectively). They represent the transient responses of flows in tubes that are very long compared with the entrance length, a condition that is seldom satisfied in biomedical tube networks. This study establishes the entrance (development) length and development time of starting laminar flow in a round tube of finite length driven by a piston pump that produces a step change from zero flow to a constant volume flux for Reynolds numbers between 500 and 3,000. The flows are examined experimentally, using stereographic particle image velocimetry and computationally using computational fluid dynamics, and are then compared with the known analytical solutions for fully developed flow conditions in infinitely long tubes. Results show that step function volume flux start-up flows reach steady state and fully developed flow five times more quickly than those driven by a step function pressure gradient, a 500 % change when compared with existing estimates. Based on these results, we present new, simple guidelines for achieving experimental flows that are fully developed in space and time in realistic (finite) tube geometries. To a first approximation, the time to achieve steady spatially developing flow is nearly equal to the time needed to achieve steady, fully developed flow. Conversely, the entrance length needed to achieve fully developed transient flow is approximately equal to the length needed to achieve fully developed steady flow. Beyond this level of description, the numerical results reveal interaction between the effects of space and time development and nonlinear Reynolds number effects.

Numerical simulation of pounding damage to caisson under storm surge

NASA Astrophysics Data System (ADS)

Yu, Chen

2018-06-01

In this paper, a new method for the numerical simulation of structural model is proposed, which is employed to analyze the pounding response of caissons subjected to storm surge loads. According to the new method, the simulation process is divided into two steps. Firstly, the wave propagation caused by storm surge is simulated by the wave-generating tool of Flow-3D, and recording the wave force time history on the caisson. Secondly, a refined 3D finite element model of caisson is established, and the wave force load is applied on the caisson according to the measured data in the first step for further analysis of structural pounding response using the explicit solver of LSDYNA. The whole simulation of pounding response of a caisson caused by "Sha Lijia" typhoon is carried out. The results show that the different wave direction results in the different angle caisson collisions, which will lead to different failure mode of caisson, and when the angle of 60 between wave direction and front/back wall is simulated, the numerical pounding failure mode is consistent with the situation.

Revision of the documentation for a model for calculating effects of liquid waste disposal in deep saline aquifers

USGS Publications Warehouse

INTERA Environmental Consultants, Inc.

1979-01-01

The major limitation of the model arises using second-order correct (central-difference) finite-difference approximation in space. To avoid numerical oscillations in the solution, the user must restrict grid block and time step sizes depending upon the magnitude of the dispersivity.

Assessment of Linear Finite-Difference Poisson-Boltzmann Solvers

PubMed Central

Wang, Jun; Luo, Ray

2009-01-01

CPU time and memory usage are two vital issues that any numerical solvers for the Poisson-Boltzmann equation have to face in biomolecular applications. In this study we systematically analyzed the CPU time and memory usage of five commonly used finite-difference solvers with a large and diversified set of biomolecular structures. Our comparative analysis shows that modified incomplete Cholesky conjugate gradient and geometric multigrid are the most efficient in the diversified test set. For the two efficient solvers, our test shows that their CPU times increase approximately linearly with the numbers of grids. Their CPU times also increase almost linearly with the negative logarithm of the convergence criterion at very similar rate. Our comparison further shows that geometric multigrid performs better in the large set of tested biomolecules. However, modified incomplete Cholesky conjugate gradient is superior to geometric multigrid in molecular dynamics simulations of tested molecules. We also investigated other significant components in numerical solutions of the Poisson-Boltzmann equation. It turns out that the time-limiting step is the free boundary condition setup for the linear systems for the selected proteins if the electrostatic focusing is not used. Thus, development of future numerical solvers for the Poisson-Boltzmann equation should balance all aspects of the numerical procedures in realistic biomolecular applications. PMID:20063271

Equilibrium Solutions of the Logarithmic Hamiltonian Leapfrog for the N-body Problem

NASA Astrophysics Data System (ADS)

Minesaki, Yukitaka

2018-04-01

We prove that a second-order logarithmic Hamiltonian leapfrog for the classical general N-body problem (CGNBP) designed by Mikkola and Tanikawa and some higher-order logarithmic Hamiltonian methods based on symmetric multicompositions of the logarithmic algorithm exactly reproduce the orbits of elliptic relative equilibrium solutions in the original CGNBP. These methods are explicit symplectic methods. Before this proof, only some implicit discrete-time CGNBPs proposed by Minesaki had been analytically shown to trace the orbits of elliptic relative equilibrium solutions. The proof is therefore the first existence proof for explicit symplectic methods. Such logarithmic Hamiltonian methods with a variable time step can also precisely retain periodic orbits in the classical general three-body problem, which generic numerical methods with a constant time step cannot do.

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

Lott, P. Aaron; Woodward, Carol S.; Evans, Katherine J.

Performing accurate and efficient numerical simulation of global atmospheric climate models is challenging due to the disparate length and time scales over which physical processes interact. Implicit solvers enable the physical system to be integrated with a time step commensurate with processes being studied. The dominant cost of an implicit time step is the ancillary linear system solves, so we have developed a preconditioner aimed at improving the efficiency of these linear system solves. Our preconditioner is based on an approximate block factorization of the linearized shallow-water equations and has been implemented within the spectral element dynamical core within themore » Community Atmospheric Model (CAM-SE). Furthermore, in this paper we discuss the development and scalability of the preconditioner for a suite of test cases with the implicit shallow-water solver within CAM-SE.« less

Real time wave forecasting using wind time history and numerical model

NASA Astrophysics Data System (ADS)

Jain, Pooja; Deo, M. C.; Latha, G.; Rajendran, V.

Operational activities in the ocean like planning for structural repairs or fishing expeditions require real time prediction of waves over typical time duration of say a few hours. Such predictions can be made by using a numerical model or a time series model employing continuously recorded waves. This paper presents another option to do so and it is based on a different time series approach in which the input is in the form of preceding wind speed and wind direction observations. This would be useful for those stations where the costly wave buoys are not deployed and instead only meteorological buoys measuring wind are moored. The technique employs alternative artificial intelligence approaches of an artificial neural network (ANN), genetic programming (GP) and model tree (MT) to carry out the time series modeling of wind to obtain waves. Wind observations at four offshore sites along the east coast of India were used. For calibration purpose the wave data was generated using a numerical model. The predicted waves obtained using the proposed time series models when compared with the numerically generated waves showed good resemblance in terms of the selected error criteria. Large differences across the chosen techniques of ANN, GP, MT were not noticed. Wave hindcasting at the same time step and the predictions over shorter lead times were better than the predictions over longer lead times. The proposed method is a cost effective and convenient option when a site-specific information is desired.

An arbitrary-order staggered time integrator for the linear acoustic wave equation

NASA Astrophysics Data System (ADS)

Lee, Jaejoon; Park, Hyunseo; Park, Yoonseo; Shin, Changsoo

2018-02-01

We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set the modelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems.

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

Zakharov, Leonic E.; Li, Xujing

This paper formulates the Tokamak Magneto-Hydrodynamics (TMHD), initially outlined by X. Li and L.E. Zakharov [Plasma Science and Technology, accepted, ID:2013-257 (2013)] for proper simulations of macroscopic plasma dynamics. The simplest set of magneto-hydrodynamics equations, sufficient for disruption modeling and extendable to more refined physics, is explained in detail. First, the TMHD introduces to 3-D simulations the Reference Magnetic Coordinates (RMC), which are aligned with the magnetic field in the best possible way. The numerical implementation of RMC is adaptive grids. Being consistent with the high anisotropy of the tokamak plasma, RMC allow simulations at realistic, very high plasma electricmore » conductivity. Second, the TMHD splits the equation of motion into an equilibrium equation and the plasma advancing equation. This resolves the 4 decade old problem of Courant limitations of the time step in existing, plasma inertia driven numerical codes. The splitting allows disruption simulations on a relatively slow time scale in comparison with the fast time of ideal MHD instabilities. A new, efficient numerical scheme is proposed for TMHD.« less

Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, Robert A.

2014-03-01

A Jacobi-Gauss-Lobatto collocation (J-GL-C) method, used in combination with the implicit Runge-Kutta method of fourth order, is proposed as a numerical algorithm for the approximation of solutions to nonlinear Schrödinger equations (NLSE) with initial-boundary data in 1+1 dimensions. Our procedure is implemented in two successive steps. In the first one, the J-GL-C is employed for approximating the functional dependence on the spatial variable, using (N-1) nodes of the Jacobi-Gauss-Lobatto interpolation which depends upon two general Jacobi parameters. The resulting equations together with the two-point boundary conditions induce a system of 2(N-1) first-order ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve this temporal system. The proposed J-GL-C method, used in combination with the implicit Runge-Kutta method of fourth order, is employed to obtain highly accurate numerical approximations to four types of NLSE, including the attractive and repulsive NLSE and a Gross-Pitaevskii equation with space-periodic potential. The numerical results obtained by this algorithm have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively few nodes used, the absolute error in our numerical solutions is sufficiently small.

An optimal open/closed-loop control method with application to a pre-stressed thin duralumin plate

NASA Astrophysics Data System (ADS)

Nadimpalli, Sruthi Raju

The excessive vibrations of a pre-stressed duralumin plate, suppressed by a combination of open-loop and closed-loop controls, also known as open/closed-loop control, is studied in this thesis. The two primary steps involved in this process are: Step (I) with an assumption that the closed-loop control law is proportional, obtain the optimal open-loop control by direct minimization of the performance measure consisting of energy at terminal time and a penalty on open-loop control force via calculus of variations. If the performance measure also involves a penalty on closed-loop control effort then a Fourier based method is utilized. Step (II) the energy at terminal time is minimized numerically to obtain optimal values of feedback gains. The optimal closed-loop control gains obtained are used to describe the displacement and the velocity of open-loop, closed-loop and open/closed-loop controlled duralumin plate.

Centrifugal step emulsification applied for absolute quantification of nucleic acids by digital droplet RPA.

PubMed

Schuler, Friedrich; Schwemmer, Frank; Trotter, Martin; Wadle, Simon; Zengerle, Roland; von Stetten, Felix; Paust, Nils

2015-07-07

Aqueous microdroplets provide miniaturized reaction compartments for numerous chemical, biochemical or pharmaceutical applications. We introduce centrifugal step emulsification for the fast and easy production of monodisperse droplets. Homogenous droplets with pre-selectable diameters in a range from 120 μm to 170 μm were generated with coefficients of variation of 2-4% and zero run-in time or dead volume. The droplet diameter depends on the nozzle geometry (depth, width, and step size) and interfacial tensions only. Droplet size is demonstrated to be independent of the dispersed phase flow rate between 0.01 and 1 μl s(-1), proving the robustness of the centrifugal approach. Centrifugal step emulsification can easily be combined with existing centrifugal microfluidic unit operations, is compatible to scalable manufacturing technologies such as thermoforming or injection moulding and enables fast emulsification (>500 droplets per second and nozzle) with minimal handling effort (2-3 pipetting steps). The centrifugal microfluidic droplet generation was used to perform the first digital droplet recombinase polymerase amplification (ddRPA). It was used for absolute quantification of Listeria monocytogenes DNA concentration standards with a total analysis time below 30 min. Compared to digital droplet polymerase chain reaction (ddPCR), with processing times of about 2 hours, the overall processing time of digital analysis was reduced by more than a factor of 4.

The Space-Time Conservation Element and Solution Element Method-A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws. 2; Numerical Simulation of Shock Waves and Contact Discontinuities

NASA Technical Reports Server (NTRS)

Wang, Xiao-Yen; Chow, Chuen-Yen; Chang, Sin-Chung

1998-01-01

Without resorting to special treatment for each individual test case, the 1D and 2D CE/SE shock-capturing schemes described previously (in Part I) are used to simulate flows involving phenomena such as shock waves, contact discontinuities, expansion waves and their interactions. Five 1D and six 2D problems are considered to examine the capability and robustness of these schemes. Despite their simple logical structures and low computational cost (for the 2D CE/SE shock-capturing scheme, the CPU time is about 2 micro-secs per mesh point per marching step on a Cray C90 machine), the numerical results, when compared with experimental data, exact solutions or numerical solutions by other methods, indicate that these schemes can accurately resolve shock and contact discontinuities consistently.

A new numerical method for calculating extrema of received power for polarimetric SAR

USGS Publications Warehouse

Zhang, Y.; Zhang, Jiahua; Lu, Z.; Gong, W.

2009-01-01

A numerical method called cross-step iteration is proposed to calculate the maximal/minimal received power for polarized imagery based on a target's Kennaugh matrix. This method is much more efficient than the systematic method, which searches for the extrema of received power by varying the polarization ellipse angles of receiving and transmitting polarizations. It is also more advantageous than the Schuler method, which has been adopted by the PolSARPro package, because the cross-step iteration method requires less computation time and can derive both the maximal and minimal received powers, whereas the Schuler method is designed to work out only the maximal received power. The analytical model of received-power optimization indicates that the first eigenvalue of the Kennaugh matrix is the supremum of the maximal received power. The difference between these two parameters reflects the depolarization effect of the target's backscattering, which might be useful for target discrimination. ?? 2009 IEEE.

Multigrid methods for numerical simulation of laminar diffusion flames

NASA Technical Reports Server (NTRS)

Liu, C.; Liu, Z.; Mccormick, S.

1993-01-01

This paper documents the result of a computational study of multigrid methods for numerical simulation of 2D diffusion flames. The focus is on a simplified combustion model, which is assumed to be a single step, infinitely fast and irreversible chemical reaction with five species (C3H8, O2, N2, CO2 and H2O). A fully-implicit second-order hybrid scheme is developed on a staggered grid, which is stretched in the streamwise coordinate direction. A full approximation multigrid scheme (FAS) based on line distributive relaxation is developed as a fast solver for the algebraic equations arising at each time step. Convergence of the process for the simplified model problem is more than two-orders of magnitude faster than other iterative methods, and the computational results show good grid convergence, with second-order accuracy, as well as qualitatively agreement with the results of other researchers.

Mesure sans contact d'un panneau d'aile d'avion et analyse numerique pour controle dimensionnel

NASA Astrophysics Data System (ADS)

Sok, Michel Christian

During the manufacturing of the wing skin, the inspection steps are essential to ensure their conformity and thus allow the wings to ensure the required aerodynamic performances. Nowadays, considering the panel's low stiffness which prevents traditional inspection methods, this inspection is done manually with a template gauge and a jig. Iteratively, as long as form compliance is not reached, the panel goes through an additional dimensional refinement before being inspected in a second time. Because the jig is accurate, it is very expensive and furthermore, the inspection of panels is time-consuming by monopolizing the jig, which cannot be used in the meantime. Using this consideration as a starting point, this project seeks to provide a response to the practicability of a methodology based on the automation of that king of operation. This by integrating into the process non-contact measuring machines capable of acquiring numerically the geometrical shape of the panel. Moreover, the opportunity of realizing this operation without the use of a jig is also being considered, which would leave it free for other tasks. The methodology suggested use numerical simulations to check form compliance. Finally, this would provide a tool to assist the operator by allowing a semi automated inspection without jig. The methodology suggested can be describe in three steps, however it is necessary to propose an additional step to validate the results achieved with this methodology. Then, the first step consist of manually acquiring reference values which will served to be compared with the values obtained during the application of the methodology. The second step deals with the numerical acquisition, with a laser scanner, of the object to be inspected settled down on some supporting plate. The third step is the numerical reconstruction of this object with a computer-aided design software. Finally the last step consists of a numerical inspection of the object to predict the form compliance. Considering the large dimensions of the wing skins and of the jigs used in industry, the methodology suggested takes accounts of the available means in laboratory. Then, the objects used have lower dimensions than those used in the industry. That is the reason why a simplifying assumption that the shot peening operation has a negligible effect on the evolution of the thickness of the wing skin is made. Furthermore, the non-contact measurement device is also tested to know its accuracy under real conditions. Those two preliminary studies show that the thickness variation of a plate after being shot peened, with extreme parameters in terms of effects, remains negligible for the study of practicability realized in this thesis. The study on the performance of the REVscan 3D also brings to light that this variation would probably be drown in the uncertainty acquired by the device during the numerical acquisition. In this project, only the steps two and three are dealt with in depth. This study involves essentially to test the measuring device and the software about their capacity of numerically acquiring an object and then to bring it to another state of stresses with the help of a simulation. Indeed, the validation of the free state step is problematic because it is precisely a state that cannot be obtained in an experimental way. As an analogy, it is suggested to pass from a particular state of stress to another because, in a simplified way, the free state step is equivalent to a change of a state of stress. The study of the result allows to put forward a particular phenomenon linked to thin plates : it is a sudden change of the form when the plate is in a particular state of stress. The software is then no more able to predict that kind of comportment. Several tests are carried out to confirm the existence of that phenomenon and show that the stress modulus, the point of application of the stresses and the position of the support points are the more influent parameters. However, even by ensuring to avoid this phenomenon during the tests, the degree of accuracy reached by the software is far from being sufficient. Indeed, the uncertainty of the results is still too high and the next studies will have to focus on improving the results. Currently, the tests realized in this thesis are not enough to validate the steps 2 and 3 of the methodology suggested. Nevertheless, the phenomenon highlighted which can suddenly modify the comportment of thin plates and the information gathered in these tests establishes a base for further research. (Abstract shortened by UMI.).

Efficient variable time-stepping scheme for intense field-atom interactions

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

Cerjan, C.; Kosloff, R.

1993-03-01

The recently developed Residuum method [Tal-Ezer, Kosloff, and Cerjan, J. Comput. Phys. 100, 179 (1992)], a Krylov subspace technique with variable time-step integration for the solution of the time-dependent Schroedinger equation, is applied to the frequently used soft Coulomb potential in an intense laser field. This one-dimensional potential has asymptotic Coulomb dependence with a softened'' singularity at the origin; thus it models more realistic phenomena. Two of the more important quantities usually calculated in this idealized system are the photoelectron and harmonic photon generation spectra. These quantities are shown to be sensitive to the choice of a numerical integration scheme:more » some spectral features are incorrectly calculated or missing altogether. Furthermore, the Residuum method allows much larger grid spacings for equivalent or higher accuracy in addition to the advantages of variable time stepping. Finally, it is demonstrated that enhanced high-order harmonic generation accompanies intense field stabilization and that preparation of the atom in an intermediate Rydberg state leads to stabilization at much lower laser intensity.« less

A linear stability analysis for nonlinear, grey, thermal radiative transfer problems

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

Wollaber, Allan B., E-mail: wollaber@lanl.go; Larsen, Edward W., E-mail: edlarsen@umich.ed

2011-02-20

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used 'Implicit Monte Carlo' (IMC) equations, the Carter Forest (CF) equations, and the Ahrens-Larsen or 'Semi-Analog Monte Carlo' (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if {alpha}, the IMC time-discretization parameter, satisfies 0.5 < {alpha} {<=} 1. This is consistent with conventional wisdom. However, wemore » also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.« less

A linear stability analysis for nonlinear, grey, thermal radiative transfer problems

NASA Astrophysics Data System (ADS)

Wollaber, Allan B.; Larsen, Edward W.

2011-02-01

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used “Implicit Monte Carlo” (IMC) equations, the Carter Forest (CF) equations, and the Ahrens-Larsen or “Semi-Analog Monte Carlo” (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if α, the IMC time-discretization parameter, satisfies 0.5 < α ⩽ 1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.

Numerical Convergence in the Dark Matter Halos Properties Using Cosmological Simulations

NASA Astrophysics Data System (ADS)

Mosquera-Escobar, X. E.; Muñoz-Cuartas, J. C.

2017-07-01

Nowadays, the accepted cosmological model is the so called -Cold Dark Matter (CDM). In such model, the universe is considered to be homogeneous and isotropic, composed of diverse components as the dark matter and dark energy, where the latter is the most abundant one. Dark matter plays an important role because it is responsible for the generation of gravitational potential wells, commonly called dark matter halos. At the end, dark matter halos are characterized by a set of parameters (mass, radius, concentration, spin parameter), these parameters provide valuable information for different studies, such as galaxy formation, gravitational lensing, etc. In this work we use the publicly available code Gadget2 to perform cosmological simulations to find to what extent the numerical parameters of the simu- lations, such as gravitational softening, integration time step and force calculation accuracy affect the physical properties of the dark matter halos. We ran a suite of simulations where these parameters were varied in a systematic way in order to explore accurately their impact on the structural parameters of dark matter halos. We show that the variations on the numerical parameters affect the structural pa- rameters of dark matter halos, such as concentration, virial radius, and concentration. We show that these modifications emerged when structures become non- linear (at redshift 2) for the scale of our simulations, such that these variations affected the formation and evolution structure of halos mainly at later cosmic times. As a quantitative result, we propose which would be the most appropriate values for the numerical parameters of the simulations, such that they do not affect the halo properties that are formed. For force calculation accuracy we suggest values smaller or equal to 0.0001, integration time step smaller o equal to 0.005 and for gravitational softening we propose equal to 1/60th of the mean interparticle distance, these values, correspond to the smaller values in the numerical parameters variations. This is an important numerical exercise, since for instance, it is believed that galaxy structural parameters are strongly dependent on dark matter halo structural parameters.

Evaluation of the matrix exponential for use in ground-water-flow and solute-transport simulations; theoretical framework

USGS Publications Warehouse

Umari, A.M.; Gorelick, S.M.

1986-01-01

It is possible to obtain analytic solutions to the groundwater flow and solute transport equations if space variables are discretized but time is left continuous. From these solutions, hydraulic head and concentration fields for any future time can be obtained without ' marching ' through intermediate time steps. This analytical approach involves matrix exponentiation and is referred to as the Matrix Exponential Time Advancement (META) method. Two algorithms are presented for the META method, one for symmetric and the other for non-symmetric exponent matrices. A numerical accuracy indicator, referred to as the matrix condition number, was defined and used to determine the maximum number of significant figures that may be lost in the META method computations. The relative computational and storage requirements of the META method with respect to the time marching method increase with the number of nodes in the discretized problem. The potential greater accuracy of the META method and the associated greater reliability through use of the matrix condition number have to be weighed against this increased relative computational and storage requirements of this approach as the number of nodes becomes large. For a particular number of nodes, the META method may be computationally more efficient than the time-marching method, depending on the size of time steps used in the latter. A numerical example illustrates application of the META method to a sample ground-water-flow problem. (Author 's abstract)

Moho Modeling Using FFT Technique

NASA Astrophysics Data System (ADS)

Chen, Wenjin; Tenzer, Robert

2017-04-01

To improve the numerical efficiency, the Fast Fourier Transform (FFT) technique was facilitated in Parker-Oldenburg's method for a regional gravimetric Moho recovery, which assumes the Earth's planar approximation. In this study, we extend this definition for global applications while assuming a spherical approximation of the Earth. In particular, we utilize the FFT technique for a global Moho recovery, which is practically realized in two numerical steps. The gravimetric forward modeling is first applied, based on methods for a spherical harmonic analysis and synthesis of the global gravity and lithospheric structure models, to compute the refined gravity field, which comprises mainly the gravitational signature of the Moho geometry. The gravimetric inverse problem is then solved iteratively in order to determine the Moho depth. The application of FFT technique to both numerical steps reduces the computation time to a fraction of that required without applying this fast algorithm. The developed numerical producers are used to estimate the Moho depth globally, and the gravimetric result is validated using the global (CRUST1.0) and regional (ESC) seismic Moho models. The comparison reveals a relatively good agreement between the gravimetric and seismic models, with the RMS of differences (of 4-5 km) at the level of expected uncertainties of used input datasets, while without the presence of significant systematic bias.

A variational numerical method based on finite elements for the nonlinear solution characteristics of the periodically forced Chen system

NASA Astrophysics Data System (ADS)

Khan, Sabeel M.; Sunny, D. A.; Aqeel, M.

2017-09-01

Nonlinear dynamical systems and their solutions are very sensitive to initial conditions and therefore need to be approximated carefully. In this article, we present and analyze nonlinear solution characteristics of the periodically forced Chen system with the application of a variational method based on the concept of finite time-elements. Our approach is based on the discretization of physical time space into finite elements where each time-element is mapped to a natural time space. The solution of the system is then determined in natural time space using a set of suitable basis functions. The numerical algorithm is presented and implemented to compute and analyze nonlinear behavior at different time-step sizes. The obtained results show an excellent agreement with the classical RK-4 and RK-5 methods. The accuracy and convergence of the method is shown by comparing numerically computed results with the exact solution for a test problem. The presented method has shown a great potential in dealing with the solutions of nonlinear dynamical systems and thus can be utilized in delineating different features and characteristics of their solutions.

Numeric calculation of celestial bodies with spreadsheet analysis

NASA Astrophysics Data System (ADS)

Koch, Alexander

2016-04-01

The motion of the planets and moons in our solar system can easily be calculated for any time by the Kepler laws of planetary motion. The Kepler laws are a special case of the gravitational law of Newton, especially if you consider more than two celestial bodies. Therefore it is more basic to calculate the motion by using the gravitational law. But the problem is, that by gravitational law it is not possible to calculate the state of motion with only one step of calculation. The motion has to be numerical calculated for many time intervalls. For this reason, spreadsheet analysis is helpful for students. Skills in programmes like Excel, Calc or Gnumeric are important in professional life and can easily be learnt by students. These programmes can help to calculate the complex motions with many intervalls. The more intervalls are used, the more exact are the calculated orbits. The sutdents will first get a quick course in Excel. After that they calculate with instructions the 2-D-coordinates of the orbits of Moon and Mars. Step by step the students are coding the formulae for calculating physical parameters like coordinates, force, acceleration and velocity. The project is limited to 4 weeks or 8 lessons. So the calcualtion will only include the calculation of one body around the central mass like Earth or Sun. The three-body problem can only be shortly discussed at the end of the project.

A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

NASA Technical Reports Server (NTRS)

Estes, R. H.

1977-01-01

A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables.

The comparison of stepping responses following perturbations applied to pelvis during overground and treadmill walking.

PubMed

Zadravec, Matjaž; Olenšek, Andrej; Matjačić, Zlatko

2017-08-09

Treadmills are used frequently in rehabilitation enabling neurologically impaired subjects to train walking while being assisted by therapists. Numerous studies compared walking on treadmill and overground for unperturbed but not also perturbed conditions. The objective of this study was to compare stepping responses (step length, step width and step time) during overground and treadmill walking in a group of healthy subjects where balance assessment robots applied perturbing pushes to the subject's pelvis in sagittal and frontal planes. During walking in both balance assessment robots (overground and treadmill-based) with applied perturbations the stepping responses of a group of seven healthy subjects were assessed with a motion tracking camera. The results show high degree of similarity of stepping responses between overground and treadmill walking for all perturbation directions. Both devices reproduced similar experimental conditions with relatively small standard deviations in the unperturbed walking as well as in perturbed walking. Based on these results we may conclude that stepping responses following perturbations can be studied on an instrumented treadmill where ground reaction forces can be readily assessed which is not the case during perturbed overground walking.

Benchmark Results Of Active Tracer Particles In The Open Souce Code ASPECT For Modelling Convection In The Earth's Mantle

NASA Astrophysics Data System (ADS)

Jiang, J.; Kaloti, A. P.; Levinson, H. R.; Nguyen, N.; Puckett, E. G.; Lokavarapu, H. V.

2016-12-01

We present the results of three standard benchmarks for the new active tracer particle algorithm in ASPECT. The three benchmarks are SolKz, SolCx, and SolVI (also known as the 'inclusion benchmark') first proposed by Duretz, May, Gerya, and Tackley (G Cubed, 2011) and in subsequent work by Theilman, May, and Kaus (Pure and Applied Geophysics, 2014). Each of the three benchmarks compares the accuracy of the numerical solution to a steady (time-independent) solution of the incompressible Stokes equations with a known exact solution. These benchmarks are specifically designed to test the accuracy and effectiveness of the numerical method when the viscosity varies up to six orders of magnitude. ASPECT has been shown to converge to the exact solution of each of these benchmarks at the correct design rate when all of the flow variables, including the density and viscosity, are discretized on the underlying finite element grid (Krobichler, Heister, and Bangerth, GJI, 2012). In our work we discretize the density and viscosity by initially placing the true values of the density and viscosity at the intial particle positions. At each time step, including the initialization step, the density and viscosity are interpolated from the particles onto the finite element grid. The resulting Stokes system is solved for the velocity and pressure, and the particle positions are advanced in time according to this new, numerical, velocity field. Note that this procedure effectively changes a steady solution of the Stokes equaton (i.e., one that is independent of time) to a solution of the Stokes equations that is time dependent. Furthermore, the accuracy of the active tracer particle algorithm now also depends on the accuracy of the interpolation algorithm and of the numerical method one uses to advance the particle positions in time. Finally, we will present new interpolation algorithms designed to increase the overall accuracy of the active tracer algorithms in ASPECT and interpolation algotithms designed to conserve properties, such as mass density, that are being carried by the particles.

A computational method for sharp interface advection

PubMed Central

Bredmose, Henrik; Jasak, Hrvoje

2016-01-01

We devise a numerical method for passive advection of a surface, such as the interface between two incompressible fluids, across a computational mesh. The method is called isoAdvector, and is developed for general meshes consisting of arbitrary polyhedral cells. The algorithm is based on the volume of fluid (VOF) idea of calculating the volume of one of the fluids transported across the mesh faces during a time step. The novelty of the isoAdvector concept consists of two parts. First, we exploit an isosurface concept for modelling the interface inside cells in a geometric surface reconstruction step. Second, from the reconstructed surface, we model the motion of the face–interface intersection line for a general polygonal face to obtain the time evolution within a time step of the submerged face area. Integrating this submerged area over the time step leads to an accurate estimate for the total volume of fluid transported across the face. The method was tested on simple two-dimensional and three-dimensional interface advection problems on both structured and unstructured meshes. The results are very satisfactory in terms of volume conservation, boundedness, surface sharpness and efficiency. The isoAdvector method was implemented as an OpenFOAM® extension and is published as open source. PMID:28018619

Effect of reaction-step-size noise on the switching dynamics of stochastic populations

NASA Astrophysics Data System (ADS)

Be'er, Shay; Heller-Algazi, Metar; Assaf, Michael

2016-05-01

In genetic circuits, when the messenger RNA lifetime is short compared to the cell cycle, proteins are produced in geometrically distributed bursts, which greatly affects the cellular switching dynamics between different metastable phenotypic states. Motivated by this scenario, we study a general problem of switching or escape in stochastic populations, where influx of particles occurs in groups or bursts, sampled from an arbitrary distribution. The fact that the step size of the influx reaction is a priori unknown and, in general, may fluctuate in time with a given correlation time and statistics, introduces an additional nondemographic reaction-step-size noise into the system. Employing the probability-generating function technique in conjunction with Hamiltonian formulation, we are able to map the problem in the leading order onto solving a stationary Hamilton-Jacobi equation. We show that compared to the "usual case" of single-step influx, bursty influx exponentially decreases the population's mean escape time from its long-lived metastable state. In particular, close to bifurcation we find a simple analytical expression for the mean escape time which solely depends on the mean and variance of the burst-size distribution. Our results are demonstrated on several realistic distributions and compare well with numerical Monte Carlo simulations.

Analysis of stability for stochastic delay integro-differential equations.

PubMed

Zhang, Yu; Li, Longsuo

2018-01-01

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler-Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

Development of a High-Order Navier-Stokes Solver Using Flux Reconstruction to Simulate Three-Dimensional Vortex Structures in a Curved Artery Model

NASA Astrophysics Data System (ADS)

Cox, Christopher

Low-order numerical methods are widespread in academic solvers and ubiquitous in industrial solvers due to their robustness and usability. High-order methods are less robust and more complicated to implement; however, they exhibit low numerical dissipation and have the potential to improve the accuracy of flow simulations at a lower computational cost when compared to low-order methods. This motivates our development of a high-order compact method using Huynh's flux reconstruction scheme for solving unsteady incompressible flow on unstructured grids. We use Chorin's classic artificial compressibility formulation with dual time stepping to solve unsteady flow problems. In 2D, an implicit non-linear lower-upper symmetric Gauss-Seidel scheme with backward Euler discretization is used to efficiently march the solution in pseudo time, while a second-order backward Euler discretization is used to march in physical time. We verify and validate implementation of the high-order method coupled with our implicit time stepping scheme using both steady and unsteady incompressible flow problems. The current implicit time stepping scheme is proven effective in satisfying the divergence-free constraint on the velocity field in the artificial compressibility formulation. The high-order solver is extended to 3D and parallelized using MPI. Due to its simplicity, time marching for 3D problems is done explicitly. The feasibility of using the current implicit time stepping scheme for large scale three-dimensional problems with high-order polynomial basis still remains to be seen. We directly use the aforementioned numerical solver to simulate pulsatile flow of a Newtonian blood-analog fluid through a rigid 180-degree curved artery model. One of the most physiologically relevant forces within the cardiovascular system is the wall shear stress. This force is important because atherosclerotic regions are strongly correlated with curvature and branching in the human vasculature, where the shear stress is both oscillatory and multidirectional. Also, the combined effect of curvature and pulsatility in cardiovascular flows produces unsteady vortices. The aim of this research as it relates to cardiovascular fluid dynamics is to predict the spatial and temporal evolution of vortical structures generated by secondary flows, as well as to assess the correlation between multiple vortex pairs and wall shear stress. We use a physiologically (pulsatile) relevant flow rate and generate results using both fully developed and uniform entrance conditions, the latter being motivated by the fact that flow upstream of a curved artery may not have sufficient straight entrance length to become fully developed. Under the two pulsatile inflow conditions, we characterize the morphology and evolution of various vortex pairs and their subsequent effect on relevant haemodynamic wall shear stress metrics.

Numerical simulation of conservation laws

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung; To, Wai-Ming

1992-01-01

A new numerical framework for solving conservation laws is being developed. This new approach differs substantially from the well established methods, i.e., finite difference, finite volume, finite element and spectral methods, in both concept and methodology. The key features of the current scheme include: (1) direct discretization of the integral forms of conservation laws, (2) treating space and time on the same footing, (3) flux conservation in space and time, and (4) unified treatment of the convection and diffusion fluxes. The model equation considered in the initial study is the standard one dimensional unsteady constant-coefficient convection-diffusion equation. In a stability study, it is shown that the principal and spurious amplification factors of the current scheme, respectively, are structurally similar to those of the leapfrog/DuFort-Frankel scheme. As a result, the current scheme has no numerical diffusion in the special case of pure convection and is unconditionally stable in the special case of pure diffusion. Assuming smooth initial data, it will be shown theoretically and numerically that, by using an easily determined optimal time step, the accuracy of the current scheme may reach a level which is several orders of magnitude higher than that of the MacCormack scheme, with virtually identical operation count.

Caution Ahead: Overdue Investments for New York's Aging Infrastructure

ERIC Educational Resources Information Center

Forman, Adam

2014-01-01

Following the devastation of Superstorm Sandy in October 2012, New York City's essential infrastructure needs were made a top policy priority for the first time in decades. The scale and severity of the storm prompted numerous studies to assess the damage and led policymakers to take steps to shore up the city's coastal infrastructure weaknesses.…

"Redefining RA": The RA Tool Kit

ERIC Educational Resources Information Center

Wyatt, Neal

2008-01-01

No one likes being two steps behind, and the fastest way to fall off the pace is by not keeping up with major titles and hot authors. Fortunately, there are numerous resources, both prepublication and postpublication, that can help. It is best when readers' advisory (RA) librarians know what is coming out months ahead of time--in order to think…

Monte Carlo Sampling in Fractal Landscapes

NASA Astrophysics Data System (ADS)

Leitão, Jorge C.; Lopes, J. M. Viana Parente; Altmann, Eduardo G.

2013-05-01

We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.

Smoothing and the second law

NASA Technical Reports Server (NTRS)

Merriam, Marshal L.

1986-01-01

The technique of obtaining second order, oscillation free, total variation diminishing (TVD), scalar difference schemes by adding a limited diffusion flux (smoothing) to a second order centered scheme is explored. It is shown that such schemes do not always converge to the correct physical answer. The approach presented here is to construct schemes that numerically satisfy the second law of thermodynamics on a cell by cell basis. Such schemes can only converge to the correct physical solution and in some cases can be shown to be TVD. An explicit scheme with this property and second order spatial accuracy was found to have an extremely restrictive time step limitation (Delta t less than Delta x squared). Switching to an implicit scheme removed the time step limitation.

Investigation of flow-induced numerical instability in a mixed semi-implicit, implicit leapfrog time discretization

NASA Astrophysics Data System (ADS)

King, Jacob; Kruger, Scott

2017-10-01

Flow can impact the stability and nonlinear evolution of range of instabilities (e.g. RWMs, NTMs, sawteeth, locked modes, PBMs, and high-k turbulence) and thus robust numerical algorithms for simulations with flow are essential. Recent simulations of DIII-D QH-mode [King et al., Phys. Plasmas and Nucl. Fus. 2017] with flow have been restricted to smaller time-step sizes than corresponding computations without flow. These computations use a mixed semi-implicit, implicit leapfrog time discretization as implemented in the NIMROD code [Sovinec et al., JCP 2004]. While prior analysis has shown that this algorithm is unconditionally stable with respect to the effect of large flows on the MHD waves in slab geometry [Sovinec et al., JCP 2010], our present Von Neumann stability analysis shows that a flow-induced numerical instability may arise when ad-hoc cylindrical curvature is included. Computations with the NIMROD code in cylindrical geometry with rigid rotation and without free-energy drive from current or pressure gradients qualitatively confirm this analysis. We explore potential methods to circumvent this flow-induced numerical instability such as using a semi-Lagrangian formulation instead of time-centered implicit advection and/or modification to the semi-implicit operator. This work is supported by the DOE Office of Science (Office of Fusion Energy Sciences).

Mixed time integration methods for transient thermal analysis of structures

NASA Technical Reports Server (NTRS)

Liu, W. K.

1982-01-01

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 different 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.

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.

Application of Energy Function as a Measure of Error in the Numerical Solution for Online Transient Stability Assessment

NASA Astrophysics Data System (ADS)

Sarojkumar, K.; Krishna, S.

2016-08-01

Online dynamic security assessment (DSA) is a computationally intensive task. In order to reduce the amount of computation, screening of contingencies is performed. Screening involves analyzing the contingencies with the system described by a simpler model so that computation requirement is reduced. Screening identifies those contingencies which are sure to not cause instability and hence can be eliminated from further scrutiny. The numerical method and the step size used for screening should be chosen with a compromise between speed and accuracy. This paper proposes use of energy function as a measure of error in the numerical solution used for screening contingencies. The proposed measure of error can be used to determine the most accurate numerical method satisfying the time constraint of online DSA. Case studies on 17 generator system are reported.

Attempts at a numerical realisation of stochastic differential equations containing Preisach operator

NASA Astrophysics Data System (ADS)

McCarthy, S.; Rachinskii, D.

2011-01-01

We describe two Euler type numerical schemes obtained by discretisation of a stochastic differential equation which contains the Preisach memory operator. Equations of this type are of interest in areas such as macroeconomics and terrestrial hydrology where deterministic models containing the Preisach operator have been developed but do not fully encapsulate stochastic aspects of the area. A simple price dynamics model is presented as one motivating example for our studies. Some numerical evidence is given that the two numerical schemes converge to the same limit as the time step decreases. We show that the Preisach term introduces a damping effect which increases on the parts of the trajectory demonstrating a stronger upwards or downwards trend. The results are preliminary to a broader programme of research of stochastic differential equations with the Preisach hysteresis operator.

Ignition and structure of a laminar diffusion flame in the field of a vortex

NASA Technical Reports Server (NTRS)

Macaraeg, Michele G.; Jackson, T. L.; Hussaini, M. Y.

1991-01-01

The distortion of flames in flows with vortical motion is examined via asymptotic analysis and numerical simulation. The model consists of a constant density, one step, irreversible Arrhenius reaction between initially unmixed species occupying adjacent half-planes which are then allowed to mix and react in the presence of a vortex. The evolution in time of the temperature and mass fraction fields is followed. Emphasis is placed on the ignition time and location as a function of vortex Reynolds number and initial temperature differences of the reacting species. The study brings out the influence of the vortex on the chemical reaction. In all phases, good agreement is observed between asymptotic analysis and the full numerical solution of the model equations.

Full-Scale Numerical Modeling of Turbulent Processes in the Earth's Ionosphere

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

Eliasson, B.; Stenflo, L.; Department of Physics, Linkoeping University, SE-581 83 Linkoeping

2008-10-15

We present a full-scale simulation study of ionospheric turbulence by means of a generalized Zakharov model based on the separation of variables into high-frequency and slow time scales. The model includes realistic length scales of the ionospheric profile and of the electromagnetic and electrostatic fields, and uses ionospheric plasma parameters relevant for high-latitude radio facilities such as Eiscat and HAARP. A nested grid numerical method has been developed to resolve the different length-scales, while avoiding severe restrictions on the time step. The simulation demonstrates the parametric decay of the ordinary mode into Langmuir and ion-acoustic waves, followed by a Langmuirmore » wave collapse and short-scale caviton formation, as observed in ionospheric heating experiments.« less

Assessment of power step performances of variable speed pump-turbine unit by means of hydro-electrical system simulation

NASA Astrophysics Data System (ADS)

Béguin, A.; Nicolet, C.; Hell, J.; Moreira, C.

2017-04-01

The paper explores the improvement in ancillary services that variable speed technologies can provide for the case of an existing pumped storage power plant of 2x210 MVA which conversion from fixed speed to variable speed is investigated with a focus on the power step performances of the units. First two motor-generator variable speed technologies are introduced, namely the Doubly Fed Induction Machine (DFIM) and the Full Scale Frequency Converter (FSFC). Then a detailed numerical simulation model of the investigated power plant used to simulate power steps response and comprising the waterways, the pump-turbine unit, the motor-generator, the grid connection and the control systems is presented. Hydroelectric system time domain simulations are performed in order to determine the shortest response time achievable, taking into account the constraints from the maximum penstock pressure and from the rotational speed limits. It is shown that the maximum instantaneous power step response up and down depends on the hydro-mechanical characteristics of the pump-turbine unit and of the motor-generator speed limits. As a results, for the investigated test case, the FSFC solution offer the best power step response performances.

TRUST84. Sat-Unsat Flow in Deformable Media

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

Narasimhan, T.N.

1984-11-01

TRUST84 solves for transient and steady-state flow in variably saturated deformable media in one, two, or three dimensions. It can handle porous media, fractured media, or fractured-porous media. Boundary conditions may be an arbitrary function of time. Sources or sinks may be a function of time or of potential. The theoretical model considers a general three-dimensional field of flow in conjunction with a one-dimensional vertical deformation field. The governing equation expresses the conservation of fluid mass in an elemental volume that has a constant volume of solids. Deformation of the porous medium may be nonelastic. Permeability and the compressibility coefficientsmore » may be nonlinearly related to effective stress. Relationships between permeability and saturation with pore water pressure in the unsaturated zone may be characterized by hysteresis. The relation between pore pressure change and effective stress change may be a function of saturation. The basic calculational model of the conductive heat transfer code TRUMP is applied in TRUST84 to the flow of fluids in porous media. The model combines an integrated finite difference algorithm for numerically solving the governing equation with a mixed explicit-implicit iterative scheme in which the explicit changes in potential are first computed for all elements in the system, after which implicit corrections are made only for those elements for which the stable time-step is less than the time-step being used. Time-step sizes are automatically controlled to optimize the number of iterations, to control maximum change to potential during a time-step, and to obtain desired output information. Time derivatives, estimated on the basis of system behavior during the two previous time-steps, are used to start the iteration process and to evaluate nonlinear coefficients. Both heterogeneity and anisotropy can be handled.« less

Inverting pump-probe spectroscopy for state tomography of excitonic systems.

PubMed

Hoyer, Stephan; Whaley, K Birgitta

2013-04-28

We propose a two-step protocol for inverting ultrafast spectroscopy experiments on a molecular aggregate to extract the time-evolution of the excited state density matrix. The first step is a deconvolution of the experimental signal to determine a pump-dependent response function. The second step inverts this response function to obtain the quantum state of the system, given a model for how the system evolves following the probe interaction. We demonstrate this inversion analytically and numerically for a dimer model system, and evaluate the feasibility of scaling it to larger molecular aggregates such as photosynthetic protein-pigment complexes. Our scheme provides a direct alternative to the approach of determining all Hamiltonian parameters and then simulating excited state dynamics.

Faster and exact implementation of the continuous cellular automaton for anisotropic etching simulations

NASA Astrophysics Data System (ADS)

Ferrando, N.; Gosálvez, M. A.; Cerdá, J.; Gadea, R.; Sato, K.

2011-02-01

The current success of the continuous cellular automata for the simulation of anisotropic wet chemical etching of silicon in microengineering applications is based on a relatively fast, approximate, constant time stepping implementation (CTS), whose accuracy against the exact algorithm—a computationally slow, variable time stepping implementation (VTS)—has not been previously analyzed in detail. In this study we show that the CTS implementation can generate moderately wrong etch rates and overall etching fronts, thus justifying the presentation of a novel, exact reformulation of the VTS implementation based on a new state variable, referred to as the predicted removal time (PRT), and the use of a self-balanced binary search tree that enables storage and efficient access to the PRT values in each time step in order to quickly remove the corresponding surface atom/s. The proposed PRT method reduces the simulation cost of the exact implementation from {O}(N^{5/3}) to {O}(N^{3/2} log N) without introducing any model simplifications. This enables more precise simulations (only limited by numerical precision errors) with affordable computational times that are similar to the less precise CTS implementation and even faster for low reactivity systems.

A gas-kinetic BGK scheme for the compressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Xu, Kun

2000-01-01

This paper presents an improved gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK) model for the compressible Navier-Stokes equations. The current method extends the previous gas-kinetic Navier-Stokes solver developed by Xu and Prendergast by implementing a general nonequilibrium state to represent the gas distribution function at the beginning of each time step. As a result, the requirement in the previous scheme, such as the particle collision time being less than the time step for the validity of the BGK Navier-Stokes solution, is removed. Therefore, the applicable regime of the current method is much enlarged and the Navier-Stokes solution can be obtained accurately regardless of the ratio between the collision time and the time step. The gas-kinetic Navier-Stokes solver developed by Chou and Baganoff is the limiting case of the current method, and it is valid only under such a limiting condition. Also, in this paper, the appropriate implementation of boundary condition for the kinetic scheme, different kinetic limiting cases, and the Prandtl number fix are presented. The connection among artificial dissipative central schemes, Godunov-type schemes, and the gas-kinetic BGK method is discussed. Many numerical tests are included to validate the current method.

A scale-free network with limiting on vertices

NASA Astrophysics Data System (ADS)

Tang, Lian; Wang, Bin

2010-05-01

We propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity. The random graph process is defined as follows: at any time-step t, (i) with probability α(k) and independently of other time-step, each vertex vi (i≤t-1) is inactive which means it cannot be connected by more edges, where k is the degree of vi at the time-step t; (ii) a new vertex vt is added along with m edges incident with vt at one time and its neighbors are chosen in the manner of preferential attachment. We prove that the degree distribution P(k) of this random graph process satisfies P(k)∝C1k if α(ṡ) is a constant α0; and P(k)∝C2k-3 if α(ℓ)↓0 as ℓ↑∞, where C1,C2 are two positive constants. The analytical result is found to be in good agreement with that obtained by numerical simulations. Furthermore, we get the degree distributions in this model with m-varying functions by simulation.

The role of particle jamming on the formation and stability of step-pool morphology: insight from a reduced-complexity model

NASA Astrophysics Data System (ADS)

Saletti, M.; Molnar, P.; Hassan, M. A.

2017-12-01

Granular processes have been recognized as key drivers in earth surface dynamics, especially in steep landscapes because of the large size of sediment found in channels. In this work we focus on step-pool morphologies, studying the effect of particle jamming on step formation. Starting from the jammed-state hypothesis, we assume that grains generate steps because of particle jamming and those steps are inherently more stable because of additional force chains in the transversal direction. We test this hypothesis with a particle-based reduced-complexity model, CAST2, where sediment is organized in patches and entrainment, transport and deposition of grains depend on flow stage and local topography through simplified phenomenological rules. The model operates with 2 grain sizes: fine grains, that can be mobilized both my large and moderate flows, and coarse grains, mobile only during large floods. First, we identify the minimum set of processes necessary to generate and maintain steps in a numerical channel: (a) occurrence of floods, (b) particle jamming, (c) low sediment supply, and (d) presence of sediment with different entrainment probabilities. Numerical results are compared with field observations collected in different step-pool channels in terms of step density, a variable that captures the proportion of the channel occupied by steps. Not only the longitudinal profiles of numerical channels display step sequences similar to those observed in real step-pool streams, but also the values of step density are very similar when all the processes mentioned before are considered. Moreover, with CAST2 it is possible to run long simulations with repeated flood events, to test the effect of flood frequency on step formation. Numerical results indicate that larger step densities belong to system more frequently perturbed by floods, compared to system having a lower flood frequency. Our results highlight the important interactions between external hydrological forcing and internal geomorphic adjustment (e.g. jamming) on the response of step-pool streams, showing the potential of reduced-complexity models in fluvial geomorphology.

New Reduced Two-Time Step Method for Calculating Combustion and Emission Rates of Jet-A and Methane Fuel With and Without Water Injection

NASA Technical Reports Server (NTRS)

Molnar, Melissa; Marek, C. John

2004-01-01

A simplified kinetic scheme for Jet-A, and methane fuels with water injection was developed to be used in numerical combustion codes, such as the National Combustor Code (NCC) or even simple FORTRAN codes that are being developed at Glenn. The two time step method is either an initial time averaged value (step one) or an instantaneous value (step two). The switch is based on the water concentration in moles/cc of 1x10(exp -20). The results presented here results in a correlation that gives the chemical kinetic time as two separate functions. This two step method is used as opposed to a one step time averaged method previously developed to determine the chemical kinetic time with increased accuracy. The first time averaged step is used at the initial times for smaller water concentrations. This gives the average chemical kinetic time as a function of initial overall fuel air ratio, initial water to fuel mass ratio, temperature, and pressure. The second instantaneous step, to be used with higher water concentrations, gives the chemical kinetic time as a function of instantaneous fuel and water mole concentration, pressure and temperature (T4). The simple correlations would then be compared to the turbulent mixing times to determine the limiting properties of the reaction. The NASA Glenn GLSENS kinetics code calculates the reaction rates and rate constants for each species in a kinetic scheme for finite kinetic rates. These reaction rates were then used to calculate the necessary chemical kinetic times. Chemical kinetic time equations for fuel, carbon monoxide and NOx were obtained for Jet-A fuel and methane with and without water injection to water mass loadings of 2/1 water to fuel. A similar correlation was also developed using data from NASA's Chemical Equilibrium Applications (CEA) code to determine the equilibrium concentrations of carbon monoxide and nitrogen oxide as functions of overall equivalence ratio, water to fuel mass ratio, pressure and temperature (T3). The temperature of the gas entering the turbine (T4) was also correlated as a function of the initial combustor temperature (T3), equivalence ratio, water to fuel mass ratio, and pressure.

Solving the chemical master equation using sliding windows

PubMed Central

2010-01-01

Background The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species. Results In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy. Conclusions The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori. PMID:20377904

An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Pan, Liang; Xu, Kun; Li, Qibing; Li, Jiequan

2016-12-01

For computational fluid dynamics (CFD), the generalized Riemann problem (GRP) solver and the second-order gas-kinetic scheme (GKS) provide a time-accurate flux function starting from a discontinuous piecewise linear flow distributions around a cell interface. With the adoption of time derivative of the flux function, a two-stage Lax-Wendroff-type (L-W for short) time stepping method has been recently proposed in the design of a fourth-order time accurate method for inviscid flow [21]. In this paper, based on the same time-stepping method and the second-order GKS flux function [42], a fourth-order gas-kinetic scheme is constructed for the Euler and Navier-Stokes (NS) equations. In comparison with the formal one-stage time-stepping third-order gas-kinetic solver [24], the current fourth-order method not only reduces the complexity of the flux function, but also improves the accuracy of the scheme. In terms of the computational cost, a two-dimensional third-order GKS flux function takes about six times of the computational time of a second-order GKS flux function. However, a fifth-order WENO reconstruction may take more than ten times of the computational cost of a second-order GKS flux function. Therefore, it is fully legitimate to develop a two-stage fourth order time accurate method (two reconstruction) instead of standard four stage fourth-order Runge-Kutta method (four reconstruction). Most importantly, the robustness of the fourth-order GKS is as good as the second-order one. In the current computational fluid dynamics (CFD) research, it is still a difficult problem to extend the higher-order Euler solver to the NS one due to the change of governing equations from hyperbolic to parabolic type and the initial interface discontinuity. This problem remains distinctively for the hypersonic viscous and heat conducting flow. The GKS is based on the kinetic equation with the hyperbolic transport and the relaxation source term. The time-dependent GKS flux function provides a dynamic process of evolution from the kinetic scale particle free transport to the hydrodynamic scale wave propagation, which provides the physics for the non-equilibrium numerical shock structure construction to the near equilibrium NS solution. As a result, with the implementation of the fifth-order WENO initial reconstruction, in the smooth region the current two-stage GKS provides an accuracy of O ((Δx) 5 ,(Δt) 4) for the Euler equations, and O ((Δx) 5 ,τ2 Δt) for the NS equations, where τ is the time between particle collisions. Many numerical tests, including difficult ones for the Navier-Stokes solvers, have been used to validate the current method. Perfect numerical solutions can be obtained from the high Reynolds number boundary layer to the hypersonic viscous heat conducting flow. Following the two-stage time-stepping framework, the third-order GKS flux function can be used as well to construct a fifth-order method with the usage of both first-order and second-order time derivatives of the flux function. The use of time-accurate flux function may have great advantages on the development of higher-order CFD methods.

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chen, Lin-Jie; Ma, Chang-Feng

2010-01-01

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form ut + αuux + βunux + γuxx + δuxxx + ζuxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

Lattice Boltzmann model for numerical relativity.

PubMed

Ilseven, E; Mendoza, M

2016-02-01

In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.

Computation of Transonic Nozzle Sound Transmission and Rotor Problems by the Dispersion-Relation-Preserving Scheme

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.; Aganin, Alexei

2000-01-01

The transonic nozzle transmission problem and the open rotor noise radiation problem are solved computationally. Both are multiple length scales problems. For efficient and accurate numerical simulation, the multiple-size-mesh multiple-time-step Dispersion-Relation-Preserving scheme is used to calculate the time periodic solution. To ensure an accurate solution, high quality numerical boundary conditions are also needed. For the nozzle problem, a set of nonhomogeneous, outflow boundary conditions are required. The nonhomogeneous boundary conditions not only generate the incoming sound waves but also, at the same time, allow the reflected acoustic waves and entropy waves, if present, to exit the computation domain without reflection. For the open rotor problem, there is an apparent singularity at the axis of rotation. An analytic extension approach is developed to provide a high quality axis boundary treatment.

Long-Time Numerical Integration of the Three-Dimensional Wave Equation in the Vicinity of a Moving Source

NASA Technical Reports Server (NTRS)

Ryabenkii, V. S.; Turchaninov, V. I.; Tsynkov, S. V.

1999-01-01

We propose a family of algorithms for solving numerically a Cauchy problem for the three-dimensional wave equation. The sources that drive the equation (i.e., the right-hand side) are compactly supported in space for any given time; they, however, may actually move in space with a subsonic speed. The solution is calculated inside a finite domain (e.g., sphere) that also moves with a subsonic speed and always contains the support of the right-hand side. The algorithms employ a standard consistent and stable explicit finite-difference scheme for the wave equation. They allow one to calculate tile solution for arbitrarily long time intervals without error accumulation and with the fixed non-growing amount of tile CPU time and memory required for advancing one time step. The algorithms are inherently three-dimensional; they rely on the presence of lacunae in the solutions of the wave equation in oddly dimensional spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains.

Step-by-step design of a single phase 3.3 kV/200 a resistive type superconducting fault current limiter (R-SFCL) and cryostat

NASA Astrophysics Data System (ADS)

Kar, Soumen; Rao, V. V.

2018-07-01

In our first attempt to design a single phase R-SFCL in India, we have chosen the typical rating of a medium voltage level (3.3 kVrms, 200 Arms, 1Φ) R-SFCL. The step-by-step design procedure for the R-SFCL involves conductor selection, time dependent electro-thermal simulations and recovery time optimization after fault removal. In the numerical analysis, effective fault limitation for a fault current of 5 kA for the medium voltage level R-SFCL are simulated. Maximum normal state resistance and maximum temperature rise in the SFCL coil during current limitation are estimated using one-dimensional energy balance equation. Further, a cryogenic system is conceptually designed for aforesaid MV level R-SFCL by considering inner and outer vessel materials, wall-thickness and thermal insulation which can be used for R-SFCL system. Finally, the total thermal load is calculated for the designed R-SFCL cryostat to select a suitable cryo-refrigerator for LN2 re-condensation.

Adaptive dynamic programming for discrete-time linear quadratic regulation based on multirate generalised policy iteration

NASA Astrophysics Data System (ADS)

Chun, Tae Yoon; Lee, Jae Young; Park, Jin Bae; Choi, Yoon Ho

2018-06-01

In this paper, we propose two multirate generalised policy iteration (GPI) algorithms applied to discrete-time linear quadratic regulation problems. The proposed algorithms are extensions of the existing GPI algorithm that consists of the approximate policy evaluation and policy improvement steps. The two proposed schemes, named heuristic dynamic programming (HDP) and dual HDP (DHP), based on multirate GPI, use multi-step estimation (M-step Bellman equation) at the approximate policy evaluation step for estimating the value function and its gradient called costate, respectively. Then, we show that these two methods with the same update horizon can be considered equivalent in the iteration domain. Furthermore, monotonically increasing and decreasing convergences, so called value iteration (VI)-mode and policy iteration (PI)-mode convergences, are proved to hold for the proposed multirate GPIs. Further, general convergence properties in terms of eigenvalues are also studied. The data-driven online implementation methods for the proposed HDP and DHP are demonstrated and finally, we present the results of numerical simulations performed to verify the effectiveness of the proposed methods.

2–stage stochastic Runge–Kutta for stochastic delay differential equations

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

Rosli, Norhayati; Jusoh Awang, Rahimah; Bahar, Arifah

2015-05-15

This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

Double-slit interferometry with a Bose-Einstein condensate

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

Collins, L.A.; Berman, G.P.; Bishop, A.R.

2005-03-01

A Bose-Einstein 'double-slit' interferometer has been recently realized experimentally by Y. Shin et al., Phys. Rev. Lett. 92 050405 (2004). We analyze the interferometric steps by solving numerically the time-dependent Gross-Pitaevskii equation in three-dimensional space. We focus on the adiabaticity time scales of the problem and on the creation of spurious collective excitations as a possible source of the strong degradation of the interference pattern observed experimentally. The role of quantum fluctuations is discussed.

A systematic methodology for creep master curve construction using the stepped isostress method (SSM): a numerical assessment

NASA Astrophysics Data System (ADS)

Miranda Guedes, Rui

2018-02-01

Long-term creep of viscoelastic materials is experimentally inferred through accelerating techniques based on the time-temperature superposition principle (TTSP) or on the time-stress superposition principle (TSSP). According to these principles, a given property measured for short times at a higher temperature or higher stress level remains the same as that obtained for longer times at a lower temperature or lower stress level, except that the curves are shifted parallel to the horizontal axis, matching a master curve. These procedures enable the construction of creep master curves with short-term experimental tests. The Stepped Isostress Method (SSM) is an evolution of the classical TSSP method. Higher reduction of the required number of test specimens to obtain the master curve is achieved by the SSM technique, since only one specimen is necessary. The classical approach, using creep tests, demands at least one specimen per each stress level to produce a set of creep curves upon which TSSP is applied to obtain the master curve. This work proposes an analytical method to process the SSM raw data. The method is validated using numerical simulations to reproduce the SSM tests based on two different viscoelastic models. One model represents the viscoelastic behavior of a graphite/epoxy laminate and the other represents an adhesive based on epoxy resin.

Numerical Analysis on Tensile Properties of Grout-filled Splice Sleeve Rebars under ISO 834 Standard Fire

NASA Astrophysics Data System (ADS)

Liu, Yong Jun; Li, Chao; Zhou, When Jun

2018-06-01

This paper presents some numerical simulation results of tensile properties of reinforcing bars spliced by grout-filled coupling sleeves under fire conditions to identify the effect of load ratio on fire resistance time of spliced reinforcing bars, which provide a useful base for predicting structural behaviors of pre-cast reinforced concrete buildings in fires. A spliced rebar system investigated in this paper consists of two equal-diameter steel reinforcing bars with 25mm diameter and a straight coupling sleeve with 50mm outer and 45mm inner diameters. As a result, the thickness of grout between steel bars and sleeves are 20mm. Firstly, the temperature distributions in steel bars connected by grout- filled coupling sleeves exposed to ISO 834 standard fire were calculated utilizing finite element analysis software ANSYS. Secondly, the stress changes in heated steel bars connected by grout-filled coupling sleeves under different constant tensile loads were calculated step by step until the rebar system failed due to fire. Thus, the fire resistant time of rebar spliced by grout-filled coupling sleeves under different axial tensile loads can be determined, further, the relationship between fire resistance time and axial tensile loads ratio can could be obtained. Finally, the fire resistant times versus axial tensile load ratios curve of grout-filled splice sleeve rebars exposed to ISO 834 standard fire is presented.

High-order scheme for the source-sink term in a one-dimensional water temperature model

PubMed Central

Jing, Zheng; Kang, Ling

2017-01-01

The source-sink term in water temperature models represents the net heat absorbed or released by a water system. This term is very important because it accounts for solar radiation that can significantly affect water temperature, especially in lakes. However, existing numerical methods for discretizing the source-sink term are very simplistic, causing significant deviations between simulation results and measured data. To address this problem, we present a numerical method specific to the source-sink term. A vertical one-dimensional heat conduction equation was chosen to describe water temperature changes. A two-step operator-splitting method was adopted as the numerical solution. In the first step, using the undetermined coefficient method, a high-order scheme was adopted for discretizing the source-sink term. In the second step, the diffusion term was discretized using the Crank-Nicolson scheme. The effectiveness and capability of the numerical method was assessed by performing numerical tests. Then, the proposed numerical method was applied to a simulation of Guozheng Lake (located in central China). The modeling results were in an excellent agreement with measured data. PMID:28264005

High-order scheme for the source-sink term in a one-dimensional water temperature model.

PubMed

Jing, Zheng; Kang, Ling

2017-01-01

The source-sink term in water temperature models represents the net heat absorbed or released by a water system. This term is very important because it accounts for solar radiation that can significantly affect water temperature, especially in lakes. However, existing numerical methods for discretizing the source-sink term are very simplistic, causing significant deviations between simulation results and measured data. To address this problem, we present a numerical method specific to the source-sink term. A vertical one-dimensional heat conduction equation was chosen to describe water temperature changes. A two-step operator-splitting method was adopted as the numerical solution. In the first step, using the undetermined coefficient method, a high-order scheme was adopted for discretizing the source-sink term. In the second step, the diffusion term was discretized using the Crank-Nicolson scheme. The effectiveness and capability of the numerical method was assessed by performing numerical tests. Then, the proposed numerical method was applied to a simulation of Guozheng Lake (located in central China). The modeling results were in an excellent agreement with measured data.

EXPONENTIAL TIME DIFFERENCING FOR HODGKIN–HUXLEY-LIKE ODES

PubMed Central

Börgers, Christoph; Nectow, Alexander R.

2013-01-01

Several authors have proposed the use of exponential time differencing (ETD) for Hodgkin–Huxley-like partial and ordinary differential equations (PDEs and ODEs). For Hodgkin–Huxley-like PDEs, ETD is attractive because it can deal effectively with the stiffness issues that diffusion gives rise to. However, large neuronal networks are often simulated assuming “space-clamped” neurons, i.e., using the Hodgkin–Huxley ODEs, in which there are no diffusion terms. Our goal is to clarify whether ETD is a good idea even in that case. We present a numerical comparison of first- and second-order ETD with standard explicit time-stepping schemes (Euler’s method, the midpoint method, and the classical fourth-order Runge–Kutta method). We find that in the standard schemes, the stable computation of the very rapid rising phase of the action potential often forces time steps of a small fraction of a millisecond. This can result in an expensive calculation yielding greater overall accuracy than needed. Although it is tempting at first to try to address this issue with adaptive or fully implicit time-stepping, we argue that neither is effective here. The main advantage of ETD for Hodgkin–Huxley-like systems of ODEs is that it allows underresolution of the rising phase of the action potential without causing instability, using time steps on the order of one millisecond. When high quantitative accuracy is not necessary and perhaps, because of modeling inaccuracies, not even useful, ETD allows much faster simulations than standard explicit time-stepping schemes. The second-order ETD scheme is found to be substantially more accurate than the first-order one even for large values of Δt. PMID:24058276

Geometric multigrid for an implicit-time immersed boundary method

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

Guy, Robert D.; Philip, Bobby; Griffith, Boyce E.

2014-10-12

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methodsmore » require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Moreover, several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. Finally, these tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.« less

Index Fund Selections with GAs and Classifications Based on Turnover

NASA Astrophysics Data System (ADS)

Orito, Yukiko; Motoyama, Takaaki; Yamazaki, Genji

It is well known that index fund selections are important for the risk hedge of investment in a stock market. The`selection’means that for`stock index futures’, n companies of all ones in the market are selected. For index fund selections, Orito et al.(6) proposed a method consisting of the following two steps : Step 1 is to select N companies in the market with a heuristic rule based on the coefficient of determination between the return rate of each company in the market and the increasing rate of the stock price index. Step 2 is to construct a group of n companies by applying genetic algorithms to the set of N companies. We note that the rule of Step 1 is not unique. The accuracy of the results using their method depends on the length of time data (price data) in the experiments. The main purpose of this paper is to introduce a more`effective rule’for Step 1. The rule is based on turnover. The method consisting of Step 1 based on turnover and Step 2 is examined with numerical experiments for the 1st Section of Tokyo Stock Exchange. The results show that with our method, it is possible to construct the more effective index fund than the results of Orito et al.(6). The accuracy of the results using our method depends little on the length of time data (turnover data). The method especially works well when the increasing rate of the stock price index over a period can be viewed as a linear time series data.

Numerical analysis of the three-dimensional swirling flow in centrifugal compressor volutes

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

Ayder, E.; Van den Braembussche, R.

1994-07-01

The improvement of centrifugal compressor performance and the control of the radial forces acting on the impeller due to the circumferential variation of the static pressure caused by the volute require a good understanding of the flow mechanisms and an accurate prediction of the flow pattern inside the volute. A three-dimensional volute calculation method has been developed for this purpose. The volute is discretized by means of hexahedral elements. A cell vertex finite volume approach is used in combination with a time-marching procedure. The numerical procedure makes use of a central space discretization and a four-step Runge-Kutta time-stepping scheme. Themore » artificial dissipation used in the solver is based on the fourth-order differences of the conservative variables. Implicit residual smoothing improves the convergence rate. The loss model implemented in the code accounts for the losses due to internal shear and friction losses on the walls. A comparison of the calculated and measured results inside a volute with elliptical cross section reveals that the modified Euler solver accurately predicts the velocity and pressure distribution inside and upstream of the volute.« less

Anomalous barrier escape: The roles of noise distribution and correlation.

PubMed

Hu, Meng; Zhang, Jia-Ming; Bao, Jing-Dong

2017-05-28

We study numerically and analytically the barrier escape dynamics of a particle driven by an underlying correlated Lévy noise for a smooth metastable potential. A "quasi-monochrome-color" Lévy noise, i.e., the first-order derivative variable of a linear second-order differential equation subjected to a symmetric α-stable white Lévy noise, also called the harmonic velocity Lévy noise, is proposed. Note that the time-integral of the noise Green function of this kind is equal to zero. This leads to the existence of underlying negative time correlation and implies that a step in one direction is likely followed by a step in the other direction. By using the noise of this kind as a driving source, we discuss the competition between long flights and underlying negative correlations in the metastable dynamics. The quite rich behaviors in the parameter space including an optimum α for the stationary escape rate have been found. Remarkably, slow diffusion does not decrease the stationary rate while a negative correlation increases net escape. An approximate expression for the Lévy-Kramers rate is obtained to support the numerically observed dependencies.

Anomalous barrier escape: The roles of noise distribution and correlation

NASA Astrophysics Data System (ADS)

Hu, Meng; Zhang, Jia-Ming; Bao, Jing-Dong

2017-05-01

We study numerically and analytically the barrier escape dynamics of a particle driven by an underlying correlated Lévy noise for a smooth metastable potential. A "quasi-monochrome-color" Lévy noise, i.e., the first-order derivative variable of a linear second-order differential equation subjected to a symmetric α-stable white Lévy noise, also called the harmonic velocity Lévy noise, is proposed. Note that the time-integral of the noise Green function of this kind is equal to zero. This leads to the existence of underlying negative time correlation and implies that a step in one direction is likely followed by a step in the other direction. By using the noise of this kind as a driving source, we discuss the competition between long flights and underlying negative correlations in the metastable dynamics. The quite rich behaviors in the parameter space including an optimum α for the stationary escape rate have been found. Remarkably, slow diffusion does not decrease the stationary rate while a negative correlation increases net escape. An approximate expression for the Lévy-Kramers rate is obtained to support the numerically observed dependencies.

Application of an Evolution Strategy in Planetary Ephemeris Optimization

NASA Astrophysics Data System (ADS)

Mai, E.

2016-12-01

Classical planetary ephemeris construction comprises three major steps, which are performed iteratively: simultaneous numerical integration of coupled equations of motion of a multi-body system (propagator step), reduction of thousands of observations (reduction step), and optimization of various selected model parameters (adjustment step). This traditional approach is challenged by ongoing refinements in force modeling, e.g. inclusion of much more significant minor bodies, an ever-growing number of planetary observations, e.g. vast amount of spacecraft tracking data, etc. To master the high computational burden and in order to circumvent the need for inversion of huge normal equation matrices, we propose an alternative ephemeris construction method. The main idea is to solve the overall optimization problem by a straightforward direct evaluation of the whole set of mathematical formulas involved, rather than to solve it as an inverse problem with all its tacit mathematical assumptions and numerical difficulties. We replace the usual gradient search by a stochastic search, namely an evolution strategy, the latter of which is also perfect for the exploitation of parallel computing capabilities. Furthermore, this new approach enables multi-criteria optimization and time-varying optima. This issue will become important in future once ephemeris construction is just one part of even larger optimization problems, e.g. the combined and consistent determination of the physical state (orbit, size, shape, rotation, gravity,…) of celestial bodies (planets, satellites, asteroids, or comets), and if one seeks near real-time solutions. Here we outline the general idea and discuss first results. As an example, we present a simultaneous optimization of high-correlated asteroidal ring model parameters (total mass and heliocentric radius), based on simulations.

Tokamak magneto-hydrodynamics and reference magnetic coordinates for simulations of plasma disruptions

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

Zakharov, Leonid E.; Li, Xujing

This paper formulates the Tokamak Magneto-Hydrodynamics (TMHD), initially outlined by X. Li and L. E. Zakharov [Plasma Science and Technology 17(2), 97–104 (2015)] for proper simulations of macroscopic plasma dynamics. The simplest set of magneto-hydrodynamics equations, sufficient for disruption modeling and extendable to more refined physics, is explained in detail. First, the TMHD introduces to 3-D simulations the Reference Magnetic Coordinates (RMC), which are aligned with the magnetic field in the best possible way. The numerical implementation of RMC is adaptive grids. Being consistent with the high anisotropy of the tokamak plasma, RMC allow simulations at realistic, very high plasmamore » electric conductivity. Second, the TMHD splits the equation of motion into an equilibrium equation and the plasma advancing equation. This resolves the 4 decade old problem of Courant limitations of the time step in existing, plasma inertia driven numerical codes. The splitting allows disruption simulations on a relatively slow time scale in comparison with the fast time of ideal MHD instabilities. A new, efficient numerical scheme is proposed for TMHD.« less

Effects of Conjugate Gradient Methods and Step-Length Formulas on the Multiscale Full Waveform Inversion in Time Domain: Numerical Experiments

NASA Astrophysics Data System (ADS)

Liu, Youshan; Teng, Jiwen; Xu, Tao; Badal, José; Liu, Qinya; Zhou, Bing

2017-05-01

We carry out full waveform inversion (FWI) in time domain based on an alternative frequency-band selection strategy that allows us to implement the method with success. This strategy aims at decomposing the seismic data within partially overlapped frequency intervals by carrying out a concatenated treatment of the wavelet to largely avoid redundant frequency information to adapt to wavelength or wavenumber coverage. A pertinent numerical test proves the effectiveness of this strategy. Based on this strategy, we comparatively analyze the effects of update parameters for the nonlinear conjugate gradient (CG) method and step-length formulas on the multiscale FWI through several numerical tests. The investigations of up to eight versions of the nonlinear CG method with and without Gaussian white noise make clear that the HS (Hestenes and Stiefel in J Res Natl Bur Stand Sect 5:409-436, 1952), CD (Fletcher in Practical methods of optimization vol. 1: unconstrained optimization, Wiley, New York, 1987), and PRP (Polak and Ribière in Revue Francaise Informat Recherche Opertionelle, 3e Année 16:35-43, 1969; Polyak in USSR Comput Math Math Phys 9:94-112, 1969) versions are more efficient among the eight versions, while the DY (Dai and Yuan in SIAM J Optim 10:177-182, 1999) version always yields inaccurate result, because it overestimates the deeper parts of the model. The application of FWI algorithms using distinct step-length formulas, such as the direct method ( Direct), the parabolic search method ( Search), and the two-point quadratic interpolation method ( Interp), proves that the Interp is more efficient for noise-free data, while the Direct is more efficient for Gaussian white noise data. In contrast, the Search is less efficient because of its slow convergence. In general, the three step-length formulas are robust or partly insensitive to Gaussian white noise and the complexity of the model. When the initial velocity model deviates far from the real model or the data are contaminated by noise, the objective function values of the Direct and Interp are oscillating at the beginning of the inversion, whereas that of the Search decreases consistently.

A New Microscopic Model of the Rate- and State- Friction Evolution

NASA Astrophysics Data System (ADS)

Li, T.; Rubin, A. M.

2016-12-01

The Slip (Ruina) law and the Aging (Dieterich) law are the two most common descriptions of the evolution of "state" in rate- and state-dependent friction, behind which are the ideas of slip-dependent and time-dependent fault healing, respectively. Since the mid-1990's, friction experiments have been interpreted as demonstrating that fault healing in rock is primarily time-dependent, and that frictional strength is proportional to contact area (Dieterich and Kilgore, 1994; Beeler et al., 1994). However, a recent re-examination of the data of Beeler et al. (1994) suggests that the evidence for time-dependent healing is equivocal, while large step velocity decreases provide unequivocal evidence of slip-dependent healing (Bhattacharya et al., AGU 2016). Nonetheless, unlike the Aging law, for which see-through experiments showing growing contacts could serve as a physical model, there has been no corresponding physical picture for the Slip law. In this study, we develop a new microscopic model of friction in which each asperity has a heterogeneous strength, with individual portions "remembering" the velocity at which they came into existence. Such a scenario could arise via processes that are more efficient at the margin of a contact than within the interior (e.g., chemical diffusion). A numerical kernel for friction evolution is developed for arbitrary slip histories and an exponential distribution of asperity sizes. For velocity steps we derive an analytical expression that is essentially the Slip law. Numerical inversions show that this model performs as well as the Slip law when fitting velocity step data, but (unfortunately) without improving much the fit to slide-hold-slide data. Because "state" as defined by the Aging law has traditionally been interpreted as contact age, we also use our model to determine whether the "Aging law" actually tracks contact age for general velocity histories. As is traditional, we assume that strength increases logarithmically with age. For reasonable definitions of "age" we obtain results significantly different from the Aging law for velocity step increases. Interestingly, we can obtain an analytical solution for velocity steps that is very close to the Aging law if we adopt a definition of age that we consider to be non-physical.

Numerical simulation of two-phase filtration in the near well bore zone

NASA Astrophysics Data System (ADS)

Maksat, Kalimoldayev; Kalipa, Kuspanova; Kulyash, Baisalbayeva; Orken, Mamyrbayev; Assel, Abdildayeva

2018-04-01

On the basis of the fundamental laws of energy conservation, nonstationary processes of filtration of two-phase liquids in multilayered reservoirs in the near well bore zone are considered. Number of reservoirs, fluid pressure in the given reservoirs, reservoir permeability, oil viscosity, etc. are taken into account upon that. Plane-parallel flow and axisymmetric cases have been studied. In the numerical solution, non-structured meshes are used. Closer to the well, the meshes thicken. The integration step over time is defined by the generalized Courant inequality. As a result, there are no large oscillations in the numerical solutions obtained. Oil production rates, Poisson's ratios, D-diameters of the well, filter height, filter permeability, and cumulative thickness of the filter cake and the area have been taken as the main inputs in numerical simulation of non-stationary processes of two-phase filtration.

A Parallel Numerical Algorithm To Solve Linear Systems Of Equations Emerging From 3D Radiative Transfer

NASA Astrophysics Data System (ADS)

Wichert, Viktoria; Arkenberg, Mario; Hauschildt, Peter H.

2016-10-01

Highly resolved state-of-the-art 3D atmosphere simulations will remain computationally extremely expensive for years to come. In addition to the need for more computing power, rethinking coding practices is necessary. We take a dual approach by introducing especially adapted, parallel numerical methods and correspondingly parallelizing critical code passages. In the following, we present our respective work on PHOENIX/3D. With new parallel numerical algorithms, there is a big opportunity for improvement when iteratively solving the system of equations emerging from the operator splitting of the radiative transfer equation J = ΛS. The narrow-banded approximate Λ-operator Λ* , which is used in PHOENIX/3D, occurs in each iteration step. By implementing a numerical algorithm which takes advantage of its characteristic traits, the parallel code's efficiency is further increased and a speed-up in computational time can be achieved.

Galerkin v. discrete-optimal projection in nonlinear model reduction

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

Carlberg, Kevin Thomas; Barone, Matthew Franklin; Antil, Harbir

Discrete-optimal model-reduction techniques such as the Gauss{Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible ow problems where standard Galerkin techniques have failed. However, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform projection at the time-continuous level, while discrete-optimal techniques do so at the time-discrete level. This work provides a detailed theoretical and experimental comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge{Kutta schemes.more » We present a number of new ndings, including conditions under which the discrete-optimal ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and experimentally that decreasing the time step does not necessarily decrease the error for the discrete-optimal ROM; instead, the time step should be `matched' to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible- ow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the discrete-optimal reduced-order model by an order of magnitude.« less

Contact-aware simulations of particulate Stokesian suspensions

NASA Astrophysics Data System (ADS)

Lu, Libin; Rahimian, Abtin; Zorin, Denis

2017-10-01

We present an efficient, accurate, and robust method for simulation of dense suspensions of deformable and rigid particles immersed in Stokesian fluid in two dimensions. We use a well-established boundary integral formulation for the problem as the foundation of our approach. This type of formulation, with a high-order spatial discretization and an implicit and adaptive time discretization, have been shown to be able to handle complex interactions between particles with high accuracy. Yet, for dense suspensions, very small time-steps or expensive implicit solves as well as a large number of discretization points are required to avoid non-physical contact and intersections between particles, leading to infinite forces and numerical instability. Our method maintains the accuracy of previous methods at a significantly lower cost for dense suspensions. The key idea is to ensure interference-free configuration by introducing explicit contact constraints into the system. While such constraints are unnecessary in the formulation, in the discrete form of the problem, they make it possible to eliminate catastrophic loss of accuracy by preventing contact explicitly. Introducing contact constraints results in a significant increase in stable time-step size for explicit time-stepping, and a reduction in the number of points adequate for stability.

Modified Chebyshev Picard Iteration for Efficient Numerical Integration of Ordinary Differential Equations

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 presented to compare the output from the MCPI library to current state-of-practice numerical integration methods. It is shown that MCPI is capable of out-performing the state-of-practice in terms of computational cost and accuracy.

Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

PubMed Central

Kazeev, Vladimir; Khammash, Mustafa; Nip, Michael; Schwab, Christoph

2014-01-01

The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to “lift” this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sub-linearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasi-optimal. We show how the CME can be represented in QTT format, then use the exponentially-converging -discontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTT-structured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the “basis” of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational complexity. Our approach is demonstrated by applying it to three different examples from systems biology: independent birth-death process, an example of enzymatic futile cycle, and a stochastic switch model. The numerical results on these examples demonstrate that the proposed QTT method achieves dramatic speedups and several orders of magnitude storage savings over direct approaches. PMID:24626049

A 2-DOF model of an elastic rocket structure excited by a follower force

NASA Astrophysics Data System (ADS)

Brejão, Leandro F.; da Fonseca Brasil, Reyolando Manoel L. R.

2017-10-01

We present a two degree of freedom model of an elastic rocket structure excited by the follower force given by the motor thrust that is supposed to be always in the direction of the tangent to the deformed shape of the device at its lower tip. The model comprises two massless rigid pinned bars, initially in vertical position, connected by rotational springs. Lumped masses and dampers are considered at the connections. The generalized coordinates are the angular displacements of the bars with respect to the vertical. We derive the equations of motion via Lagrange’s equations and simulate its time evolution using Runge-Kutta 4th order time step-by-step numerical integration algorithm. Results indicate possible occurrence of stable and unstable vibrations, such as limit cycles.

Multidimensional FEM-FCT schemes for arbitrary time stepping

NASA Astrophysics Data System (ADS)

Kuzmin, D.; Möller, M.; Turek, S.

2003-05-01

The flux-corrected-transport paradigm is generalized to finite-element schemes based on arbitrary time stepping. A conservative flux decomposition procedure is proposed for both convective and diffusive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory low-order method is constructed by elimination of negative off-diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non-linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions.

A new fourth-order Fourier-Bessel split-step method for the extended nonlinear Schroedinger equation

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

Nash, Patrick L.

2008-01-10

Fourier split-step techniques are often used to compute soliton-like numerical solutions of the nonlinear Schroedinger equation. Here, a new fourth-order implementation of the Fourier split-step algorithm is described for problems possessing azimuthal symmetry in 3 + 1-dimensions. This implementation is based, in part, on a finite difference approximation {delta}{sub perpendicular} {sup FDA} of 1/r ({partial_derivative})/({partial_derivative}r) r({partial_derivative})/({partial_derivative}r) that possesses an associated exact unitary representation of e{sup i/2{lambda}}{sup {delta}{sub perpendicular}{sup FDA}}. The matrix elements of this unitary matrix are given by special functions known as the associated Bessel functions. Hence the attribute Fourier-Bessel for the method. The Fourier-Bessel algorithm is shown tomore » be unitary and unconditionally stable. The Fourier-Bessel algorithm is employed to simulate the propagation of a periodic series of short laser pulses through a nonlinear medium. This numerical simulation calculates waveform intensity profiles in a sequence of planes that are transverse to the general propagation direction, and labeled by the cylindrical coordinate z. These profiles exhibit a series of isolated pulses that are offset from the time origin by characteristic times, and provide evidence for a physical effect that may be loosely termed normal mode condensation. Normal mode condensation is consistent with experimentally observed pulse filamentation into a packet of short bursts, which may occur as a result of short, intense irradiation of a medium.« less

Modeling Epidemics with Dynamic Small-World Networks

NASA Astrophysics Data System (ADS)

Kaski, Kimmo; Saramäki, Jari

2005-06-01

In this presentation a minimal model for describing the spreading of an infectious disease, such as influenza, is discussed. Here it is assumed that spreading takes place on a dynamic small-world network comprising short- and long-range infection events. Approximate equations for the epidemic threshold as well as the spreading dynamics are derived and they agree well with numerical discrete time-step simulations. Also the dependence of the epidemic saturation time on the initial conditions is analysed and a comparison with real-world data is made.

Impact of implementation choices on quantitative predictions of cell-based computational models

NASA Astrophysics Data System (ADS)

Kursawe, Jochen; Baker, Ruth E.; Fletcher, Alexander G.

2017-09-01

'Cell-based' models provide a powerful computational tool for studying the mechanisms underlying the growth and dynamics of biological tissues in health and disease. An increasing amount of quantitative data with cellular resolution has paved the way for the quantitative parameterisation and validation of such models. However, the numerical implementation of cell-based models remains challenging, and little work has been done to understand to what extent implementation choices may influence model predictions. Here, we consider the numerical implementation of a popular class of cell-based models called vertex models, which are often used to study epithelial tissues. In two-dimensional vertex models, a tissue is approximated as a tessellation of polygons and the vertices of these polygons move due to mechanical forces originating from the cells. Such models have been used extensively to study the mechanical regulation of tissue topology in the literature. Here, we analyse how the model predictions may be affected by numerical parameters, such as the size of the time step, and non-physical model parameters, such as length thresholds for cell rearrangement. We find that vertex positions and summary statistics are sensitive to several of these implementation parameters. For example, the predicted tissue size decreases with decreasing cell cycle durations, and cell rearrangement may be suppressed by large time steps. These findings are counter-intuitive and illustrate that model predictions need to be thoroughly analysed and implementation details carefully considered when applying cell-based computational models in a quantitative setting.

Quantization improves stabilization of dynamical systems with delayed feedback

NASA Astrophysics Data System (ADS)

Stepan, Gabor; Milton, John G.; Insperger, Tamas

2017-11-01

We show that an unstable scalar dynamical system with time-delayed feedback can be stabilized by quantizing the feedback. The discrete time model corresponds to a previously unrecognized case of the microchaotic map in which the fixed point is both locally and globally repelling. In the continuous-time model, stabilization by quantization is possible when the fixed point in the absence of feedback is an unstable node, and in the presence of feedback, it is an unstable focus (spiral). The results are illustrated with numerical simulation of the unstable Hayes equation. The solutions of the quantized Hayes equation take the form of oscillations in which the amplitude is a function of the size of the quantization step. If the quantization step is sufficiently small, the amplitude of the oscillations can be small enough to practically approximate the dynamics around a stable fixed point.

Algorithmically scalable block preconditioner for fully implicit shallow-water equations in CAM-SE

DOE PAGES

Lott, P. Aaron; Woodward, Carol S.; Evans, Katherine J.

2014-10-19

Performing accurate and efficient numerical simulation of global atmospheric climate models is challenging due to the disparate length and time scales over which physical processes interact. Implicit solvers enable the physical system to be integrated with a time step commensurate with processes being studied. The dominant cost of an implicit time step is the ancillary linear system solves, so we have developed a preconditioner aimed at improving the efficiency of these linear system solves. Our preconditioner is based on an approximate block factorization of the linearized shallow-water equations and has been implemented within the spectral element dynamical core within themore » Community Atmospheric Model (CAM-SE). Furthermore, in this paper we discuss the development and scalability of the preconditioner for a suite of test cases with the implicit shallow-water solver within CAM-SE.« less

HARPA: A versatile three-dimensional Hamiltonian ray-tracing program for acoustic waves in the atmosphere above irregular terrain

NASA Astrophysics Data System (ADS)

Jones, R. M.; Riley, J. P.; Georges, T. M.

1986-08-01

The modular FORTRAN 77 computer program traces the three-dimensional paths of acoustic rays through continuous model atmospheres by numerically integrating Hamilton's equations (a differential expression of Fermat's principle). The user specifies an atmospheric model by writing closed-form formulas for its three-dimensional wind and temperature (or sound speed) distribution, and by defining the height of the reflecting terrain vs. geographic latitude and longitude. Some general-purpose models are provided, or users can readily design their own. In addition to computing the geometry of each raypath, HARPA can calculate pulse travel time, phase time, Doppler shift (if the medium varies in time), absorption, and geometrical path length. The program prints a step-by-step account of a ray's progress. The 410-page documentation describes the ray-tracing equations and the structure of the program, and provides complete instructions, illustrated by a sample case.

A second-order accurate immersed boundary-lattice Boltzmann method for particle-laden flows

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

Zhou, Qiang; Fan, Liang-Shih, E-mail: fan.1@osu.edu

A new immersed boundary-lattice Boltzmann method (IB-LBM) is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The immersed boundary method (IBM) recently developed by Breugem (2012) [19] is adopted in the present method, development including the retraction technique, the multi-direct forcing method and the direct account of the inertia of the fluid contained within the particles. The present IB-LBM is, however, formulated with further improvement with the implementation of the high-order Runge–Kutta schemes in the coupled fluid–particle interaction. The major challenge to implement high-order Runge–Kutta schemes in the LBM is that the flow information suchmore » as density and velocity cannot be directly obtained at a fractional time step from the LBM since the LBM only provides the flow information at an integer time step. This challenge can be, however, overcome as given in the present IB-LBM by extrapolating the flow field around particles from the known flow field at the previous integer time step. The newly calculated fluid–particle interactions from the previous fractional time steps of the current integer time step are also accounted for in the extrapolation. The IB-LBM with high-order Runge–Kutta schemes developed in this study is validated by several benchmark applications. It is demonstrated, for the first time, that the IB-LBM has the capacity to resolve the translational and rotational motion of particles with the second-order accuracy. The optimal retraction distances for spheres and tubes that help the method achieve the second-order accuracy are found to be around 0.30 and −0.47 times of the lattice spacing, respectively. Simulations of the Stokes flow through a simple cubic lattice of rotational spheres indicate that the lift force produced by the Magnus effect can be very significant in view of the magnitude of the drag force when the practical rotating speed of the spheres is encountered. This finding may lead to more comprehensive studies of the effect of the particle rotation on fluid–solid drag laws. It is also demonstrated that, when the third-order or the fourth-order Runge–Kutta scheme is used, the numerical stability of the present IB-LBM is better than that of all methods in the literature, including the previous IB-LBMs and also the methods with the combination of the IBM and the traditional incompressible Navier–Stokes solver. - Highlights: • The IBM is embedded in the LBM using Runge–Kutta time schemes. • The effectiveness of the present IB-LBM is validated by benchmark applications. • For the first time, the IB-LBM achieves the second-order accuracy. • The numerical stability of the present IB-LBM is better than previous methods.« less

A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems

NASA Astrophysics Data System (ADS)

Heinkenschloss, Matthias

2005-01-01

We study a class of time-domain decomposition-based methods for the numerical solution of large-scale linear quadratic optimal control problems. Our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete-time optimal control (DTOC) problem. The optimality conditions for this DTOC problem lead to a linear block tridiagonal system. The diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. This motivates the application of block Gauss-Seidel (GS)-type methods for the solution of the block tridiagonal systems. Numerical experiments show that the spectral radii of the block GS iteration matrices are larger than one for typical applications, but that the eigenvalues of the iteration matrices decay to zero fast. Hence, while the GS method is not expected to convergence for typical applications, it can be effective as a preconditioner for Krylov-subspace methods. This is confirmed by our numerical tests.A byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block GS method applied to the DTOC optimality system.

A study of numerical methods for hyperbolic conservation laws with stiff source terms

NASA Technical Reports Server (NTRS)

Leveque, R. J.; Yee, H. C.

1988-01-01

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a simple advection equation with a parameter-dependent source term was studied. Two approaches to incorporate the source term were utilized: MacCormack type predictor-corrector methods with flux limiters, and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Various comparisons over a wide range of parameter values were made. In the stiff case where the solution contains discontinuities, incorrect numerical propagation speeds are observed with all of the methods considered. This phenomenon is studied and explained.

Numerical study on the effect of configuration of a simple box solar cooker for boiling water

NASA Astrophysics Data System (ADS)

Ambarita, H.

2018-02-01

In this work, a numerical study is carried out to investigate the effect of configuration of a simple box solar cooker. In order to validate the numerical results, a simple a simple solar box cooker with absorber area of 0.835 m × 0.835 m is designed and fabricated. The solar box cooker is employed to boil water by exposing to the solar radiation in Medan city of Indonesia. In the numerical method, a set of transient governing equations are developed. The governing equations are solved using forward time step marching technique. The main objective is to explore the effect of double glasses cover, dimensions of the cooking vessel, and depth of the box cooker to the performance of the solar box cooker. The results show that the experimental and numerical results show good agreement. The performance of the solar box cooker strongly affected by the distance of the double glass cover, the solar cooker depth, and the solar collector length.

Comparison of deterministic and stochastic methods for time-dependent Wigner simulations

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

Shao, Sihong, E-mail: sihong@math.pku.edu.cn; Sellier, Jean Michel, E-mail: jeanmichel.sellier@parallel.bas.bg

2015-11-01

Recently a Monte Carlo method based on signed particles for time-dependent simulations of the Wigner equation has been proposed. While it has been thoroughly validated against physical benchmarks, no technical study about its numerical accuracy has been performed. To this end, this paper presents the first step towards the construction of firm mathematical foundations for the signed particle Wigner Monte Carlo method. An initial investigation is performed by means of comparisons with a cell average spectral element method, which is a highly accurate deterministic method and utilized to provide reference solutions. Several different numerical tests involving the time-dependent evolution ofmore » a quantum wave-packet are performed and discussed in deep details. In particular, this allows us to depict a set of crucial criteria for the signed particle Wigner Monte Carlo method to achieve a satisfactory accuracy.« less

The Complex-Step-Finite-Difference method

NASA Astrophysics Data System (ADS)

Abreu, Rafael; Stich, Daniel; Morales, Jose

2015-07-01

We introduce the Complex-Step-Finite-Difference method (CSFDM) as a generalization of the well-known Finite-Difference method (FDM) for solving the acoustic and elastic wave equations. We have found a direct relationship between modelling the second-order wave equation by the FDM and the first-order wave equation by the CSFDM in 1-D, 2-D and 3-D acoustic media. We present the numerical methodology in order to apply the introduced CSFDM and show an example for wave propagation in simple homogeneous and heterogeneous models. The CSFDM may be implemented as an extension into pre-existing numerical techniques in order to obtain fourth- or sixth-order accurate results with compact three time-level stencils. We compare advantages of imposing various types of initial motion conditions of the CSFDM and demonstrate its higher-order accuracy under the same computational cost and dispersion-dissipation properties. The introduced method can be naturally extended to solve different partial differential equations arising in other fields of science and engineering.

Stepwise molding, etching, and imprinting to form libraries of nanopatterned substrates.

PubMed

Zhao, Zhi; Cai, Yangjun; Liao, Wei-Ssu; Cremer, Paul S

2013-06-04

Herein, we describe a novel colloidal lithographic strategy for the stepwise patterning of planar substrates with numerous complex and unique designs. In conjunction with colloidal self-assembly, imprint molding, and capillary force lithography, reactive ion etching was used to create complex libraries of nanoscale features. This combinatorial strategy affords the ability to develop an exponentially increasing number of two-dimensional nanoscale patterns with each sequential step in the process. Specifically, dots, triangles, circles, and lines could be assembled on the surface separately and in combination with each other. Numerous architectures are obtained for the first time with high uniformity and reproducibility. These hexagonal arrays were made from polystyrene and gold features, whereby each surface element could be tuned from the micrometer size scale down to line widths of ~35 nm. The patterned area could be 1 cm(2) or even larger. The techniques described herein can be combined with further steps to make even larger libraries. Moreover, these polymer and metal features may prove useful in optical, sensing, and electronic applications.

[Mysteries of a medicine chest ].

PubMed

Lefrère, Bertrand

2017-03-01

This article retraces the history of an old medicine chest, used at the beginning of the 19th century, but probably designed earlier. Possibly made in A ustria, with a two-headed eagle lining the bottom of the lid, this first-aid kit belongs to a small group of related chests. It should be noted that these chests were used for a wide variety of different purposes over time. Also named a «droguier» in French, this light chest, made of walnut, and, according to family lore, found in Normandy, would have belonged to a doctor, as confirmed by a short invoice found among numerous documents. The identity of the supplier of numerous old medicines is shown on the labels on the flasks (many of which are intact) and other boxes (containing, in particular, herbal drugs) : «Clément, Apothicaire. Rue St Onge N°. 42. près le Bd. du Temple A Paris», whose history is recounted here step by step.

An extended Lagrangian method

NASA Technical Reports Server (NTRS)

Liou, Meng-Sing

1993-01-01

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. The present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multidimensional discontinuities with a high level of accuracy, similar to that found in 1D problems.

Efficient Machine Learning Approach for Optimizing Scientific Computing Applications on Emerging HPC Architectures

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

Arumugam, Kamesh

Efficient parallel implementations of scientific applications on multi-core CPUs with accelerators such as GPUs and Xeon Phis is challenging. This requires - exploiting the data parallel architecture of the accelerator along with the vector pipelines of modern x86 CPU architectures, load balancing, and efficient memory transfer between different devices. It is relatively easy to meet these requirements for highly structured scientific applications. In contrast, a number of scientific and engineering applications are unstructured. Getting performance on accelerators for these applications is extremely challenging because many of these applications employ irregular algorithms which exhibit data-dependent control-ow and irregular memory accesses. Furthermore,more » these applications are often iterative with dependency between steps, and thus making it hard to parallelize across steps. As a result, parallelism in these applications is often limited to a single step. Numerical simulation of charged particles beam dynamics is one such application where the distribution of work and memory access pattern at each time step is irregular. Applications with these properties tend to present significant branch and memory divergence, load imbalance between different processor cores, and poor compute and memory utilization. Prior research on parallelizing such irregular applications have been focused around optimizing the irregular, data-dependent memory accesses and control-ow during a single step of the application independent of the other steps, with the assumption that these patterns are completely unpredictable. We observed that the structure of computation leading to control-ow divergence and irregular memory accesses in one step is similar to that in the next step. It is possible to predict this structure in the current step by observing the computation structure of previous steps. In this dissertation, we present novel machine learning based optimization techniques to address the parallel implementation challenges of such irregular applications on different HPC architectures. In particular, we use supervised learning to predict the computation structure and use it to address the control-ow and memory access irregularities in the parallel implementation of such applications on GPUs, Xeon Phis, and heterogeneous architectures composed of multi-core CPUs with GPUs or Xeon Phis. We use numerical simulation of charged particles beam dynamics simulation as a motivating example throughout the dissertation to present our new approach, though they should be equally applicable to a wide range of irregular applications. The machine learning approach presented here use predictive analytics and forecasting techniques to adaptively model and track the irregular memory access pattern at each time step of the simulation to anticipate the future memory access pattern. Access pattern forecasts can then be used to formulate optimization decisions during application execution which improves the performance of the application at a future time step based on the observations from earlier time steps. In heterogeneous architectures, forecasts can also be used to improve the memory performance and resource utilization of all the processing units to deliver a good aggregate performance. We used these optimization techniques and anticipation strategy to design a cache-aware, memory efficient parallel algorithm to address the irregularities in the parallel implementation of charged particles beam dynamics simulation on different HPC architectures. Experimental result using a diverse mix of HPC architectures shows that our approach in using anticipation strategy is effective in maximizing data reuse, ensuring workload balance, minimizing branch and memory divergence, and in improving resource utilization.« less

Numerical simulations of three-dimensional laminar flow over a backward facing step; flow near side walls

NASA Technical Reports Server (NTRS)

Steinthorsson, Erlendur; Liou, Meng-Sing; Povinelli, Louis A.; Arnone, Andrea

1993-01-01

This paper reports the results of numerical simulations of steady, laminar flow over a backward-facing step. The governing equations used in the simulations are the full 'compressible' Navier-Stokes equations, solutions to which were computed by using a cell-centered, finite volume discretization. The convection terms of the governing equations were discretized by using the Advection Upwind Splitting Method (AUSM), whereas the diffusion terms were discretized using central differencing formulas. The validity and accuracy of the numerical solutions were verified by comparing the results to existing experimental data for flow at identical Reynolds numbers in the same back step geometry. The paper focuses attention on the details of the flow field near the side wall of the geometry.

Supercomputing Aspects for Simulating Incompressible Flow

NASA Technical Reports Server (NTRS)

Kwak, Dochan; Kris, Cetin C.

2000-01-01

The primary objective of this research is to support the design of liquid rocket systems for the Advanced Space Transportation System. Since the space launch systems in the near future are likely to rely on liquid rocket engines, increasing the efficiency and reliability of the engine components is an important task. One of the major problems in the liquid rocket engine is to understand fluid dynamics of fuel and oxidizer flows from the fuel tank to plume. Understanding the flow through the entire turbo-pump geometry through numerical simulation will be of significant value toward design. One of the milestones of this effort is to develop, apply and demonstrate the capability and accuracy of 3D CFD methods as efficient design analysis tools on high performance computer platforms. The development of the Message Passage Interface (MPI) and Multi Level Parallel (MLP) versions of the INS3D code is currently underway. The serial version of INS3D code is a multidimensional incompressible Navier-Stokes solver based on overset grid technology, INS3D-MPI is based on the explicit massage-passing interface across processors and is primarily suited for distributed memory systems. INS3D-MLP is based on multi-level parallel method and is suitable for distributed-shared memory systems. For the entire turbo-pump simulations, moving boundary capability and efficient time-accurate integration methods are built in the flow solver, To handle the geometric complexity and moving boundary problems, an overset grid scheme is incorporated with the solver so that new connectivity data will be obtained at each time step. The Chimera overlapped grid scheme allows subdomains move relative to each other, and provides a great flexibility when the boundary movement creates large displacements. Two numerical procedures, one based on artificial compressibility method and the other pressure projection method, are outlined for obtaining time-accurate solutions of the incompressible Navier-Stokes equations. The performance of the two methods is compared by obtaining unsteady solutions for the evolution of twin vortices behind a flat plate. Calculated results are compared with experimental and other numerical results. For an unsteady flow, which requires small physical time step, the pressure projection method was found to be computationally efficient since it does not require any subiteration procedure. It was observed that the artificial compressibility method requires a fast convergence scheme at each physical time step in order to satisfy the incompressibility condition. This was obtained by using a GMRES-ILU(0) solver in present computations. When a line-relaxation scheme was used, the time accuracy was degraded and time-accurate computations became very expensive.

Finite-difference time-domain modelling of through-the-Earth radio signal propagation

NASA Astrophysics Data System (ADS)

Ralchenko, M.; Svilans, M.; Samson, C.; Roper, M.

2015-12-01

This research seeks to extend the knowledge of how a very low frequency (VLF) through-the-Earth (TTE) radio signal behaves as it propagates underground, by calculating and visualizing the strength of the electric and magnetic fields for an arbitrary geology through numeric modelling. To achieve this objective, a new software tool has been developed using the finite-difference time-domain method. This technique is particularly well suited to visualizing the distribution of electromagnetic fields in an arbitrary geology. The frequency range of TTE radio (400-9000 Hz) and geometrical scales involved (1 m resolution for domains a few hundred metres in size) involves processing a grid composed of millions of cells for thousands of time steps, which is computationally expensive. Graphics processing unit acceleration was used to reduce execution time from days and weeks, to minutes and hours. Results from the new modelling tool were compared to three cases for which an analytic solution is known. Two more case studies were done featuring complex geologic environments relevant to TTE communications that cannot be solved analytically. There was good agreement between numeric and analytic results. Deviations were likely caused by numeric artifacts from the model boundaries; however, in a TTE application in field conditions, the uncertainty in the conductivity of the various geologic formations will greatly outweigh these small numeric errors.

Model of multistep electron transfer in a single-mode polar medium

NASA Astrophysics Data System (ADS)

Feskov, S. V.; Yudanov, V. V.

2017-09-01

A mathematical model of multistep photoinduced electron transfer (PET) in a polar medium with a single relaxation time (Debye solvent) is developed. The model includes the polarization nonequilibrity formed in the vicinity of the donor-acceptor molecular system at the initial steps of photoreaction and its influence on the subsequent steps of PET. It is established that the results from numerical simulation of transient luminescence spectra of photoexcited donor-acceptor complexes (DAC) conform to calculated data obtained on the basis of the familiar experimental technique used to measure the relaxation function of solvent polarization in the vicinity of DAC in the picosecond and subpicosecond ranges.

Modeling of the motion of automobile elastic wheel in real-time for creation of wheeled vehicles motion control electronic systems

NASA Astrophysics Data System (ADS)

Balakina, E. V.; Zotov, N. M.; Fedin, A. P.

2018-02-01

Modeling of the motion of the elastic wheel of the vehicle in real-time is used in the tasks of constructing different models in the creation of wheeled vehicles motion control electronic systems, in the creation of automobile stand-simulators etc. The accuracy and the reliability of simulation of the parameters of the wheel motion in real-time when rolling with a slip within the given road conditions are determined not only by the choice of the model, but also by the inaccuracy and instability of the numerical calculation. It is established that the inaccuracy and instability of the calculation depend on the size of the step of integration and the numerical method being used. The analysis of these inaccuracy and instability when wheel rolling with a slip was made and recommendations for reducing them were developed. It is established that the total allowable range of steps of integration is 0.001.0.005 s; the strongest instability is manifested in the calculation of the angular and linear accelerations of the wheel; the weakest instability is manifested in the calculation of the translational velocity of the wheel and moving of the center of the wheel; the instability is less at large values of slip angle and on more slippery surfaces. A new method of the average acceleration is suggested, which allows to significantly reduce (up to 100%) the manifesting of instability of the solution in the calculation of all parameters of motion of the elastic wheel for different braking conditions and for the entire range of steps of integration. The results of research can be applied to the selection of control algorithms in vehicles motion control electronic systems and in the testing stand-simulators

A numerical investigation of phytoplankton and Pseudocalanus elongatus dynamics in the spring bloom time in the Gdańsk Gulf

NASA Astrophysics Data System (ADS)

Dzierzbicka-Głowacka, Lidia

2005-01-01

A nutrient-phytoplankton-zooplankton-detritus (1D-NPZD) `phytoplankton {Phyt} and Pseudocalanus elongatus {Zoop} dynamics in the spring bloom time in the Gdańsk Gulf. The 1D-NPZD model consists of three coupled, partial second-order differential equations of the diffusion type for phytoplankton {Phyt}, zooplankton {Zoop}, nutrients {Nutr} and one ordinary first-order differential equation for benthic detritus pool {Detr}, together with initial and boundary conditions. In this model, the {Zoop} is presented by only one species of copepod ( P. elongatus) and {Zoop} is composed of six cohorts of copepods with weights ( Wi) and numbers ( Zi); where Zoop= limit∑i=16W iZ i. The calculations were made for 90 days (March, April, May) for two stations at Gdańsk Gulf with a vertical space step of 0.5m and a time step of 900 s. The flow field and water temperature used as the inputs in the biological model 1D-NPZD were reproduced by the prognostic numerical simulation technique using hydrographic climatological data. The results of the numerical investigations described here were compared with the mean observed values of surface chlorophyll- a and depth integrated P. elongatus biomass for 10 years, 1980-1990. The slight differences between the calculated and mean observed values of surface chlorophyll- a and zooplankton biomass are ca. 10-60 mg C m -3 and ca. 5-23 mg C m -2, respectively, depending on the location of the hydrographic station. The 1D-NPZD model with a high-resolution zooplankton module for P. elongatus can be used to describe the temporal patterns for phytoplankton biomass and P. elongatus in the centre of the Gdańsk Gulf.

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

NASA Astrophysics Data System (ADS)

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

1991-01-01

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

ANALYZING NUMERICAL ERRORS IN DOMAIN HEAT TRANSPORT MODELS USING THE CVBEM.

USGS Publications Warehouse

Hromadka, T.V.

1987-01-01

Besides providing an exact solution for steady-state heat conduction processes (Laplace-Poisson equations), the CVBEM (complex variable boundary element method) can be used for the numerical error analysis of domain model solutions. For problems where soil-water phase change latent heat effects dominate the thermal regime, heat transport can be approximately modeled as a time-stepped steady-state condition in the thawed and frozen regions, respectively. The CVBEM provides an exact solution of the two-dimensional steady-state heat transport problem, and also provides the error in matching the prescribed boundary conditions by the development of a modeling error distribution or an approximate boundary generation.

An Energy Decaying Scheme for Nonlinear Dynamics of Shells

NASA Technical Reports Server (NTRS)

Bottasso, Carlo L.; Bauchau, Olivier A.; Choi, Jou-Young; Bushnell, Dennis M. (Technical Monitor)

2000-01-01

A novel integration scheme for nonlinear dynamics of geometrically exact shells is developed based on the inextensible director assumption. The new algorithm is designed so as to imply the strict decay of the system total mechanical energy at each time step, and consequently unconditional stability is achieved in the nonlinear regime. Furthermore, the scheme features tunable high frequency numerical damping and it is therefore stiffly accurate. The method is tested for a finite element spatial formulation of shells based on mixed interpolations of strain tensorial components and on a two-parameter representation of director rotations. The robustness of the, scheme is illustrated with the help of numerical examples.

Peridynamic thermal diffusion

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

Oterkus, Selda; Madenci, Erdogan, E-mail: madenci@email.arizona.edu; Agwai, Abigail

This study presents the derivation of ordinary state-based peridynamic heat conduction equation based on the Lagrangian formalism. The peridynamic heat conduction parameters are related to those of the classical theory. An explicit time stepping scheme is adopted for numerical solution of various benchmark problems with known solutions. It paves the way for applying the peridynamic theory to other physical fields such as neutronic diffusion and electrical potential distribution.

Modeling the stepping mechanism in negative lightning leaders

NASA Astrophysics Data System (ADS)

Iudin, Dmitry; Syssoev, Artem; Davydenko, Stanislav; Rakov, Vladimir

2017-04-01

It is well-known that the negative leaders develop in a step manner using a mechanism of the so-called space leaders in contrary to positive ones, which propagate continuously. Despite this fact has been known for about a hundred years till now no one had developed any plausible model explaining this asymmetry. In this study we suggest a model of the stepped development of the negative lightning leader which for the first time allows carrying out the numerical simulation of its evolution. The model is based on the probability approach and description of temporal evolution of the discharge channels. One of the key features of our model is accounting for the presence of so called space streamers/leaders which play a fundamental role in the formation of negative leader's steps. Their appearance becomes possible due to the accounting of potential influence of the space charge injected into the discharge gap by the streamer corona. The model takes into account an asymmetry of properties of negative and positive streamers which is based on well-known from numerous laboratory measurements fact that positive streamers need about twice weaker electric field to appear and propagate as compared to negative ones. An extinction of the conducting channel as a possible way of its evolution is also taken into account. This allows us to describe the leader channel's sheath formation. To verify the morphology and characteristics of the model discharge, we use the results of the high-speed video observations of natural negative stepped leaders. We can conclude that the key properties of the model and natural negative leaders are very similar.

Helicopter time-domain electromagnetic numerical simulation based on Leapfrog ADI-FDTD

NASA Astrophysics Data System (ADS)

Guan, S.; Ji, Y.; Li, D.; Wu, Y.; Wang, A.

2017-12-01

We present a three-dimension (3D) Alternative Direction Implicit Finite-Difference Time-Domain (Leapfrog ADI-FDTD) method for the simulation of helicopter time-domain electromagnetic (HTEM) detection. This method is different from the traditional explicit FDTD, or ADI-FDTD. Comparing with the explicit FDTD, leapfrog ADI-FDTD algorithm is no longer limited by Courant-Friedrichs-Lewy(CFL) condition. Thus, the time step is longer. Comparing with the ADI-FDTD, we reduce the equations from 12 to 6 and .the Leapfrog ADI-FDTD method will be easier for the general simulation. First, we determine initial conditions which are adopted from the existing method presented by Wang and Tripp(1993). Second, we derive Maxwell equation using a new finite difference equation by Leapfrog ADI-FDTD method. The purpose is to eliminate sub-time step and retain unconditional stability characteristics. Third, we add the convolution perfectly matched layer (CPML) absorbing boundary condition into the leapfrog ADI-FDTD simulation and study the absorbing effect of different parameters. Different absorbing parameters will affect the absorbing ability. We find the suitable parameters after many numerical experiments. Fourth, We compare the response with the 1-Dnumerical result method for a homogeneous half-space to verify the correctness of our algorithm.When the model contains 107*107*53 grid points, the conductivity is 0.05S/m. The results show that Leapfrog ADI-FDTD need less simulation time and computer storage space, compared with ADI-FDTD. The calculation speed decreases nearly four times, memory occupation decreases about 32.53%. Thus, this algorithm is more efficient than the conventional ADI-FDTD method for HTEM detection, and is more precise than that of explicit FDTD in the late time.

Modeling of nonequilibrium space plasma flows

NASA Technical Reports Server (NTRS)

Gombosi, Tamas

1995-01-01

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Step Bunching: Influence of Impurities and Solution Flow

NASA Technical Reports Server (NTRS)

Chernov, A. A.; Vekilov, P. G.; Coriell, S. R.; Murray, B. T.; McFadden, G. B.

1999-01-01

Step bunching results in striations even at relatively early stages of its development and in inclusions of mother liquor at the later stages. Therefore, eliminating step bunching is crucial for high crystal perfection. At least 5 major effects causing and influencing step bunching are known: (1) Basic morphological instability of stepped interfaces. It is caused by concentration gradient in the solution normal to the face and by the redistribution of solute tangentially to the interface which redistribution enhances occasional perturbations in step density due to various types of noise; (2) Aggravation of the above basic instability by solution flowing tangentially to the face in the same directions as the steps or stabilization of equidistant step train if these flows are antiparallel; (3) Enhanced bunching at supersaturation where step velocity v increases with relative supersaturation s much faster than linear. This v(s) dependence is believed to be associated with impurities. The impurities of which adsorption time is comparable with the time needed to deposit one lattice layer may also be responsible for bunching; (4) Very intensive solution flow stabilizes growing interface even at parallel solution and step flows; (5) Macrosteps were observed to nucleate at crystal corners and edges. Numerical simulation, assuming step-step interactions via surface diffusion also show that step bunching may be induced by random step nucleation at the facet edge and by discontinuity in the step density (a ridge) somewhere in the middle of a face. The corresponding bunching patterns produce the ones observed in experiment. The nature of step bunching generated at the corners and edges and by dislocation step sources, as well as the also relative importance and interrelations between mechanisms 1-5 is not clear, both from experimental and theoretical standpoints. Furthermore, several laws controlling the evolution of existing step bunches have been suggested, though unambiguous conclusions are still missing. Addressing these issues is the major goal of the present project. The theory addressing the above problem, experimental methods, several figures which include: (1) the spatial wave numbers at which the system is neutrally stable as a function of growth velocity for linear kinetics and supersaturation for nonlinear kinetics; (2) a schematic of the experiment of lysozyme crystal growing under conditions of natural convection; (3) fluctuations in time, t, of the normal growth rate, R(t), vicinal slope, p(t) and Fourier Spectra of R(t), discussions and conclusions are presented.

Common Bolted Joint Analysis Tool

NASA Technical Reports Server (NTRS)

Imtiaz, Kauser

2011-01-01

Common Bolted Joint Analysis Tool (comBAT) is an Excel/VB-based bolted joint analysis/optimization program that lays out a systematic foundation for an inexperienced or seasoned analyst to determine fastener size, material, and assembly torque for a given design. Analysts are able to perform numerous what-if scenarios within minutes to arrive at an optimal solution. The program evaluates input design parameters, performs joint assembly checks, and steps through numerous calculations to arrive at several key margins of safety for each member in a joint. It also checks for joint gapping, provides fatigue calculations, and generates joint diagrams for a visual reference. Optimum fastener size and material, as well as correct torque, can then be provided. Analysis methodology, equations, and guidelines are provided throughout the solution sequence so that this program does not become a "black box:" for the analyst. There are built-in databases that reduce the legwork required by the analyst. Each step is clearly identified and results are provided in number format, as well as color-coded spelled-out words to draw user attention. The three key features of the software are robust technical content, innovative and user friendly I/O, and a large database. The program addresses every aspect of bolted joint analysis and proves to be an instructional tool at the same time. It saves analysis time, has intelligent messaging features, and catches operator errors in real time.

Capturing Pressure Oscillations in Numerical Simulations of Internal Combustion Engines

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

Gubba, Sreenivasa Rao; Jupudi, Ravichandra S.; Pasunurthi, Shyam Sundar

In an earlier publication, the authors compared numerical predictions of the mean cylinder pressure of diesel and dual-fuel combustion, to that of measured pressure data from a medium-speed, large-bore engine. In these earlier comparisons, measured data from a flush-mounted in-cylinder pressure transducer showed notable and repeatable pressure oscillations which were not evident in the mean cylinder pressure predictions from computational fluid dynamics (CFD). In this paper, the authors present a methodology for predicting and reporting the local cylinder pressure consistent with that of a measurement location. Such predictions for large-bore, medium-speed engine operation demonstrate pressure oscillations in accordance with thosemore » measured. The temporal occurrences of notable pressure oscillations were during the start of combustion and around the time of maximum cylinder pressure. With appropriate resolutions in time steps and mesh sizes, the local cell static pressure predicted for the transducer location showed oscillations in both diesel and dual-fuel combustion modes which agreed with those observed in the experimental data. Fast Fourier transform (FFT) analysis on both experimental and calculated pressure traces revealed that the CFD predictions successfully captured both the amplitude and frequency range of the oscillations. Furthermore, resolving propagating pressure waves with the smaller time steps and grid sizes necessary to achieve these results required a significant increase in computer resources.« less

Capturing Pressure Oscillations in Numerical Simulations of Internal Combustion Engines

DOE PAGES

Gubba, Sreenivasa Rao; Jupudi, Ravichandra S.; Pasunurthi, Shyam Sundar; ...

2018-04-09

In an earlier publication, the authors compared numerical predictions of the mean cylinder pressure of diesel and dual-fuel combustion, to that of measured pressure data from a medium-speed, large-bore engine. In these earlier comparisons, measured data from a flush-mounted in-cylinder pressure transducer showed notable and repeatable pressure oscillations which were not evident in the mean cylinder pressure predictions from computational fluid dynamics (CFD). In this paper, the authors present a methodology for predicting and reporting the local cylinder pressure consistent with that of a measurement location. Such predictions for large-bore, medium-speed engine operation demonstrate pressure oscillations in accordance with thosemore » measured. The temporal occurrences of notable pressure oscillations were during the start of combustion and around the time of maximum cylinder pressure. With appropriate resolutions in time steps and mesh sizes, the local cell static pressure predicted for the transducer location showed oscillations in both diesel and dual-fuel combustion modes which agreed with those observed in the experimental data. Fast Fourier transform (FFT) analysis on both experimental and calculated pressure traces revealed that the CFD predictions successfully captured both the amplitude and frequency range of the oscillations. Furthermore, resolving propagating pressure waves with the smaller time steps and grid sizes necessary to achieve these results required a significant increase in computer resources.« less

Prediction-Correction Algorithms for Time-Varying Constrained Optimization

DOE PAGES

Simonetto, Andrea; Dall'Anese, Emiliano

2017-07-26

This article develops online algorithms to track solutions of time-varying constrained optimization problems. Particularly, resembling workhorse Kalman filtering-based approaches for dynamical systems, the proposed methods involve prediction-correction steps to provably track the trajectory of the optimal solutions of time-varying convex problems. The merits of existing prediction-correction methods have been shown for unconstrained problems and for setups where computing the inverse of the Hessian of the cost function is computationally affordable. This paper addresses the limitations of existing methods by tackling constrained problems and by designing first-order prediction steps that rely on the Hessian of the cost function (and do notmore » require the computation of its inverse). In addition, the proposed methods are shown to improve the convergence speed of existing prediction-correction methods when applied to unconstrained problems. Numerical simulations corroborate the analytical results and showcase performance and benefits of the proposed algorithms. A realistic application of the proposed method to real-time control of energy resources is presented.« less

Chaotic time series prediction for prenatal exposure to polychlorinated biphenyls in umbilical cord blood using the least squares SEATR model

NASA Astrophysics Data System (ADS)

Xu, Xijin; Tang, Qian; Xia, Haiyue; Zhang, Yuling; Li, Weiqiu; Huo, Xia

2016-04-01

Chaotic time series prediction based on nonlinear systems showed a superior performance in prediction field. We studied prenatal exposure to polychlorinated biphenyls (PCBs) by chaotic time series prediction using the least squares self-exciting threshold autoregressive (SEATR) model in umbilical cord blood in an electronic waste (e-waste) contaminated area. The specific prediction steps basing on the proposal methods for prenatal PCB exposure were put forward, and the proposed scheme’s validity was further verified by numerical simulation experiments. Experiment results show: 1) seven kinds of PCB congeners negatively correlate with five different indices for birth status: newborn weight, height, gestational age, Apgar score and anogenital distance; 2) prenatal PCB exposed group at greater risks compared to the reference group; 3) PCBs increasingly accumulated with time in newborns; and 4) the possibility of newborns suffering from related diseases in the future was greater. The desirable numerical simulation experiments results demonstrated the feasibility of applying mathematical model in the environmental toxicology field.

Chaotic time series prediction for prenatal exposure to polychlorinated biphenyls in umbilical cord blood using the least squares SEATR model

PubMed Central

Xu, Xijin; Tang, Qian; Xia, Haiyue; Zhang, Yuling; Li, Weiqiu; Huo, Xia

2016-01-01

Chaotic time series prediction based on nonlinear systems showed a superior performance in prediction field. We studied prenatal exposure to polychlorinated biphenyls (PCBs) by chaotic time series prediction using the least squares self-exciting threshold autoregressive (SEATR) model in umbilical cord blood in an electronic waste (e-waste) contaminated area. The specific prediction steps basing on the proposal methods for prenatal PCB exposure were put forward, and the proposed scheme’s validity was further verified by numerical simulation experiments. Experiment results show: 1) seven kinds of PCB congeners negatively correlate with five different indices for birth status: newborn weight, height, gestational age, Apgar score and anogenital distance; 2) prenatal PCB exposed group at greater risks compared to the reference group; 3) PCBs increasingly accumulated with time in newborns; and 4) the possibility of newborns suffering from related diseases in the future was greater. The desirable numerical simulation experiments results demonstrated the feasibility of applying mathematical model in the environmental toxicology field. PMID:27118260

A quasi-Lagrangian finite element method for the Navier-Stokes equations in a time-dependent domain

NASA Astrophysics Data System (ADS)

Lozovskiy, Alexander; Olshanskii, Maxim A.; Vassilevski, Yuri V.

2018-05-01

The paper develops a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and convergence analysis of the fully discrete (finite-difference in time and finite-element in space) method. The analysis does not assume any CFL time-step restriction, it rather needs mild conditions of the form $\\Delta t\\le C$, where $C$ depends only on problem data, and $h^{2m_u+2}\\le c\\,\\Delta t$, $m_u$ is polynomial degree of velocity finite element space. Both conditions result from a numerical treatment of practically important non-homogeneous boundary conditions. The theoretically predicted convergence rate is confirmed by a set of numerical experiments. Further we apply the method to simulate a flow in a simplified model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

Corruption of accuracy and efficiency of Markov chain Monte Carlo simulation by inaccurate numerical implementation of conceptual hydrologic models

NASA Astrophysics Data System (ADS)

Schoups, G.; Vrugt, J. A.; Fenicia, F.; van de Giesen, N. C.

2010-10-01

Conceptual rainfall-runoff models have traditionally been applied without paying much attention to numerical errors induced by temporal integration of water balance dynamics. Reliance on first-order, explicit, fixed-step integration methods leads to computationally cheap simulation models that are easy to implement. Computational speed is especially desirable for estimating parameter and predictive uncertainty using Markov chain Monte Carlo (MCMC) methods. Confirming earlier work of Kavetski et al. (2003), we show here that the computational speed of first-order, explicit, fixed-step integration methods comes at a cost: for a case study with a spatially lumped conceptual rainfall-runoff model, it introduces artificial bimodality in the marginal posterior parameter distributions, which is not present in numerically accurate implementations of the same model. The resulting effects on MCMC simulation include (1) inconsistent estimates of posterior parameter and predictive distributions, (2) poor performance and slow convergence of the MCMC algorithm, and (3) unreliable convergence diagnosis using the Gelman-Rubin statistic. We studied several alternative numerical implementations to remedy these problems, including various adaptive-step finite difference schemes and an operator splitting method. Our results show that adaptive-step, second-order methods, based on either explicit finite differencing or operator splitting with analytical integration, provide the best alternative for accurate and efficient MCMC simulation. Fixed-step or adaptive-step implicit methods may also be used for increased accuracy, but they cannot match the efficiency of adaptive-step explicit finite differencing or operator splitting. Of the latter two, explicit finite differencing is more generally applicable and is preferred if the individual hydrologic flux laws cannot be integrated analytically, as the splitting method then loses its advantage.

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

NASA Astrophysics Data System (ADS)

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

2018-03-01

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

Steepest descent method implementation on unconstrained optimization problem using C++ program

NASA Astrophysics Data System (ADS)

Napitupulu, H.; Sukono; Mohd, I. Bin; Hidayat, Y.; Supian, S.

2018-03-01

Steepest Descent is known as the simplest gradient method. Recently, many researches are done to obtain the appropriate step size in order to reduce the objective function value progressively. In this paper, the properties of steepest descent method from literatures are reviewed together with advantages and disadvantages of each step size procedure. The development of steepest descent method due to its step size procedure is discussed. In order to test the performance of each step size, we run a steepest descent procedure in C++ program. We implemented it to unconstrained optimization test problem with two variables, then we compare the numerical results of each step size procedure. Based on the numerical experiment, we conclude the general computational features and weaknesses of each procedure in each case of problem.

Solution of elliptic PDEs by fast Poisson solvers using a local relaxation factor

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

1986-01-01

A large class of two- and three-dimensional, nonseparable elliptic partial differential equations (PDEs) is presently solved by means of novel one-step (D'Yakanov-Gunn) and two-step (accelerated one-step) iterative procedures, using a local, discrete Fourier analysis. In addition to being easily implemented and applicable to a variety of boundary conditions, these procedures are found to be computationally efficient on the basis of the results of numerical comparison with other established methods, which lack the present one's: (1) insensitivity to grid cell size and aspect ratio, and (2) ease of convergence rate estimation by means of the coefficient of the PDE being solved. The two-step procedure is numerically demonstrated to outperform the one-step procedure in the case of PDEs with variable coefficients.

On the limits of numerical astronomical solutions used in paleoclimate studies

NASA Astrophysics Data System (ADS)

Zeebe, Richard E.

2017-04-01

Numerical solutions of the equations of the Solar System estimate Earth's orbital parameters in the past and represent the backbone of cyclostratigraphy and astrochronology, now widely applied in geology and paleoclimatology. Given one numerical realization of a Solar System model (i.e., obtained using one code or integrator package), various parameters determine the properties of the solution and usually limit its validity to a certain time period. Such limitations are denoted here as "internal" and include limitations due to (i) the underlying physics/physical model and (ii) numerics. The physics include initial coordinates and velocities of Solar System bodies, treatment of the Moon and asteroids, the Sun's quadrupole moment, and the intrinsic dynamics of the Solar System itself, i.e., its chaotic nature. Numerical issues include solver algorithm, numerical accuracy (e.g., time step), and round-off errors. At present, internal limitations seem to restrict the validity of astronomical solutions to perhaps the past 50 or 60 myr. However, little is currently known about "external" limitations, that is, how do different numerical realizations compare, say, between different investigators using different codes and integrators? Hitherto only two solutions for Earth's eccentricity appear to be used in paleoclimate studies, provided by two different groups that integrated the full Solar System equations over the past >100 myr (Laskar and coworkers and Varadi et al. 2003). In this contribution, I will present results from new Solar System integrations for Earth's eccentricity obtained using the integrator package HNBody (Rauch and Hamilton 2002). I will discuss the various internal limitations listed above within the framework of the present simulations. I will also compare the results to the existing solutions, the details of which are still being sorted out as several simulations are still running at the time of writing.

Numerical Recovering of a Speed of Sound by the BC-Method in 3D

NASA Astrophysics Data System (ADS)

Pestov, Leonid; Bolgova, Victoria; Danilin, Alexandr

We develop the numerical algorithm for solving the inverse problem for the wave equation by the Boundary Control method. The problem, which we refer to as a forward one, is an initial boundary value problem for the wave equation with zero initial data in the bounded domain. The inverse problem is to find the speed of sound c(x) by the measurements of waves induced by a set of boundary sources. The time of observation is assumed to be greater then two acoustical radius of the domain. The numerical algorithm for sound reconstruction is based on two steps. The first one is to find a (sufficiently large) number of controls {f_j} (the basic control is defined by the position of the source and some time delay), which generates the same number of known harmonic functions, i.e. Δ {u_j}(.,T) = 0 , where {u_j} is the wave generated by the control {f_j} . After that the linear integral equation w.r.t. the speed of sound is obtained. The piecewise constant model of the speed is used. The result of numerical testing of 3-dimensional model is presented.

Numerical experiments with a symmetric high-resolution shock-capturing scheme

NASA Technical Reports Server (NTRS)

Yee, H. C.

1986-01-01

Characteristic-based explicit and implicit total variation diminishing (TVD) schemes for the two-dimensional compressible Euler equations have recently been developed. This is a generalization of recent work of Roe and Davis to a wider class of symmetric (non-upwind) TVD schemes other than Lax-Wendroff. The Roe and Davis schemes can be viewed as a subset of the class of explicit methods. The main properties of the present class of schemes are that they can be implicit, and, when steady-state calculations are sought, the numerical solution is independent of the time step. In a recent paper, a comparison of a linearized form of the present implicit symmetric TVD scheme with an implicit upwind TVD scheme originally developed by Harten and modified by Yee was given. Results favored the symmetric method. It was found that the latter is just as accurate as the upwind method while requiring less computational effort. Currently, more numerical experiments are being conducted on time-accurate calculations and on the effect of grid topology, numerical boundary condition procedures, and different flow conditions on the behavior of the method for steady-state applications. The purpose here is to report experiences with this type of scheme and give guidelines for its use.

Multistage Schemes with Multigrid for Euler and Navier-Strokes Equations: Components and Analysis

NASA Technical Reports Server (NTRS)

Swanson, R. C.; Turkel, Eli

1997-01-01

A class of explicit multistage time-stepping schemes with centered spatial differencing and multigrids are considered for the compressible Euler and Navier-Stokes equations. These schemes are the basis for a family of computer programs (flow codes with multigrid (FLOMG) series) currently used to solve a wide range of fluid dynamics problems, including internal and external flows. In this paper, the components of these multistage time-stepping schemes are defined, discussed, and in many cases analyzed to provide additional insight into their behavior. Special emphasis is given to numerical dissipation, stability of Runge-Kutta schemes, and the convergence acceleration techniques of multigrid and implicit residual smoothing. Both the Baldwin and Lomax algebraic equilibrium model and the Johnson and King one-half equation nonequilibrium model are used to establish turbulence closure. Implementation of these models is described.

A multiple-block multigrid method for the solution of the three-dimensional Euler and Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Atkins, Harold

1991-01-01

A multiple block multigrid method for the solution of the three dimensional Euler and Navier-Stokes equations is presented. The basic flow solver is a cell vertex method which employs central difference spatial approximations and Runge-Kutta time stepping. The use of local time stepping, implicit residual smoothing, multigrid techniques and variable coefficient numerical dissipation results in an efficient and robust scheme is discussed. The multiblock strategy places the block loop within the Runge-Kutta Loop such that accuracy and convergence are not affected by block boundaries. This has been verified by comparing the results of one and two block calculations in which the two block grid is generated by splitting the one block grid. Results are presented for both Euler and Navier-Stokes computations of wing/fuselage combinations.

Carrier-envelope phase stabilization with sub-10 as residual timing jitter.

PubMed

Borchers, B; Koke, S; Husakou, A; Herrmann, J; Steinmeyer, G

2011-11-01

We demonstrate carrier-envelope phase (CEP) stabilization of a mode-locked Ti:sapphire oscillator with unprecedented timing jitter of eight attoseconds. The stabilization performance is obtained by a combination of two different stabilization approaches. In a first step the drift of the CEP is stabilized with a conventional feedback loop by means of controlling the oscillator pump power with an acousto-optic modulator (AOM). In a second step we utilize a recently developed feed-forward type stabilization scheme which has a much higher control bandwith. Here an acousto-optic frequency shifter (AOFS) produces the stabilized output in the first diffraction order. Moreover, we present numerical results on the optimization of the length of the photonic crystal fiber, which is used to generate an octave-spanning spectrum, in order to optimize the sensitivity in the f-to-2f interferometers.

Stable and verifiable state estimation methods and systems with spacecraft applications

NASA Technical Reports Server (NTRS)

Li, Rongsheng (Inventor); Wu, Yeong-Wei Andy (Inventor)

2001-01-01

The stability of a recursive estimator process (e.g., a Kalman filter is assured for long time periods by periodically resetting an error covariance P(t.sub.n) of the system to a predetermined reset value P.sub.r. The recursive process is thus repetitively forced to start from a selected covariance and continue for a time period that is short compared to the system's total operational time period. The time period in which the process must maintain its numerical stability is significantly reduced as is the demand on the system's numerical stability. The process stability for an extended operational time period T.sub.o is verified by performing the resetting step at the end of at least one reset time period T.sub.r whose duration is less than the operational time period T.sub.o and then confirming stability of the process over the reset time period T.sub.r. Because the recursive process starts from a selected covariance at the beginning of each reset time period T.sub.r, confirming stability of the process over at least one reset time period substantially confirms stability over the longer operational time period T.sub.o.

Development and testing of a simple inertial formulation of the shallow water equations for flood inundation modelling

NASA Astrophysics Data System (ADS)

Fewtrell, Timothy; Bates, Paul; Horritt, Matthew

2010-05-01

This abstract describes the development of a new set of equations derived from 1D shallow water theory for use in 2D storage cell inundation models. The new equation set is designed to be solved explicitly at very low computational cost, and is here tested against a suite of four analytical and numerical test cases of increasing complexity. In each case the predicted water depths compare favourably to analytical solutions or to benchmark results from the optimally stable diffusive storage cell code of Hunter et al. (2005). For the most complex test involving the fine spatial resolution simulation of flow in a topographically complex urban area the Root Mean Squared Difference between the new formulation and the model of Hunter et al. is ~1 cm. However, unlike diffusive storage cell codes where the stable time step scales with (1-?x)2 the new equation set developed here represents shallow water wave propagation and so the stability is controlled by the Courant-Freidrichs-Lewy condition such that the stable time step instead scales with 1-?x. This allows use of a stable time step that is 1-3 orders of magnitude greater for typical cell sizes than that possible with diffusive storage cell models and results in commensurate reductions in model run times. The maximum speed up achieved over a diffusive storage cell model was 1120x in these tests, although the actual value seen will depend on model resolution and water depth and surface gradient. Solutions using the new equation set are shown to be relatively grid-independent for the conditions considered given the numerical diffusion likely at coarse model resolution. In addition, the inertial formulation appears to have an intuitively correct sensitivity to friction, however small instabilities and increased errors on predicted depth were noted when Manning's n = 0.01. These small instabilities are likely to be a result of the numerical scheme employed, whereby friction is acting to stabilise the solution although this scheme is still widely used in practice. The new equations are likely to find widespread application in many types of flood inundation modelling and should provide a useful additional tool, alongside more established model formulations, for a variety of flood risk management studies.

Numerical study on flow over stepped spillway using Lagrangian method

NASA Astrophysics Data System (ADS)

Wang, Junmin; Fu, Lei; Xu, Haibo; Jin, Yeechung

2018-02-01

Flow over stepped spillway has been studied for centuries, due to its unstable and the characteristics of cavity, the simulation of this type of spillway flow is always difficult. Most of the early studies of flow over stepped spillway are based on experiment, while in the recent decades, numerical studies of flow over stepped spillway draw most of the researchers’ attentions due to its simplicity and efficiency. In this study, a new Lagrangian based particle method is introduced to reproduce the phenomenon of flow over stepped spillway, the inherent advantages of this particle based method provide a convincing free surface and velocity profiles compared with previous experimental data. The capacity of this new method is proved and it is anticipated to be an alternative tool of traditional mesh based method in environmental engineering field such as the simulation of flow over stepped spillway.

Toward Scientific Numerical Modeling

NASA Technical Reports Server (NTRS)

Kleb, Bil

2007-01-01

Ultimately, scientific numerical models need quantified output uncertainties so that modeling can evolve to better match reality. Documenting model input uncertainties and verifying that numerical models are translated into code correctly, however, are necessary first steps toward that goal. Without known input parameter uncertainties, model sensitivities are all one can determine, and without code verification, output uncertainties are simply not reliable. To address these two shortcomings, two proposals are offered: (1) an unobtrusive mechanism to document input parameter uncertainties in situ and (2) an adaptation of the Scientific Method to numerical model development and deployment. Because these two steps require changes in the computational simulation community to bear fruit, they are presented in terms of the Beckhard-Harris-Gleicher change model.

Microfluidic step-emulsification in a cylindrical geometry

NASA Astrophysics Data System (ADS)

Chakraborty, Indrajit; Leshansky, Alexander M.

2016-11-01

The model microfluidic device for high-throughput droplet generation in a confined cylindrical geometry is investigated numerically. The device comprises of core-annular pressure-driven flow of two immiscible viscous liquids through a cylindrical capillary connected co-axially to a tube of a larger diameter through a sudden expansion, mimicking the microfluidic step-emulsifier (1). To study this problem, the numerical simulations of axisymmetric Navier-Stokes equations have been carried out using an interface capturing procedure based on coupled level set and volume-of-fluid (CLSVOF) methods. The accuracy of the numerical method was favorably tested vs. the predictions of the linear stability analysis of core-annular two-phase flow in a cylindrical capillary. Three distinct flow regimes can be identified: the dripping (D) instability near the entrance to the capillary, the step- (S) and the balloon- (B) emulsification at the step-like expansion. Based on the simulation results we present the phase diagram quantifying transitions between various regimes in plane of the capillary number and the flow-rate ratio. MICROFLUSA EU H2020 project.

Model predictive control design for polytopic uncertain systems by synthesising multi-step prediction scenarios

NASA Astrophysics Data System (ADS)

Lu, Jianbo; Xi, Yugeng; Li, Dewei; Xu, Yuli; Gan, Zhongxue

2018-01-01

A common objective of model predictive control (MPC) design is the large initial feasible region, low online computational burden as well as satisfactory control performance of the resulting algorithm. It is well known that interpolation-based MPC can achieve a favourable trade-off among these different aspects. However, the existing results are usually based on fixed prediction scenarios, which inevitably limits the performance of the obtained algorithms. So by replacing the fixed prediction scenarios with the time-varying multi-step prediction scenarios, this paper provides a new insight into improvement of the existing MPC designs. The adopted control law is a combination of predetermined multi-step feedback control laws, based on which two MPC algorithms with guaranteed recursive feasibility and asymptotic stability are presented. The efficacy of the proposed algorithms is illustrated by a numerical example.

Signatures of two-step impurity mediated vortex lattice melting in Bose-Einstein condensate

NASA Astrophysics Data System (ADS)

Dey, Bishwajyoti

2017-04-01

We study impurity mediated vortex lattice melting in a rotating two-dimensional Bose-Einstein condensate (BEC). Impurities are introduced either through a protocol in which vortex lattice is produced in an impurity potential or first creating the vortex lattice in the absence of random pinning and then cranking up the impurity potential. These two protocols have obvious relation with the two commonly known protocols of creating vortex lattice in a type-II superconductor: zero field cooling protocol and the field cooling protocol respectively. Time-splitting Crank-Nicolson method has been used to numerically simulate the vortex lattice dynamics. It is shown that the vortex lattice follows a two-step melting via loss of positional and orientational order. This vortex lattice melting process in BEC closely mimics the recently observed two-step melting of vortex matter in weakly pinned type-II superconductor Co-intercalated NbSe2. Also, using numerical perturbation analysis, we compare between the states obtained in two protocols and show that the vortex lattice states are metastable and more disordered when impurities are introduced after the formation of an ordered vortex lattice. The author would like to thank SERB, Govt. of India and BCUD-SPPU for financial support through research Grants.

Adaptive MPC based on MIMO ARX-Laguerre model.

PubMed

Ben Abdelwahed, Imen; Mbarek, Abdelkader; Bouzrara, Kais

2017-03-01

This paper proposes a method for synthesizing an adaptive predictive controller using a reduced complexity model. This latter is given by the projection of the ARX model on Laguerre bases. The resulting model is entitled MIMO ARX-Laguerre and it is characterized by an easy recursive representation. The adaptive predictive control law is computed based on multi-step-ahead finite-element predictors, identified directly from experimental input/output data. The model is tuned in each iteration by an online identification algorithms of both model parameters and Laguerre poles. The proposed approach avoids time consuming numerical optimization algorithms associated with most common linear predictive control strategies, which makes it suitable for real-time implementation. The method is used to synthesize and test in numerical simulations adaptive predictive controllers for the CSTR process benchmark. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Integral equation methods for vesicle electrohydrodynamics in three dimensions

NASA Astrophysics Data System (ADS)

Veerapaneni, Shravan

2016-12-01

In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.

Geometric integration in Born-Oppenheimer molecular dynamics.

PubMed

Odell, Anders; Delin, Anna; Johansson, Börje; Cawkwell, Marc J; Niklasson, Anders M N

2011-12-14

Geometric integration schemes for extended Lagrangian self-consistent Born-Oppenheimer molecular dynamics, including a weak dissipation to remove numerical noise, are developed and analyzed. The extended Lagrangian framework enables the geometric integration of both the nuclear and electronic degrees of freedom. This provides highly efficient simulations that are stable and energy conserving even under incomplete and approximate self-consistent field (SCF) convergence. We investigate three different geometric integration schemes: (1) regular time reversible Verlet, (2) second order optimal symplectic, and (3) third order optimal symplectic. We look at energy conservation, accuracy, and stability as a function of dissipation, integration time step, and SCF convergence. We find that the inclusion of dissipation in the symplectic integration methods gives an efficient damping of numerical noise or perturbations that otherwise may accumulate from finite arithmetics in a perfect reversible dynamics. © 2011 American Institute of Physics

Interface- and discontinuity-aware numerical schemes for plasma 3-T radiation diffusion in two and three dimensions

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

Dai, William W., E-mail: dai@lanl.gov; Scannapieco, Anthony J.

2015-11-01

A set of numerical schemes is developed for two- and three-dimensional time-dependent 3-T radiation diffusion equations in systems involving multi-materials. To resolve sub-cell structure, interface reconstruction is implemented within any cell that has more than one material. Therefore, the system of 3-T radiation diffusion equations is solved on two- and three-dimensional polyhedral meshes. The focus of the development is on the fully coupling between radiation and material, the treatment of nonlinearity in the equations, i.e., in the diffusion terms and source terms, treatment of the discontinuity across cell interfaces in material properties, the formulations for both transient and steady states,more » the property for large time steps, and second order accuracy in both space and time. The discontinuity of material properties between different materials is correctly treated based on the governing physics principle for general polyhedral meshes and full nonlinearity. The treatment is exact for arbitrarily strong discontinuity. The scheme is fully nonlinear for the full nonlinearity in the 3-T diffusion equations. Three temperatures are fully coupled and are updated simultaneously. The scheme is general in two and three dimensions on general polyhedral meshes. The features of the scheme are demonstrated through numerical examples for transient problems and steady states. The effects of some simplifications of numerical schemes are also shown through numerical examples, such as linearization, simple average of diffusion coefficient, and approximate treatment for the coupling between radiation and material.« less

Multibody Parachute Flight Simulations for Planetary Entry Trajectories Using "Equilibrium Points"

NASA Technical Reports Server (NTRS)

Raiszadeh, Ben

2003-01-01

A method has been developed to reduce numerical stiffness and computer CPU requirements of high fidelity multibody flight simulations involving parachutes for planetary entry trajectories. Typical parachute entry configurations consist of entry bodies suspended from a parachute, connected by flexible lines. To accurately calculate line forces and moments, the simulations need to keep track of the point where the flexible lines meet (confluence point). In previous multibody parachute flight simulations, the confluence point has been modeled as a point mass. Using a point mass for the confluence point tends to make the simulation numerically stiff, because its mass is typically much less that than the main rigid body masses. One solution for stiff differential equations is to use a very small integration time step. However, this results in large computer CPU requirements. In the method described in the paper, the need for using a mass as the confluence point has been eliminated. Instead, the confluence point is modeled using an "equilibrium point". This point is calculated at every integration step as the point at which sum of all line forces is zero (static equilibrium). The use of this "equilibrium point" has the advantage of both reducing the numerical stiffness of the simulations, and eliminating the dynamical equations associated with vibration of a lumped mass on a high-tension string.

The long-term motion of comet Halley

NASA Technical Reports Server (NTRS)

Yeomans, D. K.; Kiang, T.

1981-01-01

The orbital motion of comet Halley is numerically integrated back to 1404 BC. Starting with an orbit based on the 1759, 1682, and 1607 observations of the comet, the integration was run back in time with full planetary perturbations and nongravitational forces taken into account at each 0.5 day time-step. Small empirical corrections were made to the computed perihelion passage time in 837 and to the osculating orbital eccentricity in 800. In nine cases, the perihelion passage times calculated by Kiang (1971) from Chinese observations have been redetermined, and osculating orbital elements are given at each apparition from 1910 back to 1404 BC.

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

NASA Technical Reports Server (NTRS)

Bartels, Robert E.

2002-01-01

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

A new and inexpensive non-bit-for-bit solution reproducibility test based on time step convergence (TSC1.0)

NASA Astrophysics Data System (ADS)

Wan, Hui; Zhang, Kai; Rasch, Philip J.; Singh, Balwinder; Chen, Xingyuan; Edwards, Jim

2017-02-01

A test procedure is proposed for identifying numerically significant solution changes in evolution equations used in atmospheric models. The test issues a fail signal when any code modifications or computing environment changes lead to solution differences that exceed the known time step sensitivity of the reference model. Initial evidence is provided using the Community Atmosphere Model (CAM) version 5.3 that the proposed procedure can be used to distinguish rounding-level solution changes from impacts of compiler optimization or parameter perturbation, which are known to cause substantial differences in the simulated climate. The test is not exhaustive since it does not detect issues associated with diagnostic calculations that do not feedback to the model state variables. Nevertheless, it provides a practical and objective way to assess the significance of solution changes. The short simulation length implies low computational cost. The independence between ensemble members allows for parallel execution of all simulations, thus facilitating fast turnaround. The new method is simple to implement since it does not require any code modifications. We expect that the same methodology can be used for any geophysical model to which the concept of time step convergence is applicable.

Higgs compositeness in Sp(2N) gauge theories — The pure gauge model

NASA Astrophysics Data System (ADS)

Bennett, Ed; Ki Hong, Deog; Lee, Jong-Wan; David Lin, C.-J.; Lucini, Biagio; Piai, Maurizio; Vadacchino, Davide

2018-03-01

As a first step in the study of Sp(2N) composite Higgs models, we obtained a set of novel numerical results for the pure gauge Sp(4) lattice theory in 3+1 space-time dimensions. Results for the continuum extrapolations of the string tension and the glueball mass spectrum are presented and their values are compared with the same quantities in neighbouring SU(N) models.

Modifications to the Conduit Flow Process Mode 2 for MODFLOW-2005

USGS Publications Warehouse

Reimann, T.; Birk, S.; Rehrl, C.; Shoemaker, W.B.

2012-01-01

As a result of rock dissolution processes, karst aquifers exhibit highly conductive features such as caves and conduits. Within these structures, groundwater flow can become turbulent and therefore be described by nonlinear gradient functions. Some numerical groundwater flow models explicitly account for pipe hydraulics by coupling the continuum model with a pipe network that represents the conduit system. In contrast, the Conduit Flow Process Mode 2 (CFPM2) for MODFLOW-2005 approximates turbulent flow by reducing the hydraulic conductivity within the existing linear head gradient of the MODFLOW continuum model. This approach reduces the practical as well as numerical efforts for simulating turbulence. The original formulation was for large pore aquifers where the onset of turbulence is at low Reynolds numbers (1 to 100) and not for conduits or pipes. In addition, the existing code requires multiple time steps for convergence due to iterative adjustment of the hydraulic conductivity. Modifications to the existing CFPM2 were made by implementing a generalized power function with a user-defined exponent. This allows for matching turbulence in porous media or pipes and eliminates the time steps required for iterative adjustment of hydraulic conductivity. The modified CFPM2 successfully replicated simple benchmark test problems. ?? 2011 The Author(s). Ground Water ?? 2011, National Ground Water Association.

Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach

NASA Technical Reports Server (NTRS)

Kiris, Cetin; Kwak, Dochan

1999-01-01

A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.

Space-time adaptive solution of inverse problems with the discrete adjoint method

NASA Astrophysics Data System (ADS)

Alexe, Mihai; Sandu, Adrian

2014-08-01

This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The space-time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter-grid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided through a multi-level optimization strategy using discontinuous Galerkin (DG) hp-adaptive discretizations paired with Runge-Kutta (RK) time integration. We extend the concept of dual (adjoint) consistency to space-time RK-DG discretizations, which are then shown to be well suited for the adaptive solution of time-dependent inverse problems. Furthermore, we prove that DG mesh transfer operators on general meshes are also dual consistent. This allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation (AD), potentially speeding up development of large-scale simulation codes. The theoretical analysis is supported by numerical results reported for a two-dimensional non-stationary inverse problem.

Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping

DOE PAGES

Finn, John M.

2015-03-01

Properties of integration schemes for solenoidal fields in three dimensions are studied, with a focus on integrating magnetic field lines in a plasma using adaptive time stepping. It is shown that implicit midpoint (IM) and a scheme we call three-dimensional leapfrog (LF) can do a good job (in the sense of preserving KAM tori) of integrating fields that are reversible, or (for LF) have a 'special divergence-free' property. We review the notion of a self-adjoint scheme, showing that such schemes are at least second order accurate and can always be formed by composing an arbitrary scheme with its adjoint. Wemore » also review the concept of reversibility, showing that a reversible but not exactly volume-preserving scheme can lead to a fractal invariant measure in a chaotic region, although this property may not often be observable. We also show numerical results indicating that the IM and LF schemes can fail to preserve KAM tori when the reversibility property (and the SDF property for LF) of the field is broken. We discuss extensions to measure preserving flows, the integration of magnetic field lines in a plasma and the integration of rays for several plasma waves. The main new result of this paper relates to non-uniform time stepping for volume-preserving flows. We investigate two potential schemes, both based on the general method of Ref. [11], in which the flow is integrated in split time steps, each Hamiltonian in two dimensions. The first scheme is an extension of the method of extended phase space, a well-proven method of symplectic integration with non-uniform time steps. This method is found not to work, and an explanation is given. The second method investigated is a method based on transformation to canonical variables for the two split-step Hamiltonian systems. This method, which is related to the method of non-canonical generating functions of Ref. [35], appears to work very well.« less

Numerical Analysis of Ginzburg-Landau Models for Superconductivity.

NASA Astrophysics Data System (ADS)

Coskun, Erhan

Thin film conventional, as well as High T _{c} superconductors of various geometric shapes placed under both uniform and variable strength magnetic field are studied using the universially accepted macroscopic Ginzburg-Landau model. A series of new theoretical results concerning the properties of solution is presented using the semi -discrete time-dependent Ginzburg-Landau equations, staggered grid setup and natural boundary conditions. Efficient serial algorithms including a novel adaptive algorithm is developed and successfully implemented for solving the governing highly nonlinear parabolic system of equations. Refinement technique used in the adaptive algorithm is based on modified forward Euler method which was also developed by us to ease the restriction on time step size for stability considerations. Stability and convergence properties of forward and modified forward Euler schemes are studied. Numerical simulations of various recent physical experiments of technological importance such as vortes motion and pinning are performed. The numerical code for solving time-dependent Ginzburg-Landau equations is parallelized using BlockComm -Chameleon and PCN. The parallel code was run on the distributed memory multiprocessors intel iPSC/860, IBM-SP1 and cluster of Sun Sparc workstations, all located at Mathematics and Computer Science Division, Argonne National Laboratory.

Co-state initialization for the minimum-time low-thrust trajectory optimization

NASA Astrophysics Data System (ADS)

Taheri, Ehsan; Li, Nan I.; Kolmanovsky, Ilya

2017-05-01

This paper presents an approach for co-state initialization which is a critical step in solving minimum-time low-thrust trajectory optimization problems using indirect optimal control numerical methods. Indirect methods used in determining the optimal space trajectories typically result in two-point boundary-value problems and are solved by single- or multiple-shooting numerical methods. Accurate initialization of the co-state variables facilitates the numerical convergence of iterative boundary value problem solvers. In this paper, we propose a method which exploits the trajectory generated by the so-called pseudo-equinoctial and three-dimensional finite Fourier series shape-based methods to estimate the initial values of the co-states. The performance of the approach for two interplanetary rendezvous missions from Earth to Mars and from Earth to asteroid Dionysus is compared against three other approaches which, respectively, exploit random initialization of co-states, adjoint-control transformation and a standard genetic algorithm. The results indicate that by using our proposed approach the percent of the converged cases is higher for trajectories with higher number of revolutions while the computation time is lower. These features are advantageous for broad trajectory search in the preliminary phase of mission designs.

Microfluidic step-emulsification in axisymmetric geometry.

PubMed

Chakraborty, I; Ricouvier, J; Yazhgur, P; Tabeling, P; Leshansky, A M

2017-10-25

Biphasic step-emulsification (Z. Li et al., Lab Chip, 2015, 15, 1023) is a promising microfluidic technique for high-throughput production of μm and sub-μm highly monodisperse droplets. The step-emulsifier consists of a shallow (Hele-Shaw) microchannel operating with two co-flowing immiscible liquids and an abrupt expansion (i.e., step) to a deep and wide reservoir. Under certain conditions the confined stream of the disperse phase, engulfed by the co-flowing continuous phase, breaks into small highly monodisperse droplets at the step. Theoretical investigation of the corresponding hydrodynamics is complicated due to the complex geometry of the planar device, calling for numerical approaches. However, direct numerical simulations of the three dimensional surface-tension-dominated biphasic flows in confined geometries are computationally expensive. In the present paper we study a model problem of axisymmetric step-emulsification. This setup consists of a stable core-annular biphasic flow in a cylindrical capillary tube connected co-axially to a reservoir tube of a larger diameter through a sudden expansion mimicking the edge of the planar step-emulsifier. We demonstrate that the axisymmetric setup exhibits similar regimes of droplet generation to the planar device. A detailed parametric study of the underlying hydrodynamics is feasible via inexpensive (two dimensional) simulations owing to the axial symmetry. The phase diagram quantifying the different regimes of droplet generation in terms of governing dimensionless parameters is presented. We show that in qualitative agreement with experiments in planar devices, the size of the droplets generated in the step-emulsification regime is independent of the capillary number and almost insensitive to the viscosity ratio. These findings confirm that the step-emulsification regime is solely controlled by surface tension. The numerical predictions are in excellent agreement with in-house experiments with the axisymmetric step-emulsifier.

The Motor and the Brake of the Trailing Leg in Human Walking: Leg Force Control Through Ankle Modulation and Knee Covariance

PubMed Central

Toney, Megan E.; Chang, Young-Hui

2016-01-01

Human walking is a complex task, and we lack a complete understanding of how the neuromuscular system organizes its numerous muscles and joints to achieve consistent and efficient walking mechanics. Focused control of select influential task-level variables may simplify the higher-level control of steady state walking and reduce demand on the neuromuscular system. As trailing leg power generation and force application can affect the mechanical efficiency of step-to-step transitions, we investigated how joint torques are organized to control leg force and leg power during human walking. We tested whether timing of trailing leg force control corresponded with timing of peak leg power generation. We also applied a modified uncontrolled manifold analysis to test whether individual or coordinated joint torque strategies most contributed to leg force control. We found that leg force magnitude was adjusted from step-to-step to maintain consistent leg power generation. Leg force modulation was primarily determined by adjustments in the timing of peak ankle plantar-flexion torque, while knee torque was simultaneously covaried to dampen the effect of ankle torque on leg force. We propose a coordinated joint torque control strategy in which the trailing leg ankle acts as a motor to drive leg power production while trailing leg knee torque acts as a brake to refine leg power production. PMID:27334888

One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media.

PubMed

Wang, Xiang-Hua; Yin, Wen-Yan; Chen, Zhi Zhang David

2013-09-09

The one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is reformulated for simulating general electrically dispersive media. It models material dispersive properties with equivalent polarization currents. These currents are then solved with the auxiliary differential equation (ADE) and then incorporated into the one-step leapfrog ADI-FDTD method. The final equations are presented in the form similar to that of the conventional FDTD method but with second-order perturbation. The adapted method is then applied to characterize (a) electromagnetic wave propagation in a rectangular waveguide loaded with a magnetized plasma slab, (b) transmission coefficient of a plane wave normally incident on a monolayer graphene sheet biased by a magnetostatic field, and (c) surface plasmon polaritons (SPPs) propagation along a monolayer graphene sheet biased by an electrostatic field. The numerical results verify the stability, accuracy and computational efficiency of the proposed one-step leapfrog ADI-FDTD algorithm in comparison with analytical results and the results obtained with the other methods.

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

DOE PAGES

Carlberg, Kevin Thomas; Barone, Matthew F.; Antil, Harbir

2016-10-20

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of timemore » integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.« less

Galerkin v. least-squares Petrov–Galerkin projection in nonlinear model reduction

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

Carlberg, Kevin Thomas; Barone, Matthew F.; Antil, Harbir

Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of timemore » integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.« less

A Facile Two-Step Method to Implement N√ {iSWAP} and N√ {SWAP} Gates in a Circuit QED

NASA Astrophysics Data System (ADS)

Said, T.; Chouikh, A.; Bennai, M.

2018-05-01

We propose a way for implementing a two-step N√ {iSWAP} and N √ {SWAP} gates based on the qubit-qubit interaction with N superconducting qubits, by coupling them to a resonator driven by a strong microwave field. The operation times do not increase with the growth of the qubit number. Due to the virtual excitations of the resonator, the scheme is insensitive to the decay of the resonator. Numerical analysis shows that the scheme can be implemented with high fidelity. Moreover, we propose a detailed procedure and analyze the experimental feasibility. So, our proposal can be experimentally realized in the range of current circuit QED techniques.

Efficient Wideband Numerical Simulations for Nanostructures Employing a Drude-Critical Points (DCP) Dispersive Model.

PubMed

Ren, Qiang; Nagar, Jogender; Kang, Lei; Bian, Yusheng; Werner, Ping; Werner, Douglas H

2017-05-18

A highly efficient numerical approach for simulating the wideband optical response of nano-architectures comprised of Drude-Critical Points (DCP) media (e.g., gold and silver) is proposed and validated through comparing with commercial computational software. The kernel of this algorithm is the subdomain level discontinuous Galerkin time domain (DGTD) method, which can be viewed as a hybrid of the spectral-element time-domain method (SETD) and the finite-element time-domain (FETD) method. An hp-refinement technique is applied to decrease the Degrees-of-Freedom (DoFs) and computational requirements. The collocated E-J scheme facilitates solving the auxiliary equations by converting the inversions of matrices to simpler vector manipulations. A new hybrid time stepping approach, which couples the Runge-Kutta and Newmark methods, is proposed to solve the temporal auxiliary differential equations (ADEs) with a high degree of efficiency. The advantages of this new approach, in terms of computational resource overhead and accuracy, are validated through comparison with well-known commercial software for three diverse cases, which cover both near-field and far-field properties with plane wave and lumped port sources. The presented work provides the missing link between DCP dispersive models and FETD and/or SETD based algorithms. It is a competitive candidate for numerically studying the wideband plasmonic properties of DCP media.

Numerical experiment for ultrasonic-measurement-integrated simulation of three-dimensional unsteady blood flow.

PubMed

Funamoto, Kenichi; Hayase, Toshiyuki; Saijo, Yoshifumi; Yambe, Tomoyuki

2008-08-01

Integration of ultrasonic measurement and numerical simulation is a possible way to break through limitations of existing methods for obtaining complete information on hemodynamics. We herein propose Ultrasonic-Measurement-Integrated (UMI) simulation, in which feedback signals based on the optimal estimation of errors in the velocity vector determined by measured and computed Doppler velocities at feedback points are added to the governing equations. With an eye towards practical implementation of UMI simulation with real measurement data, its efficiency for three-dimensional unsteady blood flow analysis and a method for treating low time resolution of ultrasonic measurement were investigated by a numerical experiment dealing with complicated blood flow in an aneurysm. Even when simplified boundary conditions were applied, the UMI simulation reduced the errors of velocity and pressure to 31% and 53% in the feedback domain which covered the aneurysm, respectively. Local maximum wall shear stress was estimated, showing both the proper position and the value with 1% deviance. A properly designed intermittent feedback applied only at the time when measurement data were obtained had the same computational accuracy as feedback applied at every computational time step. Hence, this feedback method is a possible solution to overcome the insufficient time resolution of ultrasonic measurement.

Vibrations of a Mindlin plate subjected to a pair of inertial loads moving in opposite directions

NASA Astrophysics Data System (ADS)

Dyniewicz, Bartłomiej; Pisarski, Dominik; Bajer, Czesław I.

2017-01-01

A Mindlin plate subjected to a pair of inertial loads traveling at a constant high speed in opposite directions along arbitrary trajectory, straight or curved, is presented. The masses represent vehicles passing a bridge or track plates. A numerical solution is obtained using the space-time finite element method, since it allows a clear and simple derivation of the characteristic matrices of the time-stepping procedure. The transition from one spatial finite element to another must be energetically consistent. In the case of the moving inertial load the classical time-integration schemes are methodologically difficult, since we consider the Dirac delta term with a moving argument. The proposed numerical approach provides the correct definition of force equilibrium in the time interval. The given approach closes the problem of the numerical analysis of vibration of a structure subjected to inertial loads moving arbitrarily with acceleration. The results obtained for a massless and an inertial load traveling over a Mindlin plate at various speeds are compared with benchmark results obtained for a Kirchhoff plate. The pair of inertial forces traveling in opposite directions causes displacements and stresses more than twice as large as their corresponding quantities observed for the passage of a single mass.

Numerical analysis of laminar and turbulent incompressible flows using the finite element Fluid Dynamics Analysis Package (FIDAP)

NASA Technical Reports Server (NTRS)

Sohn, Jeong L.

1988-01-01

The purpose of the study is the evaluation of the numerical accuracy of FIDAP (Fluid Dynamics Analysis Package). Accordingly, four test problems in laminar and turbulent incompressible flows are selected and the computational results of these problems compared with other numerical solutions and/or experimental data. These problems include: (1) 2-D laminar flow inside a wall-driven cavity; (2) 2-D laminar flow over a backward-facing step; (3) 2-D turbulent flow over a backward-facing step; and (4) 2-D turbulent flow through a turn-around duct.

Three-dimensional control of crystal growth using magnetic fields

NASA Astrophysics Data System (ADS)

Dulikravich, George S.; Ahuja, Vineet; Lee, Seungsoo

1993-07-01

Two coupled systems of partial differential equations governing three-dimensional laminar viscous flow undergoing solidification or melting under the influence of arbitrarily oriented externally applied magnetic fields have been formulated. The model accounts for arbitrary temperature dependence of physical properties including latent heat release, effects of Joule heating, magnetic field forces, and mushy region existence. On the basis of this model a numerical algorithm has been developed and implemented using central differencing on a curvilinear boundary-conforming grid and Runge-Kutta explicit time-stepping. The numerical results clearly demonstrate possibilities for active and practically instantaneous control of melt/solid interface shape, the solidification/melting front propagation speed, and the amount and location of solid accrued.

Hybrid control of the Neimark-Sacker bifurcation in a delayed Nicholson's blowflies equation.

PubMed

Wang, Yuanyuan; Wang, Lisha

In this article, for delayed Nicholson's blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.

Optimal control applied to a model for species augmentation.

PubMed

Bodine, Erin N; Gross, Louis J; Lenhart, Suzanne

2008-10-01

Species augmentation is a method of reducing species loss via augmenting declining or threatened populations with individuals from captive-bred or stable, wild populations. In this paper, we develop a differential equations model and optimal control formulation for a continuous time augmentation of a general declining population. We find a characterization for the optimal control and show numerical results for scenarios of different illustrative parameter sets. The numerical results provide considerably more detail about the exact dynamics of optimal augmentation than can be readily intuited. The work and results presented in this paper are a first step toward building a general theory of population augmentation, which accounts for the complexities inherent in many conservation biology applications.

Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model

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

Yang, Xiaofeng, E-mail: xfyang@math.sc.edu; Han, Daozhi, E-mail: djhan@iu.edu

2017-02-01

In this paper, we develop a series of linear, unconditionally energy stable numerical schemes for solving the classical phase field crystal model. The temporal discretizations are based on the first order Euler method, the second order backward differentiation formulas (BDF2) and the second order Crank–Nicolson method, respectively. The schemes lead to linear elliptic equations to be solved at each time step, and the induced linear systems are symmetric positive definite. We prove that all three schemes are unconditionally energy stable rigorously. Various classical numerical experiments in 2D and 3D are performed to validate the accuracy and efficiency of the proposedmore » schemes.« less

An improved flux-split algorithm applied to hypersonic flows in chemical equilibrium

NASA Technical Reports Server (NTRS)

Palmer, Grant

1988-01-01

An explicit, finite-difference, shock-capturing numerical algorithm is presented and applied to hypersonic flows assumed to be in thermochemical equilibrium. Real-gas chemistry is either loosely coupled to the gasdynamics by way of a Gibbs free energy minimization package or fully coupled using species mass conservation equations with finite-rate chemical reactions. A scheme is developed that maintains stability in the explicit, finite-rate formulation while allowing relatively high time steps. The codes use flux vector splitting to difference the inviscid fluxes and employ real-gas corrections to viscosity and thermal conductivity. Numerical results are compared against existing ballistic range and flight data. Flows about complex geometries are also computed.

SToRM: A Model for Unsteady Surface Hydraulics Over Complex Terrain

USGS Publications Warehouse

Simoes, Francisco J.

2014-01-01

A two-dimensional (depth-averaged) finite volume Godunov-type shallow water model developed for flow over complex topography is presented. The model is based on an unstructured cellcentered finite volume formulation and a nonlinear strong stability preserving Runge-Kutta time stepping scheme. The numerical discretization is founded on the classical and well established shallow water equations in hyperbolic conservative form, but the convective fluxes are calculated using auto-switching Riemann and diffusive numerical fluxes. The model’s implementation within a graphical user interface is discussed. Field application of the model is illustrated by utilizing it to estimate peak flow discharges in a flooding event of historic significance in Colorado, U.S.A., in 2013.

A stepwise approach for introducing numerical modeling in Environmental Engineering MSc unit: The impact of clear assessment criteria and detailed feedback

NASA Astrophysics Data System (ADS)

Rosolem, R.; Pritchard, J.

2017-12-01

An important aspect for the new generation of hydrologists and water resources managers is the understanding of hydrological processes through the application of numerical environmental models. Despite its importance, teaching numerical modeling subjects to young students in our MSc Water and Environment Management programme has been difficult, for instance, due to the wide range of student background and lack or poor contact with numerical modeling tools in the past. In previous years, this numerical skills concept has been introduced as a project assignment in our Terrestrial Hydrometeorology unit. However, previous efforts have shown non-optimal engagement by students with often signs of lack of interest or anxiety. Given our initial experience with this unit, we decided to make substantial changes to the coursework format with the aim to introduce a more efficient learning environment to the students. The proposed changes include: (1) a clear presentation and discussion of the assessment criteria at the beginning of the unit, (2) a stepwise approach in which students use our learning environment to acquire knowledge for individual components of the model step-by-step, and (3) access to timely and detailed feedback allowing for particular steps to be retraced or retested. In order to understand the overall impact on assessment and feedback, we carried out two surveys at the beginning and end of the module. Our results indicate a positive impact to student learning experience, as the students have clearly benefited from the early discussion on assignment criteria and appeared to have correctly identified the skills and knowledge required to carry out the assignment. In addition, we have observed a substantial increase in the quality of the reports. Our results results support that student engagement has increased since changes to the format of the coursework were introduced. Interestingly, we also observed a positive impact on the assignment to the final exam marks, even for students who did not particularly performed well in the coursework. This indicates that despite not reaching ideal marks, students were able to use this new learning environment to acquire their knowledge of key concepts which are needed for their final exam.

Assessment of chemistry models for compressible reacting flows

NASA Astrophysics Data System (ADS)

Lapointe, Simon; Blanquart, Guillaume

2014-11-01

Recent technological advances in propulsion and power devices and renewed interest in the development of next generation supersonic and hypersonic vehicles have increased the need for detailed understanding of turbulence-combustion interactions in compressible reacting flows. In numerical simulations of such flows, accurate modeling of the fuel chemistry is a critical component of capturing the relevant physics. Various chemical models are currently being used in reacting flow simulations. However, the differences between these models and their impacts on the fluid dynamics in the context of compressible flows are not well understood. In the present work, a numerical code is developed to solve the fully coupled compressible conservation equations for reacting flows. The finite volume code is based on the theoretical and numerical framework developed by Oefelein (Prog. Aero. Sci. 42 (2006) 2-37) and employs an all-Mach-number formulation with dual time-stepping and preconditioning. The numerical approach is tested on turbulent premixed flames at high Karlovitz numbers. Different chemical models of varying complexity and computational cost are used and their effects are compared.

Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization

NASA Astrophysics Data System (ADS)

Chen, Ying; Lowengrub, John; Shen, Jie; Wang, Cheng; Wise, Steven

2018-07-01

We develop efficient energy stable numerical methods for solving isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization. The scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is constructed based on a convex splitting approach. We prove that, for the isotropic Cahn-Hilliard system with the Willmore regularization, the total free energy of the system is non-increasing for any time step and mesh sizes. A straightforward modification of the scheme is then used to solve the regularized strongly anisotropic Cahn-Hilliard system, and it is numerically verified that the discrete energy of the anisotropic system is also non-increasing, and can be efficiently solved by using the modified stable method. We present numerical results in both two and three dimensions that are in good agreement with those in earlier work on the topics. Numerical simulations are presented to demonstrate the accuracy and efficiency of the proposed methods.

The Effects of Dissipation and Coarse Grid Resolution for Multigrid in Flow Problems

NASA Technical Reports Server (NTRS)

Eliasson, Peter; Engquist, Bjoern

1996-01-01

The objective of this paper is to investigate the effects of the numerical dissipation and the resolution of the solution on coarser grids for multigrid with the Euler equation approximations. The convergence is accomplished by multi-stage explicit time-stepping to steady state accelerated by FAS multigrid. A theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy. Numerical results give grid independent convergence in one dimension. For two-dimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density.

Space-time domain solutions of the wave equation by a non-singular boundary integral method and Fourier transform.

PubMed

Klaseboer, Evert; Sepehrirahnama, Shahrokh; Chan, Derek Y C

2017-08-01

The general space-time evolution of the scattering of an incident acoustic plane wave pulse by an arbitrary configuration of targets is treated by employing a recently developed non-singular boundary integral method to solve the Helmholtz equation in the frequency domain from which the space-time solution of the wave equation is obtained using the fast Fourier transform. The non-singular boundary integral solution can enforce the radiation boundary condition at infinity exactly and can account for multiple scattering effects at all spacings between scatterers without adverse effects on the numerical precision. More generally, the absence of singular kernels in the non-singular integral equation confers high numerical stability and precision for smaller numbers of degrees of freedom. The use of fast Fourier transform to obtain the time dependence is not constrained to discrete time steps and is particularly efficient for studying the response to different incident pulses by the same configuration of scatterers. The precision that can be attained using a smaller number of Fourier components is also quantified.

On the supersonic three-dimensional flow over an axisymmetric body with a forward-facing annular step

NASA Astrophysics Data System (ADS)

Simonenko, Mikhail; Zubkov, Alexander; Kuzmin, Alexander

2018-05-01

The 3D turbulent supersonic flow over a body of revolution at various angles of attack α is studied numerically and experimentally. The body surface incorporates a forward-facing step near its midpart and a nose cone. Experiments were conducted in a wind tunnel of the Research Institute of Mechanics, Moscow State University, at the Mach number of 3 for various lengths L of the distance between the step and nose cone. Numerical simulations were performed with a finite-volume solver ANSYS CFX-15. The study reveals bands of α and L in which the pressure on the leeward side of step abruptly increases and exceeds the pressure on the windward side.

Waveform distortion by 2-step modeling ground vibration from trains

NASA Astrophysics Data System (ADS)

Wang, F.; Chen, W.; Zhang, J.; Li, F.; Liu, H.; Chen, X.; Pan, Y.; Li, G.; Xiao, F.

2017-10-01

The 2-step procedure is widely used in numerical research on ground vibrations from trains. The ground is inconsistently represented by a simplified model in the first step and by a refined model in the second step, which may lead to distortions in the simulation results. In order to reveal this modeling error, time histories of ground-borne vibrations were computed based on the 2-step procedure and then compared with the results from a benchmark procedure of the whole system. All parameters involved were intentionally set as equal for the 2 methods, which ensures that differences in the results originated from the inconsistencies of the ground model. Excited by wheel loads of low speeds such as 60 km/h and low frequencies less than 8 Hz, the computed responses of the subgrade were quite close to the benchmarks. However, notable distortions were found in all loading cases at higher frequencies. Moreover, significant underestimation of intensity occurred when load frequencies equaled 16 Hz. This occurred not only at the subgrade but also at the points 10 m and 20 m away from the track. When the load speed was increased to 350 km/h, all computed waveforms were distorted, including the responses to the loads at very low frequencies. The modeling error found herein suggests that the ground models in the 2 steps should be calibrated in terms of frequency bands to be investigated, and the speed of train should be taken into account at the same time.

Algorithm-enabled partial-angular-scan configurations for dual-energy CT.

PubMed

Chen, Buxin; Zhang, Zheng; Xia, Dan; Sidky, Emil Y; Pan, Xiaochuan

2018-05-01

We seek to investigate an optimization-based one-step method for image reconstruction that explicitly compensates for nonlinear spectral response (i.e., the beam-hardening effect) in dual-energy CT, to investigate the feasibility of the one-step method for enabling two dual-energy partial-angular-scan configurations, referred to as the short- and half-scan configurations, on standard CT scanners without involving additional hardware, and to investigate the potential of the short- and half-scan configurations in reducing imaging dose and scan time in a single-kVp-switch full-scan configuration in which two full rotations are made for collection of dual-energy data. We use the one-step method to reconstruct images directly from dual-energy data through solving a nonconvex optimization program that specifies the images to be reconstructed in dual-energy CT. Dual-energy full-scan data are generated from numerical phantoms and collected from physical phantoms with the standard single-kVp-switch full-scan configuration, whereas dual-energy short- and half-scan data are extracted from the corresponding full-scan data. Besides visual inspection and profile-plot comparison, the reconstructed images are analyzed also in quantitative studies based upon tasks of linear-attenuation-coefficient and material-concentration estimation and of material differentiation. Following the performance of a computer-simulation study to verify that the one-step method can reconstruct numerically accurately basis and monochromatic images of numerical phantoms, we reconstruct basis and monochromatic images by using the one-step method from real data of physical phantoms collected with the full-, short-, and half-scan configurations. Subjective inspection based upon visualization and profile-plot comparison reveals that monochromatic images, which are used often in practical applications, reconstructed from the full-, short-, and half-scan data are largely visually comparable except for some differences in texture details. Moreover, quantitative studies based upon tasks of linear-attenuation-coefficient and material-concentration estimation and of material differentiation indicate that the short- and half-scan configurations yield results in close agreement with the ground-truth information and that of the full-scan configuration. The one-step method considered can compensate effectively for the nonlinear spectral response in full- and partial-angular-scan dual-energy CT. It can be exploited for enabling partial-angular-scan configurations on standard CT scanner without involving additional hardware. Visual inspection and quantitative studies reveal that, with the one-step method, partial-angular-scan configurations considered can perform at a level comparable to that of the full-scan configuration, thus suggesting the potential of the two partial-angular-scan configurations in reducing imaging dose and scan time in the standard single-kVp-switch full-scan CT in which two full rotations are performed. The work also yields insights into the investigation and design of other nonstandard scan configurations of potential practical significance in dual-energy CT. © 2018 American Association of Physicists in Medicine.

Numerical modeling of the fracture process in a three-unit all-ceramic fixed partial denture.

PubMed

Kou, Wen; Kou, Shaoquan; Liu, Hongyuan; Sjögren, Göran

2007-08-01

The main objectives were to examine the fracture mechanism and process of a ceramic fixed partial denture (FPD) framework under simulated mechanical loading using a recently developed numerical modeling code, the R-T(2D) code, and also to evaluate the suitability of R-T(2D) code as a tool for this purpose. Using the recently developed R-T(2D) code the fracture mechanism and process of a 3U yttria-tetragonal zirconia polycrystal ceramic (Y-TZP) FPD framework was simulated under static loading. In addition, the fracture pattern obtained using the numerical simulation was compared with the fracture pattern obtained in a previous laboratory test. The result revealed that the framework fracture pattern obtained using the numerical simulation agreed with that observed in a previous laboratory test. Quasi-photoelastic stress fringe pattern and acoustic emission showed that the fracture mechanism was tensile failure and that the crack started at the lower boundary of the framework. The fracture process could be followed both in step-by-step and step-in-step. Based on the findings in the current study, the R-T(2D) code seems suitable for use as a complement to other tests and clinical observations in studying stress distribution, fracture mechanism and fracture processes in ceramic FPD frameworks.

Two-dimensional Euler and Navier-Stokes Time accurate simulations of fan rotor flows

NASA Technical Reports Server (NTRS)

Boretti, A. A.

1990-01-01

Two numerical methods are presented which describe the unsteady flow field in the blade-to-blade plane of an axial fan rotor. These methods solve the compressible, time-dependent, Euler and the compressible, turbulent, time-dependent, Navier-Stokes conservation equations for mass, momentum, and energy. The Navier-Stokes equations are written in Favre-averaged form and are closed with an approximate two-equation turbulence model with low Reynolds number and compressibility effects included. The unsteady aerodynamic component is obtained by superposing inflow or outflow unsteadiness to the steady conditions through time-dependent boundary conditions. The integration in space is performed by using a finite volume scheme, and the integration in time is performed by using k-stage Runge-Kutta schemes, k = 2,5. The numerical integration algorithm allows the reduction of the computational cost of an unsteady simulation involving high frequency disturbances in both CPU time and memory requirements. Less than 200 sec of CPU time are required to advance the Euler equations in a computational grid made up of about 2000 grid during 10,000 time steps on a CRAY Y-MP computer, with a required memory of less than 0.3 megawords.

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

NASA Astrophysics Data System (ADS)

Owolabi, Kolade M.

2018-03-01

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.

Explicit formulation of second and third order optical nonlinearity in the FDTD framework

NASA Astrophysics Data System (ADS)

Varin, Charles; Emms, Rhys; Bart, Graeme; Fennel, Thomas; Brabec, Thomas

2018-01-01

The finite-difference time-domain (FDTD) method is a flexible and powerful technique for rigorously solving Maxwell's equations. However, three-dimensional optical nonlinearity in current commercial and research FDTD softwares requires solving iteratively an implicit form of Maxwell's equations over the entire numerical space and at each time step. Reaching numerical convergence demands significant computational resources and practical implementation often requires major modifications to the core FDTD engine. In this paper, we present an explicit method to include second and third order optical nonlinearity in the FDTD framework based on a nonlinear generalization of the Lorentz dispersion model. A formal derivation of the nonlinear Lorentz dispersion equation is equally provided, starting from the quantum mechanical equations describing nonlinear optics in the two-level approximation. With the proposed approach, numerical integration of optical nonlinearity and dispersion in FDTD is intuitive, transparent, and fully explicit. A strong-field formulation is also proposed, which opens an interesting avenue for FDTD-based modelling of the extreme nonlinear optics phenomena involved in laser filamentation and femtosecond micromachining of dielectrics.

Global error estimation based on the tolerance proportionality for some adaptive Runge-Kutta codes

NASA Astrophysics Data System (ADS)

Calvo, M.; González-Pinto, S.; Montijano, J. I.

2008-09-01

Modern codes for the numerical solution of Initial Value Problems (IVPs) in ODEs are based in adaptive methods that, for a user supplied tolerance [delta], attempt to advance the integration selecting the size of each step so that some measure of the local error is [similar, equals][delta]. Although this policy does not ensure that the global errors are under the prescribed tolerance, after the early studies of Stetter [Considerations concerning a theory for ODE-solvers, in: R. Burlisch, R.D. Grigorieff, J. Schröder (Eds.), Numerical Treatment of Differential Equations, Proceedings of Oberwolfach, 1976, Lecture Notes in Mathematics, vol. 631, Springer, Berlin, 1978, pp. 188-200; Tolerance proportionality in ODE codes, in: R. März (Ed.), Proceedings of the Second Conference on Numerical Treatment of Ordinary Differential Equations, Humbold University, Berlin, 1980, pp. 109-123] and the extensions of Higham [Global error versus tolerance for explicit Runge-Kutta methods, IMA J. Numer. Anal. 11 (1991) 457-480; The tolerance proportionality of adaptive ODE solvers, J. Comput. Appl. Math. 45 (1993) 227-236; The reliability of standard local error control algorithms for initial value ordinary differential equations, in: Proceedings: The Quality of Numerical Software: Assessment and Enhancement, IFIP Series, Springer, Berlin, 1997], it has been proved that in many existing explicit Runge-Kutta codes the global errors behave asymptotically as some rational power of [delta]. This step-size policy, for a given IVP, determines at each grid point tn a new step-size hn+1=h(tn;[delta]) so that h(t;[delta]) is a continuous function of t. In this paper a study of the tolerance proportionality property under a discontinuous step-size policy that does not allow to change the size of the step if the step-size ratio between two consecutive steps is close to unity is carried out. This theory is applied to obtain global error estimations in a few problems that have been solved with the code Gauss2 [S. Gonzalez-Pinto, R. Rojas-Bello, Gauss2, a Fortran 90 code for second order initial value problems, ], based on an adaptive two stage Runge-Kutta-Gauss method with this discontinuous step-size policy.

Scalable explicit implementation of anisotropic diffusion with Runge-Kutta-Legendre super-time stepping

NASA Astrophysics Data System (ADS)

Vaidya, Bhargav; Prasad, Deovrat; Mignone, Andrea; Sharma, Prateek; Rickler, Luca

2017-12-01

An important ingredient in numerical modelling of high temperature magnetized astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures. Magnetohydrodynamics typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution (Δx) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time-stepping (STS) methods has been discussed in literature, but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work, we highlight a second-order (in time) Runge-Kutta-Legendre (RKL) scheme (first described by Meyer, Balsara & Aslam 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first-order STS schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than 104 processors.

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

Simonetto, Andrea; Dall'Anese, Emiliano

This article develops online algorithms to track solutions of time-varying constrained optimization problems. Particularly, resembling workhorse Kalman filtering-based approaches for dynamical systems, the proposed methods involve prediction-correction steps to provably track the trajectory of the optimal solutions of time-varying convex problems. The merits of existing prediction-correction methods have been shown for unconstrained problems and for setups where computing the inverse of the Hessian of the cost function is computationally affordable. This paper addresses the limitations of existing methods by tackling constrained problems and by designing first-order prediction steps that rely on the Hessian of the cost function (and do notmore » require the computation of its inverse). In addition, the proposed methods are shown to improve the convergence speed of existing prediction-correction methods when applied to unconstrained problems. Numerical simulations corroborate the analytical results and showcase performance and benefits of the proposed algorithms. A realistic application of the proposed method to real-time control of energy resources is presented.« less

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

NASA Astrophysics Data System (ADS)

Ghaffar, Tayabia; Yousaf, Muhammad; Qamar, Shamsul

2018-06-01

This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

Implementation of agronomical and geochemical modules into a 3D groundwater code for assessing nitrate storage and transport through unconfined Chalk aquifer

NASA Astrophysics Data System (ADS)

Picot-Colbeaux, Géraldine; Devau, Nicolas; Thiéry, Dominique; Pettenati, Marie; Surdyk, Nicolas; Parmentier, Marc; Amraoui, Nadia; Crastes de Paulet, François; André, Laurent

2016-04-01

Chalk aquifer is the main water resource for domestic water supply in many parts in northern France. In same basin, groundwater is frequently affected by quality problems concerning nitrates. Often close to or above the drinking water standards, nitrate concentration in groundwater is mainly due to historical agriculture practices, combined with leakage and aquifer recharge through the vadose zone. The complexity of processes occurring into such an environment leads to take into account a lot of knowledge on agronomy, geochemistry and hydrogeology in order to understand, model and predict the spatiotemporal evolution of nitrate content and provide a decision support tool for the water producers and stakeholders. To succeed in this challenge, conceptual and numerical models representing accurately the Chalk aquifer specificity need to be developed. A multidisciplinary approach is developed to simulate storage and transport from the ground surface until groundwater. This involves a new agronomic module "NITRATE" (NItrogen TRansfer for Arable soil to groundwaTEr), a soil-crop model allowing to calculate nitrogen mass balance in arable soil, and the "PHREEQC" numerical code for geochemical calculations, both coupled with the 3D transient groundwater numerical code "MARTHE". Otherwise, new development achieved on MARTHE code allows the use of dual porosity and permeability calculations needed in the fissured Chalk aquifer context. This method concerning the integration of existing multi-disciplinary tools is a real challenge to reduce the number of parameters by selecting the relevant equations and simplifying the equations without altering the signal. The robustness and the validity of these numerical developments are tested step by step with several simulations constrained by climate forcing, land use and nitrogen inputs over several decades. In the first time, simulations are performed in a 1D vertical unsaturated soil column for representing experimental nitrates vertical soil profiles (0-30m depth experimental measurements in Somme region). In the second time, this approach is used to simulate with a 3D model a drinking water catchment area in order to compared nitrate contents time series calculated and measured in the domestic water pumping well since 1995 (field in northern France - Avre Basin region). This numerical tool will help the decision-making in all activities in relation with water uses.

Stability of nonuniform rotor blades in hover using a mixed formulation

NASA Technical Reports Server (NTRS)

Stephens, W. B.; Hodges, D. H.; Avila, J. H.; Kung, R. M.

1980-01-01

A mixed formulation for calculating static equilibrium and stability eigenvalues of nonuniform rotor blades in hover is presented. The static equilibrium equations are nonlinear and are solved by an accurate and efficient collocation method. The linearized perturbation equations are solved by a one step, second order integration scheme. The numerical results correlate very well with published results from a nearly identical stability analysis based on a displacement formulation. Slight differences in the results are traced to terms in the equations that relate moments to derivatives of rotations. With the present ordering scheme, in which terms of the order of squares of rotations are neglected with respect to unity, it is not possible to achieve completely equivalent models based on mixed and displacement formulations. The one step methods reveal that a second order Taylor expansion is necessary to achieve good convergence for nonuniform rotating blades. Numerical results for a hypothetical nonuniform blade, including the nonlinear static equilibrium solution, were obtained with no more effort or computer time than that required for a uniform blade.

A network of spiking neurons for computing sparse representations in an energy efficient way

PubMed Central

Hu, Tao; Genkin, Alexander; Chklovskii, Dmitri B.

2013-01-01

Computing sparse redundant representations is an important problem both in applied mathematics and neuroscience. In many applications, this problem must be solved in an energy efficient way. Here, we propose a hybrid distributed algorithm (HDA), which solves this problem on a network of simple nodes communicating via low-bandwidth channels. HDA nodes perform both gradient-descent-like steps on analog internal variables and coordinate-descent-like steps via quantized external variables communicated to each other. Interestingly, such operation is equivalent to a network of integrate-and-fire neurons, suggesting that HDA may serve as a model of neural computation. We compare the numerical performance of HDA with existing algorithms and show that in the asymptotic regime the representation error of HDA decays with time, t, as 1/t. We show that HDA is stable against time-varying noise, specifically, the representation error decays as 1/t for Gaussian white noise. PMID:22920853

An automatic step adjustment method for average power analysis technique used in fiber amplifiers

NASA Astrophysics Data System (ADS)

Liu, Xue-Ming

2006-04-01

An automatic step adjustment (ASA) method for average power analysis (APA) technique used in fiber amplifiers is proposed in this paper for the first time. In comparison with the traditional APA technique, the proposed method has suggested two unique merits such as a higher order accuracy and an ASA mechanism, so that it can significantly shorten the computing time and improve the solution accuracy. A test example demonstrates that, by comparing to the APA technique, the proposed method increases the computing speed by more than a hundredfold under the same errors. By computing the model equations of erbium-doped fiber amplifiers, the numerical results show that our method can improve the solution accuracy by over two orders of magnitude at the same amplifying section number. The proposed method has the capacity to rapidly and effectively compute the model equations of fiber Raman amplifiers and semiconductor lasers.

Real Gas Computation Using an Energy Relaxation Method and High-Order WENO Schemes

NASA Technical Reports Server (NTRS)

Montarnal, Philippe; Shu, Chi-Wang

1998-01-01

In this paper, we use a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition into two parts: one part is associated with a simpler pressure law and the other part (the nonlinear deviation) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the first part. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

A network of spiking neurons for computing sparse representations in an energy-efficient way.

PubMed

Hu, Tao; Genkin, Alexander; Chklovskii, Dmitri B

2012-11-01

Computing sparse redundant representations is an important problem in both applied mathematics and neuroscience. In many applications, this problem must be solved in an energy-efficient way. Here, we propose a hybrid distributed algorithm (HDA), which solves this problem on a network of simple nodes communicating by low-bandwidth channels. HDA nodes perform both gradient-descent-like steps on analog internal variables and coordinate-descent-like steps via quantized external variables communicated to each other. Interestingly, the operation is equivalent to a network of integrate-and-fire neurons, suggesting that HDA may serve as a model of neural computation. We show that the numerical performance of HDA is on par with existing algorithms. In the asymptotic regime, the representation error of HDA decays with time, t, as 1/t. HDA is stable against time-varying noise; specifically, the representation error decays as 1/√t for gaussian white noise.

Speeding up biomolecular interactions by molecular sledding

DOE PAGES

Turkin, Alexander; Zhang, Lei; Marcozzi, Alessio; ...

2015-10-07

In numerous biological processes associations involve a protein with its binding partner, an event that is preceded by a diffusion-mediated search bringing the two partners together. Often hindered by crowding in biologically relevant environments, three-dimensional diffusion can be slow and result in long bimolecular association times. Moreover, the initial association step between two binding partners often represents a rate-limiting step in biotechnologically relevant reactions. We also demonstrate the practical use of an 11-a.a. DNA-interacting peptide derived from adenovirus to reduce the dimensionality of diffusional search processes and speed up associations between biological macromolecules. We functionalize binding partners with the peptidemore » and demonstrate that the ability of the peptide to one-dimensionally diffuse along DNA results in a 20-fold reduction in reaction time. We also show that modifying PCR primers with the peptide sled enables significant acceleration of standard PCR reactions.« less

Dynamic earthquake rupture simulation on nonplanar faults embedded in 3D geometrically complex, heterogeneous Earth models

NASA Astrophysics Data System (ADS)

Duru, K.; Dunham, E. M.; Bydlon, S. A.; Radhakrishnan, H.

2014-12-01

Dynamic propagation of shear ruptures on a frictional interface is a useful idealization of a natural earthquake.The conditions relating slip rate and fault shear strength are often expressed as nonlinear friction laws.The corresponding initial boundary value problems are both numerically and computationally challenging.In addition, seismic waves generated by earthquake ruptures must be propagated, far away from fault zones, to seismic stations and remote areas.Therefore, reliable and efficient numerical simulations require both provably stable and high order accurate numerical methods.We present a numerical method for:a) enforcing nonlinear friction laws, in a consistent and provably stable manner, suitable for efficient explicit time integration;b) dynamic propagation of earthquake ruptures along rough faults; c) accurate propagation of seismic waves in heterogeneous media with free surface topography.We solve the first order form of the 3D elastic wave equation on a boundary-conforming curvilinear mesh, in terms of particle velocities and stresses that are collocated in space and time, using summation-by-parts finite differences in space. The finite difference stencils are 6th order accurate in the interior and 3rd order accurate close to the boundaries. Boundary and interface conditions are imposed weakly using penalties. By deriving semi-discrete energy estimates analogous to the continuous energy estimates we prove numerical stability. Time stepping is performed with a 4th order accurate explicit low storage Runge-Kutta scheme. We have performed extensive numerical experiments using a slip-weakening friction law on non-planar faults, including recent SCEC benchmark problems. We also show simulations on fractal faults revealing the complexity of rupture dynamics on rough faults. We are presently extending our method to rate-and-state friction laws and off-fault plasticity.

Energy-momentum conserving higher-order time integration of nonlinear dynamics of finite elastic fiber-reinforced continua

NASA Astrophysics Data System (ADS)

Erler, Norbert; Groß, Michael

2015-05-01

Since many years the relevance of fibre-reinforced polymers is steadily increasing in fields of engineering, especially in aircraft and automotive industry. Due to the high strength in fibre direction, but the possibility of lightweight construction, these composites replace more and more traditional materials as metals. Fibre-reinforced polymers are often manufactured from glass or carbon fibres as attachment parts or from steel or nylon cord as force transmission parts. Attachment parts are mostly subjected to small strains, but force transmission parts usually suffer large deformations in at least one direction. Here, a geometrically nonlinear formulation is necessary. Typical examples are helicopter rotor blades, where the fibres have the function to stabilize the structure in order to counteract large centrifugal forces. For long-run analyses of rotor blade deformations, we have to apply numerically stable time integrators for anisotropic materials. This paper presents higher-order accurate and numerically stable time stepping schemes for nonlinear elastic fibre-reinforced continua with anisotropic stress behaviour.

Oxidation Behavior of Carbon Fiber-Reinforced Composites

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2008-01-01

OXIMAP is a numerical (FEA-based) solution tool capable of calculating the carbon fiber and fiber coating oxidation patterns within any arbitrarily shaped carbon silicon carbide composite structure as a function of time, temperature, and the environmental oxygen partial pressure. The mathematical formulation is derived from the mechanics of the flow of ideal gases through a chemically reacting, porous solid. The result of the formulation is a set of two coupled, non-linear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined at each time step using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The non-linear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual finite element method, allowing for the solution of the differential equations numerically.

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

Naughton, M.J.; Bourke, W.; Browning, G.L.

The convergence of spectral model numerical solutions of the global shallow-water equations is examined as a function of the time step and the spectral truncation. The contributions to the errors due to the spatial and temporal discretizations are separately identified and compared. Numerical convergence experiments are performed with the inviscid equations from smooth (Rossby-Haurwitz wave) and observed (R45 atmospheric analysis) initial conditions, and also with the diffusive shallow-water equations. Results are compared with the forced inviscid shallow-water equations case studied by Browning et al. Reduction of the time discretization error by the removal of fast waves from the solution usingmore » initialization is shown. The effects of forcing and diffusion on the convergence are discussed. Time truncation errors are found to dominate when a feature is large scale and well resolved; spatial truncation errors dominate for small-scale features and also for large scale after the small scales have affected them. Possible implications of these results for global atmospheric modeling are discussed. 31 refs., 14 figs., 4 tabs.« less

Distributed optimisation problem with communication delay and external disturbance

NASA Astrophysics Data System (ADS)

Tran, Ngoc-Tu; Xiao, Jiang-Wen; Wang, Yan-Wu; Yang, Wu

2017-12-01

This paper investigates the distributed optimisation problem for the multi-agent systems (MASs) with the simultaneous presence of external disturbance and the communication delay. To solve this problem, a two-step design scheme is introduced. In the first step, based on the internal model principle, the internal model term is constructed to compensate the disturbance asymptotically. In the second step, a distributed optimisation algorithm is designed to solve the distributed optimisation problem based on the MASs with the simultaneous presence of disturbance and communication delay. Moreover, in the proposed algorithm, each agent interacts with its neighbours through the connected topology and the delay occurs during the information exchange. By utilising Lyapunov-Krasovskii functional, the delay-dependent conditions are derived for both slowly and fast time-varying delay, respectively, to ensure the convergence of the algorithm to the optimal solution of the optimisation problem. Several numerical simulation examples are provided to illustrate the effectiveness of the theoretical results.

Numerical simulation code for self-gravitating Bose-Einstein condensates

NASA Astrophysics Data System (ADS)

Madarassy, Enikő J. M.; Toth, Viktor T.

2013-04-01

We completed the development of simulation code that is designed to study the behavior of a conjectured dark matter galactic halo that is in the form of a Bose-Einstein Condensate (BEC). The BEC is described by the Gross-Pitaevskii equation, which can be solved numerically using the Crank-Nicholson method. The gravitational potential, in turn, is described by Poisson’s equation, that can be solved using the relaxation method. Our code combines these two methods to study the time evolution of a self-gravitating BEC. The inefficiency of the relaxation method is balanced by the fact that in subsequent time iterations, previously computed values of the gravitational field serve as very good initial estimates. The code is robust (as evidenced by its stability on coarse grids) and efficient enough to simulate the evolution of a system over the course of 109 years using a finer (100×100×100) spatial grid, in less than a day of processor time on a contemporary desktop computer. Catalogue identifier: AEOR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOR_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5248 No. of bytes in distributed program, including test data, etc.: 715402 Distribution format: tar.gz Programming language: C++ or FORTRAN. Computer: PCs or workstations. Operating system: Linux or Windows. Classification: 1.5. Nature of problem: Simulation of a self-gravitating Bose-Einstein condensate by simultaneous solution of the Gross-Pitaevskii and Poisson equations in three dimensions. Solution method: The Gross-Pitaevskii equation is solved numerically using the Crank-Nicholson method; Poisson’s equation is solved using the relaxation method. The time evolution of the system is governed by the Gross-Pitaevskii equation; the solution of Poisson’s equation at each time step is used as an initial estimate for the next time step, which dramatically increases the efficiency of the relaxation method. Running time: Depends on the chosen size of the problem. On a typical personal computer, a 100×100×100 grid can be solved with a time span of 10 Gyr in approx. a day of running time.

A New Time-Space Accurate Scheme for Hyperbolic Problems. 1; Quasi-Explicit Case

NASA Technical Reports Server (NTRS)

Sidilkover, David

1998-01-01

This paper presents a new discretization scheme for hyperbolic systems of conservations laws. It satisfies the TVD property and relies on the new high-resolution mechanism which is compatible with the genuinely multidimensional approach proposed recently. This work can be regarded as a first step towards extending the genuinely multidimensional approach to unsteady problems. Discontinuity capturing capabilities and accuracy of the scheme are verified by a set of numerical tests.

Initial Investigation on the Aerodynamic Performance of Flapping Wings for Nano Air Vehicles

DTIC Science & Technology

2008-02-01

Experiments with Primitive Equations”, Monthly Weather Review, 93:99-164, 1963. 36. Yuan, W., Schilling, R., “Numerical Simulation of the Draft Tube and...LE and TE wake vorticity – Fully flexible wake – Linear approximation of Kutta condition yields a linear system of equations at each time step...all cases one must consider the drain rate. High drain rates substantially diminish capacity! Some batteries simply not capable of delivering

A Conserving Discretization for the Free Boundary in a Two-Dimensional Stefan Problem

NASA Astrophysics Data System (ADS)

Segal, Guus; Vuik, Kees; Vermolen, Fred

1998-03-01

The dissolution of a disk-likeAl2Cuparticle is considered. A characteristic property is that initially the particle has a nonsmooth boundary. The mathematical model of this dissolution process contains a description of the particle interface, of which the position varies in time. Such a model is called a Stefan problem. It is impossible to obtain an analytical solution for a general two-dimensional Stefan problem, so we use the finite element method to solve this problem numerically. First, we apply a classical moving mesh method. Computations show that after some time steps the predicted particle interface becomes very unrealistic. Therefore, we derive a new method for the displacement of the free boundary based on the balance of atoms. This method leads to good results, also, for nonsmooth boundaries. Some numerical experiments are given for the dissolution of anAl2Cuparticle in anAl-Cualloy.

The Evolution and Discharge of Electric Fields within a Thunderstorm

NASA Astrophysics Data System (ADS)

Hager, William W.; Nisbet, John S.; Kasha, John R.

1989-05-01

A 3-dimensional electrical model for a thunderstorm is developed and finite difference approximations to the model are analyzed. If the spatial derivatives are approximated by a method akin to the ☐ scheme and if the temporal derivative is approximated by either a backward difference or the Crank-Nicholson scheme, we show that the resulting discretization is unconditionally stable. The forward difference approximation to the time derivative is stable when the time step is sufficiently small relative to the ratio between the permittivity and the conductivity. Max-norm error estimates for the discrete approximations are established. To handle the propagation of lightning, special numerical techniques are devised based on the Inverse Matrix Modification Formula and Cholesky updates. Numerical comparisons between the model and theoretical results of Wilson and Holzer-Saxon are presented. We also apply our model to a storm observed at the Kennedy Space Center on July 11, 1978.

A three-dimensional, time-dependent model of Mobile Bay

NASA Technical Reports Server (NTRS)

Pitts, F. H.; Farmer, R. C.

1976-01-01

A three-dimensional, time-variant mathematical model for momentum and mass transport in estuaries was developed and its solution implemented on a digital computer. The mathematical model is based on state and conservation equations applied to turbulent flow of a two-component, incompressible fluid having a free surface. Thus, bouyancy effects caused by density differences between the fresh and salt water, inertia from thare river and tidal currents, and differences in hydrostatic head are taken into account. The conservation equations, which are partial differential equations, are solved numerically by an explicit, one-step finite difference scheme and the solutions displayed numerically and graphically. To test the validity of the model, a specific estuary for which scaled model and experimental field data are available, Mobile Bay, was simulated. Comparisons of velocity, salinity and water level data show that the model is valid and a viable means of simulating the hydrodynamics and mass transport in non-idealized estuaries.

Study of Unsteady Flows with Concave Wall Effect

NASA Technical Reports Server (NTRS)

Wang, Chi R.

2003-01-01

This paper presents computational fluid dynamic studies of the inlet turbulence and wall curvature effects on the flow steadiness at near wall surface locations in boundary layer flows. The time-stepping RANS numerical solver of the NASA Glenn-HT RANS code and a one-equation turbulence model, with a uniform inlet turbulence modeling level of the order of 10 percent of molecular viscosity, were used to perform the numerical computations. The approach was first calibrated for its predictabilities of friction factor, velocity, and temperature at near surface locations within a transitional boundary layer over concave wall. The approach was then used to predict the velocity and friction factor variations in a boundary layer recovering from concave curvature. As time iteration proceeded in the computations, the computed friction factors converged to their values from existing experiments. The computed friction factors, velocity, and static temperatures at near wall surface locations oscillated periodically in terms of time iteration steps and physical locations along the span-wise direction. At the upstream stations, the relationship among the normal and tangential velocities showed vortices effects on the velocity variations. Coherent vortices effect on the velocity components broke down at downstream stations. The computations also predicted the vortices effects on the velocity variations within a boundary layer flow developed along a concave wall surface with a downstream recovery flat wall surface. It was concluded that the computational approach might have the potential to analyze the flow steadiness in a turbine blade flow.

Investigating dynamic underground coal fires by means of numerical simulation

NASA Astrophysics Data System (ADS)

Wessling, S.; Kessels, W.; Schmidt, M.; Krause, U.

2008-01-01

Uncontrolled burning or smoldering of coal seams, otherwise known as coal fires, represents a worldwide natural hazard. Efficient application of fire-fighting strategies and prevention of mining hazards require that the temporal evolution of fire propagation can be sufficiently precise predicted. A promising approach for the investigation of the temporal evolution is the numerical simulation of involved physical and chemical processes. In the context of the Sino-German Research Initiative `Innovative Technologies for Detection, Extinction and Prevention of Coal Fires in North China,' a numerical model has been developed for simulating underground coal fires at large scales. The objective of such modelling is to investigate observables, like the fire propagation rate, with respect to the thermal and hydraulic parameters of adjacent rock. In the model, hydraulic, thermal and chemical processes are accounted for, with the last process complemented by laboratory experiments. Numerically, one key challenge in modelling coal fires is to circumvent the small time steps resulting from the resolution of fast reaction kinetics at high temperatures. In our model, this problem is solved by means of an `operator-splitting' approach, in which transport and reactive processes of oxygen are independently calculated. At high temperatures, operator-splitting has the decisive advantage of allowing the global time step to be chosen according to oxygen transport, so that time-consuming simulation through the calculation of fast reaction kinetics is avoided. Also in this model, because oxygen distribution within a coal fire has been shown to remain constant over long periods, an additional extrapolation algorithm for the coal concentration has been applied. In this paper, we demonstrate that the operator-splitting approach is particularly suitable for investigating the influence of hydraulic parameters of adjacent rocks on coal fire propagation. A study shows that dynamic propagation strongly depends on permeability variations. For the assumed model, no fire exists for permeabilities k < 10-10m2, whereas the fire propagation velocity ranges between 340ma-1 for k = 10-8m2, and drops to lower than 3ma-1 for k = 5 × 10-10m2. Additionally, strong temperature variations are observed for the permeability range 5 × 10-10m2 < k < 10-8m2.

One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

NASA Astrophysics Data System (ADS)

Patrone, Paul N.; Einstein, T. L.; Margetis, Dionisios

2010-12-01

We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.

A novel method to quantify and compare anatomical shape: application in cervix cancer radiotherapy

NASA Astrophysics Data System (ADS)

Oh, Seungjong; Jaffray, David; Cho, Young-Bin

2014-06-01

Adaptive radiation therapy (ART) had been proposed to restore dosimetric deficiencies during treatment delivery. In this paper, we developed a technique of Geometric reLocation for analyzing anatomical OBjects' Evolution (GLOBE) for a numerical model of tumor evolution under radiation therapy and characterized geometric changes of the target using GLOBE. A total of 174 clinical target volumes (CTVs) obtained from 32 cervical cancer patients were analyzed. GLOBE consists of three main steps; step (1) deforming a 3D surface object to a sphere by parametric active contour (PAC), step (2) sampling a deformed PAC on 642 nodes of icosahedron geodesic dome for reference frame, and step (3) unfolding 3D data to 2D plane for convenient visualization and analysis. The performance was evaluated with respect to (1) convergence of deformation (iteration number and computation time) and (2) accuracy of deformation (residual deformation). Based on deformation vectors from planning CTV to weekly CTVs, target specific (TS) margins were calculated on each sampled node of GLOBE and the systematic (Σ) and random (σ) variations of the vectors were calculated. Population based anisotropic (PBA) margins were generated using van Herk's margin recipe. GLOBE successfully modeled 152 CTVs from 28 patients. Fast convergence was observed for most cases (137/152) with the iteration number of 65 ± 74 (average ± STD) and the computation time of 13.7 ± 18.6 min. Residual deformation of PAC was 0.9 ± 0.7 mm and more than 97% was less than 3 mm. Margin analysis showed random nature of TS-margin. As a consequence, PBA-margins perform similarly to ISO-margins. For example, PBA-margins for 90% patients' coverage with 95% dose level is close to 13 mm ISO-margins in the aspect of target coverage and OAR sparing. GLOBE demonstrates a systematic analysis of tumor motion and deformation of patients with cervix cancer during radiation therapy and numerical modeling of PBA-margin on 642 locations of CTV surface.

Direct modeling for computational fluid dynamics

NASA Astrophysics Data System (ADS)

Xu, Kun

2015-06-01

All fluid dynamic equations are valid under their modeling scales, such as the particle mean free path and mean collision time scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. The current computational fluid dynamics (CFD) focuses on the numerical solution of partial differential equations (PDEs), and its aim is to get the accurate solution of these governing equations. Under such a CFD practice, it is hard to develop a unified scheme that covers flow physics from kinetic to hydrodynamic scales continuously because there is no such governing equation which could make a smooth transition from the Boltzmann to the NS modeling. The study of fluid dynamics needs to go beyond the traditional numerical partial differential equations. The emerging engineering applications, such as air-vehicle design for near-space flight and flow and heat transfer in micro-devices, do require further expansion of the concept of gas dynamics to a larger domain of physical reality, rather than the traditional distinguishable governing equations. At the current stage, the non-equilibrium flow physics has not yet been well explored or clearly understood due to the lack of appropriate tools. Unfortunately, under the current numerical PDE approach, it is hard to develop such a meaningful tool due to the absence of valid PDEs. In order to construct multiscale and multiphysics simulation methods similar to the modeling process of constructing the Boltzmann or the NS governing equations, the development of a numerical algorithm should be based on the first principle of physical modeling. In this paper, instead of following the traditional numerical PDE path, we introduce direct modeling as a principle for CFD algorithm development. Since all computations are conducted in a discretized space with limited cell resolution, the flow physics to be modeled has to be done in the mesh size and time step scales. Here, the CFD is more or less a direct construction of discrete numerical evolution equations, where the mesh size and time step will play dynamic roles in the modeling process. With the variation of the ratio between mesh size and local particle mean free path, the scheme will capture flow physics from the kinetic particle transport and collision to the hydrodynamic wave propagation. Based on the direct modeling, a continuous dynamics of flow motion will be captured in the unified gas-kinetic scheme. This scheme can be faithfully used to study the unexplored non-equilibrium flow physics in the transition regime.

Numerical calculation of primary slot leakage inductance of a Single-sided HTS linear induction motor used for linear metro

NASA Astrophysics Data System (ADS)

Li, Dong; Wen, Yinghong; Li, Weili; Fang, Jin; Cao, Junci; Zhang, Xiaochen; Lv, Gang

2017-03-01

In the paper, the numerical method calculating asymmetric primary slot leakage inductances of Single-sided High-Temperature Superconducting (HTS) Linear Induction Motor (HTS LIM) is presented. The mathematical and geometric models of three-dimensional nonlinear transient electromagnetic field are established and the boundary conditions are also given. The established model is solved by time-stepping Finite Element Method (FEM). Then, the three-phase asymmetric primary slot leakage inductances under different operation conditions are calculated by using the obtained electromagnetic field distribution. The influences of the special effects such as longitudinal end effects, transversal edge effects, etc. on the primary slot leakage inductance are investigated. The presented numerical method is validated by experiments carried out on a 3.5 kW prototype with copper wires which has the same structures with the HTS LIM.

An extended Lagrangian method

NASA Technical Reports Server (NTRS)

Liou, Meng-Sing

1992-01-01

A unique formulation of describing fluid motion is presented. The method, referred to as 'extended Lagrangian method', is interesting from both theoretical and numerical points of view. The formulation offers accuracy in numerical solution by avoiding numerical diffusion resulting from mixing of fluxes in the Eulerian description. Meanwhile, it also avoids the inaccuracy incurred due to geometry and variable interpolations used by the previous Lagrangian methods. Unlike the Lagrangian method previously imposed which is valid only for supersonic flows, the present method is general and capable of treating subsonic flows as well as supersonic flows. The method proposed in this paper is robust and stable. It automatically adapts to flow features without resorting to clustering, thereby maintaining rather uniform grid spacing throughout and large time step. Moreover, the method is shown to resolve multi-dimensional discontinuities with a high level of accuracy, similar to that found in one-dimensional problems.

Numerical simulation and comparison of nonlinear self-focusing based on iteration and ray tracing

NASA Astrophysics Data System (ADS)

Li, Xiaotong; Chen, Hao; Wang, Weiwei; Ruan, Wangchao; Zhang, Luwei; Cen, Zhaofeng

2017-05-01

Self-focusing is observed in nonlinear materials owing to the interaction between laser and matter when laser beam propagates. Some of numerical simulation strategies such as the beam propagation method (BPM) based on nonlinear Schrödinger equation and ray tracing method based on Fermat's principle have applied to simulate the self-focusing process. In this paper we present an iteration nonlinear ray tracing method in that the nonlinear material is also cut into massive slices just like the existing approaches, but instead of paraxial approximation and split-step Fourier transform, a large quantity of sampled real rays are traced step by step through the system with changing refractive index and laser intensity by iteration. In this process a smooth treatment is employed to generate a laser density distribution at each slice to decrease the error caused by the under-sampling. The characteristics of this method is that the nonlinear refractive indices of the points on current slice are calculated by iteration so as to solve the problem of unknown parameters in the material caused by the causal relationship between laser intensity and nonlinear refractive index. Compared with the beam propagation method, this algorithm is more suitable for engineering application with lower time complexity, and has the calculation capacity for numerical simulation of self-focusing process in the systems including both of linear and nonlinear optical media. If the sampled rays are traced with their complex amplitudes and light paths or phases, it will be possible to simulate the superposition effects of different beam. At the end of the paper, the advantages and disadvantages of this algorithm are discussed.

Modified unified kinetic scheme for all flow regimes.

PubMed

Liu, Sha; Zhong, Chengwen

2012-06-01

A modified unified kinetic scheme for the prediction of fluid flow behaviors in all flow regimes is described. The time evolution of macrovariables at the cell interface is calculated with the idea that both free transport and collision mechanisms should be considered. The time evolution of macrovariables is obtained through the conservation constraints. The time evolution of local Maxwellian distribution is obtained directly through the one-to-one mapping from the evolution of macrovariables. These improvements provide more physical realities in flow behaviors and more accurate numerical results in all flow regimes especially in the complex transition flow regime. In addition, the improvement steps introduce no extra computational complexity.

Distributed-observer-based cooperative control for synchronization of linear discrete-time multi-agent systems.

PubMed

Liang, Hongjing; Zhang, Huaguang; Wang, Zhanshan

2015-11-01

This paper considers output synchronization of discrete-time multi-agent systems with directed communication topologies. The directed communication graph contains a spanning tree and the exosystem as its root. Distributed observer-based consensus protocols are proposed, based on the relative outputs of neighboring agents. A multi-step algorithm is presented to construct the observer-based protocols. In light of the discrete-time algebraic Riccati equation and internal model principle, synchronization problem is completed. At last, numerical simulation is provided to verify the effectiveness of the theoretical results. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Demonstration of ultra-low NA rare-earth doped step index fiber for applications in high power fiber lasers.

PubMed

Jain, Deepak; Jung, Yongmin; Barua, Pranabesh; Alam, Shaiful; Sahu, Jayanta K

2015-03-23

In this paper, we report the mode area scaling of a rare-earth doped step index fiber by using low numerical aperture. Numerical simulations show the possibility of achieving an effective area of ~700 um² (including bend induced effective area reduction) at a bend diameter of 32 cm from a 35 μm core fiber with a numerical aperture of 0.038. An effective single mode operation is ensured following the criterion of the fundamental mode loss to be lower than 0.1 dB/m while ensuring the higher order modes loss to be higher than 10 dB/m at a wavelength of 1060 nm. Our optimized modified chemical vapor deposition process in conjunction with solution doping process allows fabrication of an Yb-doped step index fiber having an ultra-low numerical aperture of ~0.038. Experimental results confirm a Gaussian output beam from a 35 μm core fiber validating our simulation results. Fiber shows an excellent laser efficiency of ~81%and aM² less than 1.1.

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

NASA Astrophysics Data System (ADS)

Vilar, François; Shu, Chi-Wang; Maire, Pierre-Henri

2016-05-01

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non-ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89,90,22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19,64,15,82,84].

Evaluation of a hybrid kinetics/mixing-controlled combustion model for turbulent premixed and diffusion combustion using KIVA-II

NASA Technical Reports Server (NTRS)

Nguyen, H. Lee; Wey, Ming-Jyh

1990-01-01

Two-dimensional calculations were made of spark ignited premixed-charge combustion and direct injection stratified-charge combustion in gasoline fueled piston engines. Results are obtained using kinetic-controlled combustion submodel governed by a four-step global chemical reaction or a hybrid laminar kinetics/mixing-controlled combustion submodel that accounts for laminar kinetics and turbulent mixing effects. The numerical solutions are obtained by using KIVA-2 computer code which uses a kinetic-controlled combustion submodel governed by a four-step global chemical reaction (i.e., it assumes that the mixing time is smaller than the chemistry). A hybrid laminar/mixing-controlled combustion submodel was implemented into KIVA-2. In this model, chemical species approach their thermodynamics equilibrium with a rate that is a combination of the turbulent-mixing time and the chemical-kinetics time. The combination is formed in such a way that the longer of the two times has more influence on the conversion rate and the energy release. An additional element of the model is that the laminar-flame kinetics strongly influence the early flame development following ignition.

Evaluation of a hybrid kinetics/mixing-controlled combustion model for turbulent premixed and diffusion combustion using KIVA-2

NASA Technical Reports Server (NTRS)

Nguyen, H. Lee; Wey, Ming-Jyh

1990-01-01

Two dimensional calculations were made of spark ignited premixed-charge combustion and direct injection stratified-charge combustion in gasoline fueled piston engines. Results are obtained using kinetic-controlled combustion submodel governed by a four-step global chemical reaction or a hybrid laminar kinetics/mixing-controlled combustion submodel that accounts for laminar kinetics and turbulent mixing effects. The numerical solutions are obtained by using KIVA-2 computer code which uses a kinetic-controlled combustion submodel governed by a four-step global chemical reaction (i.e., it assumes that the mixing time is smaller than the chemistry). A hybrid laminar/mixing-controlled combustion submodel was implemented into KIVA-2. In this model, chemical species approach their thermodynamics equilibrium with a rate that is a combination of the turbulent-mixing time and the chemical-kinetics time. The combination is formed in such a way that the longer of the two times has more influence on the conversion rate and the energy release. An additional element of the model is that the laminar-flame kinetics strongly influence the early flame development following ignition.

New numerical approach for the modelling of machining applied to aeronautical structural parts

NASA Astrophysics Data System (ADS)

Rambaud, Pierrick; Mocellin, Katia

2018-05-01

The manufacturing of aluminium alloy structural aerospace parts involves several steps: forming (rolling, forging …etc), heat treatments and machining. Before machining, the manufacturing processes have embedded residual stresses into the workpiece. The final geometry is obtained during this last step, when up to 90% of the raw material volume is removed by machining. During this operation, the mechanical equilibrium of the part is in constant evolution due to the redistribution of the initial stresses. This redistribution is the main cause for workpiece deflections during machining and for distortions - after unclamping. Both may lead to non-conformity of the part regarding the geometrical and dimensional specifications and therefore to rejection of the part or additional conforming steps. In order to improve the machining accuracy and the robustness of the process, the effect of the residual stresses has to be considered for the definition of the machining process plan and even in the geometrical definition of the part. In this paper, the authors present two new numerical approaches concerning the modelling of machining of aeronautical structural parts. The first deals with the use of an immersed volume framework to model the cutting step, improving the robustness and the quality of the resulting mesh compared to the previous version. The second is about the mechanical modelling of the machining problem. The authors thus show that in the framework of rolled aluminium parts the use of a linear elasticity model is functional in the finite element formulation and promising regarding the reduction of computation times.

A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations

NASA Astrophysics Data System (ADS)

Ke, Rihuan; Ng, Michael K.; Sun, Hai-Wei

2015-12-01

In this paper, we study the block lower triangular Toeplitz-like with tri-diagonal blocks system which arises from the time-fractional partial differential equation. Existing fast numerical solver (e.g., fast approximate inversion method) cannot handle such linear system as the main diagonal blocks are different. The main contribution of this paper is to propose a fast direct method for solving this linear system, and to illustrate that the proposed method is much faster than the classical block forward substitution method for solving this linear system. Our idea is based on the divide-and-conquer strategy and together with the fast Fourier transforms for calculating Toeplitz matrix-vector multiplication. The complexity needs O (MNlog2 ⁡ M) arithmetic operations, where M is the number of blocks (the number of time steps) in the system and N is the size (number of spatial grid points) of each block. Numerical examples from the finite difference discretization of time-fractional partial differential equations are also given to demonstrate the efficiency of the proposed method.

Adaptive temporal refinement in injection molding

NASA Astrophysics Data System (ADS)

Karyofylli, Violeta; Schmitz, Mauritius; Hopmann, Christian; Behr, Marek

2018-05-01

Mold filling is an injection molding stage of great significance, because many defects of the plastic components (e.g. weld lines, burrs or insufficient filling) can occur during this process step. Therefore, it plays an important role in determining the quality of the produced parts. Our goal is the temporal refinement in the vicinity of the evolving melt front, in the context of 4D simplex-type space-time grids [1, 2]. This novel discretization method has an inherent flexibility to employ completely unstructured meshes with varying levels of resolution both in spatial dimensions and in the time dimension, thus allowing the use of local time-stepping during the simulations. This can lead to a higher simulation precision, while preserving calculation efficiency. A 3D benchmark case, which concerns the filling of a plate-shaped geometry, is used for verifying our numerical approach [3]. The simulation results obtained with the fully unstructured space-time discretization are compared to those obtained with the standard space-time method and to Moldflow simulation results. This example also serves for providing reliable timing measurements and the efficiency aspects of the filling simulation of complex 3D molds while applying adaptive temporal refinement.

Adaptive multi-time-domain subcycling for crystal plasticity FE modeling of discrete twin evolution

NASA Astrophysics Data System (ADS)

Ghosh, Somnath; Cheng, Jiahao

2018-02-01

Crystal plasticity finite element (CPFE) models that accounts for discrete micro-twin nucleation-propagation have been recently developed for studying complex deformation behavior of hexagonal close-packed (HCP) materials (Cheng and Ghosh in Int J Plast 67:148-170, 2015, J Mech Phys Solids 99:512-538, 2016). A major difficulty with conducting high fidelity, image-based CPFE simulations of polycrystalline microstructures with explicit twin formation is the prohibitively high demands on computing time. High strain localization within fast propagating twin bands requires very fine simulation time steps and leads to enormous computational cost. To mitigate this shortcoming and improve the simulation efficiency, this paper proposes a multi-time-domain subcycling algorithm. It is based on adaptive partitioning of the evolving computational domain into twinned and untwinned domains. Based on the local deformation-rate, the algorithm accelerates simulations by adopting different time steps for each sub-domain. The sub-domains are coupled back after coarse time increments using a predictor-corrector algorithm at the interface. The subcycling-augmented CPFEM is validated with a comprehensive set of numerical tests. Significant speed-up is observed with this novel algorithm without any loss of accuracy that is advantageous for predicting twinning in polycrystalline microstructures.

A GPU-accelerated semi-implicit fractional step method for numerical solutions of incompressible Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Ha, Sanghyun; Park, Junshin; You, Donghyun

2017-11-01

Utility of the computational power of modern Graphics Processing Units (GPUs) is elaborated for solutions of incompressible Navier-Stokes equations which are integrated using a semi-implicit fractional-step method. Due to its serial and bandwidth-bound nature, the present choice of numerical methods is considered to be a good candidate for evaluating the potential of GPUs for solving Navier-Stokes equations using non-explicit time integration. An efficient algorithm is presented for GPU acceleration of the Alternating Direction Implicit (ADI) and the Fourier-transform-based direct solution method used in the semi-implicit fractional-step method. OpenMP is employed for concurrent collection of turbulence statistics on a CPU while Navier-Stokes equations are computed on a GPU. Extension to multiple NVIDIA GPUs is implemented using NVLink supported by the Pascal architecture. Performance of the present method is experimented on multiple Tesla P100 GPUs compared with a single-core Xeon E5-2650 v4 CPU in simulations of boundary-layer flow over a flat plate. Supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (Ministry of Science, ICT and Future Planning NRF-2016R1E1A2A01939553, NRF-2014R1A2A1A11049599, and Ministry of Trade, Industry and Energy 201611101000230).

Force sum rules for stepped surfaces of jellium

NASA Astrophysics Data System (ADS)

Farjam, Mani

2007-03-01

The Budd-Vannimenus theorem for jellium surface is generalized for stepped surfaces of jellium. Our sum rules show that the average value of the electrostatic potential over the stepped jellium surface equals the value of the potential at the corresponding flat jellium surface. Several sum rules are tested with numerical results obtained within the Thomas-Fermi model of stepped surfaces.

Analytical investigations of seismic responses for reinforced concrete bridge columns subjected to strong near-fault ground motion

NASA Astrophysics Data System (ADS)

Su, Chin-Kuo; Sung, Yu-Chi; Chang, Shuenn-Yih; Huang, Chao-Hsun

2007-09-01

Strong near-fault ground motion, usually caused by the fault-rupture and characterized by a pulse-like velocity-wave form, often causes dramatic instantaneous seismic energy (Jadhav and Jangid 2006). Some reinforced concrete (RC) bridge columns, even those built according to ductile design principles, were damaged in the 1999 Chi-Chi earthquake. Thus, it is very important to evaluate the seismic response of a RC bridge column to improve its seismic design and prevent future damage. Nonlinear time history analysis using step-by-step integration is capable of tracing the dynamic response of a structure during the entire vibration period and is able to accommodate the pulsing wave form. However, the accuracy of the numerical results is very sensitive to the modeling of the nonlinear load-deformation relationship of the structural member. FEMA 273 and ATC-40 provide the modeling parameters for structural nonlinear analyses of RC beams and RC columns. They use three parameters to define the plastic rotation angles and a residual strength ratio to describe the nonlinear load-deformation relationship of an RC member. Structural nonlinear analyses are performed based on these parameters. This method provides a convenient way to obtain the nonlinear seismic responses of RC structures. However, the accuracy of the numerical solutions might be further improved. For this purpose, results from a previous study on modeling of the static pushover analyses for RC bridge columns (Sung et al. 2005) is adopted for the nonlinear time history analysis presented herein to evaluate the structural responses excited by a near-fault ground motion. To ensure the reliability of this approach, the numerical results were compared to experimental results. The results confirm that the proposed approach is valid.

Analysis of Nonlinear Periodic and Aperiodic Media: Application to Optical Logic Gates

NASA Astrophysics Data System (ADS)

Yu, Yisheng

This dissertation is about the analysis of nonlinear periodic and aperiodic media and their application to the design of intensity controlled all optical logic gates: AND, OR, and NOT. A coupled nonlinear differential equation that characterizes the electromagnetic wave propagation in a nonlinear periodic (and aperiodic) medium has been derived from the first principle. The equations are general enough that it reflects the effect of transverse modal fields and can be used to analyze both co-propagating and counter propagating waves. A numerical technique based on the finite differences method and absorbing boundary condition has been developed to solve the coupled differential equations here. The numerical method is simple and accurate. Unlike the method based on characteristics that has been reported in the literature, this method does not involve integration and step sizes of time and space coordinates are decoupled. The decoupling provides independent choice for time and space step sizes. The concept of "gap soliton" has also been re-examined. The dissertation consists of four manuscripts. Manuscript I reports on the design of all optical logic gates: AND, OR, and NOT based on the bistability property of nonlinear periodic and aperiodic waveguiding structures. The functioning of the logic gates has been shown by analysis. The numerical technique that has been developed to solve the nonlinear differential equations are addressed in manuscript II. The effect of transverse modal fields on the bistable property of nonlinear periodic medium is reported in manuscript III. The concept of "gap soliton" that are generated in a nonlinear periodic medium has been re-examined. The details on the finding of the re-examination are discussed in manuscript IV.

Improvements of the particle-in-cell code EUTERPE for petascaling machines

NASA Astrophysics Data System (ADS)

Sáez, Xavier; Soba, Alejandro; Sánchez, Edilberto; Kleiber, Ralf; Castejón, Francisco; Cela, José M.

2011-09-01

In the present work we report some performance measures and computational improvements recently carried out using the gyrokinetic code EUTERPE (Jost, 2000 [1] and Jost et al., 1999 [2]), which is based on the general particle-in-cell (PIC) method. The scalability of the code has been studied for up to sixty thousand processing elements and some steps towards a complete hybridization of the code were made. As a numerical example, non-linear simulations of Ion Temperature Gradient (ITG) instabilities have been carried out in screw-pinch geometry and the results are compared with earlier works. A parametric study of the influence of variables (step size of the time integrator, number of markers, grid size) on the quality of the simulation is presented.

Single-step methods for predicting orbital motion considering its periodic components

NASA Astrophysics Data System (ADS)

Lavrov, K. N.

1989-01-01

Modern numerical methods for integration of ordinary differential equations can provide accurate and universal solutions to celestial mechanics problems. The implicit single sequence algorithms of Everhart and multiple step computational schemes using a priori information on periodic components can be combined to construct implicit single sequence algorithms which combine their advantages. The construction and analysis of the properties of such algorithms are studied, utilizing trigonometric approximation of the solutions of differential equations containing periodic components. The algorithms require 10 percent more machine memory than the Everhart algorithms, but are twice as fast, and yield short term predictions valid for five to ten orbits with good accuracy and five to six times faster than algorithms using other methods.

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

Dall-Anese, Emiliano; Simonetto, Andrea

This paper focuses on the design of online algorithms based on prediction-correction steps to track the optimal solution of a time-varying constrained problem. Existing prediction-correction methods have been shown to work well for unconstrained convex problems and for settings where obtaining the inverse of the Hessian of the cost function can be computationally affordable. The prediction-correction algorithm proposed in this paper addresses the limitations of existing methods by tackling constrained problems and by designing a first-order prediction step that relies on the Hessian of the cost function (and do not require the computation of its inverse). Analytical results are establishedmore » to quantify the tracking error. Numerical simulations corroborate the analytical results and showcase performance and benefits of the algorithms.« less

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

NASA Astrophysics Data System (ADS)

Christlieb, Andrew J.; Feng, Xiao; Seal, David C.; Tang, Qi

2016-07-01

We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage (i.e., it has no internal stages to store), single-step (i.e., it has no time history that needs to be stored), maintains a discrete divergence-free condition on the magnetic field, and has the capacity to preserve the positivity of the density and pressure. To accomplish this, we use a Taylor discretization of the Picard integral formulation (PIF) of the finite difference WENO method proposed in Christlieb et al. (2015) [23], where the focus is on a high-order discretization of the fluxes (as opposed to the conserved variables). We use the version where fluxes are expanded to third-order accuracy in time, and for the fluid variables space is discretized using the classical fifth-order finite difference WENO discretization. We use constrained transport in order to obtain divergence-free magnetic fields, which means that we simultaneously evolve the magnetohydrodynamic (that has an evolution equation for the magnetic field) and magnetic potential equations alongside each other, and set the magnetic field to be the (discrete) curl of the magnetic potential after each time step. In this work, we compute these derivatives to fourth-order accuracy. In order to retain a single-stage, single-step method, we develop a novel Lax-Wendroff discretization for the evolution of the magnetic potential, where we start with technology used for Hamilton-Jacobi equations in order to construct a non-oscillatory magnetic field. The end result is an algorithm that is similar to our previous work Christlieb et al. (2014) [8], but this time the time stepping is replaced through a Taylor method with the addition of a positivity-preserving limiter. Finally, positivity preservation is realized by introducing a parameterized flux limiter that considers a linear combination of high and low-order numerical fluxes. The choice of the free parameter is then given in such a way that the fluxes are limited towards the low-order solver until positivity is attained. Given the lack of additional degrees of freedom in the system, this positivity limiter lacks energy conservation where the limiter turns on. However, this ingredient can be dropped for problems where the pressure does not become negative. We present two and three dimensional numerical results for several standard test problems including a smooth Alfvén wave (to verify formal order of accuracy), shock tube problems (to test the shock-capturing ability of the scheme), Orszag-Tang, and cloud shock interactions. These results assert the robustness and verify the high-order of accuracy of the proposed scheme.

Time-dependent dynamical behavior of surface tension on rotating fluids under microgravity environment

NASA Technical Reports Server (NTRS)

Hung, R. J.; Tsao, Y. D.; Hong, B. B.; Leslie, F. W.

1988-01-01

Time dependent evolutions of the profile of free surface (bubble shapes) for a cylindrical container partially filled with a Newtonian fluid of constant density, rotating about its axis of symmetry, have been studied. Numerical computations of the dynamics of bubble shapes have been carried out with the following situations: (1) linear functions of spin-up and spin-down in low and microgravity environments, (2) step functions of spin-up and spin-down in a low gravity environment, and (3) sinusoidal function oscillation of gravity environment in high and low rotating cylinder speeds.

The exact fundamental solution for the Benes tracking problem

NASA Astrophysics Data System (ADS)

Balaji, Bhashyam

2009-05-01

The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

Algorithms for elasto-plastic-creep postbuckling

NASA Technical Reports Server (NTRS)

Padovan, J.; Tovichakchaikul, S.

1984-01-01

This paper considers the development of an improved constrained time stepping scheme which can efficiently and stably handle the pre-post-buckling behavior of general structure subject to high temperature environments. Due to the generality of the scheme, the combined influence of elastic-plastic behavior can be handled in addition to time dependent creep effects. This includes structural problems exhibiting indefinite tangent properties. To illustrate the capability of the procedure, several benchmark problems employing finite element analyses are presented. These demonstrate the numerical efficiency and stability of the scheme. Additionally, the potential influence of complex creep histories on the buckling characteristics is considered.

A new and inexpensive non-bit-for-bit solution reproducibility test based on time step convergence (TSC1.0)

DOE PAGES

Wan, Hui; Zhang, Kai; Rasch, Philip J.; ...

2017-02-03

A test procedure is proposed for identifying numerically significant solution changes in evolution equations used in atmospheric models. The test issues a fail signal when any code modifications or computing environment changes lead to solution differences that exceed the known time step sensitivity of the reference model. Initial evidence is provided using the Community Atmosphere Model (CAM) version 5.3 that the proposed procedure can be used to distinguish rounding-level solution changes from impacts of compiler optimization or parameter perturbation, which are known to cause substantial differences in the simulated climate. The test is not exhaustive since it does not detect issues associatedmore » with diagnostic calculations that do not feedback to the model state variables. Nevertheless, it provides a practical and objective way to assess the significance of solution changes. The short simulation length implies low computational cost. The independence between ensemble members allows for parallel execution of all simulations, thus facilitating fast turnaround. The new method is simple to implement since it does not require any code modifications. We expect that the same methodology can be used for any geophysical model to which the concept of time step  convergence is applicable.« less

Some Key Factors in Policy Implementation.

ERIC Educational Resources Information Center

Rowen, Henry

Business policy texts identify numerous steps that make up the policy implementation process for private firms. On the surface, these steps also appear applicable to the implementation of public policies. However, the problems of carrying out these implementing steps in the public sector are significantly different than in the private sector due…

Variable-mesh method of solving differential equations

NASA Technical Reports Server (NTRS)

Van Wyk, R.

1969-01-01

Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations.

Numerical Treatment of Degenerate Diffusion Equations via Feller's Boundary Classification, and Applications

NASA Technical Reports Server (NTRS)

Cacio, Emanuela; Cohn, Stephen E.; Spigler, Renato

2011-01-01

A numerical method is devised to solve a class of linear boundary-value problems for one-dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite-difference scheme, grid, and treatment of the boundary data. Second-order accuracy, unconditional stability, and unconditional convergence of solutions of the finite-difference scheme to a constant as the time-step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration.

The development of an explicit thermochemical nonequilibrium algorithm and its application to compute three dimensional AFE flowfields

NASA Technical Reports Server (NTRS)

Palmer, Grant

1989-01-01

This study presents a three-dimensional explicit, finite-difference, shock-capturing numerical algorithm applied to viscous hypersonic flows in thermochemical nonequilibrium. The algorithm employs a two-temperature physical model. Equations governing the finite-rate chemical reactions are fully-coupled to the gas dynamic equations using a novel coupling technique. The new coupling method maintains stability in the explicit, finite-rate formulation while allowing relatively large global time steps. The code uses flux-vector accuracy. Comparisons with experimental data and other numerical computations verify the accuracy of the present method. The code is used to compute the three-dimensional flowfield over the Aeroassist Flight Experiment (AFE) vehicle at one of its trajectory points.

Modeling of atomization and distribution of drop-liquid fuel in unsteady swirling flows in a combustion chamber and free space

NASA Astrophysics Data System (ADS)

Sviridenkov, A. A.; Toktaliev, P. D.; Tretyakov, V. V.

2018-03-01

Numerical and experimental research of atomization and propagation of drop-liquid phase in swirling flow behind the frontal device of combustion chamber was performed. Numerical procedure was based on steady and unsteady Reynolds equations solution. It's shown that better agreement with experimental data could be obtained with unsteady approach. Fractional time step method was implemented to solve Reynolds equations. Models of primary and secondary breakup of liquid fuel jet in swirling flows are formulated and tested. Typical mean sizes of fuel droplets for base operational regime of swirling device and combustion chamber were calculated. Comparison of main features of internal swirling flow in combustion chamber with unbounded swirling flow was made.

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

Le, Hai P.; Cambier, Jean -Luc

Here, we present a numerical model and a set of conservative algorithms for Non-Maxwellian plasma kinetics with inelastic collisions. These algorithms self-consistently solve for the time evolution of an isotropic electron energy distribution function interacting with an atomic state distribution function of an arbitrary number of levels through collisional excitation, deexcitation, as well as ionization and recombination. Electron-electron collisions, responsible for thermalization of the electron distribution, are also included in the model. The proposed algorithms guarantee mass/charge and energy conservation in a single step, and is applied to the case of non-uniform gridding of the energy axis in the phasemore » space of the electron distribution function. Numerical test cases are shown to demonstrate the accuracy of the method and its conservation properties.« less

Machine learning of atmospheric chemistry. Applications to a global chemistry transport model.

NASA Astrophysics Data System (ADS)

Evans, M. J.; Keller, C. A.

2017-12-01

Atmospheric chemistry is central to many environmental issues such as air pollution, climate change, and stratospheric ozone loss. Chemistry Transport Models (CTM) are a central tool for understanding these issues, whether for research or for forecasting. These models split the atmosphere in a large number of grid-boxes and consider the emission of compounds into these boxes and their subsequent transport, deposition, and chemical processing. The chemistry is represented through a series of simultaneous ordinary differential equations, one for each compound. Given the difference in life-times between the chemical compounds (mili-seconds for O(1D) to years for CH4) these equations are numerically stiff and solving them consists of a significant fraction of the computational burden of a CTM.We have investigated a machine learning approach to solving the differential equations instead of solving them numerically. From an annual simulation of the GEOS-Chem model we have produced a training dataset consisting of the concentration of compounds before and after the differential equations are solved, together with some key physical parameters for every grid-box and time-step. From this dataset we have trained a machine learning algorithm (random regression forest) to be able to predict the concentration of the compounds after the integration step based on the concentrations and physical state at the beginning of the time step. We have then included this algorithm back into the GEOS-Chem model, bypassing the need to integrate the chemistry.This machine learning approach shows many of the characteristics of the full simulation and has the potential to be substantially faster. There are a wide range of application for such an approach - generating boundary conditions, for use in air quality forecasts, chemical data assimilation systems, centennial scale climate simulations etc. We discuss our approches' speed and accuracy, and highlight some potential future directions for improving this approach.

A time-based potential step analysis of electrochemical impedance incorporating a constant phase element: a study of commercially pure titanium in phosphate buffered saline.

PubMed

Ehrensberger, Mark T; Gilbert, Jeremy L

2010-05-01

The measurement of electrochemical impedance is a valuable tool to assess the electrochemical environment that exists at the surface of metallic biomaterials. This article describes the development and validation of a new technique, potential step impedance analysis (PSIA), to assess the electrochemical impedance of materials whose interface with solution can be modeled as a simplified Randles circuit that is modified with a constant phase element. PSIA is based upon applying a step change in voltage to a working electrode and analyzing the subsequent current transient response in a combined time and frequency domain technique. The solution resistance, polarization resistance, and interfacial capacitance are found directly in the time domain. The experimental current transient is numerically transformed to the frequency domain to determine the constant phase exponent, alpha. This combined time and frequency approach was tested using current transients generated from computer simulations, from resistor-capacitor breadboard circuits, and from commercially pure titanium samples immersed in phosphate buffered saline and polarized at -800 mV or +1000 mV versus Ag/AgCl. It was shown that PSIA calculates equivalent admittance and impedance behavior over this range of potentials when compared to standard electrochemical impedance spectroscopy. This current transient approach characterizes the frequency response of the system without the need for expensive frequency response analyzers or software. Copyright 2009 Wiley Periodicals, Inc.

Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant-Exner morphodynamic model

NASA Astrophysics Data System (ADS)

Carraro, F.; Valiani, A.; Caleffi, V.

2018-03-01

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant-Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.