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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 307 records · Page 17
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems.

Yee, H. C.↗
NASA NTRS1987

Three dimensional hypersonic flow simulations with the CSCM implicit upwind Navier-Stokes method

The Bardina and Lombard (1985) three-dimensional CSCM Navier-Stokes method is presently extended to the simulation of complex hypersonic reentry vehicle external flows at angle of attack. The robust stability of the method derives from the combination of conservative implicit upwind flux difference splitting with a three-dimensional diagonally-dominant approximate factorization and relaxation scheme and characteristic-based implicit boundary approximations. The method's efficiency derives from an implicit symmetric Gauss-Seidel 'method of planes' relaxation scheme with alternating directional space marching sweeps along the flow coordinate direction.

Bardina, Jorge↗
DOE OSTI2016

An implicit δf particle-in-cell method with sub-cycling and orbit averaging for Lorentz ions

A second order implicit δf Lorentz ion hybrid model with sub-cycling and orbit averaging has been developed to study low-frequency, quasi-neutral plasmas. Models using the full Lorentz force equations of motion for ions may be useful for verifying gyrokinetic ion simulation models in applications where higher order terms may be important. In the presence of a strong external magnetic field, previous Lorentz ion models are limited to simulating very short time scales due to the small time step required for resolving the ion gyromotion. Here, we use a simplified model for ion Landau damped ion acoustic waves in a uniform magnetic field as a test bed for developing efficient time stepping methods to be used with the Lorentz ion hybrid model. A detailed linear analysis of the model is derived to validate simulations and to examine the significance of ion Bernstein waves in the Lorentz ion model. Linear analysis of a gyrokinetic ion model is also performed, and excellent agreement with the dispersion results from the Lorentz ion model is demonstrated for the ion acoustic wave. The sub-cycling/orbit averaging algorithm is shown to produce accurate finite-Larmor-radius effects using large macro-time steps sizes, and numerical damping of high frequency fluctuations can be achieved by formulating the field model in terms of the perturbed flux density. Furthermore, a CPU–GPU implementation of the sub-cycling/orbit averaging is presented and is shown to achieve a significant speedup over an equivalent serial code.

Computer Science↗
NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

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

Yee, H. C.↗
NASA NTRS2006

Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation

We investigate p-multigrid as a solution method for several different discontinuous Galerkin (DG) formulations of the Poisson equation. Different combinations of relaxation schemes and basis sets have been combined with the DG formulations to find the best performing combination. The damping factors of the schemes have been determined using Fourier analysis for both one and two-dimensional problems. One important finding is that when using DG formulations, the standard approach of forming the coarse p matrices separately for each level of multigrid is often unstable. To ensure stability the coarse p matrices must be constructed from the fine grid matrices using algebraic multigrid techniques. Of the relaxation schemes, we find that the combination of Jacobi relaxation with the spectral element basis is fairly effective. The results using this combination are p sensitive in both one and two dimensions, but reasonable convergence rates can still be achieved for moderate values of p and isotropic meshes. A competitive alternative is a block Gauss-Seidel relaxation. This actually out performs a more expensive line relaxation when the mesh is isotropic. When the mesh becomes highly anisotropic, the implicit line method and the Gauss-Seidel implicit line method are the only effective schemes. Adding the Gauss-Seidel terms to the implicit line method gives a significant improvement over the line relaxation method.

Helenbrook, B. T.↗
DOE OSTI2026

Explicit Monotone Stable Super-Time-stepping Methods for Finite Time Singularities

We explore a novel way to numerically resolve the scaling behavior of finite-time singularities in solutions of nonlinear parabolic PDEs. The Runge–Kutta–Legendre (RKL) and Runge–Kutta–Gegenbauer (RKG) super-time-stepping methods were originally developed for nonlinear complex physics problems with diffusion. These are multistage single step second-order, forward-in-time methods with no implicit solves. The advantage is that the time-step size for stability scales with stage number 𝑠 as $\mathcal{O}$⁡(𝑠 2 ). Many interesting nonlinear PDEs have finite-time singularities, and the presence of diffusion often limits one to using implicit or semi-implicit time-step methods for stability constraints. Finite-time singularities are particularly challenging due to the large range of scales that one desires to resolve, often with adaptive spatial grids and adaptive time steps. Here, in this study, we show two examples of nonlinear PDEs for which the self-similar singularity structure has time and space scales that are resolvable using the RKL and RKG methods, without forcing even smaller time steps. Compared to commonly used implicit numerical methods, we achieve a significantly smaller run time while maintaining comparable accuracy. We also prove numerical monotonicity for both the RKL and RKG methods under their linear stability conditions for the constant coefficient heat equation, in the case of infinite domain and periodic boundary condition, leading to a theoretical guarantee of the superiority of the RKL and RKG methods over traditional super-time-stepping methods, such as the Runge-Kutta-Chebyshev and the orthogonal Runge-Kutta-Chebyshev methods. Code can be found at https://github.com/ZT220501/SRK-Singularity.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Yee, H. C.↗
NASA NTRSDate not supplied

Updates to Implicit Edge-Based Gradient Methods

In this paper, we report updates to the implicit edge-based gradient methods originally introduced in [H. Nishikawa, AIAA Paper 2020-3048, 2020]. First, we clarify the relationship between gradient accuracy and truncation error and show that the quadratic method involves a free parameter. Then, we provide a complete description and a simplified matrix form of the implicit gradient systems including a consistent boundary treatment, and derive a set of parameters for achieving fourth-order gradient accuracy on regular tetrahedral grids. A stability analysis is performed for a relaxation scheme used to solve the implicit gradient systems, and the result serves as a guide for choosing parameters. Numerical results are shown for accuracy verification and also for realistic inviscid flow problems in three dimensions, including flows with shock waves.

Weighted Least-Squares↗
NASA NTRS1977

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

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.

Davis, J. E.↗
DOE OSTI2021

Tusas: A fully implicit parallel approach for coupled phase-field equations

In this study, we develop a fully-coupled, fully-implicit approach for phase-field modeling of solidification in metals and alloys. Predictive simulation of solidification in pure metals and metal alloys remains a significant challenge in the field of materials science, as microstructure formation during the solidification process plays a critical role in the properties and performance of the solid material. Our simulation approach consists of a finite element spatial discretization of the fully-coupled nonlinear system of partial differential equations at the microscale, which is treated implicitly in time with a preconditioned Jacobian-free Newton-Krylov method. The approach is algorithmically scalable as well as efficient due to an effective preconditioning strategy based on algebraic multigrid and block factorization. We implement this approach in the open-source Tusas framework, which is a general, flexible tool developed in C++ for solving coupled systems of nonlinear partial differential equations. The performance of our approach is analyzed in terms of algorithmic scalability and efficiency, while the computational performance of Tusas is presented in terms of parallel scalability and efficiency on emerging heterogeneous architectures. We demonstrate that modern algorithms, discretizations, and computational science, and heterogeneous hardware provide a robust route for predictive phase-field simulation of microstructure evolution during additive manufacturing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Implicit extensions of an explicit multirate Runge–Kutta scheme

In this work, we propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable for time-stepping adaptive mesh refinement PDE discretizations with different degrees of stiffness. A numerical example with an advection-diffusion problem illustrates the new method's properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Implicit sampling combined with reduced order modeling for the inversion of vadose zone hydrological data

Bayesian inverse modeling techniques are computationally expensive because many forward simulations are needed when sampling the posterior distribution of the parameters. Here, in this paper, we combine the implicit sampling method and generalized polynomial chaos expansion (gPCE) to significantly reduce the computational cost of performing Bayesian inverse modeling. There are three steps in this approach: (1) find the maximizer of the likelihood function using deterministic approaches; (2) construct a gPCE-based surrogate model using the results from a limited number of forward simulations; and (3) efficiently sample the posterior distribution of the parameters using implicit sampling method. The cost of constructing the gPCE-based surrogate model is further decreased by using sparse Bayesian learning to reduce the number of gPCE coefficients that have to be determined. We demonstrate the approach for a synthetic ponded infiltration experiment simulated with TOUGH2. The surrogate model is highly accurate with mean relative error that is < 0.035% in predicting saturation and < 0.25% in predicting the likelihood function. The posterior distribution of the parameters obtained using our proposed technique is nearly indistinguishable from the results obtained from either an implicit sampling method or a Markov chain Monte Carlo method utilizing the full mode.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. In this work, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1994

Steady potential solver for unsteady aerodynamic analyses

Development of a steady flow solver for use with LINFLO was the objective of this report. The solver must be compatible with LINFLO, be composed of composite mesh, and have transonic capability. The approaches used were: (1) steady flow potential equations written in nonconservative form; (2) Newton's Method; (3) implicit, least-squares, interpolation method to obtain finite difference equations; and (4) matrix inversion routines from LINFLO. This report was given during the NASA LeRC Workshop on Forced Response in Turbomachinery in August of 1993.

Hoyniak, Dan↗
NASA NTRS1995

Nozzle flow with vibrational nonequilibrium

This research concerns the modeling and numerical solutions of the coupled system of compressible Navier-Stokes equations in cylindrical coordinates under conditions of equilibrium and nonequilibrium thermodynamics. The problem considered was the modeling of a high temperature diatomic gas N2 flowing through a converging-diverging high expansion nozzle. The problem was modeled in two ways. The first model uses a single temperature with variable specific heats as functions of this temperature. For the second model we assume that the various degrees of freedom all have a Boltzmann distribution and that there is a continuous redistribution of energy among the various degrees of freedom as the gas passes through the nozzle. Each degree of freedom is assumed to have its own temperature and, consequently, each system state can be characterized by these temperatures. This suggests that formulation of a second model with a vibrational degree of freedom along with a rotational-translation degree of freedom, each degree of freedom having its own temperature. Initially the vibrational degree of freedom is excited by heating the gas to a high temperature. As the high temperature gas passes through the nozzle throat there is a sudden drop in temperature along with a relaxation time for the vibrational degree of freedom to achieve equilibrium with the rotational-translation degree of freedom. That is, we assume that the temperature change upon passing through the throat is so great that the changes in the vibrational degree of freedom occur at a much slower pace and consequently lags behind the rotational-translational energy changes. This lag results in a finite relaxation time. In this context the term nonequilibrium is used to denote the fact that the energy content of the various degrees of freedom are characterized by two temperatures. We neglect any chemical reactions which could also add nonequilibrium effects. We develop the energy equations for the nonequilibrium model from first principles. The resulting equations, which model the nozzle flow, can be expressed in various forms. In most forms the resulting equations are coupled systems of nonlinear partial differential equations subject to certain boundary conditions. To solve the resulting coupled system of nonlinear partial differential equations, several numerical techniques were investigated: (1) the explicit MacCormack method, (2) the explicit-implicit MacCormack method, (3) the method of operator splitting, (4) factorization schemes, and (5) the Steger-Warming scheme.

Heinbockel, J. H.↗
DOE OSTI2017

Memory-optimized shift operator alternating direction implicit finite difference time domain method for plasma

Through introducing the alternating direction implicit (ADI) technique and the memory-optimized algorithm to the shift operator (SO) finite difference time domain (FDTD) method, the memory-optimized SO-ADI FDTD for nonmagnetized collisional plasma is proposed and the corresponding formulae of the proposed method for programming are deduced. In order to further the computational efficiency, the iteration method rather than Gauss elimination method is employed to solve the equation set in the derivation of the formulae. Complicated transformations and convolutions are avoided in the proposed method compared with the Z transforms (ZT) ADI FDTD method and the piecewise linear JE recursive convolution (PLJERC) ADI FDTD method. The numerical dispersion of the SO-ADI FDTD method with different plasma frequencies and electron collision frequencies is analyzed and the appropriate ratio of grid size to the minimum wavelength is given. The accuracy of the proposed method is validated by the reflection coefficient test on a nonmagnetized collisional plasma sheet. The testing results show that the proposed method is advantageous for improving computational efficiency and saving computer memory. The reflection coefficient of a perfect electric conductor (PEC) sheet covered by multilayer plasma and the RCS of the objects coated by plasma are calculated by the proposed method and the simulation results are analyzed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1985

Three dimensional CSCM method for the compressible Navier-Stokes equations with application to a multi-nozzle exhaust flowfield

The effective design of three-dimensional flow systems such as a clustered multinozzle shrouded exhaust system requires detailed analysis of the internal flowfield. It is pointed out that new numerical techniques on the most modern vector computers are making these analyses practicable. In this paper, the upwind differenced implicit conservative supracharacteristics method (CSCM) for three-dimensional viscous flow is presented. The new method is an implicit 'method of planes' symmetric Gauss-Seidel relaxation scheme. The data is conveniently stored on successive planes along the streamwise coordinate, and the system of equations is solved twice in each successive plane of the streamwise coordinate. The developed new three-dimensional compressible Navier-Stokes algorithm combines the best features of the storage and computationally efficient space marching procedures with the generality of time dependent techniques to solve flows with elliptic and streamwise separated flows.

Bardina, J.↗