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Results for “explicit and implicit methods”

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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

Computational methods for structural load and resistance modeling

An automated capability for computing structural reliability considering uncertainties in both load and resistance variables is presented. The computations are carried out using an automated Advanced Mean Value iteration algorithm (AMV +) with performance functions involving load and resistance variables obtained by both explicit and implicit methods. A complete description of the procedures used is given as well as several illustrative examples, verified by Monte Carlo Analysis. In particular, the computational methods described in the paper are shown to be quite accurate and efficient for a material nonlinear structure considering material damage as a function of several primitive random variables. The results show clearly the effectiveness of the algorithms for computing the reliability of large-scale structural systems with a maximum number of resolutions.

Thacker, B. H.↗
NASA NTRS1979

Generalized disks of contractivity for explicit and implicit Runge-Kutta methods

The A-contractivity of Runge-Kutta methods with respect to an inner product norm was investigated thoroughly by Butcher and Burrage (who used the term B-stability). Their theory is extended to contractivity in a region bounded by a circle through the origin. The largest possible circle is calculated for many known explicit Runge-Kutta methods. As a rule it is considerably smaller than the stability region, and in several cases it degenerates to a point. It is shown that an explicit Runge-Kutta method cannot be contractive in any circle of this class if it is more than fourth order accurate.

Dahlquist, G.↗
DOE OSTI2021

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. Here this work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1972

Computer simulation of impurity diffusion in silicon, part 1

The elementary classical models for idealized diffusion conditions are described, and the principles are then used in developing more realistic models. The practical models require some type of numerical analysis. The numerical techniques are outlined and details concerning their implementation are given. Some results are presented which were obtained with the computer programs implementing the numerical techniques with implicit and explicit methods. Special problems of impurity-rich interlayers forming between an oxide and silicon are considered. A set of computed curves for sheet resistance, junction depth, and oxide thickness for different diffusion schedules is included.

Gassaway, J. D.↗
NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗
NASA NTRS1989

An expert system for setting time steps in dynamic finite element programs

An expert system, ETUDES - Expert Time integration control Using Deep and Surface Knowledge System, which addresses the determination of the timestep for time integration of linear structural dynamic equations is described. This timestep may also be applicable for a moderately nonlinear simulation of the same structure. The program also determines whether an explicit or implicit method is most efficient for the particular simulation. A production rule programming system written in OPS5 is used for the implementation of this prototype expert system. Issues relating to the expert system architecture for this application, such as knowledge representation and structure, as well as domain knowledge are discussed. The prototype is evaluated by measuring its performance in various benchmark model problems.

Ramirez, Martin R.↗
NASA NTRS1990

Mapping unstructured grid problems to the connection machine

We present a highly parallel graph mapping technique that enables one to solve unstructured grid problems on massively parallel computers. Many implicit and explicit methods for solving discretizated partial differential equations require each point in the discretization to exchange data with its neighboring points every time step or iteration. The time spent communicating can limit the high performance promised by massively parallel computing. To eliminate this bottleneck, we map the graph of the irregular problem to the graph representing the interconnection topology of the computer such that the sum of the distances that the messages travel is minimized. We show that, in comparison to a naive assignment of processors, our heuristic mapping algorithm significantly reduces the communication time on the Connection Machine, CM-2.

Hammond, Steven W.↗
NASA NTRS1991

An assessment of computational fluid dynamic techniques in the analysis and design of turbomachinery - The 1990 Freeman Scholar Lecture

Various computational fluid dynamic techniques are reviewed focusing on the Euler and Navier-Stokes solvers with a brief assessment of boundary layer solutions, and quasi-3D and quasi-viscous techniques. Particular attention is given to a pressure-based method, explicit and implicit time marching techniques, a pseudocompressibility technique for incompressible flow, and zonal techniques. Recommendations are presented with regard to the most appropriate technique for various flow regimes and types of turbomachinery, incompressible and compressible flows, cascades, rotors, stators, liquid-handling, and gas-handling turbomachinery.

Lakshminarayana, B.↗
NASA NTRS1990

Solving unstructured grid problems on massively parallel computers

A highly parallel graph mapping technique that enables one to efficiently solve unstructured grid problems on massively parallel computers is presented. Many implicit and explicit methods for solving discretized partial differential equations require each point in the discretization to exchange data with its neighboring points every time step or iteration. The cost of this communication can negate the high performance promised by massively parallel computing. To eliminate this bottleneck, the graph of the irregular problem is mapped into the graph representing the interconnection topology of the computer such that the sum of the distances that the messages travel is minimized. It is shown that using the heuristic mapping algorithm significantly reduces the communication time compared to a naive assignment of processes to processors.

Hammond, Steven W.↗
NASA NTRS1997

Flowfield-Dependent Mixed Explicit-Implicit (FDMEL) Algorithm for Computational Fluid Dynamics

Despite significant achievements in computational fluid dynamics, there still remain many fluid flow phenomena not well understood. For example, the prediction of temperature distributions is inaccurate when temperature gradients are high, particularly in shock wave turbulent boundary layer interactions close to the wall. Complexities of fluid flow phenomena include transition to turbulence, relaminarization separated flows, transition between viscous and inviscid incompressible and compressible flows, among others, in all speed regimes. The purpose of this paper is to introduce a new approach, called the Flowfield-Dependent Mixed Explicit-Implicit (FDMEI) method, in an attempt to resolve these difficult issues in Computational Fluid Dynamics (CFD). In this process, a total of six implicitness parameters characteristic of the current flowfield are introduced. They are calculated from the current flowfield or changes of Mach numbers, Reynolds numbers, Peclet numbers, and Damkoehler numbers (if reacting) at each nodal point and time step. This implies that every nodal point or element is provided with different or unique numerical scheme according to their current flowfield situations, whether compressible, incompressible, viscous, inviscid, laminar, turbulent, reacting, or nonreacting. In this procedure, discontinuities or fluctuations of an variables between adjacent nodal points are determined accurately. If these implicitness parameters are fixed to certain numbers instead of being calculated from the flowfield information, then practically all currently available schemes of finite differences or finite elements arise as special cases. Some benchmark problems to be presented in this paper will show the validity, accuracy, and efficiency of the proposed methodology.

Garcia, S. M.↗
NASA NTRS1982

The SPAR thermal analyzer: Present and future

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

Marlowe, M. B.↗
NASA NTRS2014

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

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.

Nielsen, Tanner B.↗
DOE OSTI2019

Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods

We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Nørsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numerical and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.

42 ENGINEERING↗
DOE OSTI2018

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

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, 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.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1982

Mixed time integration methods for transient thermal analysis of structures

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.

Liu, W. K.↗
NASA NTRS1983

Mixed time integration methods for transient thermal analysis of structures

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.

Liu, W. K.↗
NASA NTRS1986

Report on the feasibility of hypercube concurrent processing systems in computational fluid dynamics

The feasibility of using hypercube-connected concurrent processor systems for problems in computational fluid dynamics is studied. Both explicit and implicit numerical methods are considered and several alternative implementations of these methods are evaluated on concurrent processor systems. A Lax-Wendroff explicit method was designed and implemented for the Navier-Stokes equations. The code runs on the Intel iPSC concurrent processor system. Tests of this code show that it is reasonably efficient. The Beam and Warming implicit factored method was designed and implemented for Berger's equation. Preliminary tests show that the efficiency of code is poor.

Bruno, J.↗
NASA NTRS1987

A barely implicit correction for flux-corrected transport

In virtue of adding one elliptic equation that must be implicitly solved, the 'barely implicit correction' method presented removes the stringent sidestep limit imposed by sound speed in the explicit methods. The barely implicit method is presently combined with a flux-corrected transport algorithm, which has been rendered fourth-order by the appropriate subtraction of corrected fluxes, in order to accurately represent the sharp gradients in subsonic flows. The multigrid MGRID technique (DeVore, 1984) is used to efficiently solve the elliptic pressure equation.

Patnaik, G.↗