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All sourcesNASA NTRSDOE OSTI
NASA NTRS2002

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

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.

Yee, H. C.↗
NASA NTRS2004

Reliability of Complex Nonlinear Numerical Simulations

This work describes some of the procedure 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.

Yee, H. C.↗
NASA NTRS2005

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

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.

Yee, H. C.↗
NASA NTRS1993

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Tsai, Wen-Lang↗
NASA NTRS1992

Non-local sub-characteristic zones of influence in unsteady interactive boundary-layers

The properties of incompressible, unsteady, interactive, boundary layers are examined for a model hypersonic boundary layer and internal flow past humps or, equivalently, external flow past short-scaled humps. Using a linear high frequency analysis, it is shown that the domains of dependence within the viscous sublayer may be a strong function of position within the sublayer and may be strongly influenced by the pressure displacement interaction, or the prescribed displacement condition. Detailed calculations are presented for the hypersonic boundary layer. This effect is found to carry over directly to the fully viscous problem as well as the nonlinear problem. In the fully viscous problem, the non-local character of the domains of dependence manifests itself in the sub-characteristics. Potential implications of the domain of dependence structure on finite difference computations of unsteady boundary layers are briefly discussed.

Rothmayer, A. P.↗
NASA NTRS2012

The Transport of Mass, Energy, and Entropy in Cryogenic Support Struts for Engineering Design

Engineers working to understand and reduce cryogenic boil-off must solve a. variety of transport problems. An important class of nonlinear problems involves the thermal and mechanical design of cryogenic struts. These classic problems are scattered about the literature and typically require too many resources to obtain. So, to save time for practicing engineers, the author presents this essay. Herein, a variety of new, old, and revisited analytical and finite difference solutions of the thermal problem are covered in this essay, along with commentary on approach and assumptions, This includes a few thermal radiation and conduction combined mode solution with a discussion on insulation, optimum emissivity, and geometrical phenomenon. Solutions to cooling and heat interception problems are also presented, including a discussion of the entropy generation. And the literature on the combined mechanical and thermal design of cryogenic support struts is reviewed with an introduction to the associated numerical methods.

Elchert, J. P.↗
NASA NTRS2012

The Transport of Mass, Energy, and Entropy in Cryogenic Support Struts for Engineering Design

Engineers working to understand and reduce cryogenic boil-off must solve a variety of transport problems. An important class of nonlinear problems involves the thermal and mechanical design of cryogenic struts. These classic problems are scattered about the literature and typically require too many resources to obtain. So, to save time for practicing engineers, the author presents this essay. Herein, a variety of new, old, and revisited analytical and finite difference solutions of the thermal problem are covered in this essay, along with commentary on approach and assumptions. This includes a few thermal radiation and conduction combined mode solutions with a discussion on insulation, optimum emissivity, and geometrical phenomenon. Solutions to cooling and heat interception problems are also presented, including a discussion of the entropy generation. The literature on the combined mechanical and thermal design of cryogenic support struts is reviewed with an introduction to the associated numerical methods.

Elchert, J. P.↗
NASA NTRS1986

Multi-dimensional initial-boundary value problems with strong nonlinearities

Initial boundary value problems for the general scalar singularly perturbed parabolic equation in a cylindrical domain are studied. The one-dimensional reaction-diffusion equation is considered in the absence of convection. Then, the general one-dimensional and N-dimensional reaction-convection-diffusion equations are analyzed.

Howes, F. A.↗
NASA NTRS1973

Processor for high-density digital tape-recorded signals

Linear filter and detection theory can bear on problem of reconstructing recorded bit stream. Problem can be taken from realm of nonlinear problems even though basic record process is still recognized as highly nonlinear. Digital tape recorder can be modeled as particular type of linear communication channel with intersymbol interference.

Ashlock, J. C.↗
DOE OSTI2025

Explicit Runge–Kutta Methods that Alleviate Order Reduction

Explicit Runge–Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to an initial boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method’s order, weak stage order, and number of stages. Furthermore, we derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.

explicit Runge–Kutta↗
NASA NTRS1983

Spectral methods for time dependent partial differential equations

The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

Gottlieb, D.↗
NASA NTRS1989

Finite element methods in fracture mechanics

Finite-element methodology specific to the analysis of fracture mechanics problems is reviewed. Primary emphasis is on the important algorithmic developments which have enhanced the numerical modeling of fracture processes. Methodologies to address elastostatic problems in two and three dimensions, elastodynamic problems, elastoplastic problems, special considerations for three-dimensional nonlinear problems, and the modeling of stable crack growth are reviewed. In addition, the future needs of the fracture community are discussed and open questions are identified.

Liebowitz, H.↗
NASA NTRS1994

Finite dimensional approximation of a class of constrained nonlinear optimal control problems

An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, and approximate problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second, the Ginzburg-Landau equations of superconductivity, and the third, the Navier-Stokes equations for incompressible, viscous flows.

Gunzburger, Max D.↗
NASA NTRS1985

Iterative methods for mixed finite element equations

Iterative strategies for the solution of indefinite system of equations arising from the mixed finite element method are investigated in this paper with application to linear and nonlinear problems in solid and structural mechanics. The augmented Hu-Washizu form is derived, which is then utilized to construct a family of iterative algorithms using the displacement method as the preconditioner. Two types of iterative algorithms are implemented. Those are: constant metric iterations which does not involve the update of preconditioner; variable metric iterations, in which the inverse of the preconditioning matrix is updated. A series of numerical experiments is conducted to evaluate the numerical performance with application to linear and nonlinear model problems.

Nakazawa, S.↗
NASA NTRS1997

Solution of Aeroacoustic Problems by a Nonlinear, Hybrid Method

Category 1, problem 3 (scattering of sound by a sphere) and category 2, problem 1 (spherical source in a cylindrical duct subject to uniform flow) are solved in generalized coordinates using the nonlinear Euler equations together with nonreflecting boundary conditions. A temporally and spatially fourth-order accurate finite-difference, Runge-Kutta time-marching technique is employed for the near-field calculations and a Kirchhoff method is employed for the prediction of far-field sound. Computations are all performed on parallel processors using the data-parallel paradigm.

Oezyoeruek, Yusuf↗
NASA NTRS1987

Model-size reduction for the analysis of symmetric structures with asymmetric boundary conditions

A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (1) restructuring the governing finite-element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (2) successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite-element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving nonlinear problems is discussed.

Noor, Ahmed K.↗