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At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1979

Multi-level adaptive finite element methods. 1: Variation problems

A general numerical strategy for solving partial differential equations and other functional problems by cycling between coarser and finer levels of discretization is described. Optimal discretization schemes are provided together with very fast general solvers. It is described in terms of finite element discretizations of general nonlinear minimization problems. The basic processes (relaxation sweeps, fine-grid-to-coarse-grid transfers of residuals, coarse-to-fine interpolations of corrections) are directly and naturally determined by the objective functional and the sequence of approximation spaces. The natural processes, however, are not always optimal. Concrete examples are given and some new techniques are reviewed. Including the local truncation extrapolation and a multilevel procedure for inexpensively solving chains of many boundary value problems, such as those arising in the solution of time-dependent problems.

Brandt, A.↗
NASA NTRS1984

Computational problems on composite grids

Most currently used algorithms for the numerical solution of the partial differential equations encountered in fluid flow problems can be implemented on composite grid systems. Finite volume formulations are easier to derive on composite grids, and may in principle be derived for partial differential equations of all types. Except when using Alternating Difference Implicit-type schemes, the overlapping of grids is an alternative to the more common grid construction procedure where grid lines continue smoothly from one subregion to the next. In the solution of model problems, the correct choice of an interpolation formula has been found able to reduce errors by a factor of two.

Mastin, C. W.↗
NASA NTRS1989

Goertler vortices in supersonic and hypersonic boundary layers

The problem of Goertler vortices in compressible boundary layers over concave walls is considered. At O(1) wavelengths, the instability is governed by parabolic partial differential equations that are solved numerically to determine the effect of various initial conditions on the development of Goertler vortex instability in compressible boundary layers. The results show that both the velocity and temperature fluctuations may lead to a Goertler vortex. The vortex growth rates determined from the present method are found to differ somewhat from those given by a normal mode solution. At both the supersonic and hypersonic Mach numbers, cooling has a small destabilizing effect. In addition, the most unstable disturbances shift toward lower wavelengths because of thinning of the boundary layer. The results also show that compressibility has a stabilizing effect on the Goertler instability, while the effect of an adverse pressure gradient is found to be destabilizing. The behavior of the Goertler vortex structure with Mach number is also examined. At hypersonic Mach numbers, vortices are located near the edge of the boundary layer for adiabatic wall conditions. However, the entire boundary layer is affected when the wall is cooled.

Spall, R. E.↗
NASA NTRS1990

On the receptivity and non-parallel stability of travelling disturbances in rotating disk flow

The generation and evolution of small amplitude wavelength traveling disturbances in rotating disk flow is discussed. The steady rotational speed of the disk is perturbed so as to introduce high frequency oscillations in the flow field. Secondly, surface imperfections are introduced on the disk such as roughness elements. The interaction of these two disturbances will generate the instability waves whose evolution is governed by parabolic partial differential equations that are solved numerically. For the class of disturbances considered (wavelength on the order of Reynolds number), it is found that eigensolutions exist which decay or grow algebraically in the radial direction. However, these solutions grow only for frequencies larger than 4.58 times the steady rotational speed of the disk. The computed receptivity coefficient shows that there is an optimum size of roughness for which these modes are excited the most. The width of these roughness elements in the radial direction is about .1 r(sub 0) where r(sub 0) is the radial location of the roughness. It is also found that the receptivity coefficient is larger for a negative spanwise wavenumber than for a positive one. Typical wave angles found for these disturbances are about -26 degrees.

Balakumar, P.↗
NASA NTRS1991

Mapping implicit spectral methods to distributed memory architectures

Spectral methods were proven invaluable in numerical simulation of PDEs (Partial Differential Equations), but the frequent global communication required raises a fundamental barrier to their use on highly parallel architectures. To explore this issue, a 3-D implicit spectral method was implemented on an Intel hypercube. Utilization of about 50 percent was achieved on a 32 node iPSC/860 hypercube, for a 64 x 64 x 64 Fourier-spectral grid; finer grids yield higher utilizations. Chebyshev-spectral grids are more problematic, since plane-relaxation based multigrid is required. However, by using a semicoarsening multigrid algorithm, and by relaxing all multigrid levels concurrently, relatively high utilizations were also achieved in this harder case.

Overman, Andrea L.↗
NASA NTRS1990

Fast secant methods for the iterative solution of large nonsymmetric linear systems

A family of secant methods based on general rank-1 updates was revisited in view of the construction of iterative solvers for large non-Hermitian linear systems. As it turns out, both Broyden's good and bad update techniques play a special role, but should be associated with two different line search principles. For Broyden's bad update technique, a minimum residual principle is natural, thus making it theoretically comparable with a series of well known algorithms like GMRES. Broyden's good update technique, however, is shown to be naturally linked with a minimum next correction principle, which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized partial differential equations of convection diffusion type in 2-D with integral layers give a first impression of the possible power of the derived good Broyden variant.

Deuflhard, Peter↗
NASA NTRS1991

On the receptivity and nonparallel stability of traveling disturbances in rotating-disk flow

The generation and evolution of small amplitude long wavelength traveling disturbances in rotating-disk flow are discussed. The steady rotational speed of the disk is perturbed so as to introduce high-frequency oscillations in the flow field. Secondly, surface imperfections on the disk such as roughness elements are introduced. The interaction of these two disturbances will generate the instability waves whose evolution is governed by parabolic partial differential equations which are solved numerically. It is found that, for the class of disturbances considered here (wavelength on the order of Reynolds number), eigensolutions exist which decay or grow algebraically in the radial direction. However, these solutions grow only for frequencies larger than 4.58 times the steady rotational speed of the disk. The computed receptivity coefficient shows that there is an optimum size of roughness for which these modes are preferentially excited.

Balakumar, P.↗
NASA NTRS1995

The initial-value problem for viscous channel flows

Plane viscous channel flows are perturbed and the ensuing initial-value problems are investigated in detail. Unlike traditional methods where traveling wave normal modes are assumed for solution, this works offers a means whereby completely arbitrary initial input can be specified without having to resort to eigenfunction expansions. The full temporal behavior, including both early time transients and the long time asymptotics, can be determined for any initial disturbance. Effects of three-dimensionality can be assessed. The bases for the analysis are: (a) linearization of the governing equations; (b) Fourier decomposition in the spanwise and streamwise directions of the flow; and (c) direct numerical integration of the resulting partial differential equations. All of the stability data that are known for such flows can be reproduced. Also, the optimal initial condition can be determined in a straight forward manner and such optimal conditions clearly reflect transient growth data that is easily determined by a rational choice of a basis for the initial conditions. Although there can be significant transient growth for subcritical values of the Reynolds number using this approach it does not appear possible that arbitrary initial conditions will lead to the exceptionally large transient amplitudes that have been determined by optimization of normal modes. The approach is general and can be applied to other classes of problems where only a finite discrete spectrum exists, such as the boundary layer for example.

Criminale, W. O.↗
NASA NTRS2002

Elliptic Grid Generation of Spiral-Bevel Pinion Gear Typical of OH-58 Helicopter Transmission

This paper discusses the source term treatment in the numerical solution of elliptic partial differential equations for an interior grid generation problem in generalized curvilinear coordinates. The geometry considered is that of a planar cross-section of a generic spiral-bevel gear tooth typical of a pinion in the OH-58 helicopter transmission. The source terms used are appropriate for an interior grid domain where all the boundaries are prescribed via a combination of Dirichlet and Neumann boundary conditions. New constraints based on the Green's Theorem are derived which uniquely determine the coefficients in the source terms. These constraints are designed for boundary clustered grids where gradients in physical quantities need to be resolved adequately. However, it is seen that the present formulation works satisfactorily for mild clustering also. Thus, a fully automated elliptic grid generation technique is made possible where there is no need for a parametric study of these parameters since the new relations fix these free parameters uniquely.

Kaul, Upender K.↗
NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations, the steady Euler equations, using Newton's linearization procedure is presented. A theorem indicating quadratic convergence for the case of differential equations is demonstrated. A condition for the domain of quadratic convergence Omega(2) is obtained which indicates that whether an approximation lies in Omega(2) depends on the rate of change and the smoothness of the flow vectors, and hence is problem-dependent. The choice of spatial differencing, of particular importance for the present method, is discussed. The treatment of boundary conditions is addressed, and the system of equations resulting from the foregoing analysis is summarized and solution strategies are discussed. The convergence of calculated solutions is demonstrated by comparing them with exact solutions to one and two-dimensional problems.

Liou, M.-S.↗
NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Liou, M. S.↗
NASA NTRS1987

Numerical observations of dynamic behavior in two-dimensional compressible convection

Using numerical simulations of the anelastic partial differential equations, convective flow in a two-dimensional polytropic atmosphere is investigated. Fixing the aspect ratio at unity and choosing the viscosity as the order parameter, the numerical solutions have indicated a variety of dynamic behavior including (1) oscillatory motion with a period of order the eddy turnover time, (2) quasi-periodic motion with a long period much longer than an eddy turnover time, and (3) at least two branches of solutions in certain regimes.

Ginet, G. P.↗
NASA NTRS1980

An alternative approach to the numerical simulation of steady inviscid flow

A numerical procedure for the efficient simulation of steady inviscid flow is described and its utility demonstrated. Application of the surrogate equation technique allows the formulation of stable, fully conservative, type dependent finite difference equations for use in obtaining numerical solutions to systems of first order partial differential equations, such as the steady state Euler equations or their various approximations. Computational results are presented for the full Euler equations and for the transonic disturbance equations. For the latter case, a computational efficiency greater than that obtained by means of the standard perturbation potential approach is indicated.

Johnson, G. M.↗
NASA NTRS1981

Surrogate-equation technique for simulation of steady inviscid flow

A numerical procedure for the iterative solution of inviscid flow problems is described, and its utility for the calculation of steady subsonic and transonic flow fields is demonstrated. Application of the surrogate equation technique defined herein allows the formulation of stable, fully conservative, type dependent finite difference equations for use in obtaining numerical solutions to systems of first order partial differential equations, such as the steady state Euler equations. Steady, two dimensional solutions to the Euler equations for both subsonic, rotational flow and supersonic flow and to the small disturbance equations for transonic flow are presented.

Johnson, G. M.↗
DOE OSTI2023

Staggered scheme for the compressible fluctuating hydrodynamics of multispecies fluid mixtures

Here, we present a numerical formulation for the solution of nonisothermal, compressible Navier-Stokes equations with thermal fluctuations to describe mesoscale transport phenomena in multispecies fluid mixtures. The novelty of our numerical method is the use of staggered grid momenta along with a finite volume discretization of the thermodynamic variables to solve the resulting stochastic partial differential equations. The key advantages of the numerical scheme are that it significantly simplifies the discretization of diffusive and stochastic momentum fluxes into a more compact form, and it provides an unambiguous prescription of boundary conditions involving pressure. The staggered grid scheme more accurately reproduces the equilibrium static structure factor of hydrodynamic fluctuations in gas mixtures compared to a collocated scheme described previously by Balakrishnan et al. [Phys. Rev. E 89, 013017 (2014)1539-375510.1103/PhysRevE.89.013017]. The numerical method is tested for ideal noble gases mixtures under various nonequilibrium conditions, such as applied thermal and concentration gradients, to assess the role of cross-diffusion effects, such as Soret and Dufour, on the long-ranged correlations of hydrodynamic fluctuations, which are also more accurately reproduced compared to the collocated scheme. We numerically study giant nonequilibrium fluctuations driven by concentration gradients and fluctuation-driven Rayleigh-Taylor instability in gas mixtures. Wherever applicable, excellent agreement is observed with theory and measurements from the direct simulation Monte Carlo method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗
NASA NTRS1980

An interaction algorithm for prediction of mean and fluctuating velocities in two-dimensional aerodynamic wake flows

A theoretical analysis is presented yielding sets of partial differential equations for determination of turbulent aerodynamic flowfields in the vicinity of an airfoil trailing edge. A four phase interaction algorithm is derived to complete the analysis. Following input, the first computational phase is an elementary viscous corrected two dimensional potential flow solution yielding an estimate of the inviscid-flow induced pressure distribution. Phase C involves solution of the turbulent two dimensional boundary layer equations over the trailing edge, with transition to a two dimensional parabolic Navier-Stokes equation system describing the near-wake merging of the upper and lower surface boundary layers. An iteration provides refinement of the potential flow induced pressure coupling to the viscous flow solutions. The final phase is a complete two dimensional Navier-Stokes analysis of the wake flow in the vicinity of a blunt-bases airfoil. A finite element numerical algorithm is presented which is applicable to solution of all partial differential equation sets of inviscid-viscous aerodynamic interaction algorithm. Numerical results are discussed.

Baker, A. J.↗
NASA NTRS1985

Generation of Surface Coordinates by Elliptic Partial Differential Equations

The problem of generating spatial coordinates by numerical methods through carefully selected mathematical models is of current interest both in mechanics and physics. The problem of generation of a desired system of coordinates in a given surface was considered, which essentially is an effort directed to the problem of grid generation in a two-dimensional non-Euclidean space. The mathematical model selected for this purpose is based on the formulae of Gauss for a surface. The proposed equations can be used to generate a new coordinate system from the data of an already given coordinate system in a surface. If the coefficients of the first and second fundamental forms have been given, then the proposed equations can be used to generate a surface satisfying the given data (surface fitting). The proposed equations can also be used to generate surfaces in the space between two arbitrary given surfaces, thus providing 3D grids in an Euclidean space.

Warsi, Z. U. A.↗