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At least 793 records · Page 44

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NASA NTRS1993

Automatic partitioning of unstructured meshes for the parallel solution of problems in computational mechanics

Most of the recently proposed computational methods for solving partial differential equations on multiprocessor architectures stem from the 'divide and conquer' paradigm and involve some form of domain decomposition. For those methods which also require grids of points or patches of elements, it is often necessary to explicitly partition the underlying mesh, especially when working with local memory parallel processors. In this paper, a family of cost-effective algorithms for the automatic partitioning of arbitrary two- and three-dimensional finite element and finite difference meshes is presented and discussed in view of a domain decomposed solution procedure and parallel processing. The influence of the algorithmic aspects of a solution method (implicit/explicit computations), and the architectural specifics of a multiprocessor (SIMD/MIMD, startup/transmission time), on the design of a mesh partitioning algorithm are discussed. The impact of the partitioning strategy on load balancing, operation count, operator conditioning, rate of convergence and processor mapping is also addressed. Finally, the proposed mesh decomposition algorithms are demonstrated with realistic examples of finite element, finite volume, and finite difference meshes associated with the parallel solution of solid and fluid mechanics problems on the iPSC/2 and iPSC/860 multiprocessors.

Farhat, Charbel↗

NASA NTRS1986

Application of viscous-inviscid interaction methods to transonic turbulent flows

Two different viscous-inviscid interaction schemes were developed for the analysis of steady, turbulent, transonic, separated flows over axisymmetric bodies. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity approach. In the semi-inverse interaction scheme, the viscous and inviscid equations are solved in an explicitly separate manner and the displacement thickness distribution is iteratively updated by a simple coupling algorithm. In the simultaneous interaction method, local solutions of viscous and inviscid equations are treated simultaneously, and the displacement thickness is treated as an unknown and is obtained as a part of the solution through a global iteration procedure. The inviscid flow region is described by a direct finite-difference solution of a velocity potential equation in conservative form. The potential equation is solved on a numerically generated mesh by an approximate factorization (AF2) scheme in the semi-inverse interaction method and by a successive line overrelaxation (SLOR) scheme in the simultaneous interaction method. The boundary-layer equations are used for the viscous flow region. The continuity and momentum equations are solved inversely in a coupled manner using a fully implicit finite-difference scheme.

Lee, D.↗

NASA NTRS1977

Computation of unsteady transonic flows by the indicial method

The indicial method is investigated for the computation of unsteady transonic force and moment coefficients for use in flutter analyses. This approach has the advantage that solutions for all reduced frequencies for a given mode of motion can be obtained from a single finite-difference flowfield computation. Comparisons of indicial and time-integration computations for oscillating airfoil and flap motions help define limits on the motion amplitude for the applicability of the indicial method to transonic flows. Within these limits, solutions for various motion modes can be superposed to obtain solutions for multiple-degree-of-freedom aeroelastic systems. Also, a simple aeroelastic problem is solved by an alternative approach in which the structural motion and flowfield equations are integrated simultaneously using a time-integration finite-difference procedure.

Ballhaus, W. F.↗

NASA NTRS1993

Parallel solution of high-order numerical schemes for solving incompressible flows

A new parallel numerical scheme for solving incompressible steady-state flows is presented. The algorithm uses a finite-difference approach to solving the Navier-Stokes equations. The algorithms are scalable and expandable. They may be used with only two processors or with as many processors as are available. The code is general and expandable. Any size grid may be used. Four processors of the NASA LeRC Hypercluster were used to solve for steady-state flow in a driven square cavity. The Hypercluster was configured in a distributed-memory, hypercube-like architecture. By using a 50-by-50 finite-difference solution grid, an efficiency of 74 percent (a speedup of 2.96) was obtained.

Milner, Edward J.↗

NASA NTRS1973

Numerical marching techniques for fluid flows with heat transfer

The finite difference formulation and method of solution is presented for a wide variety of fluid flow problems with associated heat transfer. Only a few direct results from these formulations are given as examples, since the book is intended primarily to serve a discussion of the techniques and as a starting point for further investigations; however, the formulations are sufficiently complete that a workable computer program may be written from them. In the appendixes a number of topics are discussed which are of interest with respect to the finite difference equations presented. These include a very rapid method for solving certain sets of linear algebraic equations, a discussion of numerical stability, the inherent error in flow rate for confined flow problems, and a method for obtaining high accuracy with a relatively small number of mesh points.

Hornbeck, R. W.↗

NASA NTRSDate not supplied

Predicting SLS Launch Environment using a Novel Multiphase Formulation

Powerful acoustic waves generated during ignition of launch vehicles may be dangerous to the vehicle, its payload, or the surrounding structures. The water-based Ignition Overpressure and Sound Suppression (IOP/SS) system at Kennedy Space Center’s (KSC) Launch Complex 39B (LC-39B) will be used to protect the Space Launch System (SLS) from the acoustic vibrations generated during launch. The IOP/SS system uses enormous amounts of water to dampen and attenuate these sound waves. To better understand the launch environment risks and to study the effectiveness of the IOP/SS system it is desirable to have time-accurate unsteady simulations of the vehicle ignition with water-based sound suppression. This paper presents results obtained with a novel, high-order accurate, and robust numerical method designed for simulating compressible multiphase flows. A positivity-preserving finite difference scheme is utilized which is formally high-order accurate and also provably robust. Robustness is critical due to the extreme nature of the flow which exhibits highly nonlinear shock and rarefaction waves interacting with liquid-gas interfaces with density ratios of the order of 1000:1. Furthermore, the high-order accuracy (and the high resolution property) is desirable for predicting wave phenomena like IOP waves since the signal can be resolved accurately and propagated long distances with fewer grid points. This finite-difference method was developed using NASA’s Launch, Ascent, and Vehicle Aerodynamics (LAVA) Cartesian immersed boundary framework. We present a validation case by applying our solver to the SLS Scale Model Acoustic Test (SMAT). The SLS SMAT is a well-instrumented 5% scale model test meant to represent the SLS at NASA KSC’s LC-39B pad. Scale IOP tests were performed with and without the sound suppression water and included many sensors which recorded the pressure waves produced during ignition. For this validation case we conduct two simulations, likewise with and without sound suppression water, and compare the SLS SMAT pressure sensor signals with our numerical signals at identical locations. Following this validation case we present a study of the SLS launch environment to examine engineering safety concerns about the mobile launch pad. Engineers at KSC redesigned the main flame deflector at LC-39B anticipating the increased loads from the SLS and to repair damage from prior Shuttle missions. This deflector redesign made use of surface pressure and temperature data from LAVA full-scale SLS simulations without the sound suppression system. The engineers were questioning the possibility of increased pressure loads on the underside of the mobile launcher due to the water in the flame trench. Based on the results established in our simulations of the SLS SMAT, we performed updated calculations for SLS at LC-39B with and without water systems active to assess the readiness of the launch pad for Artemis I launch. Our results show that the IOP/SS system is effective at reducing the overpressure signal and overall sound pressure levels felt by the vehicle and additionally that the pressure loads experienced by the mobile launcher (ML) during engine startup is not increased by the presence of water.

EGS↗

NASA NTRS1981

A diagonal form of an implicit approximate-factorization algorithm

A modification of an implicit approximate-factorization finite-difference algorithm applied to partial differential equations is presented. This algorithm is applied to the two- and three-dimensional Euler equations in general curvilinear coordinates. The modification transforms the coupled system of equations into an uncoupled diagonal form that requires less computational work. For steady-state applications, the resulting diagonal algorithm retains the stability and accuracy characteristics of the original algorithm. The diagonal algorithm reduces the storage requirement of the implicit solution process and therefore has an important effect on the application of implicit finite-difference schemes to vector processors. Results are presented for realistic two-dimensional transonic flow fields about airfoils. Computation costs are reduced to 24-34%.

Pulliam, T. H.↗

NASA NTRS1981

A Diagonal Form of an Implicit Approximate-Factorization Algorithm

Pulliam, T. H.↗

NASA NTRS1971

Transpiration and film cooling boundary layer computer program. Volume 1: Numerical solutions of the turbulent boundary layer equations with equilibrium chemistry

A finite difference turbulent boundary layer computer program has been developed. The program is primarily oriented towards the calculation of boundary layer performance losses in rocket engines; however, the solution is general, and has much broader applicability. The effects of transpiration and film cooling as well as the effect of equilibrium chemical reactions (currently restricted to the H2-O2 system) can be calculated. The turbulent transport terms are evaluated using the phenomenological mixing length - eddy viscosity concept. The equations of motion are solved using the Crank-Nicolson implicit finite difference technique. The analysis and computer program have been checked out by solving a series of both laminar and turbulent test cases and comparing the results to data or other solutions. These comparisons have shown that the program is capable of producing very satisfactory results for a wide range of flows. Further refinements to the analysis and program, especially as applied to film cooling solutions, would be aided by the acquisition of a firm data base.

Levine, J. N.↗

NASA NTRS1986

Numerical study of hydrogen-air supersonic combustion by using elliptic and parabolized equations

The two-dimensional Navier-Stokes and species continuity equations are used to investigate supersonic chemically reacting flow problems which are related to scramjet-engine configurations. A global two-step finite-rate chemistry model is employed to represent the hydrogen-air combustion in the flow. An algebraic turbulent model is adopted for turbulent flow calculations. The explicit unsplit MacCormack finite-difference algorithm is used to develop a computer program suitable for a vector processing computer. The computer program developed is then used to integrate the system of the governing equations in time until convergence is attained. The chemistry source terms in the species continuity equations are evaluated implicitly to alleviate stiffness associated with fast chemical reactions. The problems solved by the elliptic code are re-investigated by using a set of two-dimensional parabolized Navier-Stokes and species equations. A linearized fully-coupled fully-implicit finite difference algorithm is used to develop a second computer code which solves the governing equations by marching in spce rather than time, resulting in a considerable saving in computer resources. Results obtained by using the parabolized formulation are compared with the results obtained by using the fully-elliptic equations. The comparisons indicate fairly good agreement of the results of the two formulations.

Chitsomboon, T.↗

NASA NTRS1983

User's manual for a TEACH computer program for the analysis of turbulent, swirling reacting flow in a research combustor

Described is a computer program for the analysis of the subsonic, swirling, reacting turbulent flow in an axisymmetric, bluff-body research combustor. The program features an improved finite-difference procedure designed to reduce the effects of numerical diffusion and a new algorithm for predicting the pressure distribution within the combustor. A research version of the computer program described in the report was supplied to United Technologies Research Center by Professor A. D. Gosman and his students, R. Benodeker and R. I. Issa, of Imperial College, London. The Imperial College staff also supplied much of the program documentation. Presented are a description of the mathematical model for flow within an axisymmetric bluff-body combustor, the development of the finite-difference procedure used to represent the system of equations, an outline of the algorithm for determining the static pressure distribution within the combustor, a description of the computer program including its input format, and the results for representative test cases.

Chiappetta, L. M.↗

NASA NTRS2001

An Implicit Characteristic Based Method for Electromagnetics

An implicit characteristic-based approach for numerical solution of Maxwell's time-dependent curl equations in flux conservative form is introduced. This method combines a characteristic based finite difference spatial approximation with an implicit lower-upper approximate factorization (LU/AF) time integration scheme. This approach is advantageous for three-dimensional applications because the characteristic differencing enables a two-factor approximate factorization that retains its unconditional stability in three space dimensions, and it does not require solution of tridiagonal systems. Results are given both for a Fourier analysis of stability, damping and dispersion properties, and for one-dimensional model problems involving propagation and scattering for free space and dielectric materials using both uniform and nonuniform grids. The explicit Finite Difference Time Domain Method (FDTD) algorithm is used as a convenient reference algorithm for comparison. The one-dimensional results indicate that for low frequency problems on a highly resolved uniform or nonuniform grid, this LU/AF algorithm can produce accurate solutions at Courant numbers significantly greater than one, with a corresponding improvement in efficiency for simulating a given period of time. This approach appears promising for development of dispersion optimized LU/AF schemes for three dimensional applications.

Beggs, John H.↗

NASA NTRS1975

Numerical computation of three-dimensional blunt body flow fields with an impinging shock

A time-marching finite-difference method was used to solve the compressible Navier-Stokes equations for the three-dimensional wing-leading-edge shock impingement problem. The bow shock was treated as a discontinuity across which the exact shock jump conditions were applied. All interior shock layer detail such as shear layers, shock waves, jets, and the wall boundary layer were automatically captured in the solution. The impinging shock was introduced by discontinuously changing the freestream conditions across the intersection line at the bow shock. A special storage-saving procedure for sweeping through the finite-difference mesh was developed which reduces the required amount of computer storage by at least a factor of two without sacrificing the execution time. Numerical results are presented for infinite cylinder blunt body cases as well as the three-dimensional shock impingement case. The numerical results are compared with existing experimental and theoretical results.

Holst, T. L.↗

DOE OSTI2023

Inverse modeling of circular lattices via orbit response measurements in the presence of degeneracy

The number and location of beam position monitors (BPMs) and steerers with respect to the quadrupoles in a circular lattice can lead to degeneracy in the context of fitting linear optics and extracting lattice information from measured closed orbits. Furthermore, the measurement uncertainties due to the imperfection of BPMs and steerers can be propagated by the fitting process in ways that prohibit the successful extraction of discrepancies between lattice elements in the real machine and their description in the corresponding model. We systematically studied the influence of the placement of BPMs and steerers on the reconstruction of linear optics and corresponding lattice information. The derivative of orbit response coefficients with respect to the quadrupole strengths, the Jacobian, is derived as an analytical formula. This analytical version of the Jacobian is used to further derive the theoretical limitations of fitting linear optics from closed orbits in terms of the placement of BPMs and steerers. It is further demonstrated that when evaluating the Jacobian during the fitting procedure, the analytical version can be used in place of the conventional finite-difference computation. This allows for greatly improved efficiency when computing the Jacobian during each iteration of the fitting procedure. The approach is tested with large-scale simulations and the findings are verified by measurement data taken on SIS18 synchrotron at GSI Helmholtz Centre for Heavy Ion Research. The presented methods are of general nature and can be applied to other accelerator lattices as well. The fitting procedure by using the analytical Jacobian is tested in conjunction with various methods for mitigating quasidegeneracy and the results agree with those obtained by using the conventional Jacobian via finite-difference approximation.

47 OTHER INSTRUMENTATION↗

NASA NTRS1985

Shock capturing

Recent developments which have improved the understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations are discussed. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. A conservative difference scheme is defined. Conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, the reason for this excellent shock resolution can be determined. This result is used to provide a mechanism for improving the resolution of two dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. Total variation diminishing (TVD) finite difference schemes and flux limiters are introduced to deal with money shocks and contact discontinuities.

Davis, S. F.↗

NASA NTRS1979

Explicit large time-step schemes for the shallow water equations

Modifications to explicit finite difference schemes for solving the shallow water equations for meteorological applications by increasing the time step for the fast gravity waves are analyzed. Terms associated with the gravity waves in the shallow water equations are treated on a coarser grid than those associated with the slow Rossby waves, which contain much more of the available energy and must be treated with higher accuracy, enabling a several-fold increase in time step without degrading the accuracy of the solution. The method is presented in Cartesian and spherical coordinates for a rotating earth, using generalized leapfrog, frozen coefficient, and Fourier filtering finite difference schemes. Computational results verify the numerical stability of the approach.

Turkel, E.↗

NASA NTRS1978

Numerical solution of the Navier-Stokes equations for super-sonic flows with strong shocks

The numerical solution of the full Navier-Stokes Equations for viscous flows with high Mach numbers and a strong detached bow shock was obtained. Two dimensional flows around a circular cylinder, and a circular cylinder with an aft-body in the form of a fairing, were considered. The solution of the compressible N.S. equations was accomplished by the method of finite differences. An implicit scheme of solution, the S.O.R., was used with the optimum acceleration parameters determined by trial and error. The tensor notation was used in writing the N-S Equations transformed into general curvilinear coordinates. The equations for the generation of the coordinate system were solved, followed by the solution of the N.S. equations, at the end of a set of given number of time steps. "Wiggles", constituted the one major problem that needed to be overcome. These oscillations give rise to quantities such as negative temperatures, which ultimately caused the computational program to break down. Certain dissipative finite-difference schemes damped these oscillations.

Devarayalu, K.↗

NASA NTRS1990

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

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.

Yee, H. C.↗