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NASA NTRS1981

Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods

The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included.

Steger, J. L.↗

NASA NTRS1989

An adaptive implicit/explicit finite element scheme for compressible viscous high speed flow

An adaptive implicit/explicit finite element procedure for the solution of three-dimensional problems of steady compressible viscous high-speed flows is presented. In the vicinity of solid walls, a grid-exhibiting structure in the normal direction is employed, while away from this region the grid is totally unstructured. The implicit form of the algorithm is used near solid walls, with the grid structure being utilized in an equation solution approach based upon line relaxation. The explicit form of the algorithm is used elsewhere. Grid adaptation is achieved by means of adaptive remeshing. To illustrate the performance of the proposed method, solutions are obtained for the problems of shock-boundary layer interaction and shock-shock interaction on a swept cylindrical leading edge. Comparisons are made with experimental observations.

Hassan, O.↗

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 NTRS1984

Vectorized schemes for conical potential flow using the artificial density method

A method is developed to determine solutions to the full-potential equation for steady supersonic conical flow using the artificial density method. Various update schemes used generally for transonic potential solutions are investigated. The schemes are compared for speed and robustness. All versions of the computer code have been vectorized and are currently running on the CYBER-203 computer. The update schemes are vectorized, where possible, either fully (explicit schemes) or partially (implicit schemes). Since each version of the code differs only by the update scheme and elements other than the update scheme are completely vectorizable, comparisons of computational effort and convergence rate among schemes are a measure of the specific scheme's performance. Results are presented for circular and elliptical cones at angle of attack for subcritical and supercritical crossflows.

Bradley, P. F.↗

NASA NTRS1981

Accuracy and stability of time-split finite-difference schemes

In a recently published work by Abarbanel and Gottlieb (1980), a new class of explicit time-split algorithms designed for application to the compressible Navier-Stokes equations was developed. These algorithms, which utilize locally-one-dimensional (LOD) spatial steps, were shown to possess stability characteristics superior to those of other time-split schemes. In the present work, the properties of an implicit LOD method, analogous to the Abarbanel-Gottlieb algorithm, are examined using the two-dimensional heat conduction equation as the test problem. Both temporal and spatial inconsistencies inherent in the scheme are identified, and a new consistent, implicit splitting approach is developed and applied to the linear Burgers' equation. The relationship between this new method and other time-split implicit schemes is explained and stability problems encountered with the method in three dimensions are discussed.

Dwoyer, D. L.↗

NASA NTRS1989

An efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

Radespiel, R.↗

NASA NTRS1997

Visualization of Unsteady Computational Fluid Dynamics

The current compute environment that most researchers are using for the calculation of 3D unsteady Computational Fluid Dynamic (CFD) results is a super-computer class machine. The Massively Parallel Processors (MPP's) such as the 160 node IBM SP2 at NAS and clusters of workstations acting as a single MPP (like NAS's SGI Power-Challenge array and the J90 cluster) provide the required computation bandwidth for CFD calculations of transient problems. If we follow the traditional computational analysis steps for CFD (and we wish to construct an interactive visualizer) we need to be aware of the following: (1) Disk space requirements. A single snap-shot must contain at least the values (primitive variables) stored at the appropriate locations within the mesh. For most simple 3D Euler solvers that means 5 floating point words. Navier-Stokes solutions with turbulence models may contain 7 state-variables. (2) Disk speed vs. Computational speeds. The time required to read the complete solution of a saved time frame from disk is now longer than the compute time for a set number of iterations from an explicit solver. Depending, on the hardware and solver an iteration of an implicit code may also take less time than reading the solution from disk. If one examines the performance improvements in the last decade or two, it is easy to see that depending on disk performance (vs. CPU improvement) may not be the best method for enhancing interactivity. (3) Cluster and Parallel Machine I/O problems. Disk access time is much worse within current parallel machines and cluster of workstations that are acting in concert to solve a single problem. In this case we are not trying to read the volume of data, but are running the solver and the solver outputs the solution. These traditional network interfaces must be used for the file system. (4) Numerics of particle traces. Most visualization tools can work upon a single snap shot of the data but some visualization tools for transient problems require dealing with time.

Haimes, Robert↗

NASA NTRS1987

Supercomputing of supersonic flows using upwind relaxation and MacCormack schemes

The performance of two numerical solution schemes, (1) an implicit upwind relaxation with a finite-volume discretization (Thomas and Walters, 1985) and (2) an explicit-implicit MacCormack (1981) scheme with a finite-difference discretization, is compared in two-dimensional simulations of supersonic flow past a flat plate with leading edge, a rearward-facing step, a 10-deg compression corner, a NACA 0012 airfoil at high angle of attack, and a cavity. The algebraic turbulence model, the solution methods, and the boundary conditions and SIMD coding are explained, and the results are presented in tables and graphs and characterized with reference to published experimental data. Scheme (1) is found to converge more rapidly and to give more accurate results than (2) in a wide range of problem types.

Baysal, O.↗

NASA NTRS1988

Space Station Active Thermal Control System modeling

The Space Station Active Thermal Control System (ATCS) has been modeled using modified SINDA/SINFLO programs to solve two-phase Thermo-fluid problems. The modifications include changes in several subroutines to incorporate implicit solution which allows larger time step as compared to that for explicit solutions. Larger time step saves computer time but involves larger computational error. Several runs were made using various time steps for the ATCS model. It has been found that for a reasonable approach, three times larger time step as compared to that used in explicit method is a good value which will reduce the computer time by approximately 50 percent and still maintain the accuracy of the output data to within 90 percent of the explicit values.

Hye, Abdul↗

NASA NTRS1991

Derivative interface conditions for multiblock grids

Methods are developed for computing numerical solutions along block boundaries, even when there is a discontinuity in the grid lines or slopes. The technique is based on matching derivatives and does not require overlapping and interpolation of solution values at block boundaries. The comparison of block boundary values is implicit and has proven to be stable for both implicit and explicit numerical algorithms. Examples are included for the numerical solution of the Euler equations for compressible flow on grids with both grid line discontinuities and discontinuous slopes at block boundaries.

Mastin, C. W.↗

NASA NTRS1982

Numerical boundary condition procedures and multigrid methods; Proceedings of the Symposium, NASA Ames Research Center, Moffett Field, CA, October 19-22, 1981

Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.

Source record↗

NASA NTRS1985

Implicit finite-difference methods for the Euler equations

The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.

Pulliam, T. H.↗

NASA NTRS1990

Thermal instabilities in protogalactic clouds

The means by which a protogalaxy can fragment to form the first generation of stars and globular clusters remains an important problem in astrophysics. Gravitational instabilities grow on timescales too long to drive fragmentation before the background density grows by many orders of magnitude (see Murray and Lin 1989a, and references therein). Thermal instability provides a much more likely mechanism. After its initial collapse, a protogalactic cloud is expected to be shock heated to its virial temperature approx. 10(exp 6) K. Cooling by H and He+ below 10(exp 6) K has a negative slope, so that the cloud is subject to strong thermal instabilities. Density enhancements may then grow rapidly, fragmenting the protogalaxy as it cools to lower temperatures. The role of dynamical effects upon the growth of perturbations is considered here. The method used is similar to that used in Murray and Lin (1989a; see also the Erratum to appear September 15), which examined the growth of thermal instabilities with a one-dimensional Lagrangian hydrodynamics code, written for spherical symmetry. Perturbed regions therefore take the form of shells. The dynamical variables are integrated explicitly, while the temperature, ionization fraction, and molecular fraction are integrated implicitly, and account is taken for non-equilibrium values of these quantities.

Murray, S. D.↗

NASA NTRS1983

A time-split finite-volume algorithm for three-dimensional flow-field simulation

A general finite-volume algorithm is developed for solving three-dimensional, time-dependent, compressible Navier-Stokes equations for high Reynolds number flows over an arbitrary geometry. This algorithm adapts MacCormack's (1982) explicit-implicit scheme to a time-split, three-dimensional finite-volume concept in a general coordinate system. It is shown that the thin-layer approximation in all three spatial directions significantly reduces the evaluation of viscous terms and allows the algorithm to solve more complicated geometries with all boundaries in two or all three directions. The calculated results using this method are found to be in good agreement with the experimental measurements of a blunt-fin induced shock wave and boundary-layer interaction problems. Observations of the existence of peak pressure, primary horseshoe and secondary vortices, and reversed supersonic zones show that computational fluid dynamics can effectively supplement the wind tunnel tests for aerodynamic design as well as for understanding basic fluid dynamics.

Hung, C. M.↗

NASA NTRS1994

Time integration algorithms for the two-dimensional Euler equations on unstructured meshes

Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.

Slack, David C.↗

NASA NTRS1982

An approximation concepts method for space frame synthesis

A method is presented for the minimum mass design of three dimensional space frames constructed of thin walled rectangular cross-section members. Constraints on nodal displacements and rotations, material stress, local buckling, and cross sectional dimensions are included. A high quality separable approximate problem is formed in terms of the reciprocals of the four section properties of the frame element cross section, replacing all implicit functions with simplified explicit relations. The cross sectional dimensions are efficiently calculated without using multilevel techniques. Several test problems are solved, demonstrating that a series of approximate problem solutions converge rapidly to an optimal design.

Mills-Curran, W. C.↗

NASA NTRS2007

DNA Recombinase Proteins, their Function and Structure in the Active Form, a Computational Study

Homologous recombination is a crucial sequence of reactions in all cells for the repair of double strand DNA (dsDNA) breaks. While it was traditionally considered as a means for generating genetic diversity, it is now known to be essential for restart of collapsed replication forks that have met a lesion on the DNA template (Cox et al., 2000). The central stage of this process requires the presence of the DNA recombinase protein, RecA in bacteria, RadA in archaea, or Rad51 in eukaryotes, which leads to an ATP-mediated DNA strand-exchange process. Despite many years of intense study, some aspects of the biochemical mechanism, and structure of the active form of recombinase proteins are not well understood. Our theoretical study is an attempt to shed light on the main structural and mechanistic issues encountered on the RecA of the e-coli, the RecA of the extremely radio resistant Deinococcus Radiodurans (promoting an inverse DNA strand-exchange repair), and the homolog human Rad51. The conformational changes are analyzed for the naked enzymes, and when they are linked to ATP and ADP. The average structures are determined over 2ns time scale of Langevian dynamics using a collision frequency of 1.0 ps(sup -1). The systems are inserted in an octahedron periodic box with a 10 Angstrom buffer of water molecules explicitly described by the TIP3P model. The corresponding binding free energies are calculated in an implicit solvent using the Poisson-Boltzmann solvent accessible surface area, MM-PBSA model. The role of the ATP is not only in stabilizing the interaction RecA-DNA, but its hydrolysis is required to allow the DNA strand-exchange to proceed. Furthermore, we extended our study, using the hybrid QM/MM method, on the mechanism of this chemical process. All the calculations were performed using the commercial code Amber 9.

Carra, Claudio↗

NASA NTRS1993

Three Dimensional Aerodynamic Analysis of a High-Lift Transport Configuration

Two computational methods, a surface panel method and an Euler method employing unstructured grid methodology, were used to analyze a subsonic transport aircraft in cruise and high-lift conditions. The computational results were compared with two separate sets of flight data obtained for the cruise and high-lift configurations. For the cruise configuration, the surface pressures obtained by the panel method and the Euler method agreed fairly well with results from flight test. However, for the high-lift configuration considerable differences were observed when the computational surface pressures were compared with the results from high-lift flight test. On the lower surface of all the elements with the exception of the slat, both the panel and Euler methods predicted pressures which were in good agreement with flight data. On the upper surface of all the elements the panel method predicted slightly higher suction compared to the Euler method. On the upper surface of the slat, pressure coefficients obtained by both the Euler and panel methods did not agree with the results of the flight tests. A sensitivity study of the upward deflection of the slat from the 40 deg. flap setting suggested that the differences in the slat deflection between the computational model and the flight configuration could be one of the sources of this discrepancy. The computation time for the implicit version of the Euler code was about 1/3 the time taken by the explicit version though the implicit code required 3 times the memory taken by the explicit version.

Dodbele, Simha S.↗