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NASA NTRS1994

Preconditioned conjugate gradient methods for the Navier-Stokes equations

A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulation. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a conventional implicit solver, namely line Gauss-Seidel relaxation (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU time on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow case are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner.

Ajmani, Kumud↗

NASA NTRS1994

Parallel implicit unstructured grid Euler solvers

A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on a multiple-instruction/multiple-data stream parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed: one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All of the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the Intel iPSC/860.

TRT-THEORETICAL↗

NASA NTRS1995

A steady-state Kalman filter for assimilating data from a single polar orbiting satellite

A steady-state scheme for data assimilation in the context of a single, short period (relative to a day), sun-synchronous, polar-orbiting satellite is examined. If the satellite takes observations continuously, the gains, which are the weights for blending observations and predictions together, are steady in time. For a linear system forced by random noise, the optimal steady-state gains (Wiener gains) are equivalent to those of a Kalman filter. Computing the Kalman gains increases the computational cost of the model by a large factor, but computing the Wiener gains does not. The latter are computed by iteration using prior estimates of the gains to assimilate simulated observations of one run of the model, termed 'truth' into another run termed 'prediction'. At each stage, the prediction errors form the basis for the next estimate of the gains. Steady state is achieved after three or four iterations. Further simplification is achieved by making the gains depend on longitudinal distance from the observation point, not on absolute longitude. For a single-layer primitive equation model, the scheme works well even if only the mass field is observed but not the velocity field. Although the scheme was developed for Mars Observer, it should be applicable to data retrieved from Earth atmosphere satellites, for example, UARS.

Banfield, Don↗

NASA NTRS1996

Apparatus Tests Peeling Of Bonded Rubbery Material

Instrumented hydraulic constrained blister-peel apparatus obtains data on degree of bonding between specimen of rubbery material and rigid plate. Growth of blister tracked by video camera, digital clock, pressure transducer, and piston-displacement sensor. Cylinder pressure controlled by hydraulic actuator system. Linear variable-differential transformer (LVDT) and float provide second, independent measure of change in blister volume used as more precise volume feedback in low-growth-rate test.

Crook, Russell A.↗

NASA NTRS1996

Multiple Coarse Grid Multigrid Methods for Solving Elliptic Problems

In this paper we describe some classes of multigrid methods for solving large linear systems arising in the solution by finite difference methods of certain boundary value problems involving Poisson's equation on rectangular regions. If parallel computing systems are used, then with standard multigrid methods many of the processors will be idle when one is working at the coarsest grid levels. We describe the use of Multiple Coarse Grid MultiGrid (MCGMG) methods. Here one first constructs a periodic set of equations corresponding to the given system. One then constructs a set of coarse grids such that for each grid corresponding to the grid size h there are four grids corresponding to the grid size 2*h. Multigrid operations such as restriction of residuals and interpolation of corrections are done in parallel at each grid level. For suitable choices of the multigrid operators the MCGMG method is equivalent to the Parallel Superconvergent MultiGrid (PSMG) method of Frederickson and McBryan. The convergence properties of MCGMG methods can be accurately analyzed using spectral methods.

Xiao, Shengyou↗

NASA NTRS1996

Preserving Symmetry in Preconditioned Krylov Subspace Methods

We consider the problem of solving a linear system Ax = b when A is nearly symmetric and when the system is preconditioned by a symmetric positive definite matrix M. In the symmetric case, one can recover symmetry by using M-inner products in the conjugate gradient (CG) algorithm. This idea can also be used in the nonsymmetric case, and near symmetry can be preserved similarly. Like CG, the new algorithms are mathematically equivalent to split preconditioning, but do not require M to be factored. Better robustness in a specific sense can also be observed. When combined with truncated versions of iterative methods, tests show that this is more effective than the common practice of forfeiting near-symmetry altogether.

Chan, Tony F.↗

NASA NTRS1996

Optimum Suction Distribution for Transition Control

The optimum suction distribution which gives the longest laminar region for a given total suction is computed. The goal here is to provide the designer with a method to find the best suction distribution subject to some overall constraint applied to the suction. We formulate the problem using the Lagrangian multiplier method with constraints. The resulting non-linear system of equations is solved using the Newton-Raphson technique. The computations are performed for a Blasius boundary layer on a flat-plate and crossflow cases. For the Blasius boundary layer, the optimum suction distribution peaks upstream of the maximum growth rate region and remains flat in the middle before it decreases to zero at the end of the transition point. For the stationary and travelling crossflow instability, the optimum suction peaks upstream of the maximum growth rate region and decreases gradually to zero.

Balakumar, P.↗

NASA NTRS1996

Asymptotic solution of the turbulent mixing layer for velocity ratio close to unity

The equations describing the first two terms of an asymptotic expansion of the solution of the planar turbulent mixing layer for values of the velocity ratio close to one are obtained. The first term of this expansion is the solution of the well-known time-evolving problem and the second, which includes the effects of the increase of the turbulence scales in the stream-wise direction, obeys a linear system of equations. Numerical solutions of these equations for a two-dimensional reacting mixing layer show that the correction to the time-evolving solution may explain the asymmetry of the entrainment and the differences in product generation observed in flip experiments.

Higuera, F. J.↗

NASA NTRS1996

The incremental unknowns-a multilevel scheme for the simulation of turbulent channel flows

In numerical simulation of complex flows, it is important to identify different length scales of the flow and treat them differently. In this report, we introduce a new multilevel scheme for simulating turbulent channel flows. Two different versions of the scheme, namely the spectral and finite difference versions, are presented. The spectral version of the scheme is based on a spectral-Galerkin formulation which provides a natural decomposition of the flow into small and large wavelength parts, and which leads to linear systems that can be solved with quasi-optimal computational complexity. In the finite difference version, the Incremental Unknown (IU) is used to separate the length scales. Preliminary numerical results indicate that the scheme is well suited for turbulence computations and provides results which are comparable to that by Direct Numerical Simulation (DNS) but with significantly less CPU time.

Chen, M.↗

NASA NTRS1997

A Multiblock Approach for Calculating Incompressible Fluid Flows on Unstructured Grids

A multiblock approach is presented for solving two-dimensional incompressible turbulent flows on unstructured grids. The artificial compressibility form of the governing equations is solved by a vertex-centered, finite-volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work introduces a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements while not increasing the CPU time. Results presented in this work shows that the current multiblock algorithm requires 70% less memory than the single block algorithm.

Sheng, Chunhua↗

NASA NTRS1997

CGRO Guest Investigator Program

The following are highlights from the research supported by this grant: (1) Theory of gamma-ray blazars: We studied the theory of gamma-ray blazars, being among the first investigators to propose that the GeV emission arises from Comptonization of diffuse radiation surrounding the jet, rather than from the synchrotron-self-Compton mechanism. In related work, we uncovered possible connections between the mechanisms of gamma-ray blazars and those of intraday radio variability, and have conducted a general study of the role of Compton radiation drag on the dynamics of relativistic jets. (2) A Nonlinear Monte Carlo code for gamma-ray spectrum formation: We developed, tested, and applied the first Nonlinear Monte Carlo (NLMC) code for simulating gamma-ray production and transfer under much more general (and realistic) conditions than are accessible with other techniques. The present version of the code is designed to simulate conditions thought to be present in active galactic nuclei and certain types of X-ray binaries, and includes the physics needed to model thermal and nonthermal electron-positron pair cascades. Unlike traditional Monte-Carlo techniques, our method can accurately handle highly non-linear systems in which the radiation and particle backgrounds must be determined self-consistently and in which the particle energies span many orders of magnitude. Unlike models based on kinetic equations, our code can handle arbitrary source geometries and relativistic kinematic effects In its first important application following testing, we showed that popular semi-analytic accretion disk corona models for Seyfert spectra are seriously in error, and demonstrated how the spectra can be simulated if the disk is sparsely covered by localized 'flares'.

Begelman, Mitchell C.↗

NASA NTRS1995

A Game Theoretic Fault Detection Filter

The fault detection process is modelled as a disturbance attenuation problem. The solution to this problem is found via differential game theory, leading to an H(sub infinity) filter which bounds the transmission of all exogenous signals save the fault to be detected. For a general class of linear systems which includes some time-varying systems, it is shown that this transmission bound can be taken to zero by simultaneously bringing the sensor noise weighting to zero. Thus, in the limit, a complete transmission block can he achieved, making the game filter into a fault detection filter. When we specialize this result to time-invariant system, it is found that the detection filter attained in the limit is identical to the well known Beard-Jones Fault Detection Filter. That is, all fault inputs other than the one to be detected (the "nuisance faults") are restricted to an invariant subspace which is unobservable to a projection on the output. For time-invariant systems, it is also shown that in the limit, the order of the state-space and the game filter can be reduced by factoring out the invariant subspace. The result is a lower dimensional filter which can observe only the fault to be detected. A reduced-order filter can also he generated for time-varying systems, though the computational overhead may be intensive. An example given at the end of the paper demonstrates the effectiveness of the filter as a tool for fault detection and identification.

Chung, Walter H.↗

NASA NTRS1999

The X-33 Flight Test Challenge

Low cost access to space has eluded present launch system technologies. Our objective is to reduce the cost of putting a payload into space from $10,000 per pound to $1000 per pound. In July 1996, a cooperative agreement was initiated between the Lockheed Martin Skunk Works and NASA to help accomplish this goal. The X-33 is the first step in the process to make low cost space access a reality. The X-33 is a suborbital, hypersonic lifting body, proof of concept of a reusable launch vehicle. The X-33 flight test program will validate technologies such as a metallic thermal protection system, Linear Aerospike Engines, use of tanks and struts as fundamental structural elements, as well as quick turnaround time. Flight testing will begin in July 2000, with launches originating from Edwards Air Force Base and initial landings at Michael Army Airfield in Utah. Data collected from these flight tests will aid in the decision to build an economically viable single stage to orbit reusable launch vehicle. This paper will explore the technical challenges facing the X-33 Flight Test Team.

Borden, David↗

NASA NTRS1989

A Computational Fluid Dynamics Algorithm on a Massively Parallel Computer

The discipline of computational fluid dynamics is demanding ever-increasing computational power to deal with complex fluid flow problems. We investigate the performance of a finite-difference computational fluid dynamics algorithm on a massively parallel computer, the Connection Machine. Of special interest is an implicit time-stepping algorithm; to obtain maximum performance from the Connection Machine, it is necessary to use a nonstandard algorithm to solve the linear systems that arise in the implicit algorithm. We find that the Connection Machine ran achieve very high computation rates on both explicit and implicit algorithms. The performance of the Connection Machine puts it in the same class as today's most powerful conventional supercomputers.

Jespersen, Dennis C.↗

NASA NTRS1999

Large-Scale Parallel Viscous Flow Computations using an Unstructured Multigrid Algorithm

The development and testing of a parallel unstructured agglomeration multigrid algorithm for steady-state aerodynamic flows is discussed. The agglomeration multigrid strategy uses a graph algorithm to construct the coarse multigrid levels from the given fine grid, similar to an algebraic multigrid approach, but operates directly on the non-linear system using the FAS (Full Approximation Scheme) approach. The scalability and convergence rate of the multigrid algorithm are examined on the SGI Origin 2000 and the Cray T3E. An argument is given which indicates that the asymptotic scalability of the multigrid algorithm should be similar to that of its underlying single grid smoothing scheme. For medium size problems involving several million grid points, near perfect scalability is obtained for the single grid algorithm, while only a slight drop-off in parallel efficiency is observed for the multigrid V- and W-cycles, using up to 128 processors on the SGI Origin 2000, and up to 512 processors on the Cray T3E. For a large problem using 25 million grid points, good scalability is observed for the multigrid algorithm using up to 1450 processors on a Cray T3E, even when the coarsest grid level contains fewer points than the total number of processors.

Mavriplis, Dimitri J.↗

NASA NTRS1999

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

Kouatchou, Jules↗

NASA NTRS2000

Three-Dimensional Nacelle Aeroacoustics Code With Application to Impedance Education

A three-dimensional nacelle acoustics code that accounts for uniform mean flow and variable surface impedance liners is developed. The code is linked to a commercial version of the NASA-developed General Purpose Solver (for solution of linear systems of equations) in order to obtain the capability to study high frequency waves that may require millions of grid points for resolution. Detailed, single-processor statistics for the performance of the solver in rigid and soft-wall ducts are presented. Over the range of frequencies of current interest in nacelle liner research, noise attenuation levels predicted from the code were in excellent agreement with those predicted from mode theory. The equation solver is memory efficient, requiring only a small fraction of the memory available on modern computers. As an application, the code is combined with an optimization algorithm and used to reduce the impedance spectrum of a ceramic liner. The primary problem with using the code to perform optimization studies at frequencies above I1kHz is the excessive CPU time (a major portion of which is matrix assembly). The research recommends that research be directed toward development of a rapid sparse assembler and exploitation of the multiprocessor capability of the solver to further reduce CPU time.

Watson, Willie R.↗

NASA NTRS2000

Three-Dimensional Incompressible Navier-Stokes Flow Computations about Complete Configurations Using a Multiblock Unstructured Grid Approach

A multiblock unstructured grid approach is presented for solving three-dimensional incompressible inviscid and viscous turbulent flows about complete configurations. The artificial compressibility form of the governing equations is solved by a node-based, finite volume implicit scheme which uses a backward Euler time discretization. Point Gauss-Seidel relaxations are used to solve the linear system of equations at each time step. This work employs a multiblock strategy to the solution procedure, which greatly improves the efficiency of the algorithm by significantly reducing the memory requirements by a factor of 5 over the single-grid algorithm while maintaining a similar convergence behavior. The numerical accuracy of solutions is assessed by comparing with the experimental data for a submarine with stem appendages and a high-lift configuration.

Sheng, Chunhua↗