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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 NTRS1978

Non-linear effects on sound propagation in a variable area duct carrying high subsonic Mach number flows

The general one-dimensional non-linear equation for the acoustic velocity potential in a variable area duct carrying high subsonic Mach number flows is presented and solved numerically using an implicit finite difference scheme. For the linearized equation with no flow, the present scheme is compared with the exact solution and the fourth-order Runge-Kutta method with excellent agreement for dimensionless time periods. Non-linear solutions are more sensitive to Mach numbers and exciting amplitudes and less sensitive to exciting frequencies. In general, non-linear effects can be safely neglected for low Mach number flows but must be accounted for when high Mach number flows are encountered.

Lumsdaine, E.↗

NASA NTRS1979

Computations of unsteady transonic flow governed by the conservative full potential equation using an alternating direction implicit algorithm

A development was the time linearization of the density function. This linearization reduces the solution process from solving just a single equation. Two sample cases were computed. First, a one dimensional traveling shock wave was computed and compared with the analytic solution. Second, a two dimensional case was calculated for a flow field which resulted from a thickening and subsequently, thinning airfoil. The resulting flow field, which included a traveling shock wave, was compared to the flow field obtained from the low frequency, small disturbance, transonic equation.

Goorjian, P. M.↗

NASA NTRS1979

Modern estimation techniques applied to microwave sensing of the marine boundary layer

Previous efforts in the area of satellite microwave sensing of the marine boundary layer have relied upon linear regression techniques to extract geophysical parameters from the microwave measurement data. The approach used in the present paper shifts emphasis away from the generation of regression weighting matrices which implicitly assume that the data are linear in the parameters to be determined and that the problem is statistically stationary. The idea is simply to employ modern computational estimation techniques to obtain parameter estimates from nonlinear noisy measurements. The approach is limited only to the region of validity of Grody's (1976) model. Attention is focused on documenting how estimation techniques, in particular the square root information filter (SRIF), are used to solve a nonlinear function optimization problem.

Bierman, G. J.↗

NASA NTRS1991

Exponential integration algorithms applied to viscoplasticity

Four, linear, exponential, integration algorithms (two implicit, one explicit, and one predictor/corrector) are applied to a viscoplastic model to assess their capabilities. Viscoplasticity comprises a system of coupled, nonlinear, stiff, first order, ordinary differential equations which are a challenge to integrate by any means. Two of the algorithms (the predictor/corrector and one of the implicits) give outstanding results, even for very large time steps.

Freed, Alan D.↗

NASA NTRS1992

Grid generation and flow solution method for Euler equations on unstructured grids

A grid generation and flow solution algorithm for the Euler equations on unstructured grids is presented. The grid generation scheme, which uses Delaunay triangulation, generates the field points for the mesh based on cell aspect ratios and allows clustering of grid points near solid surfaces. The flow solution method is an implicit algorithm in which the linear set of equations arising at each time step is solved using a Gauss-Seidel procedure that is completely vectorizable. Also, a study is conducted to examine the number of subiterations required for good convergence of the overall algorithm. Grid generation results are shown in two dimensions for an NACA 0012 airfoil as well as a two element configuration. Flow solution results are shown for a two dimensional flow over the NACA 0012 airfoil and for a two element configuration in which the solution was obtained through an adaptation procedure and compared with an exact solution. Preliminary three dimensional results also are shown in which the subsonic flow over a business jet is computed.

Anderson, W. Kyle↗

NASA NTRS1993

Multigrid properties of upwind-biased data reconstructions

The multigrid properties of two data reconstruction methods used for achieving second-order spatial accuracy when solving the two-dimensional Euler equations are examined. The data reconstruction methods are used with an implicit upwind algorithm which uses linearized backward-Euler time-differencing. The solution of the resulting linear system is performed by an iterative procedure. In the present study only regular quadrilateral grids are considered, so a red-black Gauss-Seidel iteration is used. Although the Jacobian is approximated by first-order upwind extrapolation, two alternative data reconstruction techniques for the flux integral that yield higher-order spatial accuracy at steady state are examined. The first method, probably most popular for structured quadrilateral grids, is based on estimating the cell gradients using one-dimensional reconstruction along curvilinear coordinates. The second method is based on Green's theorem. Analysis and numerical results for the two dimensional Euler equations show that data reconstruction based on Green's theorem has superior multigrid properties as compared to the one-dimensional data reconstruction method.

Warren, Gary P.↗

NASA NTRS1994

A grid generation and flow solution method for the Euler equations on unstructured grids

A grid generation and flow solution algorithm for the Euler equations on unstructured grids is presented. The grid generation scheme utilizes Delaunay triangulation and self-generates the field points for the mesh based on cell aspect ratios and allows for clustering near solid surfaces. The flow solution method is an implicit algorithm in which the linear set of equations arising at each time step is solved using a Gauss Seidel procedure which is completely vectorizable. In addition, a study is conducted to examine the number of subiterations required for good convergence of the overall algorithm. Grid generation results are shown in two dimensions for a National Advisory Committee for Aeronautics (NACA) 0012 airfoil as well as a two-element configuration. Flow solution results are shown for two-dimensional flow over the NACA 0012 airfoil and for a two-element configuration in which the solution has been obtained through an adaptation procedure and compared to an exact solution. Preliminary three-dimensional results are also shown in which subsonic flow over a business jet is computed.

Anderson, W. Kyle↗

NASA NTRS1996

Scalable Solutions to Integral Equation and Finite Element Simulations

The application of advanced computer architecture and software to a broad range of electromagnetic problems had allowed more accurate simulations of electrically larger and more complex components and systems...This paper overviews solutions to Maxwell's equations implicitly defined through systems of linear equations.

parallel↗

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 NTRS1975

The structure of robust observers

Conventional observers for linear time-invariant systems are shown to be structurally inadequate from a sensitivity standpoint. It is proved that if a linear dynamic system is to provide observer action despite arbitrary small perturbations in a specified subset of its parameters, it must: (1) be a closed loop system, be driven by the observer error, (2) possess redundancy, the observer must be generating, implicitly or explicitly, at least one linear combination of states that is already contained in the measurements, and (3) contain a perturbation-free model of the portion of the system observable from the external input to the observer. The procedure for design of robust observers possessing the above structural features is established and discussed.

Bhattacharyya, S. P.↗

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 NTRS1978

On the accuracy and convergence of implicit numerical integration of finite element generated ordinary differential equations

A study of accuracy and convergence of linear functional finite element solution to linear parabolic and hyperbolic partial differential equations is presented. A variable-implicit integration procedure is employed for the resultant system of ordinary differential equations. Accuracy and convergence is compared for the consistent and two lumped assembly procedures for the identified initial-value matrix structure. Truncation error estimation is accomplished using Richardson extrapolation.

Baker, A. J.↗

NASA NTRS2008

A Verification-Driven Approach to Control Analysis and Tuning

This paper proposes a methodology for the analysis and tuning of controllers using control verification metrics. These metrics, which are introduced in a companion paper, measure the size of the largest uncertainty set of a given class for which the closed-loop specifications are satisfied. This framework integrates deterministic and probabilistic uncertainty models into a setting that enables the deformation of sets in the parameter space, the control design space, and in the union of these two spaces. In regard to control analysis, we propose strategies that enable bounding regions of the design space where the specifications are satisfied by all the closed-loop systems associated with a prescribed uncertainty set. When this is unfeasible, we bound regions where the probability of satisfying the requirements exceeds a prescribed value. In regard to control tuning, we propose strategies for the improvement of the robust characteristics of a baseline controller. Some of these strategies use multi-point approximations to the control verification metrics in order to alleviate the numerical burden of solving a min-max problem. Since this methodology targets non-linear systems having an arbitrary, possibly implicit, functional dependency on the uncertain parameters and for which high-fidelity simulations are available, they are applicable to realistic engineering problems..

Crespo, Luis G.↗

NASA NTRS2016

An Inviscid Decoupled Method for the Roe FDS Scheme in the Reacting Gas Path of FUN3D

An approach is described to decouple the species continuity equations from the mixture continuity, momentum, and total energy equations for the Roe flux difference splitting scheme. This decoupling simplifies the implicit system, so that the flow solver can be made significantly more efficient, with very little penalty on overall scheme robustness. Most importantly, the computational cost of the point implicit relaxation is shown to scale linearly with the number of species for the decoupled system, whereas the fully coupled approach scales quadratically. Also, the decoupled method significantly reduces the cost in wall time and memory in comparison to the fully coupled approach. This work lays the foundation for development of an efficient adjoint solution procedure for high speed reacting flow.

Thompson, Kyle B.↗

NASA NTRS2018

Applications of Phase-Based Motion Processing

Image pyramids provide useful information in determining structural response at low cost using commercially available cameras. The current effort applies previous work on the complex steerable pyramid to analyze and identify imperceptible linear motions in video. Instead of implicitly computing motion spectra through phase analysis of the complex steerable pyramid and magnifying the associated motions, instead present a visual technique and the necessary software to display the phase changes of high frequency signals within video. The present technique quickly identifies regions of largest motion within a video with a single phase visualization and without the artifacts of motion magnification, but requires use of the computationally intensive Fourier transform. While Riesz pyramids present an alternative to the computationally intensive complex steerable pyramid for motion magnification, the Riesz formulation contains significant noise, and motion magnification still presents large amounts of data that cannot be quickly assessed by the human eye. Thus, user-friendly software is presented for quickly identifying structural response through optical flow and phase visualization in both Python and MATLAB.

Branch, Nicholas A.↗

NASA NTRSDate not supplied

Recent Progress in OVERFLOW Convergence Improvements

Improvements have been made to the implicit symmetric successive overrelaxation algorithm in the OVERFLOW 2.3 structured, overset grid, computational fluid dynamics flow solver. These improvements, consisting of implicit boundary conditions, improved flux Jacobian linearizations, and CFL number ramping, are a series of evolutionary changes to the linear solver that have resulted in increased nonlinear convergence rates and faster time to solution. A series of test cases are presented that demonstrate the effect of the changes through comparison with the original SSOR path and other linear solver implementations within OVERFLOW.

Joseph M Derlaga↗

NASA NTRS1993

Analysis of implicit second-order upwind-biased stencils

Truncation error and stability properties of several implicit upwind schemes for the two-dimensional Euler equations are examined. The schemes use linear data reconstruction methods to achieve second-order flux integrations where the implicit Jacobian operators are first order. The stability properties of the schemes are examined by a Von Neumann analysis of the linearized, constant-coefficient Euler equations. The choice of the data reconstruction method used to evaluate the flux integral has a dramatic effect on the convergence properties of the implicit solution method. In particular, the typical one-dimensional data reconstruction methods used with structured grids exhibit poor convergence properties compared to the unstructured grid method considered. Of the schemes examined, the one with the superior convergence properties is well-suited for both unstructured and structured grids, which has important implications for the design of implicit methods.

Roberts, Thomas W.↗