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NASA NTRS1974

Principal components colour display of ERTS imagery

In the technique presented, colours are not derived from single bands, but rather from independent linear combinations of the bands. Using a simple model of the processing done by the visual system, three informationally independent linear combinations of the four ERTS bands are mapped onto the three visual colour dimensions of brightness, redness-greenness and blueness-yellowness. The technique permits user-specific transformations which enhance particular features, but this is not usually needed, since a single transformation provides a picture which conveys much of the information implicit in the ERTS data. Examples of experimental vector images with matched individual band images are shown.

Taylor, M. M.↗

NASA NTRS1990

Solution algorithms for the two-dimensional Euler equations on unstructured meshes

The objective of the study was to analyze implicit techniques employed in structured grid algorithms for solving two-dimensional Euler equations and extend them to unstructured solvers in order to accelerate convergence rates. A comparison is made between nine different algorithms for both first-order and second-order accurate solutions. Higher-order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The discussion is illustrated by results for flow over a transonic circular arc.

Whitaker, D. L.↗

NASA NTRS1989

Numerical investigation of hypersonic exhaust plume/afterbody flow fields

An upwind, implicit Navier-Stokes computer program has been applied to hypersonic exhaust plume/afterbody flowfields. The sensitivity of gross thrust to operating conditions has been assessed through parametric variations. Comparison of the numerical results with available experimental data shows good agreement in all cases investigated. Results show that, for moderately underexpanded jets, the afterbody force varies linearly with the nozzle exit pressure. Exhaust gases with low isentropic exponents (gamma) were found to contribute up to 25 percent more afterbody force than high-gamma exhaust gases. Modifying the nozzle geometry influenced the exhaust plume development, which had a significant effect on the afterbody force. Grid density, while important to resolving the initial plume/afterbody interaction, had only a minor impact on the resultant afterbody force.

Edwards, T. A.↗

NASA NTRS1991

Multigrid calculation of three-dimensional viscous cascade flows

A 3-D code for viscous cascade flow prediction was developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full multigrid method. The Baldwin-Lomax eddy viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Arnone, A.↗

NASA NTRS1998

Parallel 3D Multi-Stage Simulation of a Turbofan Engine

A 3D multistage simulation of each component of a modern GE Turbofan engine has been made. An axisymmetric view of this engine is presented in the document. This includes a fan, booster rig, high pressure compressor rig, high pressure turbine rig and a low pressure turbine rig. In the near future, all components will be run in a single calculation for a solution of 49 blade rows. The simulation exploits the use of parallel computations by using two levels of parallelism. Each blade row is run in parallel and each blade row grid is decomposed into several domains and run in parallel. 20 processors are used for the 4 blade row analysis. The average passage approach developed by John Adamczyk at NASA Lewis Research Center has been further developed and parallelized. This is APNASA Version A. It is a Navier-Stokes solver using a 4-stage explicit Runge-Kutta time marching scheme with variable time steps and residual smoothing for convergence acceleration. It has an implicit K-E turbulence model which uses an ADI solver to factor the matrix. Between 50 and 100 explicit time steps are solved before a blade row body force is calculated and exchanged with the other blade rows. This outer iteration has been coined a "flip." Efforts have been made to make the solver linearly scaleable with the number of blade rows. Enough flips are run (between 50 and 200) so the solution in the entire machine is not changing. The K-E equations are generally solved every other explicit time step. One of the key requirements in the development of the parallel code was to make the parallel solution exactly (bit for bit) match the serial solution. This has helped isolate many small parallel bugs and guarantee the parallelization was done correctly. The domain decomposition is done only in the axial direction since the number of points axially is much larger than the other two directions. This code uses MPI for message passing. The parallel speed up of the solver portion (no 1/0 or body force calculation) for a grid which has 227 points axially.

Turner, Mark G.↗

NASA NTRS1978

A high order accurate finite element algorithm for high Reynolds number flow prediction

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Baker, A. J.↗

NASA NTRS1993

MARIKA - A model revision system using qualitative analysis of simulations

This paper describes portions of a novel system called MARIKA (Model Analysis and Revision of Implicit Key Assumptions) to automatically revise a model of the normal human orientation system. The revision is based on analysis of discrepancies between experimental results and computer simulations. The discrepancies are calculated from qualitative analysis of quantitative simulations. The experimental and simulated time series are first discretized in time segments. Each segment is then approximated by linear combinations of simple shapes. The domain theory and knowledge are represented as a constraint network. Incompatibilities detected during constraint propagation within the network yield both parameter and structural model alterations. Interestingly, MARIKA diagnosed a data set from the Massachusetts Eye and Ear Infirmary Vestibular Laboratory as abnormal though the data was tagged as normal. Published results from other laboratories confirmed the finding. These encouraging results could lead to a useful clinical vestibular tool and to a scientific discovery system for space vestibular adaptation.

Groleau, Nicolas↗

NASA NTRS2002

Higher Order Time Integration Schemes for the Unsteady Navier-Stokes Equations on Unstructured Meshes

The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.

Jothiprasad, Giridhar↗

NASA NTRS1983

Symmetric linear systems - An application of algebraic systems theory

Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.

Hazewinkel, M.↗

NASA NTRS1991

Multigrid calculation of three-dimensional viscous cascade flows

A three-dimensional code for viscous cascade flow prediction has been developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four-stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full-multigrid method. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large-scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Arnone, A.↗

NASA NTRS1984

The application of satellite data to study the effects of latent heat release on cyclones

Generalized energetics were studied for nonlinear inviscid symmetric instability (SI). It was found that the linear theory fails to predict the stability in certain cases where the basic state is transitional between stability and instability. The initial growth of the SI perturbations can be fairly well approximated by linear theory, but the long time nonlinear evaluations will be bonded energetically if the SI region is finite. However, a further extension of the energetics to conditional symmetric instability (CSI) shows that the nonlinear evolution of circulation will energetically depend much more on the precipitation in a complicated way. By treating the latent heat as a source which is implicitly related to the motion field, the existence, uniqueness and stability of steady viscous (CSI) circulations are studied. Viscous CSI circulations are proved to be unique and asymptotically stable when the heat sources are weak and less sensitive to the motion perturbations. By considering the fact that moist updrafts are narrow and using eddy viscosity of 0(1,000 m squared/s) the stability criterion suggests that some frontal rainbands were probably dominated by the CSI mechanism even in their mature quasi-steady stage.

Clark, J. H. E.↗

NASA NTRS2018

Design of a Modular Monolithic Implicit Solver for Multi-Physics Applications

The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high- fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-re effective boundary conditions and wall modeling. The solver is based on high-order space-time - finite element methods. Continuous, discontinuous and C1-discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system.

Carton De Wiart, Corentin↗

NASA NTRS1972

A computer program for the geometrically nonlinear static and dynamic analysis of arbitrarily loaded shells of revolution, theory and users manual

A digital computer program known as SATANS (static and transient analysis, nonlinear, shells) for the geometrically nonlinear static and dynamic response of arbitrarily loaded shells of revolution is presented. Instructions for the preparation of the input data cards and other information necessary for the operation of the program are described in detail and two sample problems are included. The governing partial differential equations are based upon Sanders' nonlinear thin shell theory for the conditions of small strains and moderately small rotations. The governing equations are reduced to uncoupled sets of four linear, second order, partial differential equations in the meridional and time coordinates by expanding the dependent variables in a Fourier sine or cosine series in the circumferential coordinate and treating the nonlinear modal coupling terms as pseudo loads. The derivatives with respect to the meridional coordinate are approximated by central finite differences, and the displacement accelerations are approximated by the implicit Houbolt backward difference scheme with a constant time interval. The boundaries of the shell may be closed, free, fixed, or elastically restrained. The program is coded in the FORTRAN 4 language and is dimensioned to allow a maximum of 10 arbitrary Fourier harmonics and a maximum product of the total number of meridional stations and the total number of Fourier harmonics of 200. The program requires 155,000 bytes of core storage.

Ball, R. E.↗

NASA NTRS2003

High Performance Parallel Methods for Space Weather Simulations

This is the final report of our NASA AISRP grant entitled 'High Performance Parallel Methods for Space Weather Simulations'. The main thrust of the proposal was to achieve significant progress towards new high-performance methods which would greatly accelerate global MHD simulations and eventually make it possible to develop first-principles based space weather simulations which run much faster than real time. We are pleased to report that with the help of this award we made major progress in this direction and developed the first parallel implicit global MHD code with adaptive mesh refinement. The main limitation of all earlier global space physics MHD codes was the explicit time stepping algorithm. Explicit time steps are limited by the Courant-Friedrichs-Lewy (CFL) condition, which essentially ensures that no information travels more than a cell size during a time step. This condition represents a non-linear penalty for highly resolved calculations, since finer grid resolution (and consequently smaller computational cells) not only results in more computational cells, but also in smaller time steps.

Hunter, Paul↗

NASA NTRS1992

Development of iterative techniques for the solution of unsteady compressible viscous flows

The development of efficient iterative solution methods for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations is discussed. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. In this work, another approach based on the classical conjugate gradient method, known as the Generalized Minimum Residual (GMRES) algorithm is investigated. The GMRES algorithm has been used in the past by a number of researchers for solving steady viscous and inviscid flow problems. Here, we investigate the suitability of this algorithm for solving the system of non-linear equations that arise in unsteady Navier-Stokes solvers at each time step.

Sankar, Lakshmi N.↗

NASA NTRS1978

Large eddy simulation of incompressible turbulent channel flow

The three-dimensional, time-dependent primitive equations of motion were numerically integrated for the case of turbulent channel flow. A partially implicit numerical method was developed. An important feature of this scheme is that the equation of continuity is solved directly. The residual field motions were simulated through an eddy viscosity model, while the large-scale field was obtained directly from the solution of the governing equations. An important portion of the initial velocity field was obtained from the solution of the linearized Navier-Stokes equations. The pseudospectral method was used for numerical differentiation in the horizontal directions, and second-order finite-difference schemes were used in the direction normal to the walls. The large eddy simulation technique is capable of reproducing some of the important features of wall-bounded turbulent flows. The resolvable portions of the root-mean square wall pressure fluctuations, pressure velocity-gradient correlations, and velocity pressure-gradient correlations are documented.

Moin, P.↗

NASA NTRS1988

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Mostrel, M. M.↗

NASA NTRS1995

Sensitivity analysis of the non-linear dynamic viscoplastic response of 2-d structures with respect to material parameters

A computational procedure is presented for evaluating the sensitivity coefficients of the viscoplastic response of structures subjected to dynamic loading. A state of plane stress is assumed to exist in the structure, a velocity strain-Cauchy stress formulation is used, and the geometric non-linearities arising from large strains are incorporated. The Jaumann rate is used as a frame indifferent stress rate. The material model is chosen to be isothermal viscoplasticity, and an associated flow rule is used with a von Mises effective stress. The equations of motion emanating from a finite element semi-discretization are integrated using an explicit central difference scheme with an implicit stress update. The sensitivity coefficients are evaluated using a direct differentiation approach. Since the domain of integration is the current configuration, the sensitivity coefficients of the spatial derivatives of the shape functions must be included. Numerical results are presented for a thin plate with a central cutout subjected to an in-plane compressive loading. The sensitivity coefficients are generated by evaluating the derivatives of the response quantities with respect to Young's modulus, and two of the material parameters characterizing the viscoplastic response. Time histories of the response and sensitivity coefficients, and spatial distributions at selected times are presented.

Kulkarni, Makarand↗