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NASA NTRS1989

A survey on the structured singular value

The structured singular value, U, is an important linear algebra tool to study a class of matrix perturbation problems. It is useful for analyzing the robustness of stability and performance of uncertain, (nominally) linear systems. Computation of (M) is difficult, and usually, upper and lower bounds are all that can be reliably computed. Upper bounds give conservative estimates of the sizes of allowable perturbations. The maximum singular value of a matrix M is an upper bound for (M). As an upper bound, it can be improved by finding a transformations to the data (i.e. M) which do not change the structured singular value, but do reduce the maximum singular value. Typically, upper bound algorithms involve searches over sets of transformations to yield the tightest bound. Lower bound algorithms are intelligent searches for minimum-norm solutions to multivariable polynomial equations, and are based on various optimality conditions that hold at the global (and, unfortunately, some local) minima. The current methods to compute both of these types of bounds are reviewed. Theoretical justification and extensive numerical experience with the various algorithms are covered.

Packard, Andy↗

NASA NTRS1993

Quaternion normalization in spacecraft attitude determination

Attitude determination of spacecraft usually utilizes vector measurements such as Sun, center of Earth, star, and magnetic field direction to update the quaternion which determines the spacecraft orientation with respect to some reference coordinates in the three dimensional space. These measurements are usually processed by an extended Kalman filter (EKF) which yields an estimate of the attitude quaternion. Two EKF versions for quaternion estimation were presented in the literature; namely, the multiplicative EKF (MEKF) and the additive EKF (AEKF). In the multiplicative EKF, it is assumed that the error between the correct quaternion and its a-priori estimate is, by itself, a quaternion that represents the rotation necessary to bring the attitude which corresponds to the a-priori estimate of the quaternion into coincidence with the correct attitude. The EKF basically estimates this quotient quaternion and then the updated quaternion estimate is obtained by the product of the a-priori quaternion estimate and the estimate of the difference quaternion. In the additive EKF, it is assumed that the error between the a-priori quaternion estimate and the correct one is an algebraic difference between two four-tuple elements and thus the EKF is set to estimate this difference. The updated quaternion is then computed by adding the estimate of the difference to the a-priori quaternion estimate. If the quaternion estimate converges to the correct quaternion, then, naturally, the quaternion estimate has unity norm. This fact was utilized in the past to obtain superior filter performance by applying normalization to the filter measurement update of the quaternion. It was observed for the AEKF that when the attitude changed very slowly between measurements, normalization merely resulted in a faster convergence; however, when the attitude changed considerably between measurements, without filter tuning or normalization, the quaternion estimate diverged. However, when the quaternion estimate was normalized, the estimate converged faster and to a lower error than with tuning only. In last years, symposium we presented three new AEKF normalization techniques and we compared them to the brute force method presented in the literature. The present paper presents the issue of normalization of the MEKF and examines several MEKF normalization techniques.

Deutschmann, J.↗

NASA NTRS2013

Scheduling Operations for Massive Heterogeneous Clusters

High-performance computing (HPC) programming has become increasingly difficult with the advent of hybrid supercomputers consisting of multicore CPUs and accelerator boards such as the GPU. Manual tuning of software to achieve high performance on this type of machine has been performed by programmers. This is needlessly difficult and prone to being invalidated by new hardware, new software, or changes in the underlying code. A system was developed for task-based representation of programs, which when coupled with a scheduler and runtime system, allows for many benefits, including higher performance and utilization of computational resources, easier programming and porting, and adaptations of code during runtime. The system consists of a method of representing computer algorithms as a series of data-dependent tasks. The series forms a graph, which can be scheduled for execution on many nodes of a supercomputer efficiently by a computer algorithm. The schedule is executed by a dispatch component, which is tailored to understand all of the hardware types that may be available within the system. The scheduler is informed by a cluster mapping tool, which generates a topology of available resources and their strengths and communication costs. Software is decoupled from its hardware, which aids in porting to future architectures. A computer algorithm schedules all operations, which for systems of high complexity (i.e., most NASA codes), cannot be performed optimally by a human. The system aids in reducing repetitive code, such as communication code, and aids in the reduction of redundant code across projects. It adds new features to code automatically, such as recovering from a lost node or the ability to modify the code while running. In this project, the innovators at the time of this reporting intend to develop two distinct technologies that build upon each other and both of which serve as building blocks for more efficient HPC usage. First is the scheduling and dynamic execution framework, and the second is scalable linear algebra libraries that are built directly on the former.

Humphrey, John↗

NASA NTRS2001

Unstructured Mesh Methods for the Simulation of Hypersonic Flows

This report describes the research work undertaken at the Massachusetts Institute of Technology. The aim of this research is to identify effective algorithms and methodologies for the efficient and routine solution of hypersonic viscous flows about re-entry vehicles. For over ten years we have received support from NASA to develop unstructured mesh methods for Computational Fluid Dynamics. As a result of this effort a methodology based on the use, of unstructured adapted meshes of tetrahedra and finite volume flow solvers has been developed. A number of gridding algorithms flow solvers, and adaptive strategies have been proposed. The most successful algorithms developed from the basis of the unstructured mesh system FELISA. The FELISA system has been extensively for the analysis of transonic and hypersonic flows about complete vehicle configurations. The system is highly automatic and allows for the routine aerodynamic analysis of complex configurations starting from CAD data. The code has been parallelized and utilizes efficient solution algorithms. For hypersonic flows, a version of the, code which incorporates real gas effects, has been produced. One of the latest developments before the start of this grant was to extend the system to include viscous effects. This required the development of viscous generators, capable of generating the anisotropic grids required to represent boundary layers, and viscous flow solvers. In figures I and 2, we show some sample hypersonic viscous computations using the developed viscous generators and solvers. Although these initial results were encouraging, it became apparent that in order to develop a fully functional capability for viscous flows, several advances in gridding, solution accuracy, robustness and efficiency were required. As part of this research we have developed: 1) automatic meshing techniques and the corresponding computer codes have been delivered to NASA and implemented into the GridEx system, 2) a finite element algorithm for the solution of the viscous compressible flow equations which can solve flows all the way down to the incompressible limit and that can use higher order (quadratic) approximations leading to highly accurate answers, and 3) and iterative algebraic multigrid solution techniques.

Peraire, Jaime↗

NASA NTRS1987

Computation of asymmetric supersonic flows around cones at large incidence

The Schiff-Steger parabolized Navier-Stokes (PNS) code has been modified to allow computation of conical flowfields around cones at high incidence. The improved algorithm of Degani and Schiff has been incorporated with the PNS code. This algorithm adds the cross derivative and circumferential viscous terms to the original PNS code and modifies the algebraic eddy viscosity turbulence model to take into account regions of so called cross-flow separation. Assuming the flowfield is conical (but not necessarily symmetric) a marching stepback procedure is used: the solution is marched one step downstream using improved PNS code and the flow variables are then scaled to place the solution back to the original station. The process is repeated until no change in the flow variables is observed with further marching. The flow variables are then constant along rays of the flowfield. The experiments obtained by Bannik and Nebbeling were chosen as a test case. In these experiments a cone of 7.5 deg. half angle at Mach number 2.94 and Reynolds number 1.372 x 10(7) was tested up 34 deg. angle of attack. At high angle of attack nonconical asymmetric leeward side vortex patterns were observed. In the first set of computations, using an earlier obtained solution of the above cone for angle of attack of 22.6 deg. and at station x=0.5 as a starting solution, the angle of attack was gradually increased up to 34 deg. During this procedure the grid was carfully adjusted to capture the bow shock. A stable, converged symmetric solution was obtained. Since the numerical code converged to a symmetric solution which is not the physical one, the stability was tested by a random perturbation at each point. The possible effect of surface roughness or non perfect body shape was also investigated. It was concluded that although the assumption of conical viscous flows can be very useful for certain cases, it can not be used for the present case. Thus the second part of the investigation attempted to obtain a marching (in space) solution with the PNS method using the conical solution as initial data. Finally, the solution of the full Navier-Stokes equations was carried out.

Degani, David↗

NASA NTRS1994

Large Eddy simulation of turbulence: A subgrid scale model including shear, vorticity, rotation, and buoyancy

The Reynolds numbers that characterize geophysical and astrophysical turbulence (Re approximately equals 10(exp 8) for the planetary boundary layer and Re approximately equals 10(exp 14) for the Sun's interior) are too large to allow a direct numerical simulation (DNS) of the fundamental Navier-Stokes and temperature equations. In fact, the spatial number of grid points N approximately Re(exp 9/4) exceeds the computational capability of today's supercomputers. Alternative treatments are the ensemble-time average approach, and/or the volume average approach. Since the first method (Reynolds stress approach) is largely analytical, the resulting turbulence equations entail manageable computational requirements and can thus be linked to a stellar evolutionary code or, in the geophysical case, to general circulation models. In the volume average approach, one carries out a large eddy simulation (LES) which resolves numerically the largest scales, while the unresolved scales must be treated theoretically with a subgrid scale model (SGS). Contrary to the ensemble average approach, the LES+SGS approach has considerable computational requirements. Even if this prevents (for the time being) a LES+SGS model to be linked to stellar or geophysical codes, it is still of the greatest relevance as an 'experimental tool' to be used, inter alia, to improve the parameterizations needed in the ensemble average approach. Such a methodology has been successfully adopted in studies of the convective planetary boundary layer. Experienc e with the LES+SGS approach from different fields has shown that its reliability depends on the healthiness of the SGS model for numerical stability as well as for physical completeness. At present, the most widely used SGS model, the Smagorinsky model, accounts for the effect of the shear induced by the large resolved scales on the unresolved scales but does not account for the effects of buoyancy, anisotropy, rotation, and stable stratification. The latter phenomenon, which affects both geophysical and astrophysical turbulence (e.g., oceanic structure and convective overshooting in stars), has been singularly difficult to account for in turbulence modeling. For example, the widely used model of Deardorff has not been confirmed by recent LES results. As of today, there is no SGS model capable of incorporating buoyancy, rotation, shear, anistropy, and stable stratification (gravity waves). In this paper, we construct such a model which we call CM (complete model). We also present a hierarchy of simpler algebraic models (called AM) of varying complexity. Finally, we present a set of models which are simplified even further (called SM), the simplest of which is the Smagorinsky-Lilly model. The incorporation of these models into the presently available LES codes should begin with the SM, to be followed by the AM and finally by the CM.

Canuto, V. M.↗