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NASA NTRS1993

Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems

In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss-Seidel iterations. We also study smoothers based entirely on the original operator. One is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi, and Gauss-Seidel. We show that the uniform estimates for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).

Bramble, James H.↗

NASA NTRS1993

The molecular matching problem

Molecular chemistry contains many difficult optimization problems that have begun to attract the attention of optimizers in the Operations Research community. Problems including protein folding, molecular conformation, molecular similarity, and molecular matching have been addressed. Minimum energy conformations for simple molecular structures such as water clusters, Lennard-Jones microclusters, and short polypeptides have dominated the literature to date. However, a variety of interesting problems exist and we focus here on a molecular structure matching (MSM) problem.

Kincaid, Rex K.↗

NASA NTRS1994

High-performance parallel analysis of coupled problems for aircraft propulsion

Applications are described of high-performance parallel, computation for the analysis of complete jet engines, considering its multi-discipline coupled problem. The coupled problem involves interaction of structures with gas dynamics, heat conduction and heat transfer in aircraft engines. The methodology issues addressed include: consistent discrete formulation of coupled problems with emphasis on coupling phenomena; effect of partitioning strategies, augmentation and temporal solution procedures; sensitivity of response to problem parameters; and methods for interfacing multiscale discretizations in different single fields. The computer implementation issues addressed include: parallel treatment of coupled systems; domain decomposition and mesh partitioning strategies; data representation in object-oriented form and mapping to hardware driven representation, and tradeoff studies between partitioning schemes and fully coupled treatment.

Felippa, C. A.↗

NASA NTRS1994

High Performance Parallel Analysis of Coupled Problems for Aircraft Propulsion

In order to predict the dynamic response of a flexible structure in a fluid flow, the equations of motion of the structure and the fluid must be solved simultaneously. In this paper, we present several partitioned procedures for time-integrating this focus coupled problem and discuss their merits in terms of accuracy, stability, heterogeneous computing, I/O transfers, subcycling, and parallel processing. All theoretical results are derived for a one-dimensional piston model problem with a compressible flow, because the complete three-dimensional aeroelastic problem is difficult to analyze mathematically. However, the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two-dimensional transient aeroelastic response of a flexible panel in a transonic nonlinear Euler flow regime.

Felippa, C. A.↗

NASA NTRS1994

Automation of reverse engineering process in aircraft modeling and related optimization problems

During the year of 1994, the engineering problems in aircraft modeling were studied. The initial concern was to obtain a surface model with desirable geometric characteristics. Much of the effort during the first half of the year was to find an efficient way of solving a computationally difficult optimization model. Since the smoothing technique in the proposal 'Surface Modeling and Optimization Studies of Aerodynamic Configurations' requires solutions of a sequence of large-scale quadratic programming problems, it is important to design algorithms that can solve each quadratic program in a few interactions. This research led to three papers by Dr. W. Li, which were submitted to SIAM Journal on Optimization and Mathematical Programming. Two of these papers have been accepted for publication. Even though significant progress has been made during this phase of research and computation times was reduced from 30 min. to 2 min. for a sample problem, it was not good enough for on-line processing of digitized data points. After discussion with Dr. Robert E. Smith Jr., it was decided not to enforce shape constraints in order in order to simplify the model. As a consequence, P. Dierckx's nonparametric spline fitting approach was adopted, where one has only one control parameter for the fitting process - the error tolerance. At the same time the surface modeling software developed by Imageware was tested. Research indicated a substantially improved fitting of digitalized data points can be achieved if a proper parameterization of the spline surface is chosen. A winning strategy is to incorporate Dierckx's surface fitting with a natural parameterization for aircraft parts. The report consists of 4 chapters. Chapter 1 provides an overview of reverse engineering related to aircraft modeling and some preliminary findings of the effort in the second half of the year. Chapters 2-4 are the research results by Dr. W. Li on penalty functions and conjugate gradient methods for quadratic programming problems.

Li, W.↗

NASA NTRS1995

The Benard problem: A comparison of finite difference and spectral collocation eigen value solutions

The application of spectral methods, using a Chebyshev collocation scheme, to solve hydrodynamic stability problems is demonstrated on the Benard problem. Implementation of the Chebyshev collocation formulation is described. The performance of the spectral scheme is compared with that of a 2nd order finite difference scheme. An exact solution to the Marangoni-Benard problem is used to evaluate the performance of both schemes. The error of the spectral scheme is at least seven orders of magnitude smaller than finite difference error for a grid resolution of N = 15 (number of points used). The performance of the spectral formulation far exceeded the performance of the finite difference formulation for this problem. The spectral scheme required only slightly more effort to set up than the 2nd order finite difference scheme. This suggests that the spectral scheme may actually be faster to implement than higher order finite difference schemes.

Skarda, J. Raymond Lee↗

NASA NTRS1995

Pseudo-time methods for constrained optimization problems governed by PDE

In this paper we present a novel method for solving optimization problems governed by partial differential equations. Existing methods are gradient information in marching toward the minimum, where the constrained PDE is solved once (sometimes only approximately) per each optimization step. Such methods can be viewed as a marching techniques on the intersection of the state and costate hypersurfaces while improving the residuals of the design equations per each iteration. In contrast, the method presented here march on the design hypersurface and at each iteration improve the residuals of the state and costate equations. The new method is usually much less expensive per iteration step since, in most problems of practical interest, the design equation involves much less unknowns that that of either the state or costate equations. Convergence is shown using energy estimates for the evolution equations governing the iterative process. Numerical tests show that the new method allows the solution of the optimization problem in a cost of solving the analysis problems just a few times, independent of the number of design parameters. The method can be applied using single grid iterations as well as with multigrid solvers.

Taasan, Shlomo↗

NASA NTRS1994

Test problems for radiation and radiation-hydrodynamics codes

A number of test problems for radiation and radiation-hydrodynamics computer codes are described. These include evolution to radiative equilibrium, cooling from radiative equilibrium, subcritical and supercritical radiating shocks, and a radiating blast wave in a power-law density distribution. For each test problem, example input parameters and plots of the results are presented. Some test problems for pure hydrodynamics are also suggested. The radiation-hydrodynamics code used to perform the example test problems and the equations it solves are described in some detail.

Ensman, Lisa↗

NASA NTRS1995

Multigrid Methods for Aerodynamic Problems in Complex Geometries

Work has been directed at the development of efficient multigrid methods for the solution of aerodynamic problems involving complex geometries, including the development of computational methods for the solution of both inviscid and viscous transonic flow problems. The emphasis is on problems of complex, three-dimensional geometry. The methods developed are based upon finite-volume approximations to both the Euler and the Reynolds-Averaged Navier-Stokes equations. The methods are developed for use on multi-block grids using diagonalized implicit multigrid methods to achieve computational efficiency. The work is focused upon aerodynamic problems involving complex geometries, including advanced engine inlets.

Caughey, David A.↗

NASA NTRS1995

Enroute flight planning: Evaluating design concepts for the development of cooperative problem-solving systems

There are many problem-solving tasks that are too complex to fully automate given the current state of technology. Nevertheless, significant improvements in overall system performance could result from the introduction of well-designed computer aids. We have been studying the development of cognitive tools for one such problem-solving task, enroute flight path planning for commercial airlines. Our goal has been two-fold. First, we have been developing specific system designs to help with this important practical problem. Second, we have been using this context to explore general design concepts to guide in the development of cooperative problem-solving systems. These design concepts are described below, along with illustrations of their application.

Smith, Philip J.↗

NASA NTRS1992

Scare Tactics: Evaluating Problem Decompositions Using Failure Scenarios

Our interest is in the design of multi-agent problem-solving systems, which we refer to as composite systems. We have proposed an approach to composite system design by decomposition of problem statements. An automated assistant called Critter provides a library of reusable design transformations which allow a human analyst to search the space of decompositions for a problem. In this paper we describe a method for evaluating and critiquing problem decompositions generated by this search process. The method uses knowledge stored in the form of failure decompositions attached to design transformations. We suggest the benefits of our critiquing method by showing how it could re-derive steps of a published development example. We then identify several open issues for the method.

Helm, B. Robert↗

NASA NTRS1996

A History of Aerospace Problems, Their Solutions, Their Lessons

The positive aspect of problem occurrences is the opportunity for learning and a challenge for innovation. The learning aspect is not restricted to the solution period of the problem occurrence, but can become the beacon for problem prevention on future programs. Problems/failures serve as a point of departure for scaling to new designs. To ensure that problems/failures and their solutions guide the future programs, a concerted effort has been expended to study these problems, their solutions, their derived lessons learned, and projections for future programs. This includes identification of technology thrusts, process changes, codes development, etc. However, they must not become an excuse for adding layers upon layers of standards, criteria, and requirements, but must serve as guidelines that assist instead of stifling engineers. This report is an extension of prior efforts to accomplish this task. Although these efforts only scratch the surface, it is a beginning that others must complete.

Ryan, R. S.↗

NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗

NASA NTRS1997

Computational Issues in Damping Identification for Large Scale Problems

Two damping identification methods are tested for efficiency in large-scale applications. One is an iterative routine, and the other a least squares method. Numerical simulations have been performed on multiple degree-of-freedom models to test the effectiveness of the algorithm and the usefulness of parallel computation for the problems. High Performance Fortran is used to parallelize the algorithm. Tests were performed using the IBM-SP2 at NASA Ames Research Center. The least squares method tested incurs high communication costs, which reduces the benefit of high performance computing. This method's memory requirement grows at a very rapid rate meaning that larger problems can quickly exceed available computer memory. The iterative method's memory requirement grows at a much slower pace and is able to handle problems with 500+ degrees of freedom on a single processor. This method benefits from parallelization, and significant speedup can he seen for problems of 100+ degrees-of-freedom.

Pilkey, Deborah L.↗

NASA NTRS1985

An Algorithm for Efficient Maximum Likelihood Estimation and Confidence Interval Determination in Nonlinear Estimation Problems

An algorithm for maximum likelihood (ML) estimation is developed with an efficient method for approximating the sensitivities. The algorithm was developed for airplane parameter estimation problems but is well suited for most nonlinear, multivariable, dynamic systems. The ML algorithm relies on a new optimization method referred to as a modified Newton-Raphson with estimated sensitivities (MNRES). MNRES determines sensitivities by using slope information from local surface approximations of each output variable in parameter space. The fitted surface allows sensitivity information to be updated at each iteration with a significant reduction in computational effort. MNRES determines the sensitivities with less computational effort than using either a finite-difference method or integrating the analytically determined sensitivity equations. MNRES eliminates the need to derive sensitivity equations for each new model, thus eliminating algorithm reformulation with each new model and providing flexibility to use model equations in any format that is convenient. A random search technique for determining the confidence limits of ML parameter estimates is applied to nonlinear estimation problems for airplanes. The confidence intervals obtained by the search are compared with Cramer-Rao (CR) bounds at the same confidence level. It is observed that the degree of nonlinearity in the estimation problem is an important factor in the relationship between CR bounds and the error bounds determined by the search technique. The CR bounds were found to be close to the bounds determined by the search when the degree of nonlinearity was small. Beale's measure of nonlinearity is developed in this study for airplane identification problems; it is used to empirically correct confidence levels for the parameter confidence limits. The primary utility of the measure, however, was found to be in predicting the degree of agreement between Cramer-Rao bounds and search estimates.

Murphy, Patrick Charles↗

NASA NTRS1999

Geodynamic Effects of Ocean Tides: Progress and Problems

Satellite altimetry, particularly Topex/Poseidon, has markedly improved our knowledge of global tides, thereby allowing significant progress on some longstanding problems in geodynamics. This paper reviews some of that progress. Emphasis is given to global-scale problems, particularly those falling within the mandate of the new IERS Special Bureau for Tides: angular momentum, gravitational field, geocenter motion. For this discussion I use primarily the new ocean tide solutions GOT99.2, CSR4.0, and TPXO.4 (for which G. Egbert has computed inverse-theoretic error estimates), and I concentrate on new results in angular momentum and gravity and their solid-earth implications. One example is a new estimate of the effective tidal Q at the M_2 frequency, based on combining these ocean models with tidal estimates from satellite laser ranging. Three especially intractable problems are also addressed: (1) determining long-period tides in the Arctic [large unknown effect on the inertia tensor, particularly for Mf]; (2) determining the global psi_l tide [large unknown effect on interpretations of gravimetry for the near-diurnal free wobble]; and (3) determining radiational tides [large unknown temporal variations at important frequencies]. Problems (2) and (3) are related.

Richard, Ray↗

NASA NTRS2000

Numerical Solutions to the Third CAA Workshop Benchmark Problems

This paper presents numerical solutions to the problems of propagation of sound waves through a transonic nozzle, shock-sound interactions, and automobile noise involving feedback of the third NASA Computational Aeroacoustics (CAA) Workshop on benchmark problems. The numerical algorithm is based on a dual time scheme for temporal discretization and a third-order finite volume scheme for spatial discretization. The aims of this study are to apply a dual time stepping scheme to treat aeroacoustic problems of sound propagation and to validate our CAA solver with the benchmark problems for developing a numerical tool for noise analysis and control.

Loh, Roy H.↗

NASA NTRS1997

Alternative Perspectives on Risk: Individual Differences in Problem Structuring

Team decision making involves contributions of multiple players toward a common goal. While much has been written about the importance of developing shared mental models in order for teams to work together effectively, little has been done to determine the value of alternative perspectives on problem solving and decision making. Early studies of expertise contrasted experts with novices and noted that the two groups differ in the way they structure problems and in their selection of information as salient. Little attention has been given to differences among experts who differ in their specializations. A series of experiments was conducted to determine: (1) what dimensions of flight-related problem situations pilots judge to be most important when making flight-relevant decisions; and (2) whether pilots in different crew positions differ in the way they interpret problems relating to flight decisions. A sorting task was used to identify underlying dimensions judged as salient to individual pilots. Captains, first officers, and flight engineers from two major carriers participated in the study. Twenty-two flight scenarios were developed based on ASRS reports. Pilots were required to make judgments about how they would respond in each case and to sort the scenarios on the basis of similarity of decision factors. They were also asked to provide a verbal label that described each of their sorted categories. A second study required a different group of pilots (also captains, first officers and flight engineers) to sort on predetermined bases.

Orasanu, Judith↗