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At least 721 records · Page 40
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NASA NTRS1985

A Survey of Composite Grid Generation for General Three-dimensional Sections

The generation and use of composite grids for general three-dimensional physical boundary configurations is discussed, and the availability of several codes or procedures is noted. With the composite framework, the physical region is segmented into sub-regions, each bounded by six curved sides, and a grid is generated in each sub-region. These grids may be joined at the interfaces between the sub-regions with various degrees of continuity. This structure allows codes to be constructed to operate on rectangular blocks in computational space, so that existing solution procedures can be readily incorporated in the construction of codes for general configurations. Numerical grid generation is an integral part of the numerical solution of partial differential equations and is one of the pacing items in the development of codes for general configurations. The numerically generated grid frees the computational simulation from restriction to certain boundary shapes and allows general codes to be written in which the boundary shape is specified simply by input. The numerically generated grid allows all computation to be done on a fixed square grid in the computational space, which is always rectangular by construction.

Thompson, J. F.↗
NASA NTRS1985

Shock capturing

Recent developments which have improved the understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations are discussed. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. A conservative difference scheme is defined. Conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, the reason for this excellent shock resolution can be determined. This result is used to provide a mechanism for improving the resolution of two dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. Total variation diminishing (TVD) finite difference schemes and flux limiters are introduced to deal with money shocks and contact discontinuities.

Davis, S. F.↗
NASA NTRS1985

Finite Control in Underdamped Distributed Parameter Systems

Specifically, a class of distributed parameter systems is defined using Hilbert space methods based on a partial differential equation model of a structure. In this space a simple, easy to check definition of underdamping is constructed based on the well known single degree of freedom concept of critical damping. It is then shown that if a given distributed parameter system satisfies this definition, each mode of the modal expansion of the solution is in fact an underdamped function of time. It is noted that a distributed parameter system which is underdamped is also uniformly exponentially stable. By appealing to the work of Gibson, a finite dimensional model of the system will yield satisfactory control laws. Furthermore, it is shown that the assumption of the underdamping also allows straightforward computation of rough bounds on the magnitude of the unmodeled or residual modes for a modal truncation scheme. Hence, underdamped systems are precisely that class of distributed parameter systems which are not likely to have spillover problems and which will yield convergence of finite dimensional control laws to control laws which are optimal for the full distributed parameter model of the system.

Inman, D. J.↗
NASA NTRS1985

New Directions in Asymptotically Stable Finite-dimensional Adaptive Control of Linear Distributed Parameter Systems

Distributed Parameter Systems (DPS), such as systems described by partial differential equations, require infinite-dimensional state space descriptions to correctly model their dynamical behavior. However, any adaptive control algorithm must be finite-dimensional in order to be implemented via on-line digital computers. Finite-dimensional adaptive control of linear DPS requires stability analysis of nonlinear, time-varying, infinite-dimensional systems. The structure of nonadaptive finite-dimensional control of linear DPS is summarized as it relates to the existence of limiting systems for adaptive control. Two candidate schemes for finite-dimensional adaptive control of DPS are described and critical issues in infinite-dimensional stability analysis are discussed, in particular, the invariance principle, center manifold theory, and relationships between input-output and internal stability.

Balas, M. J.↗
NASA NTRS1985

On a variational approach to some parameter estimation problems

Examples (1-D seismic, large flexible structures, bioturbation, nonlinear population dispersal) in which a variation setting can provide a convenient framework for convergence and stability arguments in parameter estimation problems are considered. Some of these examples are 1-D seismic, large flexible structures, bioturbation, and nonlinear population dispersal. Arguments for convergence and stability via a variational approach of least squares formulations of parameter estimation problems for partial differential equations is one aspect of the problem considered.

Banks, H. T.↗
NASA NTRS1984

A review of spectral methods

An outline for spectral methods for partial differential equations is presented. The basic spectral algorithm is defined, collocation are emphasized and the main advantage of the method, the infinite order of accuracy in problems with smooth solutions are discussed. Examples of theoretical numerical analysis of spectral calculations are presented. An application of spectral methods to transonic flow is presented. The full potential transonic equation is among the best understood among nonlinear equations.

Lustman, L.↗
NASA NTRS1984

The influence of wind-tunnel walls on discrete frequency noise

This paper describes an analytical model that can be used to examine the effects of wind-tunnel walls on discrete frequency noise. First, a complete physical model of an acoustic source in a wind tunnel is described, and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. Second, the simplified physical model is formulated as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. The integral equation has been solved with a panel program on a computer. Preliminary results from a simple model problem will be shown and compared with the approximate analytic solution.

Mosher, M.↗
NASA NTRS1984

Multigrid applications to three-dimensional elliptic coordinate generation

The multigrid algorithm was applied to solve the coupled set of elliptic quasilinear partial differential equations associated with three-dimensional coordinate generation. The results indicate that the multigrid scheme is more than twice as fast as conventional relaxation schemes on moderate-size grids. Convergence factors of order 0.90 per work unit were achieved on 36,000-point grids. The paper covers the form of transformation, develops the set of generation equations, and gives details on the multigrid approach used. Included are a development of the full-approximation storage scheme, details of the smoothing-rate analysis, and a section devoted to rational programming techniques applicable to the multigrid algorithm.

Thames, F. C.↗
NASA NTRS1983

Linear approximations of nonlinear systems

A method for designing an automatic flight controller for short and vertical takeoff aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system, called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation of an exact transformation for the nonlinear system can be found. It is shown that a canonical expansion in Lie brackets about the point x(0) yields the same modified tangent model.

Hunt, L. R.↗
NASA NTRS1985

Vibration response of geometrically nonlinear elastic beams subjected to pulse and impact loading

An understanding of the response of composite laminates to foreign object impact is of fundamental importance to the design and the accurate evaluation of aircraft and aerospace structures. A characterization of the basic deformation response of the system is needed for the accurate and consistent modeling of the damage caused by the impact. The response to low velocity impact of foreign objects (less than 10 m/sec at 10 J of impact energy or less) is of particular interest because it corresponds to tool drop problems. An investigation of the vibration response of a simple square beam was conducted, taking into account both sharp initial velocity pulse conditions and simulated impact conditions. Attention is given to nonlinear beam theory, boundary conditions and materials modeling, initial conditions and impact modeling, and a solution of the obtained system of partial differential equations by the finite difference approach.

Moyer, E. T., Jr.↗
NASA NTRS1985

A factored implicit scheme for numerical weather prediction

An implicit method is proposed to factor the nonlinear partial differential equations governing fast and slow modes of dynamic motion in numerical weather prediction schemes. The method permits separate factorization of the slow and fast modes of the implicit operator. A simple two-dimensional version of the system of three-dimensional equations governing atmospheric dynamics over shallow water was analyzed to assess the accuracy of the proposed method. It is shown that the method has a small error which is comparable to other discretization errors in the overall scheme.

Augenbaum, J. M.↗
NASA NTRS1985

Passive damping concepts for slender columns in space structures

An experimental and theoretical study of three different passive damping concepts is conducted for a slender member with partial rotational end restraints. Over a hundred full-scale natural vibration experiments were conducted to evaluate the effectiveness of mass-string, polyethylene tubing, and chain damping concepts. The damping properties obtained from the experiments were used in the approximate analyses based on the partial differential equation of motion for the problem. The comparison of the experimental and the theoretical deflection-time relations shows that the velocity-dependent damping model used in the theory is adequate. From the experimental results, the effect of end connection friction and induced axial forces on damping is identified. The definition of an efficiency index is proposed based on the damping ratio and the mass of a given passive damping device. Using this definition, the efficiencies of the three damping devices are compared. The polyethylene tubing concept resulted into a low damping efficiency.

Razzaq, Z.↗
NASA NTRS1985

Bridgman growth of semiconductors

The purpose of this study was to improve the understanding of the transport phenomena which occurs in the directional solidification of alloy semiconductors. In particular, emphasis was placed on the strong role of convection in the melt. Analytical solutions were not deemed possible for such an involved problem. Accordingly, a numerical model of the process was developed which simulated the transport. This translates into solving the partial differential equations of energy, mass, species, and momentum transfer subject to various boundary and initial conditions. A finite element method with simple elements was initially chosen. This simulation tool will enable the crystal grower to systematically identify and modify the important design factors within her control to produce better crystals.

Carlson, F. M.↗
NASA NTRS1985

A method for determining acoustic-liner admittance in ducts with sheared flow in two-cross-sectional directions

A method is developed for determining the acoustic admittance of a test liner installed in the wall of a grazing flow impedance tube. The mean flow is permitted flow gradients in both cross-sectional directions of the tube. The unknown admittance value is obtained by solving an eigenvalue problem. This eigenvalue problem results from the application of the finite-element method to the partial differential equation and boundary conditions governing the acoustic field. The credibility of the method is established by comparing results with exact solutions obtained for a constant mean-flow profile and with previous results for cases involving shear in only one cross-sectional direction. Excellent comparisons are obtained in both cases. The analysis is used in conjunction with a limited amount of experimental data and shows that the flow must be accurately modeled in order to determine the acoustic-liner properties.

Watson, W. R.↗
NASA NTRS1983

Finite element or Galerkin type semidiscrete schemes

A finite element of Galerkin type semidiscrete method is proposed for numerical solution of a linear hyperbolic partial differential equation. The question of stability is reduced to the stability of a system of ordinary differential equations for which Dahlquist theory applied. Results of separating the part of numerical solution which causes the spurious oscillation near shock-like response of semidiscrete scheme to a step function initial condition are presented. In general all methods produce such oscillatory overshoots on either side of shocks. This overshoot pathology, which displays a behavior similar to Gibb's phenomena of Fourier series, is explained on the basis of dispersion of separated Fourier components which relies on linearized theory to be satisfactory. Expository results represented.

Durgun, K.↗
NASA NTRS1985

PIFEX: An advanced programmable pipelined-image processor

PIFEX is a pipelined-image processor being built in the JPL Robotics Lab. It will operate on digitized raster-scanned images (at 60 frames per second for images up to about 300 by 400 and at lesser rates for larger images), performing a variety of operations simultaneously under program control. It thus is a powerful, flexible tool for image processing and low-level computer vision. It also has applications in other two-dimensional problems such as route planning for obstacle avoidance and the numerical solution of two-dimensional partial differential equations (although its low numerical precision limits its use in the latter field). The concept and design of PIFEX are described herein, and some examples of its use are given.

Gennery, D. B.↗
NASA NTRS1985

Interface procedures for overlapping grids

Interpolation at grid boundaries is studied for the purpose of solving partial differential equations using either implicit or conservative explicit finite-difference methods on multi-component overlapping grid systems.

Mastin, C. W.↗
NASA NTRS1985

A partitioning strategy for nonuniform problems on multiprocessors

The partitioning of a problem on a domain with unequal work estimates in different subddomains is considered in a way that balances the work load across multiple processors. Such a problem arises for example in solving partial differential equations using an adaptive method that places extra grid points in certain subregions of the domain. A binary decomposition of the domain is used to partition it into rectangles requiring equal computational effort. The communication costs of mapping this partitioning onto different microprocessors: a mesh-connected array, a tree machine and a hypercube is then studied. The communication cost expressions can be used to determine the optimal depth of the above partitioning.

Berger, M. J.↗