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At least 235 records · Page 13

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NASA NTRS2001

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗

NASA NTRS1987

Robust eigensystem assignment for flexible structures

An improved method is developed for eigenvalues and eigenvectors placement of a closed-loop control system using either state or output feedback. The method basically consists of three steps. First, the singular value of QR decomposition is used to generate an orthonormal basis that spans admissible eigenvector space corresponding to each assigned eigenvalue. Secondly, given a unitary matrix, the eigenvector set which best approximates the given matrix in the least-square sense and still satisfy eigenvalue cosntraints is determined. Thirdly, a unitary matrix is sought to minimize the error between the unitary matrix and the assignable eigenvector matrix. For use as the desired eigenvector set, two matrices, namely, the open-loop eigenvector matrix and its closest unitary matrix are proposed. The latter matrix generally encourages both minimum conditioning and control gains. In addition, the algorithm is formulated in real arithmetic for efficient implementation. To illustrate the basic concepts, numerical examples are included.

Juang, Jer-Nan↗

NASA NTRS1991

An intrinsically n-dimensional generalized flux vector splitting implicit finite element Euler algorithm

A generalized flux-vector splitting implicit Galerkin finite-element algorithm for the Euler equations in curvilinear coordinates for ideal and reacting gases is derived. For an arbitrary equation of state, the curvilinear-coordinate flux vector is split in kinematic and kinetic components, and the associated jacobian matrix eigenvalues explicitly depend on the metric data. After directional semidiscretization, the terminal ordinary differential-equation system is solved via a nonlinearly stable implicit Runge-Kutta scheme in concert with an accurate tensor matrix product factorization. The results for selected two-dimensional supersonic and axisymmetric hypersonic flows validate the algorithm and verify its robustness for curvilinear-coordinate computations. The evolution towards a steady state is achieved for large Courant numbers without indication of numerical instabilities.

Iannelli, G. S.↗

NASA NTRS1984

Vibration and buckling of general periodic lattice structures

A method is presented for vibration and buckling analysis of arbitrary lattice structures having repetitive geometry in any combination of coordinate directions. The approach is based on exact member theory for representing the stiffness of an individual member subject to axial load, and in the case of vibration, undergoing harmonic oscillation. The method is an extension of previous work that was limited to specific geometries. The resulting eigenvalue problem is of the size associated with the repeating element of the structure. A computer program has been developed incorporating the theory and results are given for vibration of rectangular platforms and a large antenna structure having rotational symmetry. Buckling and vibration results for cable-stiffened rings are also given.

Anderson, M. S.↗

NASA NTRS1981

Vibration of prestressed periodic lattice structures

Equations are developed for vibration of general lattice structures that have repetitive geometry. The method of solution is an extension of a previous paper for buckling of similar structures. The theory is based on representing each member of the structure with the exact dynamic stiffness matrix and taking advantage of the repetitive geometry to obtain an eigenvalue problem involving the degrees-of-freedom at a single node in the lattice. Results are given for shell-and beam-like lattice structures and for rings stiffened with tension cables and a central mast. The variation of frequency with external loading and the effect of local member vibration on overall modes is shown.

Anderson, M. S.↗

NASA NTRS1985

Instability analysis procedure for 3-level multi-bearing rotor-foundation systems

A procedure for the instability analysis of a three-level multispan rotor systems is described. This procedure is based on a distributed mass elastic representation of the rotor system in several eight-coefficient bearings. Each bearing is supported from an elastic foundation on damped, elastic pedestals. The foundation is represented as a general distributed mass elastic structure on discrete supports, which may have different stiffness and damping properties in the horizontal and vertical directions. This system model is suited to studies of instability threshold conditions for multirotor turbomachines on either massive or flexible foundations. The instability conditions is found by obtaining the eigenvalues of the system determinant, which is obtained by the transfer matrix method from the three-level system model. The stability determinant is solved for the lowest rotational speed at which the system damping becomes zero in the complex eigenvalue, and for the whirl frequency corresponding to the natural frequency of the unstable mode. An efficient algorithm for achieving this is described. Application of this procedure to a rigid rotor in two damped-elastic bearings and flexible supports is described. A second example discusses a flexible rotor with four damped-elastic bearings. The third case compares the stability of a six-bearing 300 Mw turbine generator unit, using two different bearing types. These applications validate the computer program and various aspects of the analysis.

Zhou, S.↗

NASA NTRS1986

Methods for the solution of radiative transfer equation

To obtain an exact solution of the radiative-transfer equation in media where both absorption and scattering are significant, the usual approach is to use a numerical method. Three methods are known in the literature: invariant imbedding, eigenvalue-eigenfunction, and matrix doubling. This paper examines the practical application of these methods to the problem of emission from an inhomogeneous (Rayleigh) layer, the effects of layer parameters on the stability. It is found that invariant imbedding is most suitable for computing emission from an inhomogeneous layer with a temperature profile but tends to be unstable as the optical thickness of the layer increases beyond 0.5. On the other hand, the matrix-doubling method is stable for arbitrary optical thickness but is not suitable for handling multilayers. The eigenvalue-eigenfunction method is more stable than the invariant imbedding as optical thickness increases up to 2.0. It also permits temperature profile in the layer, but the computation is much more complicated. It is less stable than the matrix-doubling method when optical thickness is larger than 2.0. In general, the choice of a method is dependent on the nature of the problem.

Chen, M. F.↗

NASA NTRS1988

More on exact state reconstruction in deterministic digital control systems

Presented is a special form of the Ideal State Reconstructor for deterministic digital control systems which is simpler to implement than the most general form. The Ideal State Reconstructor is so named because, if the plant parameters are known exactly, its output will exactly equal, not just approximate, the true state of the plant and accomplish this without any knowledge of the plant's initial state. Besides this, it adds no new states or eigenvalues to the system. Nor does it affect the plant equation for the system in any way; it affects the measurement equation only. It is characterized by the fact that discrete measurements are generated every T/N seconds and input into a multi-input/multi-output moving-average (MA) process. The output of this process is sampled every T seconds and utilized in reconstructing the state of the system.

Polites, Michael E.↗

NASA NTRS1984

A generalized multiple-input, multiple-output modal parameter estimation algorithm

A new method for experimental determination of the modal parameters of a structure is presented. The method allows for multiple input forces to be applied simultaneously, and for an arbitrary number of acceleration response measurements to be employed. These data are used to form the equations of motion for a damped linear elastic structure. The modal parameters are then obtained through an eigenvalue technique. In conjunction with the development of the equations, an extensive computer simulation study was performed. The results of the study show a marked improvement in the mode shape identification for closely-spaced modes as the number of applied forces is increased. Also demonstrated is the influence of noise on the method's ability to identify accurate modal parameters. Here again, an increase in the number of exciters leads to a significant improvement in the identified parameters.

Craig, R. R., Jr.↗

NASA NTRS1986

Wave-packet evolution and quantization

A general relationship is established between wave-packet evolution and quantization in classically integrable systems. Because of wave-packet spreading, one cannot simply take the Fourier transform of the time evolution of a wave-packet. Instead, one must propagate the wave-packet using the actions as Hamiltonians. The energy eigenvalues which result are the Einstein-Brillouin-Keller values, and new forms for the eigenfunctions appear. These are free of caustic singularities, and represent averages of wave packets over the invariant torus.

Littlejohn, R. G.↗

NASA NTRS1990

Partitioning sparse matrices with eigenvectors of graphs

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorithms for computing separators. Finally, the time required to compute the Laplacian eigenvector is reported, and the accuracy with which the eigenvector must be computed to obtain good separators is considered. The spectral algorithm has the advantage that it can be implemented on a medium-size multiprocessor in a straightforward manner.

Pothen, Alex↗

NASA NTRS1975

Modal coupling procedures adapted to NASTRAN analysis of the 1/8-scale shuttle structural dynamics model. Volume 1: Technical report

A dynamic substructuring analysis, utilizing the component modes technique, of the 1/8 scale space shuttle orbiter finite element model is presented. The analysis was accomplished in 3 phases, using NASTRAN RIGID FORMAT 3, with appropriate Alters, on the IBM 360-370. The orbiter was divided into 5 substructures, each of which was reduced to interface degrees of freedom and generalized normal modes. The reduced substructures were coupled to yield the first 23 symmetric free-free orbiter modes, and the eigenvectors in the original grid point degree of freedom lineup were recovered. A comparison was made with an analysis which was performed with the same model using the direct coordinate elimination approach. Eigenvalues were extracted using the inverse power method.

Zalesak, J.↗

NASA NTRS1981

A new payload integration method

This paper introduces a new payload integration method. The system equations of motion are derived in terms of interface-restrained booster and payload modes. These system equations are directly integrated, avoiding the solution of an expensive eigenvalue problem. A modified Newmark-Chan-Beta numerical scheme is used to perform this integration. Due to the unique form of the system equations it is possible to achieve significant savings. A similar savings is feasible for the computation of the internal loads. This new approach does not involve approximations. Although completely general, the presented technique is most effective when the number of interface degrees of freedom is relatively small. Both determinate and indeterminate interfaces are allowed.

Engels, R. C.↗

NASA NTRS2016

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances.

Kennedy, Christopher A.↗

NASA NTRS1991

Modeling the spatiotemporal organization of velocity storage in the vestibuloocular reflex by optokinetic studies

1. A generalized three-dimensional state space model of visual vestibular interaction was developed. Matrix and dynamical system operators associated with inputs from the semicircular canals, otolith velocity estimator, and the visual system have been incorporated into the model, which focus on their relationship to the velocity storage integrator. 2. A relationship was postulated between the eigenvalues and the direction of the eigenvectors of the system matrix and the orientation of the spatial vertical. It was assumed that the system matrix for a tilted position was a composition of two linear transformations of the system matrix for the upright position. One transformation modifies the eigenvalues of the system matrix, whereas another rotates the eigenvectors of the system matrix. The pitch and roll eigenvectors rotate with the head, whereas the yaw axis eigenvector remains approximately spatially invariant. 3. Based on the three-dimensional model, a computational procedure was formulated to identify the eigenvalues and eigenvectors of the system matrix with the use of a modification of the marquardt algorithm. With the use of data obtained from a monkey, it was shown that the three-dimensional behavior of velocity storage cannot be predicted solely in terms of its time constants, i.e., the inverse of its eigenvalues. With the use of the same eigenvalues the data could either be fit or not fit, depending on the eigenvector directions. Therefore, it is necessary to specify eigenvector directions when characterizing velocity storage in three dimensions. 4. Parameters found with the use of the Marquardt algorithm were incorporated into the model. Diagonal matrices in a head coordinate frame were introduced for coupling the visual system to the integrator and to the direct optokinetic pathway. Simulations of optokinetic nystagmus (OKN) and optokinetic after-nystagmus (OKAN) were run. The model predicted the behavior of yaw and pitch OKN and OKAN when the animal is upright. It also predicted the cross-coupling in the side down position. The trajectories in velocity space were also accurately simulated. 5. One of the predictions of the model is that when the stimulus direction is along an eigenvector, the trajectory in velocity space is a straight line. Using the "spectral width" of the residuals from a straight line sequence during OKAN, we developed a methodology to estimate how close the OKAN decay was to an eigenvector trajectory. 6. Thus we have developed a model-based approach for studying and interpreting the response characteristics of velocity storage in three dimensions.(ABSTRACT TRUNCATED AT 400 WORDS).

Non-NASA Center↗

NASA NTRS1973

Evaluation of control laws and actuator locations for control systems applicable to deformable astronomical telescope mirrors

Some of the major difficulties associated with large orbiting astronomical telescopes are the cost of manufacturing the primary mirror to precise tolerances and the maintaining of diffraction-limited tolerances while in orbit. One successfully demonstrated approach for minimizing these problem areas is the technique of actively deforming the primary mirror by applying discrete forces to the rear of the mirror. A modal control technique, as applied to active optics, has previously been developed and analyzed. The modal control technique represents the plant to be controlled in terms of its eigenvalues and eigenfunctions which are estimated via numerical approximation techniques. The report includes an extension of previous work using the modal control technique and also describes an optimal feedback controller. The equations for both control laws are developed in state-space differential form and include such considerations as stability, controllability, and observability. These equations are general and allow the incorporation of various mode-analyzer designs; two design approaches are presented. The report also includes a technique for placing actuator and sensor locations at points on the mirror based upon the flexibility matrix of the uncontrolled or unobserved modes of the structure. The locations selected by this technique are used in the computer runs which are described. The results are based upon three different initial error distributions, two mode-analyzer designs, and both the modal and optimal control laws.

Ostroff, A. J.↗

NASA NTRS1971

Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Rosenbaum, J. S.↗

NASA NTRS1975

Development and applications of two computational procedures for determining the vibration modes of structural systems

Two computational procedures for analyzing complex structural systems for their natural modes and frequencies of vibration are presented. Both procedures are based on a substructures methodology and both employ the finite-element stiffness method to model the constituent substructures. The first procedure is a direct method based on solving the eigenvalue problem associated with a finite-element representation of the complete structure. The second procedure is a component-mode synthesis scheme in which the vibration modes of the complete structure are synthesized from modes of substructures into which the structure is divided. The analytical basis of the methods contains a combination of features which enhance the generality of the procedures. The computational procedures exhibit a unique utilitarian character with respect to the versatility, computational convenience, and ease of computer implementation. The computational procedures were implemented in two special-purpose computer programs. The results of the application of these programs to several structural configurations are shown and comparisons are made with experiment.

Kvaternik, R. G.↗