Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “matrix decomposition”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Review of NASTRAN development relative to efficiency of execution

This paper reviews the development of NASTRAN relative to the efficiency of execution, with particular emphasis on those items which have changed significantly since the original release of NASTRAN. Features discussed include main and secondary storage utilization, matrix packing, matrix assembly, matrix multiplication, matrix decomposition and equation solution. Also a brief look into the future discusses the questions of faster arithmetic units and more effective storage utilization. In some cases the improvements in NASTRAN efficiency have resulted from taking advantage of hardware developments, while in other cases increased efficiency has resulted from improvements in the state of the art for data processing or matrix operations. The modular design of NASTRAN has made it possible to improve the efficiency in many parts of NASTRAN without changing the basic design of the program.

Mccormick, C. W.↗
NASA NTRS1978

Reduction of matrix wavefront for NASTRAN

The three grid point resequencing algorithms most often run by NASTRAN users are compared for their ability to reduce matrix root-mean-square (rms) wavefront, which is the most critical parameter in determining matrix decomposition time in NASTRAN. The three algorithms are Cuthill-McKee (CM), Gibbs-Poole-Stockmeyer (GPS), and Levy. The first two (CM and GPS) are in the BANDIT program, and the Levy algorithm is in WAVEFRONT. Results are presented for a diversified collection of 30 test problems ranging in size from 59 to 2680 nodes. It is concluded that GPS is exceptionally fast and, for the conditions under which the test was made, the algorithm best able to reduce rms wavefront consistently well.

Everstine, G. C.↗
NASA NTRS1981

Mathematical specifications of the Onboard Navigation Package (ONPAC) simulator (revision 1)

The mathematical theory of the computational algorithms employed in the onboard navigation package system is described. This system, which simulates an onboard navigation processor, was developed to aid in the design and evaluation of onboard navigation software. The mathematical formulations presented include the factorized UDU(T) form of the extended Kalman filter, the equations of motion of the user satellite, the user clock equations, the observation equations and their partial derivatives, the coodinate transformations, and the matrix decomposition algorithms.

Dunham, J. B.↗
NASA NTRS1983

BANDIT in NASTRAN

BANDIT has been implemented into the NASTRAN April '83 release. It is now a permanent feature in NASTRAN and will be included in all future releases for all four computing machines (IBM, CDC, UNIVAC, and VAX). Originally BANDIT operated as a preprocessor of NASTRAN. It read the NASTRAN input cards and produced a set of resequencing (SEOGP) cards that would greatly reduce the computational time required by the matrix decomposition module of NASTRAN for a large structure. In the past, many computer centers had installed BANDIT in their systems together with the NASTRAN program. The user would run the BANDIT program and NASTRAN as if they were one program (or two separated programs depending on how the two programs were actually tied together in the computer systems). In some cases, the user was required to pass the input cards and the output SEQGP cards between the two programs, and n others, the data was manipulated through the use of cataloged disc files. Although there is nothing wrong with this BANDIT-NASTRAN arrangement, there are, however, several shortcomings which are mentioned in this paper. The BANDIT in the NASTRAN April '83 release has removed all the deficiencies, and it comes in one version applicable to the four computing machines.

Chan, G. C.↗
NASA NTRS1988

Recursive least squares estimation and Kalman filtering by systolic arrays

One of the most promising new directions for high-throughput-rate problems is that based on systolic arrays. In this paper, using the matrix-decomposition approach, a systolic Kalman filter is formulated as a modified square-root information filter consisting of a whitening filter followed by a simple least-squares operation based on the systolic QR algorithm. By proper skewing of the input data, a fully pipelined time and measurement update systolic Kalman filter can be achieved with O(n squared) processing cells, resulting in a system throughput rate of O (n).

Chen, M. J.↗
NASA NTRS1990

An efficient design sensitivity analysis of eigenvectors

Subspace iteration has been a major advance in solving large eigen problems when only a subset of eigen-pairs is required. The essence of this method is a transformation from displacement coordinates of an n-th order eigensystem to generalized coordinates of a smaller q-th order. The eigenvalue problem is then solved in the reduced space. The method was first developed by Clint and Jennings for real symmetric systems and was then called 'simultaneous iteration'. The success of the method prompted further research along this line and there have been many improved algorithms developed. This approach has been widely used by structural engineers for extracting the most useful natural frequencies and mode shapes of large-scale dynamic systems. This paper exploits into a new direction which is in the form of iterative process for simultaneously calculating eigenvector derivatives of many eigenvectors with respect to multi-variables. The method fully uses all the available information from preceding eigenvalue solution and, thus, effectively economizes computational efforts. It iterates through two equations derived from the first variation of the two fundamental equations used in subspace iteration method. There is no expensive large matrix decomposition required and the process converges to acceptable solution in a finite number of iterations. Therefore, the procedure increases its efficiency superiority over the others as the system size or the number of interested eigenvectors become larger and larger.

Ting, T.↗
NASA NTRS2015

Representing Matrix Cracks Through Decomposition of the Deformation Gradient Tensor in Continuum Damage Mechanics Methods

A method is presented to represent the large-deformation kinematics of intraply matrix cracks and delaminations in continuum damage mechanics (CDM) constitutive material models. The method involves the additive decomposition of the deformation gradient tensor into 'crack' and 'bulk material' components. The response of the intact bulk material is represented by a reduced deformation gradient tensor, and the opening of an embedded cohesive interface is represented by a normalized cohesive displacement-jump vector. The rotation of the embedded interface is tracked as the material deforms and as the crack opens. The distribution of the total local deformation between the bulk material and the cohesive interface components is determined by minimizing the difference between the cohesive stress and the bulk material stress projected onto the cohesive interface. The improvements to the accuracy of CDM models that incorporate the presented method over existing approaches are demonstrated for a single element subjected to simple shear deformation and for a finite element model of a unidirectional open-hole tension specimen. The material model is implemented as a VUMAT user subroutine for the Abaqus/Explicit finite element software. The presented deformation gradient decomposition method reduces the artificial load transfer across matrix cracks subjected to large shearing deformations, and avoids the spurious secondary failure modes that often occur in analyses based on conventional progressive damage models.

Leone, Frank A., Jr.↗
NASA NTRS1976

On the computation and updating of the modified Cholesky decomposition of a covariance matrix

Methods for obtaining and updating the modified Cholesky decomposition (MCD) for the particular case of a covariance matrix when one is given only the original data are described. These methods are the standard method of forming the covariance matrix K then solving for the MCD, L and D (where K=LDLT); a method based on Householder reflections; and lastly, a method employing the composite-t algorithm. For many cases in the analysis of remotely sensed data, the composite-t method is the superior method despite the fact that it is the slowest one, since (1) the relative amount of time computing MCD's is often quite small, (2) the stability properties of it are the best of the three, and (3) it affords an efficient and numerically stable procedure for updating the MCD. The properties of these methods are discussed and FORTRAN programs implementing these algorithms are listed.

Vanrooy, D. L.↗
NASA NTRS1982

Assessing the quality of curvilinear coordinate meshes by decomposing the Jacobian matrix

An algebraic decomposition of the Jacobian matrix which relates physical and computational variables is presented. This invertible decomposition parameterizes the mesh by the physically intuitive qualities of cell orientation, cell orthogonality, cell volume, and cell aspect ratio. The decomposition can be used to analyze numerically generated curvilinear coordinate meshes and to assess the contribution of the mesh to the truncation error for any specific differential operator and algorithm. This is worked out in detail for Laplace's equation in nonconservative and conservative forms. The analysis is applied to the solution of the full potential code TAIR, showing grid plots, carpet plots, and truncation error for a NACA 0012 airfoil.

Kerlick, G. D.↗
NASA NTRS1975

Analysis of multiloop, multirate sampled-data systems

Based upon new identities between z-transforms at a basis rate, z-transforms at faster rates, and modified z-transforms, the equivalence between the frequency decomposition method and the switch decomposition method is precisely presented so that results of one method are easily obtained from results of the other. Next, a method is developed for determining the closed loop transfer function of multiloop, multirate sampled-data systems with noninteger ratio sampling rates. Previously, this process involved solving a complex system of equations with rational polynomial coefficients. Herein, this is avoided by introducing a systematic decomposition of matrix operators which naturally arise from the switch decomposition method. The matrix operators are simplified by introducing the shifted transforms of signals sampled at one of the faster rates.

Boykin, W. H.↗
NASA NTRS2009

Effects of Tantalum on the Temporal Evolution of a Model Ni-Al-Cr Superalloy During Phase Decomposition

The effects of a 2.0 at.% addition of Ta to a model Ni-10.0Al-8.5Cr (at.%) superalloy aged at 1073 K are assessed using scanning electron microscopy and atom-probe tomography. The gamma'(Ll2)-precipitate morphology that develops as a result of gamma-(fcc)matrix phase decomposition is found to evolve from a bimodal distribution of spheroidal precipitates, to {001}-faceted cuboids and parallelepipeds aligned along the elastically soft {001}-type directions. The phase compositions and the widths of the gamma'-precipitate/gamma-matrix heterophase interfaces evolve temporally as the Ni-Al-Cr-Ta alloy undergoes quasi-stationary state coarsening after 1 h of aging. Tantalum is observed to partition preferentially to the gamma'-precipitate phase, and suppresses the mobility of Ni in the gamma-matrix sufficiently to cause an accumulation of Ni on the gamma-matrix side of the gamma'/gamma interface. Additionally, computational modeling, employing Thermo-Calc, Dictra and PrecipiCalc, is employed to elucidate the kinetic pathways that lead to phase decomposition in this concentrated Ni-Al-Cr-Ta alloy.

Booth, Morrison, Christopher↗
NASA NTRS1986

Solving Large Systems of Normal Equations

SOLVE II program combines any number of sets of normal equations and obtains solution vector and related statistics. Normal equations of square, nonnegative definite matrix form. Program utilizes only upper symmetric portion of matrix. Program uses partitioned Cholesky decomposition method for matrix inversion to accommodate large parameter systems.

Putney, B.↗
NASA NTRS2015

Carbon Structure Hazard Control

Carbon composite structures are widely used in virtually all advanced technology industries for a multitude of applications. The high strength-to-weight ratio and resistance to aggressive service environments make them highly desirable. Automotive, aerospace, and petroleum industries extensively use, and will continue to use, this enabling technology. As a result of this broad range of use, field and test personnel are increasingly exposed to hazards associated with these structures. No single published document exists to address the hazards and make recommendations for the hazard controls required for the different exposure possibilities from damaged structures including airborne fibers, fly, and dust. The potential for personnel exposure varies depending on the application or manipulation of the structure. The effect of exposure to carbon hazards is not limited to personnel, protection of electronics and mechanical equipment must be considered as well. The various exposure opportunities defined in this document include pre-manufacturing fly and dust, the cured structure, manufacturing/machining, post-event cleanup, and post-event test and/or evaluation. Hazard control is defined as it is applicable or applied for the specific exposure opportunity. The carbon exposure hazard includes fly, dust, fiber (cured/uncured), and matrix vapor/thermal decomposition products. By using the recommendations in this document, a high level of confidence can be assured for the protection of personnel and equipment.

Yoder, Tommy↗
NASA NTRS1980

Research on the application of a decoupling algorithm for structure analysis

The mathematical theory for decoupling mth-order matrix differential equations is presented. It is shown that the decoupling precedure can be developed from the algebraic theory of matrix polynomials. The role of eigenprojectors and latent projectors in the decoupling process is discussed and the mathematical relationships between eigenvalues, eigenvectors, latent roots, and latent vectors are developed. It is shown that the eigenvectors of the companion form of a matrix contains the latent vectors as a subset. The spectral decomposition of a matrix and the application to differential equations is given.

Denman, E. D.↗
NASA NTRS1984

Approaches to optimization of SS/TDMA time slot assignment

Reduction techniques for traffic matrices are explored in some detail. These matrices arise in satellite switched time-division multiple access (SS/TDMA) techniques whereby switching of uplink and downlink beams is required to facilitate interconnectivity of beam zones. A traffic matrix is given to represent that traffic to be transmitted from n uplink beams to n downlink beams within a TDMA frame typically of 1 ms duration. The frame is divided into segments of time and during each segment a portion of the traffic is represented by a switching mode. This time slot assignment is characterized by a mode matrix in which there is not more than a single non-zero entry on each line (row or column) of the matrix. Investigation is confined to decomposition of an n x n traffic matrix by mode matrices with a requirement that the decomposition be 100 percent efficient or, equivalently, that the line(s) in the original traffic matrix whose sum is maximal (called critical line(s)) remain maximal as mode matrices are subtracted throughout the decomposition process. A method of decomposition of an n x n traffic matrix by mode matrices results in a number of steps that is bounded by n(2) - 2n + 2. It is shown that this upper bound exists for an n x n matrix wherein all the lines are maximal (called a quasi doubly stochastic (QDS) matrix) or for an n x n matrix that is completely arbitrary. That is, the fact that no method can exist with a lower upper bound is shown for both QDS and arbitrary matrices, in an elementary and straightforward manner.

Wade, T. O.↗
NASA NTRS1983

On obtaining the forward phase functions of Saturn ring features from radio occultation observations

It is noted that the near-forward scattering functions of particles in Saturn ring features are related to 3.6 cm radio occultation power spectra by a Fredholm integral equation of the first kind. The equation reduces to an algebraic system of equation whose solution by usual inversion techniques (that is, least mean squares) is ruled out by the near singularity of the forward transformation matrix. A combination of constrained linear inversion and a filtering algorithm based on eigenvector decomposition of the matrix reduces the instabilities; this yields derived phase functions valid over the range of zero to about 12 mrad. It is noted that these functions represent the collective forward diffraction lobe of particles greater than about 1 m in radius. Since multiple scattering of the signal is a significant effect, the measured phase functions must be adjusted to obtain the singly scattered component. This single-scattering correction is examined for two physical models, namely the monolayer and the classical discrete random slab; in addition, the fraction of opacity in submeter particles for each model for particular ring features is estimated.

Zebker, H. A.↗
NASA NTRS1990

Input/output system identification - Learning from repeated experiments

The paper describes three approaches and possible variations for the determination of the Markov parameters for forced response data using general inputs. It is shown that, when the parameters in the solution procedure are bootstrapped, the results can be obtained very efficiently, but the errors propagate throughout all parameters. By arranging the data in a different form and using singular value decomposition, the resulting identified parameters are more accurate, in the least number of successive experiments, at the expense of a large matrix singular value decomposition. When a recursive procedure is employed, the calculations can be performed very efficiently, but the number of repetitions of the experiments is much greater for a given accuracy than for any of the previous approaches. An alternative formulation is proposed to combine the advantages of each of the approaches.

Juang, Jer-Nan↗