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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.

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NASA NTRS1990

A Shifted Block Lanczos Algorithm 1: The Block Recurrence

In this paper we describe a block Lanczos algorithm that is used as the key building block of a software package for the extraction of eigenvalues and eigenvectors of large sparse symmetric generalized eigenproblems. The software package comprises: a version of the block Lanczos algorithm specialized for spectrally transformed eigenproblems; an adaptive strategy for choosing shifts, and efficient codes for factoring large sparse symmetric indefinite matrices. This paper describes the algorithmic details of our block Lanczos recurrence. This uses a novel combination of block generalizations of several features that have only been investigated independently in the past. In particular new forms of partial reorthogonalization, selective reorthogonalization and local reorthogonalization are used, as is a new algorithm for obtaining the M-orthogonal factorization of a matrix. The heuristic shifting strategy, the integration with sparse linear equation solvers and numerical experience with the code are described in a companion paper.

Grimes, Roger G.↗
NASA NTRS2012

A New and General Formulation of the Parametric HFGMC Micromechanical Method for Three-Dimensional Multi-Phase Composites

The recent two-dimensional (2-D) parametric formulation of the high fidelity generalized method of cells (HFGMC) reported by the authors is generalized for the micromechanical analysis of three-dimensional (3-D) multiphase composites with periodic microstructure. Arbitrary hexahedral subcell geometry is developed to discretize a triply periodic repeating unit-cell (RUC). Linear parametric-geometric mapping is employed to transform the arbitrary hexahedral subcell shapes from the physical space to an auxiliary orthogonal shape, where a complete quadratic displacement expansion is performed. Previously in the 2-D case, additional three equations are needed in the form of average moments of equilibrium as a result of the inclusion of the bilinear terms. However, the present 3-D parametric HFGMC formulation eliminates the need for such additional equations. This is achieved by expressing the coefficients of the full quadratic polynomial expansion of the subcell in terms of the side or face average-displacement vectors. The 2-D parametric and orthogonal HFGMC are special cases of the present 3-D formulation. The continuity of displacements and tractions, as well as the equilibrium equations, are imposed in the average (integral) sense as in the original HFGMC formulation. Each of the six sides (faces) of a subcell has an independent average displacement micro-variable vector which forms an energy-conjugate pair with the transformed average-traction vector. This allows generating symmetric stiffness matrices along with internal resisting vectors for the subcells which enhances the computational efficiency. The established new parametric 3-D HFGMC equations are formulated and solution implementations are addressed. Several applications for triply periodic 3-D composites are presented to demonstrate the general capability and varsity of the present parametric HFGMC method for refined micromechanical analysis by generating the spatial distributions of local stress fields. These applications include triply periodic composites with inclusions in the form of a cavity, spherical inclusion, ellipsoidal inclusion, discontinuous aligned short fiber. A 3-D repeating unit-cell for foam material composite is simulated.

Haj-Ali, Rami↗
NASA NTRS1989

Communication requirements of sparse Cholesky factorization with nested dissection ordering

Load distribution schemes for minimizing the communication requirements of the Cholesky factorization of dense and sparse, symmetric, positive definite matrices on multiprocessor systems are presented. The total data traffic in factoring an n x n sparse symmetric positive definite matrix representing an n-vertex regular two-dimensional grid graph using n exp alpha, alpha not greater than 1, processors are shown to be O(n exp 1 + alpha/2). It is O(n), when n exp alpha, alpha not smaller than 1, processors are used. Under the conditions of uniform load distribution, these results are shown to be asymptotically optimal.

Naik, Vijay K.↗
NASA NTRS1988

Data traffic reduction schemes for sparse Cholesky factorizations

Load distribution schemes are presented which minimize the total data traffic in the Cholesky factorization of dense and sparse, symmetric, positive definite matrices on multiprocessor systems with local and shared memory. The total data traffic in factoring an n x n sparse, symmetric, positive definite matrix representing an n-vertex regular 2-D grid graph using n (sup alpha), alpha is equal to or less than 1, processors are shown to be O(n(sup 1 + alpha/2)). It is O(n(sup 3/2)), when n (sup alpha), alpha is equal to or greater than 1, processors are used. Under the conditions of uniform load distribution, these results are shown to be asymptotically optimal. The schemes allow efficient use of up to O(n) processors before the total data traffic reaches the maximum value of O(n(sup 3/2)). The partitioning employed within the scheme, allows a better utilization of the data accessed from shared memory than those of previously published methods.

Naik, Vijay K.↗
NASA NTRS1978

A split band-Cholesky equation solving strategy for finite element analysis of transient field problems

The paper describes the split-Cholesky strategy for banded matrices arising from the large systems of equations in certain fluid mechanics problems. The basic idea is that for a banded matrix the computation can be carried out in pieces, with only a small portion of the matrix residing in core. Mesh considerations are discussed by demonstrating the manner in which the assembly of finite element equations proceeds for linear trial functions on a triangular mesh. The FORTRAN code which implements the out-of-core decomposition strategy for banded symmetric positive definite matrices (mass matrices) of a coupled initial value problem is given.

Cooke, C. H.↗
NASA NTRS1977

NASTRAN hydroelastic modal studies. Volume 3: NASTRAN 3-D hydroelastic analysis and MESHGEN user's manuals

The method used to formulate the fluid/structure equations is described in the updates to the NASTRAN Theoretical Manual. The basis for defining the fluid are three dimensional finite elements connected to fluid grid points defining the Eulerian pressure at a point fixed in space. The use of a pressure single degree of freedom at each point rather than three displacements allows a finer mesh of elements with a reasonable matrix order. In the formulation of the fluid/structure system the interior fluid degrees of freedom are transformed and removed from the solution matrices. The eigenvalues of the combination are extracted from small, fully dense, symmetric mass and stiffness matrices, efficiently processed with the Givens method. The solution matrices are defined by the free surface displacements and the reduced structure coordinates.

Source record↗
NASA NTRS1983

Finite element analysis of fluid-filled elastic piping systems

Two finite element procedures are described for predicting the dynamic response of general 3-D fluid-filled elastic piping systems. The first approach, a low frequency procedure, models each straight pipe or elbow as a sequence of beams. The contained fluid is modeled as a separate coincident sequence axial members (rods) which are tied to the pipe in the lateral direction. The model includes the pipe hoop strain correction to the fluid sound speed and the flexibility factor correction to the elbow flexibility. The second modeling approach, an intermediate frequency procedure, follows generally the original Zienkiewicz-Newton scheme for coupled fluid-structure problems except that the velocity potential is used as the fundamental fluid unknown to symmetrize the coefficient matrices. From comparisons of the beam model predictions to both experimental data and the 3-D model, the beam model is validated for frequencies up to about two-thirds of the lowest fluid-filled labor pipe mode. Accurate elbow flexibility factors are seen to be crucial for effective beam modeling of piping systems.

Everstine, G. C.↗
NASA NTRS1982

A minimal residual method for transonic potential flows

For transonic flow calculations, a combination of the successive line over-relaxation (SLOR) and the preconditioned conjugate gradient (CG) method has been suggested by Wong and Hafez (1981). This paper studies the method of minimal residual (MR) which avoids a combined iteration. This method is closely related to the CG method, may be regarded as a first-order gradient method, and is applicable to symmetric and nonsymmetric matrices. The problem is formulated as a nonlinear mixed elliptic-hyperbolic partial differential equation which includes an artificial viscosity and a switching function which is zero in subsonic regions and nonzero in supersonic regions. Alternatives to the SLOR method which provide faster convergence rates are introduced. The preconditioned MR algorithm is developed, and transonic potential flows around NACA 0012 airfoil are calculated for different Mach numbers and angles of attack. Preliminary results are presented, demonstrating that the MR algorithm requires no parameter estimation and rapidly converges for subsonic flows.

Wong, Y. S.↗
NASA NTRS1985

Some iterative schemes for transonic potential flows

The minimal residual (MR) method for the numerical solution of transonic potential flows is closely related to the conjugate gradient method, which has found widespread use in the solution of large sparse, symmetric, and positive-definite linear equations. The primary advantage of the MR method is its applicability to both symmetric and nonsymmetric matrices.

Wong, Y. S.↗
NASA NTRS1989

Data traffic reduction schemes for Cholesky factorization on asynchronous multiprocessor systems

Communication requirements of Cholesky factorization of dense and sparse symmetric, positive definite matrices are analyzed. The communication requirement is characterized by the data traffic generated on multiprocessor systems with local and shared memory. Lower bound proofs are given to show that when the load is uniformly distributed the data traffic associated with factoring an n x n dense matrix using n to the alpha power (alpha less than or equal 2) processors is omega(n to the 2 + alpha/2 power). For n x n sparse matrices representing a square root of n x square root of n regular grid graph the data traffic is shown to be omega(n to the 1 + alpha/2 power), alpha less than or equal 1. Partitioning schemes that are variations of block assignment scheme are described and it is shown that the data traffic generated by these schemes are asymptotically optimal. The schemes allow efficient use of up to O(n to the 2nd power) processors in the dense case and up to O(n) processors in the sparse case before the total data traffic reaches the maximum value of O(n to the 3rd power) and O(n to the 3/2 power), respectively. It is shown that the block based partitioning schemes allow a better utilization of the data accessed from shared memory and thus reduce the data traffic than those based on column-wise wrap around assignment schemes.

Naik, Vijay K.↗
NASA NTRS1991

Data volume reduction for single-look polarimetric imaging radar data

An 8-b quantization scheme to reduce the data volume for single-look complex scattering matrix data measured by polarimetric imaging radar systems is described. The scattering matrices are not symmetrized before compression, thereby retaining information about background noise and system effects. The data volume is reduced by a factor of 3.2. It is shown, with measured data, that the signal to quantization noise ratio for the compression scheme is more than 35 dB for the cross-polarized channels, and more than 45 dB for the copolarized channels.

Van Zyl, Jakob J.↗
NASA NTRS1999

An Object-Oriented Collection of Minimum Degree Algorithms: Design, Implementation, and Experiences

The multiple minimum degree (MMD) algorithm and its variants have enjoyed 20+ years of research and progress in generating fill-reducing orderings for sparse, symmetric positive definite matrices. Although conceptually simple, efficient implementations of these algorithms are deceptively complex and highly specialized. In this case study, we present an object-oriented library that implements several recent minimum degree-like algorithms. We discuss how object-oriented design forces us to decompose these algorithms in a different manner than earlier codes and demonstrate how this impacts the flexibility and efficiency of our C++ implementation. We compare the performance of our code against other implementations in C or Fortran.

Kumfert, Gary↗
NASA NTRS1977

On the representation matrices of the spin permutation group

A method is presented for the determination of the representation matrices of the spin permutation group (symmetric group), a detailed knowledge of these matrices being required in the study of the electronic structure of atoms and molecules. The method is characterized by the use of two different coupling schemes. Unlike the Yamanouchi spin algebraic scheme, the method is not recursive. The matrices for the fundamental transpositions can be written down directly in one of the two bases. The method results in a computationally significant reduction in the number of matrix elements that have to be stored when compared with, say, the standard Young tableaux group theoretical approach.

Wilson, S.↗
NASA NTRS2000

Model Refinement Using Eigensystem Assignment

IA novel approach for the refinement of finite-element-based analytical models of flexible structures is presented. The proposed approach models the possible refinements in the mass, damping, and stiffness matrices of the finite element model in the form of a constant gain feedback with acceleration, velocity, and displacement measurements, respectively. Once the free elements of the structural matrices have been defined, the problem of model refinement reduces to obtaining position, velocity, and acceleration gain matrices with appropriate sparsity that reassign a desired subset of the eigenvalues of the model, along with partial mode shapes, from their baseline values to those obtained from system identification test data. A sequential procedure is used to assign one conjugate pair of eigenvalues at each step using symmetric output feedback gain matrices, and the eigenvectors are partially assigned, while ensuring that the eigenvalues assigned in the previous steps are not disturbed. The procedure can also impose that gain matrices be dissipative to guarantee the stability of the refined model. A numerical example, involving finite element model refinement for a structural testbed at NASA Langley Research Center (Controls-Structures-Interaction Evolutionary model) is presented to demonstrate the feasibility of the proposed approach.

Maghami, Peiman G.↗
NASA NTRS1998

Model Refinement Using Eigensystem Assignment

This paper presented a novel approach for the refinement of finite-element-based analytical models of flexible structures is presented. The proposed approach models the possible refinements in the mass, damping, and stiffness matrices of the finite element model in the form of a constant gain feedback with acceleration, velocity, and displacement measurements, respectively. Once, the free elements of the structural matrices have been defined, the problem of model refinement reduces to obtaining position, velocity, and acceleration gain matrices, which reassign a desired subset of the eigenvalues of the model, along with partial mode shapes, from their baseline values to those obtained from system identification test data. A sequential procedure is used to assign one self-conjugate pair of closed-loop eigenvalues at each step using symmetric output feedback gain matrices, and the closed-loop eigenvectors are partially assigned, while ensuring that the eigenvalues assigned in the previous steps are not disturbed. The procedure can also impose that gain matrices be dissipative in order to guarantee the stability of the refined model. A numerical example, involving finite element model refinement for a structural testbed at NASA Langley (CSI Evolutionary Model) is presented to demonstrate the feasibility of the proposed approach.

Maghami, Pieman G.↗
NASA NTRS1970

Sorting and ordering sparse linear systems

Transformations of sparse linear systems by row-column permutations are considered and various algorithms are constructed to transform arbitrary symmetric positive definite sparse matrices, as well as matrices in band form, doubly bordered band form, and doubly bordered block diagonal form.

Tewarson, R. P.↗
NASA NTRS1973

Free vibration analysis of spinning structural systems.

This article presents an efficient digital computer procedure, along with the complete listing of the associated computer program, which may be conveniently utilized for the accurate solution of a wide range of practical eigenvalue problems. Important applications of the present work are envisaged in the natural frequency analysis of spinning structures discretized by the finite element technique, and in the determination of transfer functions associated with the dynamic blocks of control systems of spacecraft utilizing gas jets or reaction wheels for attitude control, as well as of spin-stabilized and dual-spin-stabilized satellites. The validity of the Sturm sequence property is first established for the related matrix formulation involving Hermitian and real symmetric, positive-definite matrices, both being usually of highly banded configuration. A numerically stable algorithm based on the Sturm sequence method is then developed which fully exploits the banded form of the associated matrices.

Gupta, K. K.↗
NASA NTRS1993

A Spectral Algorithm for Envelope Reduction of Sparse Matrices

The problem of reordering a sparse symmetric matrix to reduce its envelope size is considered. A new spectral algorithm for computing an envelope-reducing reordering is obtained by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. This Laplacian eigenvector solves a continuous relaxation of a discrete problem related to envelope minimization called the minimum 2-sum problem. The permutation vector computed by the spectral algorithm is a closest permutation vector to the specified Laplacian eigenvector. Numerical results show that the new reordering algorithm usually computes smaller envelope sizes than those obtained from the current standard algorithms such as Gibbs-Poole-Stockmeyer (GPS) or SPARSPAK reverse Cuthill-McKee (RCM), in some cases reducing the envelope by more than a factor of two.

Barnard, Stephen T.↗