Dynamic analysis of structures using lanczos co‐ordinates

Nour-Omid, B.; Clough, Ray W.

doi:10.1002/eqe.4290120410

Cited by 140 publications

(26 citation statements)

References 6 publications

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“…This algorithm has also been used for a wide range of other calculations, such as solving linear systems [20,25], computing the action of the matrix exponential on a vector, see, e.g. [23,27], and even solving differential equations using exponentially-fitted methods, e.g., [16,17,12]. In exact arithmetic, the algorithm can be recast as a simple three-term recurrence, namely,…”

Section: The Lanczos Algorithmmentioning

confidence: 99%

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Sidje

Saad

2010

Numer Algor

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We are interested in computing the Fermi-Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from Density Function Theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We descibe Lanczos methods and sparse direct method to address these systems. Each approach has advanatges and disadvatanges that are illustrated with experiments.

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Section: The Lanczos Algorithmmentioning

confidence: 99%

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Sidje

Saad

2010

Numer Algor

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“…scaling gives°"^T^1 /2¢ « = ¢¢/(4*; M¢;) It has been pointed out in many references (e.g., [21]) that the M-orthogonalization (eq. (2.27)) is theoretically required only with respect to the two previously computed vectors.…”

Section: Ritz-wilson-lanczos Methodsmentioning

confidence: 99%

Computational aspects of sensitivity calculations in linear transient structural analysis

Greene

Haftka

1991

Structural Optimization

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A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal of the study was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Two types of sensitivity calculation techniques were developed and evaluated.The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semianalytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. In several cases this fixed mode approach resulted in very poor approximations of the stress sensitivities. Almost all of the original modes were required for an accurate sensitivity and for small numbers of modes, the accuracy was extremely poor. To overcome this poor accuracy, two semi-analytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in accurate values of the stress sensitivities with a small number of modes. In both techniques the computational cost is much less than would result if the vibration modes were recalculated and then used in an overall finite difference method.

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“…Nour-Omid and Clough [3] proposed the Lanczos co-ordinates method that is also free from eigenanalysis. In this method, the equations of motion are transformed to reduced tridiagonal form through Lanczos vectors derived from the Lanczos algorithm and the responses of structures are expressed in Lanczos co-ordinates.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of structures using modified Lanczos co‐ordinates

Kim

Jung

Kim

et al. 2003

Earthq Engng Struct Dyn

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SUMMARYAn e cient dynamic analysis method using the modiÿed Lanczos co-ordinates is presented. The proposed method is obtained by applying the modiÿed Lanczos algorithm using Lanczos vectors that satisfy the sti ness-orthonormality condition to the conventional Lanczos co-ordinates method. The modiÿed Lanczos co-ordinates method is more e cient than the conventional method in the case of structures under multi-input loads. The e ectiveness of the modiÿed Lanczos co-ordinates method is veriÿed by analysing a numerical example.

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Dynamic analysis of structures using lanczos co‐ordinates

Cited by 140 publications

References 6 publications

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Rational approximation to the Fermi–Dirac function with applications in density functional theory

Computational aspects of sensitivity calculations in linear transient structural analysis

Dynamic analysis of structures using modified Lanczos co‐ordinates

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