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A refined semi‐analytic design sensitivity based on mode decomposition and Neumann series
Author(s) -
Cho Maenghyo,
Kim Hyungi
Publication year - 2004
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1163
Subject(s) - eigenvalues and eigenvectors , sensitivity (control systems) , mathematics , displacement (psychology) , series (stratigraphy) , finite difference , neumann series , iterative method , eigendecomposition of a matrix , mathematical analysis , mathematical optimization , engineering , physics , paleontology , electronic engineering , biology , psychology , quantum mechanics , psychotherapist
Among various sensitivity evaluation techniques, semi‐analytical method (SAM) is quite popular since this method is more advantageous than analytical method (AM) and global finite difference method (GFD). However, SAM reveals severe inaccuracy problem when relatively large rigid body motions are identified for individual elements. Such errors result from the pseudo load vector calculated by differentiation using the finite difference scheme. In the present study, an iterative refined semi‐analytical method (IRSAM) combined with mode decomposition technique is proposed to compute reliable semi‐analytical design sensitivities. The improvement of design sensitivities corresponding to the rigid body mode is evaluated by exact differentiation of the rigid body modes and the error of SAM caused by numerical difference scheme is alleviated by using a Von Neumann series approximation considering the higher order terms for the sensitivity derivatives. In eigenvalue problems, the tendency of eigenvalue sensitivity is similar to that of displacement sensitivity in static problems. Eigenvector is decomposed into rigid body mode and pure deformation mode. The present iterative SAM guarantees that the eigenvalue and eigenvector sensitivities converge to the reliable values for the wide range of perturbed size of the design variables. Accuracy and reliability of the shape design sensitivities in static problems and eigenvalue problems by the proposed method are assessed through the various numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.