Accuracy of semi-analytical sensitivities and its improvement by the “natural method”

Mlejnek, Hans-Peter

doi:10.1007/bf01759928

Cited by 34 publications

(9 citation statements)

References 4 publications

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“…This was also shown by Mlejnek (1992). This is the geometrical-physical reason why the traditional SA method becomes inaccurate for problems where the displacement field is characterized by rigid body rotations which are large relative to actual deformations of the finite eL ements, i.e.…”

Section: Free Lransverse Vibralion Frequencies Of Clamped Square Plalementioning

confidence: 79%

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

Lund

Olhoff

1994

Structural Optimization

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As has been shown in recent years, the approximate numerical differentiation of element stiffness matrices which is inherent in the semi-analytical method of finite element based design sensitivity analysis, may give rise to severely erroneous shape design sensitivities in static problems involving linearly elastic bending of beam, plate and shell structures.This paper demonstrates that the error problem also manifests itself in semi-analytical sensitivity analyses of eigenvalues of such structures and presents a method for complete elimination of the error problem. The method, which yields "exact" numerical sensitivities on the basis of simple first-order numerical differentiation, is computationally inexpensive and easy to implement as an integral part of the finite element analysis.The method is presented in terms of semi-analytical shape design sensitivity analysis of eigenvalues in the form of frequencies of free transverse vibrations of plates modelled by isoparametric Mindlin finite elements. Finally, the development is illustrated via two examples of occurrence of the error phenomenon when the traditional method is used and it is shown that the problem is completely eliminated by the application of the new method.

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Section: Free Lransverse Vibralion Frequencies Of Clamped Square Plalementioning

confidence: 79%

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

Lund

Olhoff

1994

Structural Optimization

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“…(17) respectively. These equations can be simpliÿed using (9) and making use of the consistency Equation (2). This yieldsK…”

Section: Second-order Design Sensitivitiesmentioning

confidence: 99%

“…Popular techniques are the higher order or alternative ÿnite di erence schemes, for example References [6][7][8]. The method presented in Reference [9] is based on the 'natural approach' and consistency conditions for rigid body modes (RBM). An exact formulation of the SA variant is given in References [10; 11].…”

Section: Introductionmentioning

confidence: 99%

Refined second order semi‐analytical design sensitivities

Boer

Keulen

Vervenne

2002

Numerical Meth Engineering

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SUMMARYAccurate and e cient calculation of second order design sensitivities in a ÿnite element context is often di cult. The semi-analytical (SA) method is e cient and easy to implement but has accuracy problems even for ÿrst order shape design sensitivities. To overcome accuracy problems a reÿned semianalytical (RSA) method has been developed for ÿrst order sensitivities. The present paper investigates the application of the RSA method to second order design sensitivities. It is found that second order RSA sensitivities are signiÿcantly more accurate than their SA counterparts.

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“…Through this test, they found that when relatively large rigid body motions are identified for individual elements, SAM reveals unreliable accuracy because the rigid body mode is strongly connected with the truncation error. Mlejnek used to natural modes, which basically is a way of decomposing the displacement field to calculate the derivatives of the numerical stiffness matrix [10]. Barthelemy et al, concluded that these errors are due to the numerical differentiation of the finite element stiffness matrices [11].…”

Section: Introductionmentioning

confidence: 98%

A refined semi-analytic design sensitivity based on mode decomposition and Neumann series

Cho

Kim

2004

Int. J. Numer. Meth. Engng

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SUMMARYAmong 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 semianalytical 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.

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Accuracy of semi-analytical sensitivities and its improvement by the “natural method”

Cited by 34 publications

References 4 publications

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

Refined second order semi‐analytical design sensitivities

A refined semi-analytic design sensitivity based on mode decomposition and Neumann series

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