Applying wavelets to improve the boundary element method for elasticity problems

Ebrahimnejad, Latif; Attarnejad, Reza; Ebrahimnejad, Hassan

doi:10.1016/j.enganabound.2009.10.018

Cited by 5 publications

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References 17 publications

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“…The fast wavelet transform (FWT) algorithm [14,15] takes a vector a and produces another vector a W of the same order. The FWT can be represented by a square matrix W so that a W = Wa (8) where W is a sparse matrix whose elements are dependent on the selected wavelet transform. Since Daubechie's orthogonal wavelets are used:…”

Section: Applying Fwt To Compress Bem Matricesmentioning

confidence: 99%

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A novel approach to solve linear systems arising from BEM using fast wavelet transforms

Ebrahimnejad

Attarnejad

2009

Math. Meth. Appl. Sci.

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A novel approach to solve linear systems arising from BEM using fast wavelet transforms Latif Ebrahimnejad * † and Reza Attarnejad Communicated by V. Volpert This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi-banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS-form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.Keywords: boundary element method; wavelet transform; MRA; GMRES solver; wavelet solver IntroductionBoundary element method (BEM) can be considered as the eldest mesh reduction method. BEM is an efficient boundary type method, where only the boundary of domain is required to be discretized. Then the dimension of problem is reduced by one in comparison with FE and FD [1,2]. Although total degrees of freedom for BE is much less than FE and FD, because of arising dense and fully populated matrices instead of sparse and banded matrices in FE and FD, the efficiency of BEM for large problems is decreased considerably.Wavelets were first used by Beylkin, Coifman and Rokhlin (BCR) [3] to change the dense and fully populated matrices of BEM. Then wavelet compression was applied to BEM for Laplace equation by . In the earlier works, authors have developed the method to elasticity problems [7, 8]. Permutation technique introduced by Chen [9] changes finger-like matrices of transformed space to semi-banded matrices.Applying wavelet compression to BEM can be implemented as a black box in existing BEM codes. While standard BEM uses Gaussian elimination with O(N 3 ) operations to solve algebraic linear equations, where N is total number of equations, iterative solvers such as Lanczos, CG and GMRES need O(N 2 ) operations per iteration [10--13].Wavelet solver can reduce complexity order to O((N Log N) ), which is impressively motivating. In this paper Daubechies orthogonal wavelets are used to transform original system matrix to a semi-banded sparse one, which then is solved by using wavelet solver. This solver solves linear algebraic systems using NS-form of transformed matrix, hierarchical nature of multiresolution analysis (MRA) and GMRES algorithm.In what follows, required equations used in BEM for elastic...

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Section: Applying Fwt To Compress Bem Matricesmentioning

confidence: 99%

“…Then wavelet compression was applied to BEM for Laplace equation by . In the earlier works, authors have developed the method to elasticity problems [7,8]. Permutation technique introduced by Chen [9] changes finger-like matrices of transformed space to semi-banded matrices.…”

Section: Introductionmentioning

confidence: 99%

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

Ebrahimnejad

Attarnejad

2009

Math. Meth. Appl. Sci.

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“…In [19], the procedure of finiteelement discretization was applied to study stress concentration in a finite plate with circular or elliptic holes. Papers [13,17] are devoted to the improvement of the boundary-element method in its application to threedimensional problems of the theory of elasticity.By using the Lur'e method of homogeneous solutions, it is possible to construct a set of partial solutions for a layer (cylinder) under any conditions imposed on its base but, in the case of mixed-type boundary conditions (sliding fastening of the end faces or the case where the bases of the layer are covered with a diaphragm rigid in its own plane and flexible in the perpendicular direction), the boundary-value problem is somewhat simplified. Problems of this kind are also called periodic in one of the coordinates [5,7,12,15,16].…”

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confidence: 99%

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Tension of a Thick-Walled Shell of Finite Length Whose End Faces are Covered with A Diaphragm

Kovalev

Shramko

2015

J Math Sci

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We consider a symmetric problem of the theory of elasticity for the stressed state of a thick-walled shell whose end faces are covered with a diaphragm. The boundary-value problem is reduced to an infinite system of singular integral equations of the second kind. The expressions for stresses characterizing the stressed state of this shell are deduced. Based on the developed analytic algorithm, we performed numerical experiments whose results are represented in the graphical form and contain new quantitative and qualitative data about the stressed state of a thick-walled shell depending on its geometric parameters and Poisson's ratios of its materials.The problem of stress concentration is quite urgent for the contemporary machine building because it determines the reliability and durability of designed structures or separate structural elements. Stress concentrators either appear in structures due to the imperfect quality of materials used for their manufacturing (cavities, cracks, foreign inclusions) or may be caused by the technological or design needs (holes, notches, etc.). In both cases, it is important to study the influence either of a single stress concentrator on the stressed state of designed products or of several stress concentrators with regard for their mutual influence. The necessity of getting more accurate descriptions of the stressed state of machine components near stress concentrators forces the researchers to try to describe the stressed state of structural elements in the three-dimensional statement [4,[8][9][10][11][12].Among the efficient methods aimed at the solution of three-dimensional problems for a layer (cylinder), we should especially mention the Lur'e method of homogeneous solutions [9]. The problems of stressed state for a layer weakened by various stress concentrators were investigated with the help of this method in [2,8]. Another efficient method for the analysis of three-dimensional problems posed for layers is the method of vector eigenfunctions. In [4], this method was used to study the Kirsch problem. For a cylinder with loaded lateral surface, the problem was solved by the method of eigenfunctions in [11]. An approach based on equations of three-dimensional theory of elasticity was proposed in [3] for the solution of the problem of stressed states of thick-walled orthotropic cylinders. The boundary-and finite-element methods are alternative methods used for the solution of three-dimensional problems of the theory of elasticity. In [19], the procedure of finiteelement discretization was applied to study stress concentration in a finite plate with circular or elliptic holes. Papers [13,17] are devoted to the improvement of the boundary-element method in its application to threedimensional problems of the theory of elasticity.By using the Lur'e method of homogeneous solutions, it is possible to construct a set of partial solutions for a layer (cylinder) under any conditions imposed on its base but, in the case of mixed-type boundary conditions (sliding fastening of the end fa...

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Bending Vibration of a Thick-Walled Shell of Finite Length with Sliding Fixation of Its Ends

Kovalev

2016

J Math Sci

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Applying wavelets to improve the boundary element method for elasticity problems

Cited by 5 publications

References 17 publications

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

A novel approach to solve linear systems arising from BEM using fast wavelet transforms

Tension of a Thick-Walled Shell of Finite Length Whose End Faces are Covered with A Diaphragm

Bending Vibration of a Thick-Walled Shell of Finite Length with Sliding Fixation of Its Ends

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