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Revisit of the Dual Bem using SVD Updating Technique

Published online by Cambridge University Press:  10 April 2015

J.-T. Chen ,
W.-S. Huang ,
Y. Fan  and
S.-K. Kao
J.-T. Chen*
Affiliation:
Department of Harbor and River Engineering National Taiwan Ocean University Keelung, Taiwan Department of Mechanical and Mechatronic Engineering National Taiwan Ocean University Keelung, Taiwan
W.-S. Huang
Affiliation:
Department of Harbor and River Engineering National Taiwan Ocean University Keelung, Taiwan
Y. Fan
Affiliation:
Department of Harbor and River Engineering National Taiwan Ocean University Keelung, Taiwan
S.-K. Kao
Affiliation:
Department of Harbor and River Engineering National Taiwan Ocean University Keelung, Taiwan
*
* Corresponding author ([email protected])
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Abstract

The boundary element method (BEM) is easier than the finite element method (FEM) on the viewpoint of the discretization of one dimension reduction rather than the domain discretization of finite element method. The disadvantage of BEM is the rank deficiency in the influence matrix, e.g., degenerate boundary, degenerate scale, spurious eigenvalues and fictitious frequencies, which do not occur in the FEM. The conventional BEM can not be straightforward applied to solve a problem which contains a degenerate boundary without decomposing the domain to multi-regions. A hypersingular integral equation is used to ensure a unique solution for the problem containing a degenerate boundary. By combining the singular and hypersingular equations, it’s termed the dual BEM due to its dual frame. Following the successful experience on the retrieval of information using the singular value decomposition (SVD) updating term and updating document, this technique is also used to extract out the degenerate-boundary information and the rigid-body information in the dual BEM. It is interesting to find that true information due to a rigid-body mode in physics is found in the right singular vector with respect to the corresponding zero singular value while the degenerate-boundary mode (geometry degeneracy) in mathematics is imbedded in the left singular vector with respect to the corresponding zero singular value. The role of the common right and left singular vectors of SVD for the four influence matrices in the dual BEM is also discussed in this paper. Two examples, a potential flow problem across a cutoff wall and a cracked bar under torsion were demonstrated to see the mathematical SVD structure of four influence matrices in the dual BEM.

Keywords

Degenerate boundaryDual BEMSVD updating technique
Type
Research Article
Information
Journal of Mechanics , Volume 31 , Issue 5 , October 2015 , pp. 505 - 514
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2015 

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