Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-13T05:55:16.636Z Has data issue: false hasContentIssue false
  • English
  • Français

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 (jtchen@mail.ntou.edu.tw)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Kuo, S. R., Chen, J. T. and Kao, S. K., “Linkage Between the Unit Logarithmic Capacity in the Theory of Complex Variables and the Degenerate Scale in the BEM/Biems,” Applied Mathematics Letters, 29, pp. 929–938 (2013).CrossRefGoogle Scholar
2.Chen, J. T., Lin, S. R., Chen, K. H., Chen, I. L. and Chyuam, S. W., “Eigenanalysis for Membranes with Stringers Using Conventional BEM in Conjunction with SVD Technique,” Computer Methods in Applied Mechanics and Engineering, 192, pp. 1299–1322 (2003).Google Scholar
3.Chen, J. T., Chen, I. L. and Chen, K. H., “Treatment of Rank-Deficiency in Acoustics Using SVD,” Journal of Computational Acoustics, 157, pp. 157–183 (2006).Google Scholar
4.Kinoshita, N. and Mura, T., “On Boundary Value Problem of Elasticity,” Research Report of the Faculty of Engineering, Meiji Univ., 8, pp. 56–82 (1956).Google Scholar
5.Rizzo, F. J., “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics,” Quarterly of Applied Mathematics, 25, pp. 83–95 (1967).Google Scholar
6.Blandford, G. E., Ingraffea, A. R. and Liggett, J. A., “Two-Dimensional Stress Intensity Factor Computations Using the Boundary Element Method,” International Journal for Numerical Methods in Engineering, 17, pp. 387–404 (1981).Google Scholar
7.Hong, H.-K. and Chen, J. T., “Derivation of Integral Equations of Elasticity,” Journal of Engineering Mechanics, 114, pp. 1028–1044 (1988).CrossRefGoogle Scholar
8.Portela, A., Aliabadi, M. H. and Rooke, D. P., “The Dual Boundary Element Method: Effective Implementation for Crack Problems,” International Journal for Numerical Methods in Engineering, 33, pp. 1269–1287 (1992).Google Scholar
9.Lafe, O. E., Montes, J. S., Cheng, A. H. D., Liggett, J. A. and Liu, P. L. F., “Singularity in Darcy Flow Through Porous Media,” Journal of the Hydraulics Divisio, 106, pp. 977–997 (1980).Google Scholar
10.Chen, J. T., Hong, H. K. and Chuan, S. W., “Boundary Element Analysis and Design in Seepage Flow Problems with Sheet Piles,” Finite Elements in Analysis and Design, 17, pp. 1–20 (1994).Google Scholar
11.Chen, J. T. and Chen, Y. W., “Dual Boundary Elements Analysis Using Complex Variables for Potential Problems with or Without a Degenerate Boundary,” Engineering Analysis with Boundary Elements, 24, pp. 671–684 (2000).CrossRefGoogle Scholar
12.Chen, J. T., Chen, K. H. and Yeih, W., “Dual Boundary Element Analysis for Cracked Bars Under Torsion,” Engineering Computations, 15, pp. 732–749 (1998).Google Scholar
13.Berry, M. W., Dumais, S. T. and O’Brien, G. W., “Using Linear Algebra for Intelligent Information Retrieval,” SIAM Review, 37, pp. 573–595 (1995).CrossRefGoogle Scholar
14.Chen, J. T., Lee, C. F. and Lin, S. Y., “A New Point of View for the Polar Decomposition Using Singular Value Decomposition,” International Journal of Computational and Numerical Analysis and Applications, 2, pp. 257–264 (2002).Google Scholar
15.Lebedev, N. N., Skalskaya, I. P. and Uflyand, Y. S., Worked Problems in Applied Mathematics, Dover Publications, New York (1965).Google Scholar
16.Boresi, A. P., Chong, K. and Lee, J. D., Elasticity in Engineering Mechanics, Wiley-Interscience Publication, New Jersey (2010).Google Scholar