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Revisit of the Dual Bem using SVD Updating Technique

  • J.-T. Chen (a1) (a2), W.-S. Huang (a1), Y. Fan (a1) and S.-K. Kao (a1)


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.


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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. 929938 (2013).
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. 12991322 (2003).
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. 157183 (2006).
4.Kinoshita, N. and Mura, T., “On Boundary Value Problem of Elasticity,” Research Report of the Faculty of Engineering, Meiji Univ., 8, pp. 5682 (1956).
5.Rizzo, F. J., “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics,” Quarterly of Applied Mathematics, 25, pp. 8395 (1967).
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. 387404 (1981).
7.Hong, H.-K. and Chen, J. T., “Derivation of Integral Equations of Elasticity,” Journal of Engineering Mechanics, 114, pp. 10281044 (1988).
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. 12691287 (1992).
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. 977997 (1980).
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. 120 (1994).
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. 671684 (2000).
12.Chen, J. T., Chen, K. H. and Yeih, W., “Dual Boundary Element Analysis for Cracked Bars Under Torsion,” Engineering Computations, 15, pp. 732749 (1998).
13.Berry, M. W., Dumais, S. T. and O’Brien, G. W., “Using Linear Algebra for Intelligent Information Retrieval,” SIAM Review, 37, pp. 573595 (1995).
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. 257264 (2002).
15.Lebedev, N. N., Skalskaya, I. P. and Uflyand, Y. S., Worked Problems in Applied Mathematics, Dover Publications, New York (1965).
16.Boresi, A. P., Chong, K. and Lee, J. D., Elasticity in Engineering Mechanics, Wiley-Interscience Publication, New Jersey (2010).



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