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Obtaining the upper bound of discretization error and critical boundary integrals of circular arc boundary element method. (English) Zbl 1255.76087

Summary: The boundary element method (BEM) is a popular method of solving linear partial differential equations, and it can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics. The circular arc element (CAE) method is a scheme to discretize the boundary of problems arising in the BEM. M. Dehghan and H. Hosseinzadeh [Comput. Math. Appl. 62, No. 12, 4461–4471 (2011; Zbl 1236.65139)] the order of the convergence of CAE discretization was obtained for the 2D Laplace equation. The current work extends the formulation developed in [loc. cit.] to convert CAE to a robust discretization method. In the present paper the upper bound of the CAE’s discretization error is determined theoretically for the 2D Laplace equation. Also we present a new method based on the complex space \(\mathbb{C}\) to obtain CAE’s boundary integrals without facing singularity and near singularity. Since there is no efficient approach to treat the near singular integrals of CAE in the BEM literature, the new scheme presented in this paper enhances the CAE discretization significantly. Several test problems are given and the numerical simulations are obtained which confirm the theoretical results.

MSC:

76M15 Boundary element methods applied to problems in fluid mechanics
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
65N38 Boundary element methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Citations:

Zbl 1236.65139

Software:

BEMLIB
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pozrikidis, C., A Practical Guide to Boundary Element Methods with the Software Library Bemlib (2002), Chapman & Hall/CRC · Zbl 1019.65097
[2] Ang, W. T., A Beginner’s Course in Boundary Element Methods (2007), Universal Publishers: Universal Publishers Boca Raton, USA
[3] Stih, Z.; Trkulja, B., Application of bicubic splines in the boundary element method, Eng. Anal. Bound. Elem., 33, 896-900 (2009) · Zbl 1244.78033
[4] Katsikadelis, J. T., Boundary element methods, (Theory and Application (2002), Elsevier) · Zbl 0825.73913
[5] Munteanu, C.; Topa, V.; Simion, E., Multi-terminal resistances optimal design using spline boundary elements and genetic algorithms, Int. J. Comp. Math. Elec. Elec. Eng., 20, 279-292 (2001)
[6] Ming, Yu.; Kuffel, E., Spline element for boundary element method, IEEE Trans. Magn., 30, 2900-2917 (1994)
[7] Ye, W.; Fei, Y., On the convergence of the panel methods with non smooth domains, Eng. Anal. Bound. Elem., 33, 837-844 (2009) · Zbl 1244.65159
[8] Chen, Z.; Xiao, H.; Yang, X., Error analysis and novel near-field preconditioning techniques for Taylor series multipole-BEM, Eng. Anal. Bound. Elem., 34, 173-181 (2010) · Zbl 1244.74147
[9] Kamiya, N.; Kawaguchi, K., Error analysis and adaptive refinement of boundary elements in elastic problem, Adv. Eng. Softw., 15, 223-230 (1992) · Zbl 0770.73093
[10] Dehghan, M.; Hosseinzadeh, H., Development of boundary circular elements, Eng. Anal. Bound. Elem., 35, 543-549 (2011) · Zbl 1259.65180
[11] Sladek, V.; Sladek, J., Singular Integrals in Boundary Element Methods (1998), Southampton: Comput. Mech. Publ. · Zbl 0961.74072
[12] Zhou, H.; Niu, Z.; Cheng, C.; Guan, Z., Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems, Comput. Struct., 86, 1656-1671 (2008)
[13] Zhou, H.; Niu, Z.; Cheng, C.; Guan, Z., Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies, Eng. Anal. Bound. Elem., 31, 739-748 (2007) · Zbl 1195.74271
[14] Gao, X.; Yang, K.; Wang, J., An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals, Eng. Anal. Bound. Elem., 32, 692-696 (2008) · Zbl 1244.65199
[15] Lachat, J.; J, Watson, Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics, Int. J. Num. Meth. Eng., 10, 991-1005 (1976) · Zbl 0332.73022
[16] Telles, J. C.F., A self-adaptive coordinate transformation for efficient numerical evaluations of general boundary element integrals, Int. J. Num. Meth. Eng., 24, 959-973 (1987) · Zbl 0622.65014
[17] Fratantonioa, M.; Rencis, J. J., Exact boundary element integrations for two-dimensional Laplace equation, Eng. Anal. Bound. Elem., 24, 325-342 (2000) · Zbl 0959.65134
[18] Zhang, X. S.; Zhang, X. X., Exact integrations of two-dimensional highorder discontinuous boundary elements of elastostatic problems, Eng. Anal. Bound. Elem., 28, 725-732 (2004) · Zbl 1130.74479
[19] Tomioka, S.; Nishiyama, S., Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method, Eng. Anal. Bound. Elem., 34, 393-404 (2010) · Zbl 1244.65208
[20] Maddi, J. R.; Vable, M., An hpr-mesh refinement algorithm for BEM, Eng. Anal. Bound. Elem., 34, 549-556 (2010) · Zbl 1267.74004
[21] Dehghan, M.; Mirzaei, D., The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation, Comput. Methods Appl. Mech. Eng., 197, 476-486 (2008) · Zbl 1169.76401
[22] Dehghan, M.; Mirzaei, D., A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn-Hilliard equation, Eng. Anal. Bound. Elem., 33, 522-528 (2009) · Zbl 1244.74149
[23] Dehghan, M.; Ghesmati, A., Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Eng. Anal. Bound. Elem., 34, 51-59 (2010) · Zbl 1244.65137
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