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A combined conformal and sinh-sigmoidal transformations method for nearly singular boundary element integrals. (English) Zbl 1403.65013

Summary: Accurate and efficient evaluation of nearly singular integrals is a major concern in 3D BEM. Most existing widely-used non-linear transformations are only performed in radial direction. Actually, the near singularity may derive from three aspects: element shape, radial direction and angular direction. In this paper, a combined conformal and sinh-sigmoidal transformations method is proposed to evaluate nearly singular integrals arising in 3D BEM. The method can be decomposed into three steps: firstly, a conformal transformation is introduced to eliminate the shape effect caused by large aspect ratios and peak/big obtuse angles; secondly, the classical sinh transformation is applied in radial direction to cluster more Gaussian points towards the nearly singular point; finally, an improved sigmoidal transformation is utilized to rearrange Gaussian points in angular direction more reasonably. Extensive numerical examples including unit triangular element, elements with different aspect ratios, elements with different angles and curved triangular element are given to verify the robustness and competitiveness of presented method.

MSC:

65D30 Numerical integration
65N38 Boundary element methods for boundary value problems involving PDEs
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[1] Liu, Y. J., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Int J Numer Methods Eng, 41, 3, 541-558, (1998) · Zbl 0910.73068
[2] Dirgantara, T.; Aliabadi, M. H., Crack growth analysis of plates loaded by bending and tension using dual boundary element method, Int J Fract, 105, 1, 27-47, (2000)
[3] Aliabadi, M. H.; Martin, D., Boundary element hyper-singular formulation for elastoplastic contact problems, Int J Numer Methods Eng, 48, 7, 995-1014, (2000) · Zbl 0974.74072
[4] Zhang, D.; Rizzo, F. J.; Rudolphi, T. J., Stress intensity sensitivities via hypersingular boundary integral equations, Comput Mech, 23, 5-6, 389-396, (1999) · Zbl 0967.74075
[5] Jun, L.; Beer, G.; Meek, J. L., Efficient evaluation of integrals of order 1/r, 1/r^{2}, 1/r^{3} using Gauss quadrature, Eng Anal, 2, 3, 118-123, (1985)
[6] Lachat, J. C.; Watson, J. O., Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics, Int J Numer Methods Eng, 10, 5, 991-1005, (1976) · Zbl 0332.73022
[7] Sladek, V.; Sladek, J.; Tanaka, M., Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, Int J Numer Methods Eng, 36, 10, 1609-1628, (1993) · Zbl 0772.73091
[8] Chen, H. B.; Lu, P.; Schnack, E., Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM, Eng Anal Bound Elem, 25, 10, 851-876, (2001) · Zbl 1051.74050
[9] Zhou, H. L.; Niu, Z. R.; Cheng, C. Z.; Guan, Z. W., Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems, Comput Struct, 86, 15, 1656-1671, (2008)
[10] Zhou, H. L.; Niu, Z. R.; Cheng, C. Z.; Guan, Z. W., Analytical integral algorithm in the BEM for orthotropic potential problems of thin bodies, Eng Anal Bound Elem, 31, 9, 739-748, (2007) · Zbl 1195.74271
[11] Niu, Z. R.; Wendland, W. L.; Wang, X. X.; Wang, H. L., A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods, Comput Methods Appl Mech Eng, 194, 9, 1057-1074, (2005) · Zbl 1113.74084
[12] Telles, J. C.F., A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J Numer Methods Eng, 24, 5, 959-973, (1987) · Zbl 0622.65014
[13] Hayami, K., Variable transformations for nearly singular integrals in the boundary element method, Publ Res Inst Math Sci, 41, 4, 821-842, (2005) · Zbl 1100.65109
[14] Ma, H.; Kamiya, N., Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Eng Anal Bound Elem, 26, 4, 329-339, (2002) · Zbl 1003.65133
[15] Ma, H.; Kamiya, N., A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two-and three-dimensional elasticity, Comput Mech, 29, 4-5, 277-288, (2002) · Zbl 1128.74343
[16] Miao, Y.; Li, W.; Lv, J. H.; Long, X. H., Distance transformation for the numerical evaluation of nearly singular integrals on triangular elements, Eng Anal Bound Elem, 37, 10, 1311-1317, (2013) · Zbl 1287.65124
[17] Johnston, P. R.; Elliott, D., A sinh transformation for evaluating nearly singular boundary element integrals, Int J Numer Methods Eng, 62, 4, 564-578, (2005) · Zbl 1119.65318
[18] Elliott, D.; Johnston, P. R., The iterated sinh transformation, Int J Numer Methods Eng, 75, 1, 43-57, (2008) · Zbl 1195.65183
[19] Lv, J. H.; Miao, Y.; Zhu, H. P., The distance sinh transformation for the numerical evaluation of nearly singular integrals over curved surface elements, Comput Mech, 53, 2, 359-367, (2014)
[20] Zhang, Y. M.; Gu, Y.; Chen, J. T., Boundary layer effect in BEM with high order geometry elements using transformation, Comput Model Eng Sci, 45, 3, 227, (2009) · Zbl 1357.74072
[21] Xie, G. Z.; Zhang, J. M.; Dong, Y. Q.; Hunag, C.; Li, G. Y., An improved exponential transformation for nearly singular boundary element integrals in elasticity problems, Int J Solids Struct, 51, 6, 1322-1329, (2014)
[22] Scuderi, L., On the computation of nearly singular integrals in 3D BEM collocation, Int J Numer Methods Eng, 74, 11, 1733-1770, (2008) · Zbl 1195.74256
[23] Johnston, B. M.; Johnston, P. R.; Elliott, D., A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method, J Comput Appl Math, 245, 148-161, (2013) · Zbl 1262.65043
[24] Johnston, P. R., Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals, Int J Numer Methods Eng, 45, 10, 1333-1348, (1999) · Zbl 0935.65130
[25] Qin, X. Y.; Zhang, J. M.; Xie, G. Z., A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element, J Comput Appl Math, 235, 14, 4174-4186, (2011) · Zbl 1219.65031
[26] Rong, J. J.; Wen, L. H.; Xiao, J. Y., Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements, Eng Anal Bound Elem, 38, 83-93, (2014) · Zbl 1287.65126
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