×

zbMATH — the first resource for mathematics

New variable transformations for evaluating nearly singular integrals in 3D boundary element method. (English) Zbl 1287.65127
Summary: This work presents new variable transformations for accurate evaluation of the nearly singular integrals arising in the 3D boundary element method (BEM). The proposed method is an extension of the variable transformation method in [G. Xie et al., Eng. Anal. Bound. Elem. 35, No. 6, 811–817 (2011; Zbl 1259.65185)] for 2D BEM to 3D BEM. In this paper, first a new system denoted as \((\alpha,\beta)\) is introduced compared with the polar coordinate system. So the original transformations in [loc. cit.] can be developed to 3D in \((\alpha,\beta)\) or the polar coordinate system. Then, the new transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, a new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. To perform integration on irregular elements, an adaptive integration scheme combined with the transformations is applied. Numerical examples compared with other methods are presented. The results demonstrate that our method is accurate and effective.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
65D30 Numerical integration
45E05 Integral equations with kernels of Cauchy type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cruse, T. A.; Aithal, R., Non-singular boundary integral equation implementation, Int J Numer Methods Eng, 36, 237-254, (1993)
[2] Liu, YJ., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Int J Numer Methods Eng, 41, 541-558, (1998) · Zbl 0910.73068
[3] Krishnasamy, G.; Rizzo, F. J.; Liu, Y. J., Boundary integral equations for thin bodies, Int J Numer Methods Eng, 37, 107-121, (1994) · Zbl 0795.73076
[4] Xie, G. Z.; Zhang, J. M.; Qin, X. Y.; Li, G. Y., New variable transformations for evaluating nearly singular integrals in 2D boundary element method, Eng Anal Boundary Elem, 35, 811-817, (2011) · Zbl 1259.65185
[5] Dirgantara, T.; Aliabadi, M. H., Crack growth analysis of plates loaded by bending and tension using dual boundary element method, Int J Fract, 105, 27-47, (2000)
[6] Aliabadi, M. H.; Martin, D., Boundary element hypersingular formulation for elasto-plastic contact problems, Int J Numer Methods Eng, 48, 995-1014, (2000) · Zbl 0974.74072
[7] Zhang, D.; Rizzo, F. J.; Rudolphi, T. J., Stress intensity sensitivities via hypersingular boundary integral equations, Comput Mech, 23, 389-396, (1999) · Zbl 0967.74075
[8] Zhang, J. M.; Qin, X. Y.; Han, X.; Li, G. Y., A boundary face method for potential problems in three dimensions, Int J NumerMethods Eng, 80, 320-337, (2009) · Zbl 1176.74212
[9] Qin, X. Y.; Zhang, J. M.; Li, G. Y., An element implementation of the boundary face method for 3D potential problems, Eng Anal Boundary Elem, 34, 934-943, (2010) · Zbl 1244.74182
[10] Zhou, F. L.; Zhang, J. M.; Sheng, X. M.; Li, G. Y., Shape variable radial basis function and its application in dual reciprocity boundary face method, Eng Anal Boundary Elem, 35, 244-252, (2011) · Zbl 1259.65188
[11] Qin, X. Y.; Zhang, J. M.; Xie, G. Z.; Zhou, F. L., A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element, J Comput Appl Math, 235, 4174-4186, (2011) · Zbl 1219.65031
[12] Zhou, F. L.; Zhang, J. M.; Sheng, X. M.; Li, G. Y., A dual reciprocity boundary face method for 3D non-homogeneous elasticity problems, Eng Anal Boundary Elem, 36, 1301-1310, (2012) · Zbl 1351.74088
[13] Zhou, F. L.; Xie, G. Z.; Zhang, J. M.; Zheng, X. S., Transient heat conduction analysis of solids with small open-ended tubular cavities by boundary face method, Eng Anal Boundary Elem, 37, 542-550, (2013) · Zbl 1297.80004
[14] Chen, H. B.; Lu, P.; Huang, M. G.; Williams, F. W., An effective method for finding values on and near boundaries in the elastic BEM, Comput Struct, 69, 421-431, (1998) · Zbl 0941.74075
[15] Sladek, N.; Sladek, J.; Tanaka, M., Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, Int J NumerMethods Eng, 36, 1609-1628, (1993) · Zbl 0772.73091
[16] Liu, Y. J.; Rudolphi, T. J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, ComputMech, 24, 286-292, (1999) · Zbl 0969.74073
[17] Liu, Y. J., On the simple solution and non-singular nature of the BIE/BEM—a review and some new results, Eng Anal Boundary Elem, 24, 789-795, (2000) · Zbl 0974.65110
[18] Krishnasamy, G.; Schmerr, L. W.; Rudolphi, T. J.; Rizzo, F. J., Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, J Appl Mech, 57, 404-414, (1990) · Zbl 0729.73251
[19] Niu, Z. R.; W. L., Wendland; Wang, X. X.; Zhou, H. L., A sim-analytic algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods, Comput Methods Appl Mech Eng, 31, 949-964, (2005)
[20] 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, 1656-1671, (2008)
[21] Johnston, Peter R.; Elliott, David, A sinh transformation for evaluating nearly singular boundary element integrals, Int J Numer Methods Eng, 62, 564-578, (2005) · Zbl 1119.65318
[22] Johnston, Barbara M.; Johnston, Peter R.; Elliott, David, A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals, Int J Numer Methods Eng, 69, 1460-1479, (2007) · Zbl 1194.65143
[23] Elliott, David; Johnston, Peter R., Error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals, J Comput Appl Math, 203, 103-124, (2007) · Zbl 1116.65032
[24] Tells, J. C.F., A self adaptive coordinate transformation for efficient numerical evaluations of general boundary element integrals, Int J NumerMethods Eng, 24, 959-973, (1987) · Zbl 0622.65014
[25] Gao, X. W.; Davies, T. G., Adaptive integration in elasto-plastic boundary element analysis, JChin Inst Eng, 23, 349-356, (2000)
[26] Lachat, J. C.; Waston, J. O., Effective numerical treatment of boundary integral equations: a formulation for three-dimensional elastostatics, Int J Numer Methods Eng, 10, 273-289, (1976)
[27] 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, 277-288, (2002) · Zbl 1128.74343
[28] Ma, H.; Kamiya, N., A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM, Eng Anal Boundary Elem, 25, 833-841, (2001) · Zbl 1042.74054
[29] Ma, H.; Kamiya, N., Distance transformation for the numerical evaluation of near singular boundary integrals with various kernels in boundary element method, Eng Anal Boundary Elem, 26, 329-339, (2002) · Zbl 1003.65133
[30] Ma, H.; Kamiya, N., Nearly singular approximations of CPV integrals with end and corner- singularities for the numerical solution of hypersingular boundary integral equations, Eng Anal Boundary Elem, 27, 625-637, (2003) · Zbl 1181.74147
[31] Hayami, K.; Matsumoto, H., A numerical quadrature for nearly singular boundary element integrals, Eng Anal Boundary Elem, 13, 143-154, (1994)
[32] Hayami, K., Variable transformations for nearly singular integrals in the boundary element method, 41, 821-842, (2005), Publications of Research Institute for Mathematical Sciences Kyoto University · Zbl 1100.65109
[33] Hayami, K.; Brebbia, C. A., Quadrature methods for singular and nearly singular integrals in 3-D boundary element method, (Brebbia, C. A., Boundary elements X, (1988), Spring-Verlag), 237-264
[34] Zhang, Y. M.; Gu, Y.; Chen, J. T., Boundary layer effect in BEM with high order geometry elements using transformation, Comput Modeling Eng Sci, 45, 227-247, (2009) · Zbl 1357.74072
[35] Zhang, Y. M.; Gu, Y.; Chen, J. T., Boundary element analysis of the thermal behaviour in thin- coated cutting tools, Eng Anal Boundary Elem, 34, 775-784, (2010) · Zbl 1244.74203
[36] Scuderi, Letizia, On the computation of nearly singular integrals in 3D BEM collocation, Int J Numer Methods Eng, 74.11, 1733-1770, (2008) · Zbl 1195.74256
[37] Hayami K, Brebbia CA. A new coordinate transformation method for singular and nearly singular integrals over general curved boundary elements. In: Boundary Elements IX, Proceedings of the 9th International Conference on Boundary Elements, Stuttgart, Brebbia CA, Wendland WL, Kuhn G (editors), vol. 1. A Computational Mechanics Publications with Springer-Verlag: Berlin; 1987. p. 375-399.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.