×

zbMATH — the first resource for mathematics

New variable transformations for evaluating nearly singular integrals in 2D boundary element method. (English) Zbl 1259.65185
Summary: This work presents a further development of the distance transformation technique for accurate evaluation of the nearly singular integrals arising in the 2D boundary element method (BEM). The traditional technique separates the nearly hypersingular integral into two parts: a near strong singular part and a nearly hypersingular part. The near strong singular part with the one-ordered distance transformation is evaluated by the standard Gaussian quadrature and the nearly hypersingular part still needs to be transformed into an analytical form. In this paper, the distance transformation is performed by four steps in case the source point coincides with the projection point or five steps otherwise. For each step, new transformation is proposed based on the approximate distance function, so that all steps can finally be unified into a uniform formation. With the new formulation, the nearly hypersingular integral can be dealt with directly and the near singularity separation and the cumbersome analytical deductions related to a specific fundamental solution are avoided. Numerical examples and comparisons with the existing methods on straight line elements and curved elements demonstrate that our method is accurate and effective.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
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, 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, 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] Zhang, J.M.; Yao, Z.H., Meshless regular hybrid boundary node method, Comput modeling eng sci, 2, 307-318, (2001) · Zbl 0991.65129
[5] Zhang, J.M.; Yao, Z.H.; Tanaka, M., The meshless regular hybrid boundary node method for 2-D linear elasticity, Eng anal bound elem, 27, 259-268, (2003) · Zbl 1112.74556
[6] Zhang, J.M.; Yao, Z.H., Analysis of 2-D thin structures by the meshless regular hybrid boundary node method, Acta mech solida sinica, 15, 36-44, (2002)
[7] Dirgantara, T.; Aliabadi, M.H., Crack growth analysis of plates loaded by bending and tension using dual boundary element method, Int J fatigue, 105, 27-47, (2000)
[8] 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
[9] 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
[10] 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
[11] Sladek, N.; Sladek, J.; Tanaka, M., Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity, Int J numer methods eng, 36, 1609-1628, (1993) · Zbl 0772.73091
[12] Liu, Y.J.; Rudolphi, T.J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput mech, 24, 286-292, (1999) · Zbl 0969.74073
[13] Liu, Y.J., On the simple solution and non-singular nature of the BIE/BEM-a review and some new results, Eng anal bound elem, 24, 789-795, (2000) · Zbl 0974.65110
[14] 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
[15] Niu, Z.R.; Wendland, W.L.; 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)
[16] 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)
[17] 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
[18] 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
[19] 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
[20] Tells, J.C.F., A self adaptive coordinate transformation for efficient numerical evaluations of general boundary element integrals, Int J numer methods eng, 24, 959-973, (1987) · Zbl 0622.65014
[21] Zhang, J.M.; Qin, X.Y.; Han, X.; Li, G.Y., A boundary face method for potential problems in three dimensions, Int J numer methods eng, 80, 320-337, (2009) · Zbl 1176.74212
[22] Qin, X.Y.; Zhang, J.M.; Li, G.Y., An element implementation of the boundary face method for 3D potential problems, Eng anal bound elem, 34, 934-943, (2010) · Zbl 1244.74182
[23] Gao, X.W.; Davies, T.G., Adaptive integration in elasto-plastic boundary element analysis, J chin inst eng, 23, 349-356, (2000)
[24] 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
[25] Ma, H.; Kamiya, N., A general algorithm for accurate computation of field variables and its derivatives near boundary in BEM, Eng anal bound elem, 25, 833-841, (2001) · Zbl 1042.74054
[26] 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, 329-339, (2002) · Zbl 1003.65133
[27] 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 bound elem, 27, 625-637, (2003) · Zbl 1181.74147
[28] Hayami, K.; Matsumoto, H., A numerical quadrature for nearly singular boundary element integrals, Eng anal bound elem, 13, 143-154, (1994)
[29] Hayami, K., Variable transformations for nearly singular integrals in the boundary element method, 41, (2005), Publications of Research Institute for Mathematical Sciences Kyoto University, p. 821-42 · Zbl 1100.65109
[30] 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
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.