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A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element. (English) Zbl 1219.65031
Summary: A general numerical method is proposed to compute nearly singular integrals arising in the boundary integral equations. The method provides a new implementation of the conventional distance transformation technique to make the result stable and accurate no matter where the projection point is located. The distance functions are redefined in two local coordinate systems. A new system denoted as $$(\alpha ,\beta )$$ is introduced here firstly. Its implementation is simpler than that of the polar system and it also performs efficiently. Then a new distance transformation is developed to remove or weaken the near singularities. To perform integration on irregular elements, an adaptive integration scheme is applied. Numerical examples are presented for both planar and curved surface elements. The results demonstrate that our method can provide accurate results even when the source point is very close to the integration element, and can keep reasonable accuracy on very irregular elements. Furthermore, the accuracy of our method is much less sensitive to the position of the projection point than the conventional method.

##### MSC:
 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures 65N38 Boundary element methods for boundary value problems involving PDEs
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##### References:
  Zhang, J.M.; Qin, X.Y.; Han, X.; Li, G.Y., A boundary face method for potential problems in three dimensions, Internat. J. numer. methods engrg., 80, 320-337, (2009) · Zbl 1176.74212  Liu, Y.J., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Internat. J. numer. methods engrg., 41, 541-558, (1998) · Zbl 0910.73068  Liu, Y.J.; Fan, H., Analysis of the thin piezoelectric solids by the boundary element method, Comput. methods appl. mech. engrg., 191, 2297-2315, (2002) · Zbl 1131.74342  Gao, X.W.; Davies, T.G., Adaptive integration in elasto-plastic boundary element analysis, J. chin. inst. eng, 23, 349-356, (2000)  Letizia, Scuderi, On the computation of nearly singular integrals in 3D BEM collocation, Internat. J. numer. methods engrg., 74, 1733-1770, (2007) · Zbl 1195.74256  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, 118-123, (1985)  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. engrg., 31, 949-964, (2005)  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)  Telles, J.C.F., A self-adaptive co-ordinates transformation for efficient numerical evaluation of general boundary element integrals, Internat. J. numer. methods engrg., 24, 959-973, (1987) · Zbl 0622.65014  Chen, X.L.; Liu, Y.J., An advanced 3-D boundary element method for characterizations of composite materials, Eng. anal. bound. elem., 29, 513-523, (2005) · Zbl 1182.74212  Hayami, K., Variable transformations for nearly singular integrals in the boundary element method, Publ. res. inst. math. sci., 41, 821-842, (2005) · Zbl 1100.65109  Johnston, B.M.; Johnston, P.R.; Elliott, D., A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals, Internat. J. numer. methods engrg., 69, 1460-1479, (2007) · Zbl 1194.65143  Johnston, P.R., Application of sigmoidal transformation to weakly singular and nearly-singular boundary element integrals, Internat. J. numer. methods engrg., 45, 1333-1348, (1999) · Zbl 0935.65130  Zhang, Y.M.; Gu, Y.; Chen, J.T., Boundary layer effect in BEM with high order geometry elements using transformation, (CMES) comput. mod. eng. sci., 45, 227-247, (2009) · Zbl 1357.74072  Zhang, Y.M.; Gu, Y.; Chen, J.T., Boundary element analysis of the thermal behaviour in thin- coated cutting tools, Eng. anal. bound. elem., 34, 775-784, (2010) · Zbl 1244.74203  Wu, S., On the evaluation of nearly singular kernel integrals in boundary element analysis, Numer. method eng., 11, 331-337, (1995) · Zbl 0821.73077  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  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  Guiggiani, M.; Gigante, N., A general algorithm for multi-dimensional Cauchy principal value integrals in the boundary element method, (ASME) J. appl. mech., 57, 906-915, (1990) · Zbl 0735.73084  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
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