# zbMATH — the first resource for mathematics

The semi-analytical evaluation for nearly singular integrals in isogeometric elasticity boundary element method. (English) Zbl 1403.65199
Summary: Benefiting from improvement of accuracy in modeling complex geometry and integrity of discretization and simulation, the isogeometric analysis in the boundary element method (IGABEM) has now been implemented by several groups. However, the difficulty of evaluating the nearly singular integral in IGABEM for elasticity has not yet been effectively solved, which will hinder the application of IGABEM in engineering structure analysis. Herein, the nearly singular integrals in IGABEM are separated to the non-singular part and singular part by the subtraction technique. The integral kernels in singular part are approximated by the Taylor series polynomial expressions, in which different orders of derivatives are interpolated by the non-uniform rational B-splines (NURBS). Furthermore, the analytical formulations for the singular part with the approximated kernels are derived by a series of integration by parts, while the non-singular part is calculated with Gaussian quadrature. In this way, a semi-analytical method is proposed for the nearly singular integrals in the IGABEM. Comparing with the conventional IGABEM, the present method can yield accurate displacement and stress for inner points much closer to the boundary. It can obtain effective results with fewer elements than the finite element method because of the precise simulation of geometry and boundary-only discretization.

##### MSC:
 65N38 Boundary element methods for boundary value problems involving PDEs 65D30 Numerical integration
INSANE
Full Text:
##### References:
 [1] Hughes, TJR; Cottrell, JA; Bazilevs, Y, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput Method Appl Mech Eng, 194, 39, 4135-4195, (2005) · Zbl 1151.74419 [2] Cottrell, JA; Hughes, TJR; Reali, A, Studies of refinement and continuity in isogeometric structural analysis, Comput Method Appl Mech Eng, 196, 41, 4160-4183, (2007) · Zbl 1173.74407 [3] Cottrell, JA; Reali, A; Bazilevs, Y; Hughes, TJR, Isogeometric analysis of structural vibrations, Comput Method Appl Mech Eng, 195, 41, 5257-5296, (2006) · Zbl 1119.74024 [4] Bazilevs, Y; Calo, VM; Zhang, Y; Hughes, TJR, Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Comput Mech, 38, 4-5, 310-322, (2006) · Zbl 1161.74020 [5] Bazilevs, Y; Beirao, DVL; Cottrell, JA; Hughes, TJR; Sangalli, G, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Math Mod Meth Appl S, 16, 07, 1031-1090, (2006) · Zbl 1103.65113 [6] Gu, J; Zhang, J; Li, G, Isogeometric analysis in BIE for 3-D potential problem, Eng Anal Bound Elem, 36, 5, 858-865, (2012) · Zbl 1352.65585 [7] Simpson, RN; Bordas, SPA; Trevelyan, J; Rabczuk, T, A two-dimensional isogeometric boundary element method for elastostatic analysis, Comput Method Appl Mech Eng, 209, 87-100, (2012) · Zbl 1243.74193 [8] Simpson, RN; Bordas, SPA; Lian, H; Trevelyan, J, An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects, Comput Struct, 118, 2-12, (2013) [9] Simpson, RN; Scott, MA; Taus, M; Thomas, DC; Lian, H, Acoustic isogeometric boundary element analysis, Comput Method Appl Mech Eng, 269, 265-290, (2014) · Zbl 1296.65175 [10] Peng, X; Atroshchenko, E; Kerfriden, P; Bordas, SPA, Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment, Int J Fracture, 204, 1, 55-78, (2017) [11] Peng, X; Atroshchenko, E; Kerfriden, P; Bordas, SPA, Isogeometric boundary element methods for three dimensional static fracture and fatigue crack growth, Comput Method Appl Mech Eng, 316, 151-185, (2017) [12] Sladek, V; Sladek, J; Tanaka, M, Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems, Acta Mech, 99, 1, 173-190, (1993) · Zbl 0774.73077 [13] Liu, CS; Chang, CW; Chang, JR, A new shooting method for solving boundary layer equations in fluid mechanics, CMES-Comp Model Eng, 32, 1, 1-15, (2008) [14] Mukherjee, S; Chati, MK; Shi, X, Evaluation of nearly singular integrals in boundary element contour and node methods for three-dimensional linear elasticity, Int J Solids Struct, 37, 51, 7633-7654, (2000) · Zbl 0993.74077 [15] Jorge, AB; Ribeiro, GO; Cruse, TA; Fisher, TS, Self-regular boundary integral equation formulations for Laplace’s equation in 2-D, Int J Numer Meth Eng, 51, 1, 1-29, (2001) [16] Hosseini, S; Malekan, M; Pitangueira, RL; Silva, RP, Imposition of Dirichlet boundary conditions in element free Galerkin method through an object-oriented implementation, Lat Am J Solids Struc, 14, 6, 1017-1039, (2017) [17] Anacleto, FES; Ribeiro, TSA; Ribeiro, GO; Pitangueira, RLS; Penna, SS, An object-oriented tridimensional self-regular boundary element method implementation, Eng Anal Bound Elem, 37, 10, 1276-1284, (2013) · Zbl 1287.65120 [18] Gao, XW; Yang, K; Wang, J, An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals, Eng Anal Bound Elem, 32, 8, 692-696, (2008) · Zbl 1244.65199 [19] Telles, JCF, A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals, Int J Numer Meth Eng, 24, 5, 959-973, (1987) · Zbl 0622.65014 [20] Lutz, E, Exact Gaussian quadrature methods for near-singular integrals in the boundary element method, Eng Anal Bound Elem, 9, 3, 233-245, (1992) [21] Cerrolaza, M; Alarcon, E, A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods, Int J Numer Meth Eng, 28, 5, 987-999, (1989) · Zbl 0679.73040 [22] Scuderi, L, A new smoothing strategy for computing nearly singular integrals in 3D Galerkin BEM, J Comput Appl Math, 225, 2, 406-427, (2009) · Zbl 1159.65314 [23] Johnston, BM; Johnston, PR; 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, BM; Johnston, PR; Elliott, D, A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals, Int J Numer Meth Eng, 69, 7, 1460-1479, (2007) · Zbl 1194.65143 [25] Lv, JH; Miao, Y; Gong, WH; Zhu, HP, The sinh transformation for curved elements using the general distance function, Comput Model Eng Sci, 93, 2, 113-131, (2013) · Zbl 1357.65286 [26] Zhang, YM; Gu, Y; Chen, JT, Internal stress analysis for single and multilayered coating systems using the boundary element method, Eng Anal Bound Elem, 35, 4, 708-717, (2011) · Zbl 1259.74076 [27] Xie, G; Zhou, F; Zhang, J; Zheng, X; Huang, C, New variable transformations for evaluating nearly singular integrals in 3D boundary element method, Eng Anal Bound Elem, 37, 9, 1169-1178, (2013) · Zbl 1287.65127 [28] Zhang, J; Wang, P; Lu, C; Dong, Y, A spherical element subdivision method for the numerical evaluation of nearly singular integrals in 3D BEM, Eng Computation, 34, 6, 2074-2087, (2017) [29] Qin, X; Zhang, J; Xie, G; Zhou, F; Li, G, 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 [30] Xie, G; Zhang, J; Dong, Y; Huang, C; Li, G, An improved exponential transformation for nearly singular boundary element integrals in elasticity problems, Int J Solids Struct, 51, 6, 1322-1329, (2014) [31] Fata, SN, Semi-analytic treatment of nearly-singular Galerkin surface integrals, Appl Numer Math, 60, 10, 974-993, (2010) · Zbl 1198.65238 [32] Niu, Z; Wendland, WL; Wang, X; Zhou, H, A semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods, Comput Method Appl Mech Eng, 194, 9, 1057-1074, (2005) · Zbl 1113.74084 [33] Niu, ZR; Cheng, CZ; Zhou, HL; Hu, ZJ, Analytic formulations for calculating nearly singular integrals in two-dimensional BEM, Eng Anal Bound Elem, 31, 12, 949-964, (2007) · Zbl 1259.74056 [34] Niu, ZR; Hu, ZJ; Cheng, CZ; Zhou, HL, A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM, Eng Anal Bound Elem, 61, 42-51, (2015) · Zbl 1403.74204 [35] Gong, YP; Dong, CY; Bai, Y, Evaluation of nearly singular integrals in isogeometric boundary element method, Eng Anal Bound Elem, 75, 21-35, (2017) · Zbl 1403.65198 [36] Keuchel, S; Hagelstein, NC; Zaleski, O; Estorff, O, Evaluation of hypersingular and nearly singular integrals in the isogeometric boundary element method for acoustics, Comput Method Appl Mech Eng, 325, 488-504, (2017) [37] Krishnasamy, G; Rizzo, FJ; Liu, Y, Boundary integral equations for thin bodies, Int J Numer Meth Eng, 37, 107-121, (1994) · Zbl 0795.73076 [38] Liu, YJ; Zhang, DM; Rizzo, FJ, Nearly singular and hypersingular integrals in the boundary element method, WIT Trans Model Simul, (1970) [39] Liu, YJ., Analysis of shell-like structures by the boundary element method based on 3-D elasticity: formulation and verification, Int J Numer Meth Eng, 41, 541-558, (1998) · Zbl 0910.73068 [40] Luo, JF; Liu, YJ; Berger, EJ, Analysis of two-dimensional thin structures (from micro-to nano-scales) using the boundary element method, Comput Mech, 22, 404-412, (1998) · Zbl 0938.74075 [41] Chen, XL; Liu, YJ, An advanced 3D boundary element method for characterizations of composite materials, Eng Anal Bound Elem, 29, 513-523, (2005) · Zbl 1182.74212
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.