Huang, Junhao; Wu, Zhipeng; Chen, Yongqiang A new error upper bound formula for Gaussian integration in boundary integral equations. (English) Zbl 1464.65233 Eng. Anal. Bound. Elem. 112, 39-45 (2020). Summary: This paper proposes a new error upper bound formula for the Gaussian integration of the near-singular integral using the Boundary Element Method. First, this study found through numerical tests that the maximum relative error of the Gaussian integration has a downward concave shape but an approximately linear relationship with the relative distance, which is defined as the ratio of the distance from the source point to the element over the element length in a semi-logarithmic plot. Thus, the error upper bound can be defined as a line that closely approaches the computed error data points from the upper side. This line can be obtained by connecting two specified data points that are located outside, but very close to, the considered error range. Further research indicates that one parameter of the fitted line has a linear relationship with the number of Gaussian integration points and singularity orders and the other parameter can be treated as a constant, which together make the proposed Gaussian integration error upper bound formula widely applicable. Compared to the Lachat and Watson criterion, the proposed formula requires fewer integration points when the source point is very close to the element and thus serves to improve computational efficiency. The proposed formula also avoids calculation failure that can occur when using the Davies and Bu criterion. The numerical example results show that the proposed error upper bound formula can evaluate the integration accuracy well and improve computational efficiency when using an adaptive Gaussian integration method. MSC: 65N38 Boundary element methods for boundary value problems involving PDEs 65R20 Numerical methods for integral equations 65D32 Numerical quadrature and cubature formulas Keywords:upper bound of error; near-singular integral; adaptive Gaussian integration; boundary element method (BEM) PDFBibTeX XMLCite \textit{J. Huang} et al., Eng. Anal. Bound. Elem. 112, 39--45 (2020; Zbl 1464.65233) Full Text: DOI References: [1] Yao, ZH; Wang, HT, Boundary element method (2010), Higher Education Press: Higher Education Press Beijing [2] Gu, Y.; Fan, CM; Xu, RP, Localized method of fundamental solutions for large-scale modeling of two-dimensional elasticity problems, Appl Math Lett, 93, 8-14 (2019) · Zbl 1458.74143 [3] Johnston, P. R.; Elliott, D., Transformations for evaluating singular boundary element integrals, J Comput Appl Math, 146, 2, 231-251 (2002) · Zbl 1014.65019 [4] Telles, JCF, 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 [5] Khodakarami, M. I.; Khaji, N., Analysis of elastostatic problems using a semi-analytical method with diagonal coefficient matrices, Eng Anal Bound Elem, 35, 12, 1288-1296 (2011) · Zbl 1259.74085 [6] Padhi, GS; Shenoi, RA; Moy, SSJ; McCarthy, MA, Analytic integration of kernel shape function product integrals in the boundary element method, Comput Struct, 79, 14, 1325-1333 (2001) [7] Zhang, Y. M.; Sun, H. C., Analytical treatment of boundary integrals in direct boundary element analysis of plan potential and elasticity problems, Appl Math Mech, 22, 6, 664-673 (2001) · Zbl 1017.74080 [8] Zhang, Y. M.; Sun, H. C.; Yang, J. X., Theoretic analysis on virtual boundary element, Chin J Comput Mech, 17, 1, 56-62 (2000) [9] 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 [10] Rudolphi, T. J., The use of simple solutions in the regularization of hypersingular boundary integral equations, Math Comput Mod, 15, 3-5, 269-278 (1991) · Zbl 0728.73081 [11] Higashimach, T., Interactive structural analysis system using the advanced BEM, (Proceedings of the fifth International Conference BEM (1983)) [12] Zhang, YM; Liu, ZY; Gu, Y., Transform method for boundary layer effect in boundary element method for two-dimensional elastic problems, Chin J Comput Mech, 27, 5, 775-780 (2010) [13] Ma, H.; Kamiya, N., Domain supplemental approach to avoid boundary layer effect of BEM in elasticity, Eng Anal Bound Elem, 23, 3, 281-284 (1999) · Zbl 0963.74566 [14] Zhou, HL; Niu, ZR; Cheng, CZ; Guan, ZW, Analytical integral algorithm applied to boundary layer effect and thin body effect in BEM for anisotropic potential problems, Comput Struct, 86, 15-16, 1656-1671 (2008) [15] Hu, ZJ, A new regularization algorithm for singular integration of higher order element in boundary element method and its application, ((2012), Hefei University of Technology) [16] Liu, Y.; Fan, H., Analysis of thin piezoelectric solids by the boundary element method, Comput Methods Appl Mech Eng, 191, 21-22, 2297-2315 (2002) · Zbl 1131.74342 [17] Zhang, YM; Gu, Y.; Chen, JT, Boundary element analysis of the thermal behaviour in thin-coated cutting tools, Eng Anal Bound Elem, 34, 9, 775-784 (2010) · Zbl 1244.74203 [18] Zhang, Y. M.; Gu, Y.; Chen, J. T., Boundary element analysis of 2D thin walled structures with high-order geometry elements using transformation, Eng Anal Bound Elem, 35, 3, 581-586 (2011) · Zbl 1259.74075 [19] 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 [20] Portela, A.; Aliabadi, MH; Rooke, DP, The dual boundary element method: effective implementation for crack problem, Int J Numer Methods Eng, 33, 6, 1269-1287 (2010) · Zbl 0825.73908 [21] Aour, B.; Rahmani, O.; Nait-Abdelaziz, M., A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics, Int J Solids Struct, 44, 7-8, 2523-2539 (2007) · Zbl 1276.74034 [22] Hu, ZJ; Niu, ZR; Cheng, CZ, Semi-analytic method in almost singular integral for higher order elements of two-dimensional potential problem using boundary element method, Chin J Comput Mech, 6, 763-768 (2014) [23] Gao, XW; Davies, TG, Adaptive integration in elasto-plastic boundary element analysis, J Chin Inst Eng, 23, 3, 349-356 (2000) [24] Johnston, BM; Johnston, PR; Elliott, D., A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals, Int J Numer Methods Eng, 69, 7, 1460-1479 (2007) · Zbl 1194.65143 [25] Telles, JCF; Oliveira, RF, Third degree polynomial transformation for boundary element integrals: further improvements, Eng Anal Bound Elem, 13, 2, 135-141 (1994) [26] Yun, B. I., A non‐linear co‐ordinate transformation for accurate numerical evaluation of weakly singular integrals, Commun Numer Methods Eng, 20, 5, 401-417 (2010) · Zbl 1052.65014 [27] Yun, BI, An extended sigmoidal transformation technique for evaluating weakly singular integrals without splitting the integration interval, SIAM J Sci Comput, 25, 1, 284-301 (2003) · Zbl 1048.65028 [28] Johnston, PR, 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 [29] Elliott, D.; Johnston, PR, Error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals, J Comput Appl Math, 203, 1, 103-124 (2007) · Zbl 1116.65032 [30] Hu, Z. J.; Liu, C.; Niu, Z. R., Semi-analytical method in almost singular integral for higher-order elements of three-dimensional sound field using boundary elements, Chin J Appl Mech, 32, 5, 743-749 (2015) [31] Luo, J. F.; Liu, Y. J.; Berger, E. J., Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method, Comput Mech, 22, 5, 404-412 (1998) · Zbl 0938.74075 [32] Granados, J. J.; Gallego, R., Regularization of nearly hypersingular integrals in the boundary element method, Eng Anal Bound Elem, 25, 3, 165-184 (2001) · Zbl 1015.74073 [33] Davies, T. G.; Bu, S., Effective evaluation of non-singular integrals in 3D BEM, Adv Eng Softw, 23, 2, 121-128 (1995) [34] 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 [35] Mustoe, GGW, Advanced integration schemes over boundary elements and volume cells for two-and three-dimensional non-linear analysis, Developments in boundary element methods, 3, 213-270 (1984) [36] Gu, Y.; Gao, H.; Chen, W.; Zhang, C., A general algorithm for evaluating nearly singular integrals in anisotropic three-dimensional boundary element analysis, Comput Methods Appl Mech Eng, 308, 483-498 (2016) · Zbl 1439.65209 [37] Liu, YJ, Fast multipole boundary element method: theory and applications in engineering (2009), Cambridge University Press [38] Brebbia, CA; Telles, JCF; Wrobel, LC, Boundary element techniques: theory and applications in engineering (2012), Springer Science & Business Media [39] Stroud, AH; Secrest, D., Gaussian quadrature formulas (1966), Prentice-Hall · Zbl 0156.17002 [40] Yao, Z. H., A new type of high-accuracy BEM and local stress analysis of real beam, plate and shell structures, Eng Anal Bound Elem, 65, 1-17 (2016) · Zbl 1403.74260 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.