An adaptive scheme for the efficient evaluation of integrals in two-dimensional boundary element method

Iranian Journal of Numerical Analysis and Optimization

مقاله 5، دوره 15، Issue 4 - شماره پیاپی 35، اسفند 2025، صفحه 1420-1463   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2025.93526.1647

نویسندگان

R. Si Hadj Mohand^* ؛ Y. Belkacemi؛ S. Rechak

Laboratory of Green Mechanics and Development (LGMD), Department of mechanical engineering, Ecole Nationale Polytechnique, Algiers, Algeria.

چکیده

An efficient analysis with the boundary element method requires an accurate evaluation of all the boundary integrals. Typically, nonsingular integrals are solved numerically using Gauss quadrature. Therefore, the development of criteria and schemes that determine the appropriate Gauss order while maintaining a balance between accuracy and performance is of great importance. In the present work, an adaptive integration criterion tailored for two-dimensional elasticity problems is introduced and verified. This criterion is formulated as an empirical formula, incorporating a parameter ranging from zero to unity. This parameter enables control over computational effort, making the criterion very efficient across a wide range of applications, from thick structures to extremely thin ones where near-singularities are pronounced. The proposed integration criterion is tested on a very thin structure, where it showed a high degree of accuracy and effectiveness in solving problems with a very pronounced boundary layer effect. Additionally, the criterion demonstrated its advantage by reducing and moderating compu-tational overhead in the case of pre-treatment of near-singularities by a semi-analytical technique or a variable transformation technique. 

کلیدواژه‌ها

Boundary integrals؛ Near-singularity؛ Gauss quadrature؛ Integration criterion؛ Thin structures

مراجع

[1] Aliabadi, M.H., The boundary element method – applications in solids and structures, Wiley, Volume 2. UK, Chichester, 2002.
[2] Aliabadi, M.H., Hall, W.S., and Phemister, T.G., Taylor expansions for singular kernels in the boundary element method, Int. J. Numer. Methods Eng. 21 (1985), 2221–2236.
[3] Araújo, F.C., and Gray, L.J., Analysis of thin-walled structural elements via 3D standard BEM with generic substructuring, Comput. Mech. 41 (2008), 633–645.
[4] Bu, S., and Davies, T.G., Effective evaluation of non-singular integrals in 3D BEM, Adv. Eng. Softw. 23 (2) (1995), 121–128.
[5] Chang, C., and Mear, M.E., A boundary element method for two-dimensional linear elastic fracture analysis, Int. J. Fract. 74 (1995), 219–25.
[6] Chen, H.B., Lu, P., Huang, M.G., and Williams, F.W., An effective method for finding values on and near boundaries in the elastic BEM, Comput. Struct. 69 (1998), 421–431.
[7] Chen, H.B., Lu, P., and Schnack, E., Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM, Eng. Anal. Bound. Elem. 25 (10) (2001), 851–876.
[8] Chernov, A., and Stephan, E.P., Adaptive BEM for Contact Problems with Friction, Peter Wriggers and Udo Nackenhorst (eds.),In IUTAM Symposium on Computational Methods in Contact Mechanics: Proceedings of the IUTAM Symposium held in Hannover, Germany, November 5–8, 2006, pp. 113-122. Dordrecht: Springer Netherlands, 2007.
[9] Cruse, T.A., and Aithal, R., Non-singular boundary integral equation implementation, Int. J. Numer. Method. Eng. 36 (1993), 237–254.
[10] Eberwien, U., Duencer, C., and Moser, W., Efficient calculation of internal results in 2D BEM, Eng. Anal. Bound. Elem. 29 (2005), 447–453.

[11] Gao, X.W., and Davies, T.G., Adaptive integration in elasto-plastic boundary element analysis, J. Chin. Inst. Eng. 23 (3) (2000), 349–356. 
[12] Gu, Y., Chen, W., and Zhang, C., Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method, Int. J. Solids Struct. 50 (20-21) (2013), 3460–3471.
[13] Gu, Y., Zhang, C., Qu, W., and Ding, J., Investigation on near-boundary solutions for three-dimensional elasticity problems by an advanced BEM, Int. J. Mech. Sci. 142–143 (2018), 269–275.
[14] Guiggiani, M., Hypersingular formulation for boundary stress evaluation, Eng. Anal. Bound. Elem. 13 (1994), 169–179.
[15] Guiggiani, M., The evaluation of Cauchy principal value integrals in the boundary element method – A review, Math. Comput. Model. 15(3-5) (1991), 175–184.
[16] Guiggiani, M., and Casalini, P., Direct computation of Cauchy principal value integrals in advanced boundary elements, Int. J. Numer. Methods Eng. 24 (1987), 1711–1720.
[17] Guiggiani, M., and Casalini, P., Rigid body translation with curved boundary elements, Appl. Math. Model. 13(6) (1989), 365–368.
[18] Gwinner, J. and Stephan, E.P., A boundary element procedure for contact problems in plane linear elastostatics, ESAIM: Math. Model. Numer. Anal. 27 (4) (1994), 457–480.
[19] Jing, R., Yu, B., Ren, S., and Yao, W., A novel SCTBEM with inversion-free Padé series expansion for 3D transient heat transfer analysis in FGMs, Comput. Method. Appl. Mech. Eng. 433 (2025), 117546.
[20] Johnston, P.R., and Elliott, D., Transformations for evaluating singular boundary element integrals, J. Comput. Appl. Math. 146 (2) (2002), 231–251.
[21] Johnston, P.R., and Elliott, D., A sinh transformation for evaluating nearly singular boundary element integrals, Int. J. Numer. Method. Eng. 62 (2005), 564–578.

[22] Jun, L., Beer, G., and Meek, J.L., Efficient evaluation of integrals of order 1/r 1/r2, 1/r3 using Gauss quadrature, Eng. Anal. 2 (3) (1985), 118–123.
[23] Junhao, H., Zhipeng, W., and Yongqiang, C., A new error upper bound formula for Gaussian integration in boundary integral equations, Eng. Anal. Bound. Elem. 112 (2020), 39–45.
[24] Kachanov, M., Shafiro, B., and Tsukrov, I., Handbook of elasticity solutions, Springer Science+Business Media Dordrecht, B.V, 2003.
[25] Lachat, J.C., and Watson, J.O., Effective numerical treatment of boundary integral equations: A formulation for three dimensional elastostatics, Int. J. Numer. Method. Eng. 10 (1976), 991–1005.
[26] Li, S., and Mear, M.E., Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media, Int. J. Fract. 93 (1998), 87–114.
[27] Ma, H., and 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 (2002), 329–339.
[28] Mi, Y., and Aliabadi, M.H., A Taylor expansion algorithm for integration of 3D near-singular integrals, Commun. Appl. Numer. Method. Eng. 12 (1996), 51–62.
[29] Mustoe, G.G.W., Advanced integration schemes over boundary elements and volume cells for two-and three-dimensional non-linear analysis, Develop. Bound. Elem. Method. 3 (1984), 213–270.
[30] Newman-Jr, J.C., Mear, J.M., and Raju, I.S., Computer simulation of elastic stress analyses of two-dimensional multiple-cracked bodies, 37th Structure, Structural Dynamics and Materials Conference, AIAA Meeting Papers on Disc 1996, 869–876.
[31] Padhi, G.S., Shenoi, R.A., Moy, S.S.J., and McCarthy, M.A., Analytic integration of kernel shape function product integrals in the boundary element method, Comput. Struct. 79 (14) (2001), 1325–1333.

[32] Paulsen, K.D., and Lynch, D.R., Calculation of interior values by the boundary element method, Commun. Appl. Numer. Method. 5 (1989), 7–14.
[33] Portella, A., Aliabadi, M.H., and Rooke, D.P., The dual boundary element method: effective implementation for crack problems, Int. J. Numer. Method. Eng. 33 (1992), 1269–1287.
[34] Portella, A., Aliabadi, M.H., and Rooke, D.P., Dual boundary element incremental analysis of crack propagation, Comput. Struct. 46 (02) (1993), 237–247.
[35] Rudolphi, T.J., The use of simple solutions in the regularization of hypersingular boundary integral equations, Math. Comput. Model. 15 (3–5) (1991), 269–278.
[36] Salvadori, A., Analytical integrations in 2D BEM elasticity, Int. J. Numer. Method. Eng. 53 (2002), 1695–1719.
[37] Sladek, V., Sladek, J., and Tanaka, M., Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems, Acta Mech. 99 (1993), 173–190.
[38] Sladek, V., Sladek, J., and Tanaka, M., Optimal transformations of the integration variables in computation of singular integrals in BEM, Int. J. Numer. Method. Eng. 47 (2000), 1263–1283.
[39] Stroud, A.H., and Secrest, D., Gaussian quadrature formulas, Prentice-Hall Inc, Englewood Cliffs, N.J. 1966.
[40] Telles, J.C.F., A self-adaptive coordinate transformation for efficient numerical evaluation of general boundary element integrals, Int. J. Numer. Method. Eng. 24 (1987), 959–973.
[41] Timoshenko, S., and Goodier, J.N., Theory of elasticity, McGraw-Hill book company, New York, 1951.
[42] Yu, B., and Jing, R., SCTBEM: A scaled coordinate transformation boundary element method with 99-line MATLAB code for solving Pois-son’s equation, Comput. Phys. Commun. 300 (2024), 109185.

[43] Zhang, X. and Zhang, X. Exact integration in the boundary element method for two-dimensional elastostatic problems, Eng. Anal. Bound. Elem. 27 (2003), 987–997.
[44] Zhang, X. and Zhang, X. Exact integration for stress evaluation in the boundary element analysis of two-dimensional elastostatics, Eng. Anal. Bound. Elem. 28 (2004), 997–1004.
[45] Zhang, Y., Gong, Y., and Gao, X., Calculation of 2D nearly singular integrals over high-order geometry elements using the sinh transformation, Eng. Anal. Bound. Elem. 60 (2015), 144–153.
[46] Zhang, Y., and Gu, Y., An effective method in BEM for potential problems of thin bodies, J. Mar. Sci. Technol. 18 (2010), 137–144.
[47] Zhang, Y., Gu, Y., and Chen, J.T., Analysis of 2D thin-walled structures in BEM with high-order geometry elements using exact integration, Comput. Model. Eng. Sci. 50 (2009), 1–20.
[48] Zhou, W., Yang, X., and Chen, Y., Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning, Eng. Anal. Bound. Elem. 155 (2023), 197–211.