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Stress analysis for two-dimensional thin structural problems using the meshless singular boundary method. (English) Zbl 1403.74057
Summary: This short communication documents the first attempt to apply the singular boundary method (SBM) for the stress analysis of thin structural elastic problems. The troublesome nearly-singular kernels, which are crucial in the applications of the SBM to thin shapes, are dealt with efficiently by using a non-linear transformation technique. Three benchmark numerical examples, ranging from thin films, thin shell-like structures and multi-layer coating systems, are well studied to demonstrate the effectiveness of the proposed method. The advantages, disadvantages and potential applications of the method to thin structural problems, as compared with the boundary element (BEM) and finite element (FEM) methods, are also discussed.

MSC:
74K35 Thin films
74S15 Boundary element methods applied to problems in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
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[1] Qin, Q. H., Variational formulations for TFEM of piezoelectricity, Int J Solids Struct, 40, 23, 6335-6346, (2003) · Zbl 1057.74043
[2] Cheng, A. H.D.; Cheng, D. T., Heritage and early history of the boundary element method, Eng Anal Boundary Elem, 29, 3, 268-302, (2005) · Zbl 1182.65005
[3] Cheng, A. H.D.; Chen, C. S.; Golberg, M. A.; Rashed, Y. F., BEM for theomoelasticity and elasticity with body force—a revisit, Eng Anal Boundary Elem, 25, 4-5, 377-387, (2001) · Zbl 1014.74075
[4] Gao, X. W.; Wang, J., Interface integral BEM for solving multi-medium heat conduction problems, Eng Anal Boundary Elem, 33, 4, 539-546, (2009) · Zbl 1244.80010
[5] Liu, C. S., A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, Comput Model Eng Sci, 21, 1, 53-65, (2007) · Zbl 1232.65157
[6] Chen, J. T.; Lee, C. F.; Chen, I. L.; Lin, J. H., An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation, Eng Anal Boundary Elem, 26, 7, 559-569, (2002) · Zbl 1014.65125
[7] 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, 3, 541-558, (1998) · Zbl 0910.73068
[8] Joanni, A. E.; Kausel, E., Heat diffusion in layered media via the thin-layer method, Int J Numer Methods Eng, 78, 6, 692-712, (2009) · Zbl 1183.74354
[9] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv Comput Math, 9, 1, 69-95, (1998) · Zbl 0922.65074
[10] Golberg, M. A.; Chen, C. S.; Bowman, H., Some recent results and proposals for the use of radial basis functions in the BEM, Eng Anal Boundary Elem, 23, 4, 285-296, (1999) · Zbl 0948.65132
[11] Chen, C. S., The method of fundamental solutions for non-linear thermal explosions, Commun Numer Methods Eng, 11, 8, 675-681, (1995) · Zbl 0839.65143
[12] Chen, C. S.; Golberg, M. A.; Hon, Y. C., The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int J Numer Methods Eng, 43, 8, 1421-1435, (1998) · Zbl 0929.76098
[13] Wang, J.; Cui, Y.; Qin, Q. H.; Jia, J., Application of Trefftz BEM to anti-plane piezoelectric problem, Acta Mech Solida Sin, 19, 4, 352-364, (2006)
[14] Zhang, X.; Liu, Y.; Ma, S., Meshfree methods and their applications, Adv Mech, 39, 1, 1-36, (2009)
[15] Zhang, J. M.; Yao, Z. H.; Tanaka, M., The meshless regular hybrid boundary node method for 2D linear elasticity, Eng Anal Boundary Elem, 27, 3, 259-268, (2003) · Zbl 1112.74556
[16] Gu, Y.; Chen, W.; Zhang, C.-Z., Singular boundary method for solving plane strain elastostatic problems, Int J Solids Struct, 48, 18, 2549-2556, (2011)
[17] Young, D. L.; Chen, K. H.; Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, J Comput Phys, 209, 1, 290-321, (2005) · Zbl 1073.65139
[18] Sarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng Anal Boundary Elem, 33, 12, 1374-1382, (2009) · Zbl 1244.76084
[19] Marin, L., Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity, Int J Solids Struct, 47, 24, 3326-3340, (2010) · Zbl 1203.74056
[20] Liu, Y. J., A new boundary meshfree method with distributed sources, Eng Anal Boundary Elem, 34, 11, 914-919, (2010) · Zbl 1244.65189
[21] Wang, H.; Qin, Q. H., Some problems with the method of fundamental solution using radial basis functions, Acta Mech Solida Sin, 20, 1, 21-29, (2007)
[22] Karageorghis, A.; Lesnic, D.; Marin, L., A survey of applications of the MFS to inverse problems, Inverse Prob Sci Eng, 19, 3, 309-336, (2011) · Zbl 1220.65157
[23] Liu, C. S., A meshless regularized integral equation method for Laplace equation in arbitrary interior or exterior plane domains, Comput Model Eng Sci, 19, 1, 99-109, (2007) · Zbl 1184.65114
[24] Gu, Y.; Chen, W., Infinite domain potential problems by a new formulation of singular boundary method, Appl Math Model, 37, 4, 1638-1651, (2013) · Zbl 1349.65686
[25] Gu, Y.; Chen, W.; He, X. Q., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, Int J Heat Mass Transfer, 55, 17-18, 4837-4848, (2012)
[26] Gu, Y.; Chen, W.; Zhang, C.; He, X., A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media, Int J Heat Mass Transfer, 84, 0, 91-102, (2015)
[27] Simpson, R. N.; Bordas, S. P.A.; Lian, H.; Trevelyan, J., An isogeometric boundary element method for elastostatic analysis: 2D implementation aspects, Comput Struct, 118, 2-12, (2013)
[28] Peng X, Atroshchenko E, Bordas S. Damage tolerance assessment directly from CAD: (extended) isogeometric boundary element methods (XIGABEM). In: Proc. sixth international conference on advanced computational methods in engineering; 2014.
[29] Lian H, Simpson R, Bordas S. Sensitivity analysis and shape optimisation through a T-spline isogeometric boundary element method. In: Proc. international conference on computational mechanics; 2013.
[30] Natarajan, S.; Wang, J. C.; Song, C. M.; Birk, C., Isogeometric analysis enhanced by the scaled boundary finite element method, Comput Methods Appl Mech Eng, 283, 733-762, (2015) · Zbl 1425.65174
[31] Rabczuk, T.; Bordas, S.; Zi, G., On three-dimensional modelling of crack growth using partition of unity methods, Comput Struct, 88, 23-24, 1391-1411, (2010)
[32] Qin, Q. H., Postbuckling analysis of thin plates by a hybrid Trefftz finite element method, Comput Methods Appl Mech Eng, 128, 1-2, 123-136, (1995) · Zbl 0860.73071
[33] Fu, Z. J.; Qin, Q. H.; Chen, W., Hybrid-Trefftz finite element method for heat conduction in nonlinear functionally graded materials, Eng Comput, 28, 5-6, 578-599, (2011) · Zbl 1284.80009
[34] Karihaloo, B. L.; Xiao, Q. Z., Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity, Eng Fract Mech, 68, 15, 1609-1630, (2001)
[35] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math Comput Simul, 79, 3, 763-813, (2008) · Zbl 1152.74055
[36] Gu, Y.; Chen, W.; He, X. Q., Improved singular boundary method for elasticity problems, Comput Struct, 135, 73-82, (2014), 0
[37] Gu, Y.; Chen, W.; Fu, Z.-J.; Zhang, B., The singular boundary method: mathematical background and application in orthotropic elastic problems, Eng Anal Boundary Elem, 44, 152-160, (2014), 0 · Zbl 1297.74145
[38] Marin, L., An alternating iterative MFS algorithm for the Cauchy problem in two-dimensional anisotropic heat conduction, CMC: Comput Mater Contin, 12, 1, 71-100, (2009)
[39] Gu, Y.; Chen, W.; Zhang, C., The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements, Eng Anal Boundary Elem, 37, 2, 301-308, (2013) · Zbl 1352.65586
[40] 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
[41] Zhang, Y.-M.; Gu, Y.; Chen, J.-T., Internal stress analysis for single and multilayered coating systems using the boundary element method, Eng Anal Boundary Elem, 35, 4, 708-717, (2011) · Zbl 1259.74076
[42] Johnston, B. M.; Johnston, P. R.; 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), 0 · Zbl 1262.65043
[43] Liu, Y. J.; Nishimura, N.; Yao, Z. H., A fast multipole accelerated method of fundamental solutions for potential problems, Eng Anal Boundary Elem, 29, 11, 1016-1024, (2005) · Zbl 1182.74256
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