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Analysis of Two-Dimensional Thin Structures (From Micro- to Nano-Scales) Using the Singular Boundary Method

Published online by Cambridge University Press:  21 July 2015

Dejian Shen  and
Yan Gu
Dejian Shen
Affiliation:
Department of Civil Engineering, College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China College of Mathematics, Qingdao University, Qingdao 266071, China
Yan Gu*
Affiliation:
College of Mathematics, Qingdao University, Qingdao 266071, China
*
*Corresponding author. Email: guyan1913@163.com (Y. Gu)
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Abstract

This study investigates the applicability of the singular boundary method (SBM), a recent developed meshless boundary collocation method, for the analysis of two-dimensional (2D) thin structural problems. The troublesome nearly-singular kernels, which are crucial in the applications of SBM to thin shapes, are dealt with efficiently by using a non-linear transformation technique. Promising SBM results with only a small number of boundary nodes are obtained for thin structures with the thickness-to-length ratio is as small as 1E-9, which is sufficient for modeling most thin layered coating systems as used in smart materials and micro-electro-mechanical systems. The advantages, disadvantages and potential applications of the proposed method, as compared with the finite element (FEM) and boundary element methods (BEM), are also discussed.

Keywords

65N1565N38Singular boundary methodmeshless boundary collocation methodmethod of fundamental solutionssinh transformationthin structural problems
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 5 , October 2015 , pp. 597 - 609
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.CrossRefGoogle Scholar
[2]Karageorghis, A., Lesnic, D. and Marin, L., A survey of applications of the MFS to inverse problems, Inverse. Probl. Sci. Eng., 19 (2011), pp. 309–336.CrossRefGoogle Scholar
[3]Chen, C. S., Cho, H. A. and Golberg, M. A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng. Anal. Bound. Elem., 30 (2006), pp. 405–410.Google Scholar
[4]Marin, L., A meshless method for solving the cauchy problem in three-dimensional elastostatics, Comput. Math. Appl., 50 (2005), pp. 73–92.Google Scholar
[5]Chen, C. S., The method of fundamental solutions for non-linear thermal explosions, Commun. Numer. Methods. Eng., 11 (1995), pp. 675–681.Google Scholar
[6]Liu, C.-S., An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation, Eng. Anal. Bound. Elem., 36 (2012), pp. 1235–1245.Google Scholar
[7]Chen, W. and Tanaka, M., A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43 (2002), pp. 379–391.Google Scholar
[8]Young, D. L., Chen, K. H. and Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys., 209 (2005), pp. 290–321.Google Scholar
[9]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. Bound. Elem., 33 (2009), pp. 1374–1382.CrossRefGoogle Scholar
[10]Chen, J. T., Chang, M. H., Chen, K. H. and Lin, S. R., The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, J. Sound. Vib., 257 (2002), pp. 667–711.Google Scholar
[11]Liu, Y. J., A new boundary meshfree method with distributed sources, Eng. Anal. Bound. Elem., 34 (2010), pp. 914–919.Google Scholar
[12]Gu, Y., Chen, W. and Zhang, C.-Z., Singular boundary method for solving plane strain elasto-static problems, Int. J. Solids. Struct., 48 (2011), pp. 2549–2556.Google Scholar
[13]Chen, W. and Gu, Y., An improved formulation of singular boundary method, Adv. Appl. Math. Mech., 4 (2012), pp. 543–558.Google Scholar
[14]Gu, Y., Chen, W. and He, X. Q., Improved singular boundary method for elasticity problems, Comput. Struct., 135 (2014), pp. 73–82.Google Scholar
[15]Gu, Y., Chen, W., Fu, Z. J. and Zhang, B., The singular boundary method: Mathematical background and application in orthotropic elastic problems, Eng. Anal. Bound. Elem., 44 (2014), pp. 152–160.Google Scholar
[16]Chen, W. and Wang, F. Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Bound. Elem., 34 (2010), pp. 530–532.Google Scholar
[17]Gu, Y., Chen, W. and He, X. Q., Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media, Int. J. Heat. Mass. Transf., 55 (2012), pp. 4837–4848.Google Scholar
[18]Gu, Y. and Chen, W., Infinite domain potential problems by a new formulation of singular boundary method, Appl. Math. Model., 37 (2013), pp. 1638–1651.Google Scholar
[19]Gu, Y., Chen, W. and Zhang, J., Investigation on near-boundary solutions by singular boundary method, Eng. Anal. Bound. Elem., 36 (2012), pp. 1173–1182.Google Scholar
[20]Zhang, Y. M., Gu, Y. and Chen, J. T., Boundary element analysis of 2D thin walled structures with high-order geometry elements using transformation, Eng. Anal. Bound. Elem., 35 (2011), pp. 581–586.Google Scholar
[21]Johnston, B. M., Johnston, P. R. and Elliott, D., A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals, Int. J. Numer. Methods. Eng., 69 (2007), pp. 1460–1479.Google Scholar
[22]Johnston, P. R. and Elliott, D., A sinh transformation for evaluating nearly singular boundary element integrals, Int. J. Numer. Methods. Eng., 62 (2005), pp. 564–578.Google Scholar
[23]Johnston, B. M. and Johnston, P. R., A comparison of transformation methods for evaluating two-dimensional weakly singular integrals, Int. J. Numer. Methods. Eng., 56 (2003), pp. 589–607.Google Scholar
[24]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 (2013), pp. 3460–3471.CrossRefGoogle Scholar
[25]Luo, J. F., Liu, Y. J. and Berger, E. J., Analysis of two-dimensional thin structures (from micro-to nano-scales) using the boundary element method, Comput. Mech., 22 (1998), pp. 404–412.Google Scholar
[26]Liu, G. R., Achenbach, J. D., Kim, J. O. and Li, Z. L., A combined finite element method/boundary element method technique for V(z) curves of anisotropic-layer/substrate configurations, J. Acoust. Soc. Am., 92 (1992), pp. 2734–2740.CrossRefGoogle Scholar
[27]Sladek, V., Sladek, J. and Tanaka, M., Nonsingular BEM formulations for thin-walled structures and elastostatic crack problems, Acta. Mech., 99 (1993), pp. 173–190.Google Scholar