I-Shou University , Kaohsiung, Taiwan, May 30-31, 2015 第3屆台灣工業與應用數學會年會 (海報論文：碩士生組) A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems Wen-Sheng Huang (黃文生), Department of Harbor and River Engineering, National Taiwan Ocean University, Taiwan (10252025@mail.ntou.edu.tw) Advisor: Prof. Jeng-Tzong Chen (陳正宗) , Assistant Prof. Ying-Te Lee (李應德) Abstract In this work, the degenerate kernel and the direct searching technique are employed for the analytical study of the degenerate scales for two-dimensional (2-D) elasticity problem containing elliptical boundary, respectively. The Fichera’s idea is used to deal with the degenerate-scale problem. Based on the degenerate kernel, the deficiency of solution space in the degenerate scale problems for 2-D elasticity in the boundary integral equation method (BIEM) is analytically studied. In the discrete system of boundary element method (BEM), we search degenerate scales by using the technique of singular value decomposition (SVD) because of the zero-singular value of the influence matrix when a degenerate scale occurs. According to the Fichera’s idea, we enrich the conventional BEM/BIEM by adding the constants and a corresponding constraint. Finally, an example of degenerate-scale problem containing elliptical boundary for 2-D elasticity is demonstrated by using the necessary and sufficient BEM/BIEM. Problem description The closed-form fundamental solution The obtained coefficient of boundary densities The governing equation of the 2D elasticity problem as follows: Fig. 1 The sketch of the problem Degenerate kernel in terms of elliptical coordinates The exact solutions for the rigid body motion (displacement and rotation) are Boundary integral equation (Single layer) Expansion for boundary densities and boundary conditions Degenerate scales of an elliptical domain Necessary and sufficient BIE Results and discussions Fig. 3 Degenerate kernel of ( s(1.25, -1.25) ) Fig. 5 The displacement along the direction by using the conventions BEM/BIEM Fig. 7 The displacement along the direction by using Fichera’s method Field points Number of boundary elements : 100 Number of field points (n): 30 Semi-minor axis (b) : 0.78760 Semi-major axis (a) : 1.57520 Given : Fig. 2 The sketch of the problem Fig. 4 The minimum singular value versus the length of semi-minor axis by using the SVD Fig. 6 The displacement along the direction by using the conventions BEM/BIEM Fig. 7 The displacement along the direction by using Fichera’s method Conclusions We revisit the issue of the degenerate-scale problems for 2-D elasticity. Based on the addition theorem, the closed-form fundamental solution is expanded into the degenerate kernel in terms of the elliptical coordinates. Mechanism of the degenerate scale for two-dimensional elasticity problem containing elliptical boundary is analytically studied by using the degenerate kernel. A necessary and sufficient BEM/BIEM according to the Fichera’s idea is proposed to deal with the degenerate-scale problem. References [1] H.-K. Hong and J.T. Chen, Derivations of Integeral Equations of Elastity, J. Eng. Mech., 114, 1988, pp. 1028-1044. [2] J.T. Chen, S.R. Kuo and J.H. Lin, Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity, Int. J. Numer. Meth. Engng., 54, 2002, pp. 1669-1681. [3] J.T. Chen, Y.T. Lee and K.H. Chou, Revisit of two classical elasticity problems by using the null-field boundary integral equations, J. Mech., 26, 2010, pp. 393-401. [4] W.J. He, A necessary and sufficient boundary integral formulation for plane elasticity problems, Commum. Numer. Meth. Engng., 12, 1996, pp. 413-424. [5] J.T. Chen, H. Han, S.R. Kuo and S.K. Kao, Regularization methods for ill-conditioned system of the integral equation of the first kind with the logarithmic kernel, Inverse Probl. Sci. Eng., 22, 2014, pp. 1176-1195.