1. 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
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) :
Semi-major axis (a) :
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
[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
[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
[4] W.J. He, A necessary and sufficient boundary integral formulation for plane elasticity problems, Commum. Numer. Meth. Engng., 12, 1996, pp
[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