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An Improved Numerical Evaluation Scheme of the Fundamental Solution and its Derivatives for 3D Anisotropic Elasticity Based on Fourier Series

Y.C. Shiah1, C. L. Tan2, C.Y. Wang1
Dept. of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan, R.O.C.
Dept. of Mechanical & Aerospace Engineering, Carleton University, Ottawa, Canada K1S 5B6

Computer Modeling in Engineering & Sciences 2012, 87(1), 1-22. https://doi.org/10.3970/cmes.2012.087.001

Abstract

The fundamental solution, or Green's function, for 3D anisotropic elastostatics as derived by Ting and Lee (1997) [Q.J. Mech. Appl. Math.; 50: 407-426] is one that is fully explicit and algebraic in form. It has, however, only been utilized in boundary element method (BEM) formulations quite recently even though it is relatively straightforward and direct to implement. This Green's function and its derivatives are necessary items in this numerical analysis technique. By virtue of the periodic nature of the angles when it is expressed in the spherical coordinate system, the present authors have very recently represented the Green's function as a double Fourier series for their efficient numerical evaluation. The Fourier coefficients are determined only once, independent of the total number of field points involved in the BEM analysis of a problem. The derivatives of the fundamental solution can also be obtained simply by direct spatial differentiation of the double Fourier series without further numerical or significant analytical steps. This paper presents a re-formulation of the same scheme by taking advantage of some characteristic features of the Fourier coefficients and re-arranging and simplifying the terms. In so doing, the total number of terms necessary for the series summation is significantly reduced thereby enhancing further its computational efficiency, especially when large number of field points are involved as is typically the case when modeling practical engineering problems. Numerical examples are presented to demonstrate the veracity of the proposed scheme. It is also simpler to implement into BEM codes because of the formulation is considerably less elaborate as compared to the direct evaluation approaches of the exact analytical solution, its full explicit algebraic form notwithstanding.

Keywords

Boundary element method, Green's functions, boundary integral equations, anisotropic elasticity, Stroh's eigenvalues, Fourier series.

Cite This Article

Shiah, Y., Tan, C. L., Wang, C. (2012). An Improved Numerical Evaluation Scheme of the Fundamental Solution and its Derivatives for 3D Anisotropic Elasticity Based on Fourier Series. CMES-Computer Modeling in Engineering & Sciences, 87(1), 1–22.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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