TY - JOUR

T1 - Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis

AU - Shiah, Y. C.

AU - Tan, C. L.

AU - Wang, C. Y.

N1 - Funding Information:
The authors gratefully acknowledge the financial support from the National Science Council of Taiwan ( NSC99-2221-E-035-027-MY3 ) and the National Science and Engineering Research Council of Canada.

PY - 2012/12

Y1 - 2012/12

N2 - An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407-26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.

AB - An alternative scheme to compute the Green's function and its derivatives for three dimensional generally anisotropic elastic solids is presented in this paper. These items are essential in the formulation of the boundary element method (BEM); their evaluation has remained a subject of interest because of the mathematical complexity. The Green's function considered here is the one introduced by Ting and Lee [Q. J. Mech. Appl. Math. 1997; 50: 407-26] which is of real-variable, explicit form expressed in terms of Stroh's eigenvalues. It has received attention in BEM only quite recently. By taking advantage of the periodic nature of the spherical angles when it is expressed in the spherical coordinate system, it is proposed that this Green's function be represented by a double Fourier series. The Fourier coefficients are determined numerically only once for a given anisotropic material; this is independent of the number of field points in the BEM analysis. Derivatives of the Green's function can be performed by direct spatial differentiation of the Fourier series. The resulting formulations are more concise and simpler than those derived analytically in closed form in previous studies. Numerical examples are presented to demonstrate the veracity and superior efficiency of the scheme, particularly when the number of field points is very large, as is typically the case when analyzing practical three dimensional engineering problems.

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U2 - 10.1016/j.enganabound.2012.05.008

DO - 10.1016/j.enganabound.2012.05.008

M3 - Article

AN - SCOPUS:84864053588

VL - 36

SP - 1746

EP - 1755

JO - Engineering Analysis with Boundary Elements

JF - Engineering Analysis with Boundary Elements

SN - 0955-7997

IS - 12

ER -