A comparison of three evaluation methods for Green’s function and its derivatives for 3D generally anisotropic solids

A comparison of three different methods for the numerical evaluation of three-dimensional (3D) anisotropic Green’s function and its first and second derivatives is presented. The line integral expressions of the Green’s function and its derivatives are the starting point of this investigation. The conventional line integral expressions are rewritten in terms of three different kinds of line integrals. In the first method, the numerical integration is applied to the line integrals. In the second method, the residue calculus is used, which results in explicit expressions of the Green’s function and its derivatives in non-degenerate cases. In the third method, the three line integrals are expressed in terms of two elementary line integrals, and after a rewritten of the explicit expressions evaluated by the simple pole residue calculus, the final explicit expressions are applicable in both degenerate and non-degenerate cases. The three methods are implemented in FORTRAN to make a direct comparison. Using the analytical solutions, the three expressions of the Green’s function and its derivatives are proved to be correct. The numerical phenomenon of the three methods near a degenerate point is studied numerically. Besides, the efficiency of the three methods is compared through the computing CPU times.