A fast and non-degenerate scheme for the evaluation of the 3D fundamental solution and its derivatives for fully anisotropic magneto-electro-elastic materials

Abstract

A new expression for the fundamental solution is introduced, presenting three relevant characteristics: (i) it is explicit in terms of the Stroh’s eigenvalues, (ii) it remains well-defined when some Stroh’s eigenvalues are repeated, and (iii) it is exact. A fast and robust numerical scheme for the evaluation of the fundamental solution and its derivatives developed from double Fourier series representations is presented. The Fourier series representation is possible due to the periodic nature of the solution. The attractiveness of this series solution is that the information of the material properties is contained only in the Fourier coefficients, while the information of the dependence of the evaluation point is contained in simple trigonometric functions. This implies that any order derivatives can be determined by spatial differentiation of the trigonometric functions. Moreover, Fourier coefficients need to be obtained only once for a given material, leading to an efficient methodology. The robustness of the scheme arises from the properties (i) and (ii) of the new expression for the fundamental solution, which is used to compute the Fourier coefficients. The proposed approach combines the clean structure of the Stroh formalism with the simplicity of Fourier expansions, addressing the old drawbacks of anisotropic fundamental solutions.