Numerical solutions to Helmholtz equation of anisotropic functionally graded materials

Abstract

Abstract In this paper, interior 2D-BVPs for anisotropic FGMs governed by the Helmholtz equation with Dirichlet and Neumann boundary conditions are considered. The governing equation involves diffusivity and wave number coefficients which are spatially varying. The anisotropy of the material is presented in the diffusivity coefficient. And the inhomogeneity is described by both diffusivity and wave number. Three types of the gradation function considered are quadratic, exponential and trigonometric functions. A technique of transforming the variable coefficient governing equation to a constant coefficient equation is utilized for deriving a boundary integral equation. And a standard BEM is constructed from the boundary integral equation to find numerical solutions. Some particular examples of BVPs are solved to illustrate the application of the BEM. The results show the accuracy of the BEM solutions, especially for large wave numbers. They also show coherence between the flow vectors and scattering solutions, and the effect of the anisotropy and inhomogeneity of the material on the BEM solutions.

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