Numerical simulation of modified Helmholtz boundary value problems for anisotropic exponentially graded materials

Abstract

In this paper we consider the modified Helmholtz type equation governing interior two-dimensional boundary value problems (BVPs) for anisotropic functionally graded materials (FGMs) with Dirichlet and Neumann boundary conditions. Persistently spatially changing diffusivity and leakage factor coefficients are involved in the governing equation. Both the anisotropic diffusivity and leakage factor coefficients vary according to an exponential gradation function. We use a technique of transforming the variable coefficient governing equation to a constant coefficient equation for deriving a boundary integral equation. And from the boundary integral equation obtained a standard boundary element method (BEM) is constructed to find numerical solutions to the BVPs. In order to illustrate the application of the BEM, some particular examples of BVPs are solved. The results show the convergence, accuracy, consistency between the scattering and flow solutions and efficiency (less computation time) of the BEM solutions. The results also show the impact of the inhomogeneity and anisotropy of the material on the solutions.