A new boundary element method for modeling wave propagation in functionally graded materials

Abstract

In this paper, a boundary element method (BEM) based on a new boundary integral equation (BIE) formulation is proposed for modeling wave propagation problems in functionally graded materials (FGM) in the frequency domain. The material properties of the considered FGMs are assumed to be graded along spatial Cartesian coordinates, and can vary continuously either in a single axis, or in multiple axes simultaneously, according to an exponential law distribution. Similar to the Somigliana's identity, a new generalized Green's identity corresponding to the elastodynamic equations for FGMs is established first, which can be used to derive the BIE for wave propagation in FGMs for either 2-D or 3-D models. For convenience, a special case with the static and isotropic fundamental solutions are adopted in applying the Green's identity of FGMs. Finally, a boundary-domain integral equation with boundary-only solution scheme is derived. The BEM is applied to solve the BIE and quadratic elements are employed in the discretization. Several test problems in 2-D domains are studied using the BEM. The effects of the material gradient, gradation direction, as well as frequencies of the incident wave on the wave propagation in the FGMs are investigated intensively. The numerical results clearly show the effectiveness and efficiency of the developed BEM in modeling the wave propagation problems in FGMs.