An LT-BEM formulation for problems of anisotropic functionally graded materials governed by transient diffusion–convection–reaction equation

Abstract

Problems of anisotropic functionally graded media which are governed by the transient diffusion–convection–reaction equation of spatially varying coefficients are discussed in this paper. A mathematical analysis is used to transform the variable coefficient equation into a constant coefficient equation. A boundary-only integral equation is then derived from this constant coefficient equation after being Laplace transformed. Numerical solutions to the problems are sought by using a boundary element method (BEM) which is combined with the Stehfest formula for the numerical Laplace transform inversion. Some problems considered are those of compressible or incompressible flow, and of media which are quadratically, exponentially or trigonometrically graded materials. The results obtained show that the analysis used to transform the variable coefficients equation into the constant coefficients equation is valid, and the mixed LT-BEM is easy to implement and accurate for finding numerical solutions. The numerical solutions of some test problems are justified by showing their accuracy. Some non-test problems of geometrically symmetric systems are also considered to show the effect of the anisotropy and inhomogeneity of the material on the solutions by verifying the symmetry of solutions. In addition, the effect of boundary conditions is also exhibited.