Transient heat conduction analysis of functionally graded materials by a multiple reciprocity boundary face method

This paper applies the multiple reciprocity boundary face method to solve transient heat conduction problems of functionally graded materials. It is assumed that the material properties vary in z-direction exponentially or quadratically. Several variable substitutions are employed to convert this problem to standard diffusion equations. After those substitutions, however, the initial condition and the heat source density function, which lead the domain integral of the boundary integral equation, become more complicated. In this application, the Laplace transformation is used to remove the time dependence of the problem. A multiple reciprocity formulation with the modified Helmholtz fundamental solutions is used to convert the domain integral into boundary integrals and several non-integral terms. Numerical examples show that the results of our method are in good agreement with the analytical solutions or finite element method solutions at both internal and boundary points.