Application of Trefftz Method to Heat Conduction Problem in Functionally Graded Materials

Abstract

In this paper we present a Trefftz method for solving steady and transient heat conduction problems in FGMs, including nonlinear FGMs whose properties are dependent on temperature. For the case of steady heat transfer, Tcomplete solutions of governing equation for nonlinear exponential FGMs are derived by Kirchhoff transformation and coordinate transformation, and then, they can be used to model the temperature fields. For transient case, the analog equation method is used to convert the original governing equation to an equivalent Poissons equation. Then, the homogeneous solution is obtained by linear combination of a set of T-complete solutions while the radial basis functions (RBF) are employed to approximate the inhomogeneous terms. Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized by time-stepping method, a Trefftz-RBF scheme is established. The performance of the proposed methods are assessed through three numerical examples. The results are presented for illustrating the accuracy and efficacy of the proposed numerical models. Keywords: Functionally graded material (FGM), steady heat conduction, transient heat conduction, T-complete solution, Trefftz method, RBF, meshless discretization approach, T-Complete Solutions, Analog Equation Method, Time Stepping Method