Hybrid-Trefftz Finite Element Method for Heat Conduction in Functionally Graded Materials

Accurate and efficient analysis of heat conduction behaviors in functionally graded materials (FGMs) is very important for the reliable design of equipment and structures under high temperature environments. The hybrid-Trefftz finite element method (HTFEM) for steady-state and transient heat conduction analysis in FGMs is introduced in this chapter. For transient heat conduction problems, Laplace transformation technique is adopted to deal with the time-dependent terms and then one of the popular numerical inverse Laplace transformations, Stehfest algorithm, is introduced to regain the time-dependent numerical solutions. In the HTFEM model, the Trefftz functions including the functionally graded features of the FGMs can be derived via various variable transformations, which are adopted to approximate the temperature/heat flux inside the element. The related element stiffness matrix can be obtained via a variational functional based on the developed Trefftz functions. Numerical accuracy and efficiency of the proposed HTFEM model is assessed through several benchmark examples. Compared with the standard FEM, the HTFEM is a semi-analytical and efficient computational method without the sensitivity issues of mesh distortion for heat conduction in FGMs.

Keywords: Functionally graded materials, Heat conduction, Hybrid FEM, Laplace transformation, Radial Trefftz function, T-complete function, Variable transformations.