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.
