Hybrid-Trefftz finite element method for heat conduction in nonlinear functionally graded materials

Purpose - The purpose of this paper is to develop a hybrid-Trefftz (HT) finite element model (FEM) for simulating heat conduction in nonlinear functionally graded materials (FGMs) which can effectively handle continuously varying properties within an element. Design/methodology/approach - In the proposed model, a T-complete set of homogeneous solutions is first derived and used to represent the intra-element temperature fields. As a result, the graded properties of the FGMs are naturally reflected by using the newly developed Trefftz functions (T-complete functions in some literature) to model the intra-element fields. The derivation of the Trefftz functions is carried out by means of the well-known Kirchhoff transformation in conjunction with various variable transformations. Findings - The study shows that, in contrast to the conventional FEM, the HT-FEM is an accurate numerical scheme for FGMs in terms of the number of unknowns and is insensitive to mesh distortion. The method also performs very well in terms of numerical accuracy and can converge to the analytical solution when the number of elements is increased. Originality/value - The value of this paper is twofold: a T-complete set of homogeneous solutions for nonlinear FMGs has been derived and used to represent the intra-element temperature; and the corresponding variational functional and the associated algorithm has been constructed.