Hybrid FEM with Fundamental Solutions as Trial Functions for Heat Conduction Simulation

Abstract

A new type of hybrid finite element formulation with fundamental solutions as internal interpolation functions, named as HFS-FEM, is presented in this paper and used for solving two dimensional heat conduction problems in single and multi-layer materials. In the proposed approach, a new variational functional is firstly constructed for the proposed HFS-FE model and the related existence of extremum is presented. Then, the assumed internal potential field constructed by the linear combination of fundamental solutions at points outside the elemental domain under consideration is used as the internal interpolation function, which analytically satisfies the governing equation within each element. As a result, the domain integrals in the variational functional formulation can be converted into the boundary integrals which can significantly simplify the calculation of the element stiffness matrix. The independent frame field is also introduced to guarantee the inter-element continuity and the stationary condition of the new variational functional is used to obtain the final stiffness equations. The proposed method inherits the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional finite element method (FEM) and boundary element method (BEM), and avoids the difficulty in selecting appropriate terms of T-complete functions used in HT-FEM, as the fundamental solutions contain usually one term only, rather than a series containing infinitely many terms. Further, the fundamental solutions of a problem are, in general, easier to derive than the T-complete functions of that problem. Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show good numerical accuracy and remarkable insensitivity to mesh distortion.