Localized Collocation Trefftz Method in Conjunction With the Generalized Reciprocity Method for Transient Heat Conduction Analysis in Heterogeneous Materials

Abstract

This paper presents a novel localized collocation Trefftz method (LCTM) in conjunction with Laplace transformation for transient heat conduction analysis in heterogeneous materials under temperature loading. In contrast to the conventional CTM, the proposed LCTM divides the whole domain into many stencil support domains consisting of several discretization nodes. Inspired by the dual reciprocity method (DRM) and multiple reciprocity method (MRM), an efficient technique, the generalized reciprocity method (GRM), is proposed to derive the problem-dependent T-complete functions for approximating the particular solution of the nonhomogeneous heat conduction equations in the local subdomains. Based on the moving least square technique and T-complete functions, the LCTM numerical differentiation formulation at a certain node can be derived by using a linear combination of the T-complete functions at its adjacent discretization nodes in the related stencil support domain. It inherits the semi-analytical property from the conventional CTM and avoids the ill-conditioned dense matrix problem, which is present particularly in large-scale heat conduction analysis. Some numerical examples of heat conduction problems in heterogeneous materials are presented, and the numerical results demonstrate the accuracy and efficiency of the proposed LCTM in comparison with the known analytical solutions.