This paper presents a novel localized collocation Trefftz method (LCTM) for heat conduction analysis in two kinds of heterogeneous materials (functionally graded materials and multi-medium materials) under temperature loading. In contrast to the conventional collocation Trefftz method (CTM), the proposed LCTM divides the whole domain into many stencil support domains consisting of several discretization nodes. 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 equations in the stencil support domains. Based on the moving least square (MLS) 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. Besides, the quadrant criterion is applied to guarantee the stability of the proposed LCTM, and the domain decomposition method (DDM) is introduced to divide the multi-medium domains into several single-medium sub-domains. Numerical results demonstrate the accuracy and efficiency of the proposed LCTM in comparison with the known analytical solutions and the finite element method (FEM) results.

本文提出了一种新的局部配置 Trefftz 方法 (LCTM)，用于两种异质材料（功能梯度材料和多介质材料）在温度载荷下的热传导分析。与传统的搭配 Trefftz 方法 (CTM) 相比，所提出的 LCTM 将整个域划分为许多由多个离散化节点组成的模板支持域。提出了一种有效的技术，即广义互易法 (GRM)，以推导出与问题相关的 T 完全函数，用于逼近模板支持域中非齐次方程的特定解。基于移动最小二乘 (MLS) 技术和 T 完全函数，通过使用相关模板支持域中相邻离散化节点处的 T 完全函数的线性组合，可以导出特定节点处的 LCTM 数值微分公式。它继承了传统 CTM 的半解析特性，避免了病态密集矩阵问题。此外，采用象限准则来保证所提出的LCTM的稳定性，并引入域分解方法（DDM）将多媒体域划分为多个单介质子域。与已知的解析解和有限元方法 (FEM) 结果相比，数值结果证明了所提出的 LCTM 的准确性和效率。它继承了传统 CTM 的半解析特性，避免了病态密集矩阵问题。此外，采用象限准则来保证所提出的LCTM的稳定性，并引入域分解方法（DDM）将多媒体域划分为多个单介质子域。与已知的解析解和有限元方法 (FEM) 结果相比，数值结果证明了所提出的 LCTM 的准确性和效率。它继承了传统 CTM 的半解析特性，避免了病态密集矩阵问题。此外，采用象限准则来保证所提出的LCTM的稳定性，并引入域分解方法（DDM）将多媒体域划分为多个单介质子域。与已知的解析解和有限元方法 (FEM) 结果相比，数值结果证明了所提出的 LCTM 的准确性和效率。