In this study, we derived a three-dimensional scaled boundary finite element formulation for heat conduction problems. By incorporating Wachspress shape functions, a polyhedral scaled boundary finite element method (PSBFEM) was proposed to address heat conduction challenges in complex geometries. To address the complexity of traditional methods, this work introduced polygonal discretization techniques that simplified the topological structure of the polyhedral mesh and effectively integrated polyhedral and octree meshes, thereby reducing the number of element faces and enhancing mesh efficiency to accommodate intricate shapes. The developed formulation supported both steady-state and transient heat conduction analyses and was implemented in ABAQUS through a user-defined element (UEL). Through a series of numerical examples, the accuracy and convergence of the proposed method were validated. The results indicated that the PSBFEM consistently achieved higher accuracy than the FEM as the mesh was refined. The polyhedral elements offered a computationally efficient solution for complex simulations, significantly reducing computational costs.Additionally, by utilizing the octree mesh parent element acceleration technique, the computational efficiency of PSBFEM surpassed that of the FEM.

翻译：本研究推导了适用于热传导问题的三维比例边界有限元列式。通过引入Wachspress形函数，提出了一种多面体比例边界有限元法（PSBFEM）以解决复杂几何结构中的热传导难题。为应对传统方法的复杂性，本工作引入了多边形离散化技术，该技术简化了多面体网格的拓扑结构，并有效整合了多面体网格与八叉树网格，从而减少了单元面数量，提升了网格效率以适应复杂形状。所建立的列式同时支持稳态与瞬态热传导分析，并通过用户自定义单元（UEL）在ABAQUS中实现。通过一系列数值算例，验证了所提方法的精度与收敛性。结果表明：随着网格细化，PSBFEM始终能获得比传统有限元法更高的计算精度。多面体单元为复杂仿真提供了计算高效的解决方案，显著降低了计算成本。此外，通过采用八叉树网格母单元加速技术，PSBFEM的计算效率超越了传统有限元法。