A polyhedral scaled boundary finite element method solving three-dimensional heat conduction problems

TL;DR

This paper introduces a three-dimensional polyhedral scaled boundary finite element method (PSBFEM) for heat conduction, integrating Wachspress polygonal shape functions and octree mesh acceleration to efficiently handle complex geometries. By deriving the scaled boundary formulation and coupling it with polyhedral elements, the method achieves higher accuracy than conventional FEM for the same mesh density and enables significant reductions in preprocessing and computational cost, especially when using the octree parent-element acceleration. The approach is implemented in ABAQUS via a UEL and validated through patch tests, 3D beam, steady-state, transient, and complex geometry examples (including Stanford's bunny), demonstrating robust convergence and practical efficiency. The results indicate PSBFEM is particularly advantageous for large-scale, geometrically intricate heat conduction problems and lays groundwork for extensions to multiphysics and adaptive meshing. Overall, PSBFEM provides a flexible, accurate, and efficient framework for 3D heat conduction analysis in complex domains.

Abstract

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.

Figures (29)

• Figure 1: Scaled boundary coordinates ($\xi, \eta, \zeta$) representing the coordinate system centered at $O$ for an arbitrary faceted polyhedron.
• Figure 2: Polyhedral elements construction based on polygonal surfaces: (a) traditional polyhedral element; (b) polyhedral element constructed from polygonal surfaces; (c) traditional octree element; (d) octree element constructed from polygonal surfaces.
• Figure 3: Comparison of the surfaces between traditional octree mesh and polygon-based octree mesh.
• Figure 4: Numerical integration techniques for arbitrary polytopes.
• Figure 5: Barycentric coordinates: Wachspress basis function.