Steady-state and transient thermal stress analysis using a polygonal finite element method

TL;DR

The paper tackles the challenge of conducting accurate thermal-stress analyses on complex geometries with non-matching meshes by introducing a polygonal finite element method (PFEM) built on Wachspress basis functions. It develops steady-state and transient formulations, employs triangulation-based numerical integration, and integrates the method into ABAQUS via a User-Defined Element (UEL), augmented by a quadtree-based acceleration that reuses precomputed matrices. Across four benchmark problems, PFEM demonstrates superior convergence and accuracy compared to conventional FEM, especially when local refinement or irregular boundaries are present, while achieving significant computational savings with the proposed acceleration. The work highlights PFEM’s potential for large-scale thermo-mechanical simulations involving complex geometries and multi-resolution modeling, with implications for engineering design and analysis.

Abstract

This work develops a polygonal finite element method (PFEM) for the analysis of steady-state and transient thermal stresses in two dimensional continua. The method employs Wachspress rational basis functions to construct conforming interpolations over arbitrary convex polygonal meshes, providing enhanced geometric flexibility and accuracy in capturing complex boundary conditions and heterogeneous material behavior. A quadtree-based acceleration strategy is introduced to significantly reduce computational cost through the reuse of precomputed stiffness and mass matrices. The PFEM is implemented in ABAQUS via a user-defined element (UEL) framework. Comprehensive benchmark problems, including multi-scale and non-matching mesh scenarios, are conducted to verify the accuracy, convergence properties, and computational efficiency of the method. Results indicate that the proposed PFEM offers notable advantages over conventional FEM in terms of mesh adaptability, solution quality, and runtime performance. The method shows strong potential for large-scale simulations involving thermal-mechanical coupling, complex geometries, and multi-resolution modeling.

Figures (26)

• Figure 1: Construction of the polygonal FEM element; (a) Wachspress basis function; (b) numerical integration techniques for polygon.
• Figure 2: Mapping relationship between the parent element and sub-element.
• Figure 3: Discretization techniques of the quadtree mesh and polygonal mesh.
• Figure 4: Polygonal finite element method support element types.
• Figure 5: The model of the 2D ring plate; (a) geometry and boundary conditions; (b) polygonal mesh at size 0.1 m; (c) quadrilateral mesh at size 0.1 m.