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

Abstract

This work presents 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 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.