The polygonal finite element method for solving heat conduction problems

Abstract

In this paper, a general polygonal finite element method (FEM-polygon) based on the Wachspress shape function is used to solve two-dimensional heat conduction problems. The finite element variational form of the heat conduction problem is first established based on the weighted residual method. Then the formulations of the thermal stiffness matrix, thermal damping matrix, and nodal heat vector are derived using the Wachspress coordinates. The global stiffness matrix is further assembled and the well-behaved algebraic equations of the thermal system are established. Finally, intensive numerical tests are conducted to verify the super convergence in temperature and the high precision in equivalent energy and temperature gradient of the present FEM-polygon for 2D heat conduction problems. Besides, the proposed model also performs excellently for traditional triangular and quadrilateral elements.