Special polygonal elements for thermal analysis of cellular material containing elliptical holes

A special hybrid polygonal finite element (FE) is constructed for the efficient investigation of two-dimensional heat conduction problem in a cellular material containing a large number of elliptical holes. The special fundamental solution representing the thermal response of a point heat source in an infinite isotropic solid containing a centered inclined elliptical hole is derived for the construction of the special element enclosing the elliptical hole. Thus, the further fine mesh division along the rim of elliptical hole is thoroughly avoided. In the special FE formulation, the special fundamental solution is taken as trial function to construct the intra-element temperature field, which exactly satisfies the governing equation of problem and the insulating condition along the hole boundary, but not the specific temperature and heat flux boundary conditions on the outer boundary of the computing domain and the continuity condition between adjacent elements. To fix this, the temperature field along the element boundary is independently approximated by the conventional FE interpolation function. Combing these two independent fields into the modified double-variable Hellinger–Reissner variational principle leads to a system of stiffness equations including element boundary integrals only, whose computing dimension is reduced by one compared to the domain integrals. More importantly, such integration feature endows the present method great capability of constructing any shaped polygonal element with different numbers of edges and nodes in a unified form with same kernels, and extremely simplifying mesh effort around the elliptical hole. Numerical experiments are performed on special polygonal element meshes for the problems associated with one, two and multiple elliptic holes and the results show a good accuracy and convergence performance for the proposed special element.