A distortion-insensitive four-noded membrane quadrilateral that passes the patch test

摘要

It is widely believed that a four-noded flat quadrilateral membrane element that passes the patch test must he sensitive to mesh distortion as regards response to inplane bending actions. This sensitivity was observed numerically in distortion test benchmarks by the mid-1970s and stated as a proved theorem by MacNeal in 1987. This paper shows that while the theorem is indeed correct, it is possible to develop a quadrilateral, called UBOQP (for Unidirectional Bending-Optimal Quadrilateral Panel) that passes the patch test for any geometry while being bending exact along one direction. This model is directly constructed using the framework of finite element templates by energy fitting arguments. UBOQP is not obtainable by any conventional FEM formulation, which illustrates the customization power of templates. Numerical experiments in a well known mesh distortion test verify the exact response to bending for any material law, aspect ratio and skew angles.