Accurate Stress Recovery for the Two-Dimensional Fixed Grid Finite Element Method

Abstract

Finite element analyses can be rather complicated because numerous industrial problems are geometrically complex thus requiring a great amount user intervention to create an appropriate mesh for the analysis. The mesh generation task can be simplified by the use of the fixed grid finite element method (FGFEM) [1] or other similar methods where the domain to be analyzed is embedded into a simple and easy-to-mesh domain, whose mesh is independent of the geometry of the component. The need for the use of recovery techniques is especially important in these implementations of the finite element method (FEM) as the accuracy of the results can be considerably low in the elements located along the boundary [2,3] because they are only partially located in the domain. This paper shows the adaptation of a stress recovery technique, developed for the standard FEM [4] and based on the superconvergent patch recovery technique [5], to the FGFEM framework, where the elements do not fit the boundary. The proposed procedure uses constraint equations to obtain a recovered stress field that, at each patch, satisfies the equilibrium and compatibility equations, thus providing very accurate results. Different problems (with and without an analytical solution) have been used to test the accuracy of the technique for linear and quadratic quadrilateral elements. The behavior of the proposed technique has been compared with a simplified version of the recovery technique where no satisfaction of equilibrium and compatibility equations has been imposed. The numerical analysis show the robustness and high accuracy of the results provided by the technique presented in this paper which are considerably more accurate than the FE results. The recovered stresses have been used to create an accurate recovery type error estimator. The numerical results also show the high importance of enforcing the satisfaction of the equilibrium equations during the recovery process. A comparison with a commercial FE code has shown the computational efficiency of analysis tools based on the use of FGFEM-type codes together with the recovery technique proposed in the paper.