For problems exhibiting strong anisotropic features the finite element method (FEM) based on anisotropic meshes can be much more efficient than the one based on isotropic meshes, provided the meshes are properly aligned and the right aspect ratios are maintained. However, there has been little rigorous analysis about it, and most computational work has been restricted to linear elements, based on heuristic justification and ad-hoc treatments. The aim of this project is to provide the numerical analysis for the FEM on anisotropic meshes and develop effective controls for the adaptive mesh refinement. The main objectives of this project include: (i) Analyze various recovery type error estimators for the FEM on anisotropic meshes; (ii) Develop reliable mesh metrics and mesh quality measures to control the anisotropic mesh refinement process; (iii) Develop a software tool box for the post-processing of the finite element solutions, including error estimation, mesh metrics construction, and mesh quality evaluation. (iv) Provide research training to graduate students on adaptive finite element computation in science and engineering.
Error estimation and adaptive mesh refinement are two major components in practical finite element simulations. The former assesses the accuracy of the solution and provides the quality assurance when it is delivered. The latter is an indispensable tool for improving the solution when the desired accuracy has not been reached. For anisotropic FEM, the elements are allowed to be long and narrow to fit the underlining problems more effectively. They may vary in sizes, orientations, and aspect ratios. This project is aimed at providing better understanding of the behavior of anisotropic FEM and developing more efficient techniques for the FE simulations. Since the recovery type error estimation techniques are simple, extraordinary robust, and most commonly used in engineering, it will be the main focus of this research. Successful completion of this project will provide rigorous mathematical theory and more efficient algorithms for practical FE modeling and simulations in aerospace engineering, material processing, and biomedical researches.
