Recovery Type a Posteriori Estimation and Mesh Refinement for the Finite Element Method on Anisotropic Meshes

各向异性网格有限元法的恢复型后验估计和网格细化

• 批准号: 0811232
• 负责人: Weiming Cao
• 金额: $ 12.18万
• 依托单位: University of Texas at San Antonio
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811232&HistoricalAwards=false
• 关键词: Recovery; Type; Posteriori; Estimation; Mesh

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 adaptivemesh 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 thefinite 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 moreeffectively. 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.

对于表现出强各向异性特征的问题，只要网格正确对齐并保持正确的纵横比，基于各向异性网格的有限元方法 (FEM) 比基于各向同性网格的有限元方法更有效。然而，对其进行的严格分析很少，并且大多数计算工作仅限于基于启发式论证和临时处理的线性元素。该项目的目的是为各向异性网格上的有限元法提供数值分析，并为自适应网格细化开发有效的控制。该项目的主要目标包括：（i）分析各向异性网格上有限元法的各种恢复类型误差估计器； (ii) 制定可靠的网格度量和网格质量测量来控制各向异性网格细化过程； (iii) 开发用于有限元解决方案后处理的软件工具箱，包括误差估计、网格度量构建和网格质量评估。 (iv) 为研究生提供科学和工程中自适应有限元计算的研究培训。误差估计和自适应网格细化是实际有限元模拟的两个主要组成部分。前者评估解决方案的准确性并在交付时提供质量保证。当未达到所需的精度时，后者是改进解决方案不可或缺的工具。对于各向异性有限元，允许单元又长又窄，以更有效地解决潜在问题。它们的大小、方向和纵横比可能有所不同。该项目旨在更好地理解各向异性有限元的行为，并开发更有效的有限元模拟技术。由于恢复型误差估计技术简单、鲁棒性强且在工程中最常用，因此它将成为本研究的主要焦点。该项目的成功完成将为航空航天工程、材料加工和生物医学研究中的实用有限元建模和仿真提供严格的数学理论和更高效的算法。