Special finite element methods based on component mode synthesis techniques: analysis and applications

这项提案是利用2009年《美国复苏和再投资法案》(公法111-5)提供的资金授予的。研究目标是发展偏微分方程组驱动的复杂模型的新离散化方法。这些新的离散化方法将使这类复杂模型的精确、高效和健壮的求解成为可能。例如，由于问题中存在的尺度(空间、时间)的范围，现有的离散化方法受到多尺度问题的挑战。这里的目标是设计一种改进的离散化方法，在避免完全解析模拟的成本的同时捕捉小规模的空间和时间效应。重点将放在特殊的有限元方法上，表示使用特殊形函数的有限元类型的方法。这些函数通常是非多项式的，可以结合关于控制方程的专门知识(通过本征模式或特定解)。该方法将为定义精确、稳健和高效的特殊有限元方法提供一个系统的过程。该理论框架结合了区域分解和分量模式合成理论、线性算子的谱分解理论和有限元理论的知识。在该项目中开发的工具将适用于具有战略重要性的各种复杂问题。例如，了解异质结构的弹性行为，对肿瘤生长进行建模，以及模拟在多孔介质中的流动。这些问题基本上是多尺度的，在数学和计算上仍然具有挑战性。这项拟议的研究也有望成为计算数学研究生教育的沃土。研究生将参与最先进的离散化方法的分析和设计，并接受实用工程技术的培训，如区域分解和组件模式综合。他或她将获得一个难得的视角，将数学理论和实际工程知识结合在一起。

This proposal is awarded using funds made available by the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The research objective is the development of new discretization methods for complex models driven by partial differential equations. These new discretization methods will enable the accurate, efficient, and robust solution of such complex models. For example, existing discretization methods are challenged by multi-scale problems due to the range of scales (spatial, temporal) present in the problem. Here the objective would be to design an improved discretization method that captures the small-scale spatial and temporal effects while avoiding the cost of a fully resolved simulation. The focus will be on special finite element methods, denoting methods of finite element type that employ special shape functions. These functions, typically non-polynomials, can incorporate specialized knowledge about the governing equation (via eigenmodes or particular solutions). The approach will supply a systematic procedure for defining accurate, robust, and efficient special finite element methods. The theoretical frame combines knowledge from the theories of domain decomposition and component mode synthesis, of spectral decomposition for linear operators, and of finite elements. The tools developed in this project will apply to a wide range of complex problems of strategic importance. Examples include understanding the elastic behavior of heterogeneous structures, modeling tumor growth, and simulating flow through porous media. These problems are fundamentally multi-scale and remain mathematically and computationally challenging. The proposed research is also expected to be a fertile ground for graduate education in computational mathematics. A graduate student will be involved in the analysis and design of state-of-the-art discretization method and be trained in practical engineering techniques, like domain decomposition and component mode synthesis. He or she will get a rare perspective that combines knowledge from mathematical theory and practical engineering.