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

本提案使用2009年美国复苏和再投资法案（公法111-5）提供的资金进行奖励。研究目标是为偏微分方程驱动的复杂模型开发新的离散化方法。这些新的离散化方法将使此类复杂模型的精确、高效和鲁棒解成为可能。例如，现有的离散化方法受到多尺度问题的挑战，因为问题中存在的尺度范围（空间、时间）。这里的目标是设计一种改进的离散化方法，以捕获小尺度的空间和时间效应，同时避免完全解决模拟的成本。重点将放在特殊的有限元方法上，表示采用特殊形状函数的有限元类型的方法。这些函数，通常是非多项式的，可以包含有关控制方程的专业知识（通过特征模态或特解）。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程序。理论框架结合了领域分解和分量模态综合、线性算子的谱分解和有限元素的理论知识。本项目开发的工具将适用于一系列具有战略重要性的复杂问题。例子包括理解非均质结构的弹性行为、模拟肿瘤生长和模拟通过多孔介质的流动。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。该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，如领域分解和组件模式综合。他或她将获得一个罕见的视角，将数学理论知识与实际工程相结合。

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.