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

University Of Washington, Seattle WA

Investigators

Abstract

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.