Analysis of Algorithms for Simulating Complex Materials

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

The focus of this project is the development and analysis of numerical algorithms to simulate materials which exhibit intricate rheological behavior or mechanical response due to their microstructural makeup. For example, elastic properties of particulates, molecules, or cells of a material frequently influence the macroscopic properties. These materials are modeled by formidable systems of partial differential equations, and it is important to develop numerical schemes to faithfully represent the mathematical structure of these models. Tools from partial differential equations, continuum mechanics, and numerical analysis, will be used to analyze numerical schemes to simulate these systems. Past experience has shown that new mathematical tools can lead to numerical schemes which inherit important structural properties of these models, and that a deeper understanding of the current schemes frequently leads to improved and simpler algorithms. Essentially all biological and manufactured materials exhibit complex macroscopic behavior due to their fine scale makeup. Examples include micro electromechanical systems (MEMS); biological fluids; ink for ink jet devices; semiconductors; liquid crystals; and metals undergoing plastic deformation. Predicting material response is an essential technology needed to determine biological or physiological function; or to design and manufacture these materials; or for the design of the multitudes of devices which use their special properties. This research will improve our understanding of the mathematical models and the computational tools used to accomplish these tasks. The equations used to model such phenomena are complex and poorly understood, and much of the research is directed to revealing the theoretical (mathematical) properties of these models. This work complements the more practical approaches undertaken in the engineering community and at the national laboratories.