The focus of this project is the analysis of numerical schemes, and the development of algorithms, to simulate materials which exhibit intricate rheological behavior or mechanical response due to their microstructural makeup. Examples include polymers, liquid crystals, and blood, whose elastic molecules or cells influence the macroscopic properties at the macroscopic scale. These materials are modeled by formidable systems of partial differential equations whose structural properties capture important properties and physical principles, and it is important to develop numerical schemes to faithfully inherit these. This project will bring together tools from partial differential equations, continuum mechanics, and numerical analysis, to analyze numerical schemes to simulate these systems.
The ability to simulate complex materials is a key technology required for the design and development of many next generation products such as micro-mechanical devices, ink jets, bio-materials, solar energy devices, and prosthetic organs. Predicting material response is an essential tool 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. The research proposed here will result in improved understanding of the mathematical models and the computational tools used to accomplish these tasks.