Analysis of Algorithms for Continuum Models of Complex Materials

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The focus of this project is the analysis of numerical schemes, andthe development of algorithms, to simulate materials which exhibitintricate rheological behavior or mechanical response due to theirmicrostructural makeup. Examples include polymers, liquid crystals,and blood, whose elastic molecules or cells influence the macroscopicproperties at the macroscopic scale. These materials are modeled byformidable systems of partial differential equations whose structuralproperties capture important properties and physical principles, andit is important to develop numerical schemes to faithfully inheritthese. This project will bring together tools from partialdifferential equations, continuum mechanics, and numerical analysis,to analyze numerical schemes to simulate these systems. The ability to simulate complex materials is a key technology requiredfor the design and development of many next generation products suchas micro-mechanical devices, ink jets, bio-materials, solar energydevices, and prosthetic organs. Predicting material response is anessential tool needed to determine biological or physiologicalfunction; or to design and manufacture these materials; or for thedesign of the multitudes of devices which use their specialproperties. The research proposed here will result in improvedunderstanding of the mathematical models and the computational toolsused to accomplish these tasks.