Analysis of Algorithms for Simulating Macroscopic Material Response

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The very essence of science is to explain and understand natural phenomena in order to predict and forecast outcomes. The most successful predictions result when fundamental laws of nature are integrated into conceptual models of the phenomena of interest. Newton's development of mathematical tools to express many fundamental laws of nature has resulted in mathematical models with unparalleled predictive power. These models consist of complex systems of equations relating the physical quantities of interest and form the conceptual foundation of modern engineering and science. Solution of these complex systems of equations is a key technology needed to realize the potential of these theories, and the computational tools under investigation in this project are indispensable in this step of the modeling process. This project will enhance the computational tools used to simulate materials such as polymers, liquid crystals, and many biological components. Improved predictive capability of computational models will play an essential role in the development and manufacture of many next generation devices such as micro-mechanical devices, biological materials, and prosthetic organs. Predicting material response is essential to determine biological and/or physiological function, reliability, and durability of these devices. In addition to the technological developments, this project will also support the education and training of the next generation of scientists needed sustain the remarkable pace of discovery and our scientific leadership in these disciplines. The focus of this proposal is the development and analysis of numerical schemes to simulate materials whose macroscopic response depends upon the state of their fine scale structure. This scenario is typical when material particles exhibit elasticity, attraction and/or repulsion, entropic interactions which can result in phase formation, and internal dissipation. At the macroscopic scale these effects are modeled with internal variables which couple to the dynamic equations of motion. This multi-scale character gives rise to many modeling, mathematical, and numerical challenges. Models of materials with microstructure involve formidable systems of partial differential equations which inherit the delicate balance between transport and inertial effects, configurational energy, and dissipation of the physical system. While the past two decades have witnessed the development of many algorithms and codes in the engineering and scientific computing communities to solve these equations, there are many gaps in the mathematical theory and very little analysis of their fundamental properties is available. In this situation is important to develop numerical schemes which faithfully inherit the complex interactions of the physical system. Experience has shown that this paradigm can lead to a deeper understanding of the current schemes and frequently leads to improved and simpler algorithms. This project will bring together tools from partial differential equations, continuum mechanics, and numerical analysis, to develop and analyze numerical schemes which simulate these systems.