Analysis of Algorithms for Simulating Complex Materials

该项目的重点是开发和分析数值算法，以模拟由于其微观结构组成而表现出复杂的流变行为或机械响应的材料。例如，材料的微粒、分子或细胞的弹性特性经常影响其宏观特性。这些材料是由强大的偏微分方程系统建模的，重要的是要开发数值格式来忠实地表示这些模型的数学结构。工具从偏微分方程，连续力学和数值分析，将用于分析数值方案来模拟这些系统。过去的经验表明，新的数学工具可以导致继承这些模型的重要结构特性的数值方案，并且对当前方案的更深入理解通常会导致改进和更简单的算法。基本上，所有的生物材料和人造材料由于其精细的尺度构成而表现出复杂的宏观行为。例子包括微机电系统（MEMS）；生物体液;喷墨装置用油墨；半导体;液晶;以及发生塑性变形的金属。预测材料反应是确定生物或生理功能所需的基本技术；或者设计和制造这些材料；或者用于设计使用其特殊属性的众多设备。这项研究将提高我们对数学模型和用于完成这些任务的计算工具的理解。用来模拟这种现象的方程是复杂的，而且很难理解，许多研究都是为了揭示这些模型的理论（数学）特性。这项工作补充了工程界和国家实验室所采用的更实用的方法。

The focus of this project is the development and analysis of numericalalgorithms to simulate materials which exhibit intricate rheologicalbehavior or mechanical response due to their microstructuralmakeup. For example, elastic properties of particulates, molecules, orcells of a material frequently influence the macroscopic properties.These materials are modeled by formidable systems of partialdifferential equations, and it is important to develop numericalschemes to faithfully represent the mathematical structure of thesemodels. Tools from partial differential equations, continuummechanics, and numerical analysis, will be used to analyze numericalschemes to simulate these systems. Past experience has shown that newmathematical tools can lead to numerical schemes which inheritimportant structural properties of these models, and that a deeperunderstanding of the current schemes frequently leads to improved andsimpler algorithms.Essentially all biological and manufactured materials exhibit complexmacroscopic behavior due to their fine scale makeup. Examples includemicro electromechanical systems (MEMS); biological fluids; ink for inkjet devices; semiconductors; liquid crystals; and metals undergoingplastic deformation. Predicting material response is an essentialtechnology needed to determine biological or physiological function;or to design and manufacture these materials; or for the design of themultitudes of devices which use their special properties. Thisresearch will improve our understanding of the mathematical models andthe computational tools used to accomplish these tasks. The equationsused to model such phenomena are complex and poorly understood, andmuch of the research is directed to revealing the theoretical(mathematical) properties of these models. This work complements themore practical approaches undertaken in the engineering community andat the national laboratories.