Analysis of Algorithms for Simulating Complex Materials

该项目的重点是数值算法的开发和分析，以模拟由于其微观结构组成而表现出复杂流变行为或机械响应的材料。例如，一种材料的微粒、分子或细胞的弹性特性经常会影响其宏观特性。这些材料都是由强大的偏微分方程系统来模拟的，因此，发展数值方案来忠实地表示这些模型的数学结构是很重要的。 工具从偏微分方程，连续力学和数值分析，将被用来分析numericalschemes来模拟这些系统。过去的经验表明，新的数学工具可以产生继承这些模型的重要结构特性的数值方案，而对现有方案的深入理解往往会产生改进和更简单的算法。基本上所有的生物和人工材料由于其精细的尺度组成而表现出复杂的宏观行为。例子包括微机电系统（MEMS）;生物流体;喷墨设备的墨水;半导体;液晶;和经历塑性变形的金属。 预测材料响应是确定生物或生理功能，或设计和制造这些材料，或设计使用其特殊性能的设备所需的基本技术。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。 用于模拟这种现象的方程很复杂，而且人们对它的理解也很有限，许多研究都是为了揭示这些模型的理论（数学）性质。这项工作补充了工程界和国家实验室采取的更实用的方法。

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.