Analysis of Algorithms for Simulating Complex Materials

本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力学响应的材料。例如，一种材料的颗粒、分子或细胞的弹性性质经常影响宏观性质。这些材料是由强大的偏微分方程组模拟的，开发数值格式以忠实地表示这些模型的数学结构是很重要的。将使用偏微分方程组、连续介质力学和数值分析的工具来分析模拟这些系统的数值方案。过去的经验表明，新的数学工具可以产生继承了这些模型重要结构特性的数值格式，而对现有格式的深入理解往往会导致改进和更简单的算法。本质上，所有生物和人造材料由于其精细的尺度组成，都表现出复杂的宏观行为。例子包括微型机电系统(MEMS)、生物流体、喷墨设备的墨水、半导体、液晶和塑性变形的金属。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，或设计和制造利用其特殊性质的设备的高度所需的关键技术。这项研究将提高我们对用于完成这些任务的数学模型和计算工具的理解。用来模拟这种现象的方程很复杂，也很难理解，许多研究都是为了揭示这些模型的理论(数学)性质。这项工作是对工程界和国家实验室采取的更为实际的方法的补充。

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.