Analysis of Algorithms for Continuum Models of Complex Materials

本项目的重点是数值格式的分析和算法的开发，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。例如聚合物、液晶和血液，它们的弹性分子或细胞在宏观尺度上影响宏观性质。这些材料由强大的偏微分方程组模拟，这些偏微分方程组的结构性质捕捉到了重要的性质和物理原理，开发数值方案以忠实地继承这些性质是很重要的。这个项目将汇集来自偏微分方程组、连续介质力学和数值分析的工具，以分析模拟这些系统的数值方案。模拟复杂材料的能力是设计和开发许多下一代产品所需的关键技术，如微机械设备、喷墨、生物材料、太阳能设备和假体器官。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，或设计大量利用其特殊性质的设备所必需的工具。这里提出的研究将导致对用于完成这些任务的数学模型和计算工具的更好理解。

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.