Analysis of Algorithms for Continuum Models of Complex Materials

该项目的重点是数值方案的分析和算法的开发，以模拟由于其微观结构组成而具有复杂流变行为或机械响应的材料。例子包括聚合物、液晶和血液，其弹性分子或细胞在宏观尺度上影响宏观性质。 这些材料是由强大的偏微分方程系统，其结构特性捕获重要的属性和物理原理，它是重要的是开发数值方案忠实地继承这些。这个项目将汇集偏微分方程，连续介质力学和数值分析的工具，分析数值方案来模拟这些系统。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，例如微机械设备、喷墨打印机、生物材料、太阳能设备和假体器官。 预测材料的反应是一个必要的工具，需要确定生物或生理功能，或设计和制造这些材料，或设计的大量设备，利用其特殊性能。这里提出的研究将导致改进的理解的数学模型和计算工具来完成这些任务。

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.