Analysis of Algorithms for Simulating Macroscopic Material Response

科学的本质是解释和理解自然现象，以预测和预测结果。最成功的预测结果时，基本的自然规律被整合到概念模型的现象感兴趣。牛顿发展数学工具来表达许多基本的自然规律，导致数学模型具有无与伦比的预测能力。这些模型由与感兴趣的物理量相关的复杂方程组组成，构成了现代工程和科学的概念基础。 这些复杂的方程组的解决方案是实现这些理论的潜力所需的关键技术，并且在该项目中正在调查的计算工具在建模过程的这一步骤中是不可或缺的。 该项目将增强用于模拟聚合物，液晶和许多生物成分等材料的计算工具。计算模型的预测能力的提高将在许多下一代器件的开发和制造中发挥重要作用，例如微机械器件、生物材料和假体器官。预测材料反应对于确定这些器械的生物和/或生理功能、可靠性和耐久性至关重要。除了技术发展外，该项目还将支持下一代科学家的教育和培训，以维持发现的显着步伐和我们在这些学科中的科学领导地位。 该提案的重点是数值方案的开发和分析，以模拟其宏观响应取决于其细尺度结构状态的材料。当材料颗粒表现出弹性、吸引力和/或排斥力、熵相互作用（其可导致相形成和内部耗散）时，这种情况是典型的。 在宏观尺度上，这些影响与耦合到运动的动力学方程的内部变量进行建模。 这种多尺度特性给建模、数学和数值计算带来了许多挑战。 具有微观结构的材料模型涉及强大的偏微分方程系统，该系统继承了物理系统的传输和惯性效应、构型能量和耗散之间的微妙平衡。虽然在过去的二十年里，工程和科学计算界已经开发了许多算法和代码来解决这些方程，但数学理论中存在许多空白，并且对其基本属性的分析很少。 在这种情况下，重要的是开发数值方案，忠实地继承复杂的相互作用的物理系统。 经验表明，这种模式可以导致更深入地了解目前的计划，并经常导致改进和更简单的算法。 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，开发和分析模拟这些系统的数值方案。

The very essence of science is to explain and understand natural phenomena in order to predict and forecast outcomes. The most successful predictions result when fundamental laws of nature are integrated into conceptual models of the phenomena of interest. Newton's development of mathematical tools to express many fundamental laws of nature has resulted in mathematical models with unparalleled predictive power. These models consist of complex systems of equations relating the physical quantities of interest and form the conceptual foundation of modern engineering and science. Solution of these complex systems of equations is a key technology needed to realize the potential of these theories, and the computational tools under investigation in this project are indispensable in this step of the modeling process. This project will enhance the computational tools used to simulate materials such as polymers, liquid crystals, and many biological components. Improved predictive capability of computational models will play an essential role in the development and manufacture of many next generation devices such as micro-mechanical devices, biological materials, and prosthetic organs. Predicting material response is essential to determine biological and/or physiological function, reliability, and durability of these devices. In addition to the technological developments, this project will also support the education and training of the next generation of scientists needed sustain the remarkable pace of discovery and our scientific leadership in these disciplines. The focus of this proposal is the development and analysis of numerical schemes to simulate materials whose macroscopic response depends upon the state of their fine scale structure. This scenario is typical when material particles exhibit elasticity, attraction and/or repulsion, entropic interactions which can result in phase formation, and internal dissipation. At the macroscopic scale these effects are modeled with internal variables which couple to the dynamic equations of motion. This multi-scale character gives rise to many modeling, mathematical, and numerical challenges. Models of materials with microstructure involve formidable systems of partial differential equations which inherit the delicate balance between transport and inertial effects, configurational energy, and dissipation of the physical system. While the past two decades have witnessed the development of many algorithms and codes in the engineering and scientific computing communities to solve these equations, there are many gaps in the mathematical theory and very little analysis of their fundamental properties is available. In this situation is important to develop numerical schemes which faithfully inherit the complex interactions of the physical system. Experience has shown that this paradigm can lead to a deeper understanding of the current schemes and frequently leads to improved and simpler algorithms. This project will bring together tools from partial differential equations, continuum mechanics, and numerical analysis, to develop and analyze numerical schemes which simulate these systems.