Analysis of Algorithms for Simulating Macroscopic Material Response

科学的本质就是解释和理解自然现象，以便预测和预测结果。最成功的预测是将自然界的基本规律整合到感兴趣的现象的概念模型中。牛顿发展了数学工具来表达许多基本的自然定律，从而产生了具有无与伦比的预测能力的数学模型。这些模型由关联感兴趣的物理量的复杂方程系统组成，构成了现代工程和科学的概念基础。求解这些复杂的方程组是实现这些理论潜力所需的关键技术，而本项目正在研究的计算工具在建模过程的这一步是不可或缺的。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。提高计算模型的预测能力将在许多下一代设备的开发和制造中发挥重要作用，如微机械设备、生物材料和假体器官。预测材料响应对于确定这些设备的生物和/或生理功能、可靠性和耐久性至关重要。除了技术发展，该项目还将支持对下一代科学家的教育和培训，这些科学家需要保持惊人的发现速度和我们在这些学科中的科学领导地位。这一建议的重点是开发和分析数值方案来模拟材料的宏观响应取决于其精细尺度结构的状态。当材料颗粒表现出弹性、吸引和/或排斥、可能导致相形成的熵相互作用和内部耗散时，这种情况是典型的。在宏观尺度上，这些效应用耦合到动态运动方程的内部变量来模拟。这种多尺度特性带来了许多建模、数学和数值方面的挑战。具有微观结构的材料模型包括强大的偏微分方程组，这些偏微分方程组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求解这些方程，但在数学理论方面还存在许多空白，对它们的基本性质的分析也非常少。在这种情况下，开发忠实地继承物理系统的复杂相互作用的数值格式是很重要的。经验表明，这一范例可以加深对当前方案的理解，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组、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.