Analysis of Algorithms for Simulating Macroscopic Material Response

科学的本质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的基本规律融入感兴趣的现象的概念模型中。牛顿发展了数学工具来表达自然的许多基本定律，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。这些复杂方程组的求解是实现这些理论潜力所需的关键技术，而本项目所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、可靠性和耐用性至关重要。除了技术发展之外，该项目还将支持下一代科学家的教育和培训，以保持我们在这些学科中的卓越发现步伐和科学领导地位。本建议的重点是发展和分析数值方案，以模拟材料的宏观响应取决于其精细尺度结构的状态。这种情况是典型的，当材料颗粒表现出弹性，吸引和/或排斥，熵的相互作用，可以导致相的形成，和内部耗散。在宏观尺度上，这些效应是用耦合到动力学运动方程的内部变量来建模的。这种多尺度特征引起了许多建模、数学和数值方面的挑战。具有微观结构的材料模型涉及强大的偏微分方程组，它继承了物理系统的输运和惯性效应、构型能量和耗散之间的微妙平衡。虽然在过去的二十年中，工程和科学计算社区已经开发了许多算法和代码来解决这些方程，但在数学理论中存在许多空白，并且对其基本性质的分析很少。在这种情况下，制定忠实地继承物理系统复杂相互作用的数值格式是重要的。经验表明，这种范式可以导致对当前方案的更深入的理解，并经常导致改进和更简单的算法。该项目将汇集来自偏微分方程、连续介质力学和数值分析的工具，以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.