Finite element exterior calculus and applications

有限元外微积分及其应用

• 批准号: 0713568
• 负责人: Douglas Arnold
• 金额: $ 29.86万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713568&HistoricalAwards=false
• 关键词: Finite; element; exterior; calculus; applications

The first step in simulating, understanding, predicting, and eventually controlling a complex natural or man-made physical system, or even social system, is often to model the system mathematically. For a wide variety of systems--for example those based on solid mechanics, fluid mechanics, quantum mechanics, electromagnetism, gravitation, acoustics, thermodynamics, certain stochastic processes, and many others--the best mathematical models take the form of a set of partial differential equations. For the complex modern applications that arise in structural engineering, climatology, biology, or many other fields, the resulting partial differential equations can only be solved approximately using fast computers. The development of accurate algorithms to approximately solve these equations on computers remains a tremendous challenge as we tackle new and more complex applications. One of the greatest advances for the computer solution of partial differential equations came in the past century with the development of the finite element method, which has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering. A tremendous asset of finite elements is that they not only provide a methodology to develop numerical algorithms for simulation, but also a theoretical framework in which to assess the accuracy of the computed solutions. This project aims to develop new algorithms which extend the applicability of the finite element method, and to develop new tools which allow for better understanding of the performance of finite element algorithms, and, in particular, allow precise certification of their accuracy. A particular emphasis will be on the partial differential equations of elasticity, which describe the deformation and possible fracture of a solid body subject to forces like gravity, loading, and wind, and on the partial differential equations of electromagnetism, which are used in a wide variety modeling situations involving electric power transmission, transmission of light, radio waves, and magnetism. But the framework considered will be quite general and the techniques will extend to numerous other application areas.Robust and reliable methods for solving the equations of elasticity are needed in many industrial and engineering applications, especially in the most challenging design applications, for example for aircraft, advanced buildings and bridges, and offshore oil platforms. Recent design failures, some of them catastrophic, have been traced to inadequate numerical algorithms for elasticity. Similarly reliable methods for solving the equations of electromagnetism are at the basis of much of modern technology. Thus this project has the potential to contribute to national competitiveness and public safety.

模拟、理解、预测并最终控制复杂的自然或人造物理系统甚至社会系统的第一步通常是对系统进行数学建模。 对于各种各样的系统——例如基于固体力学、流体力学、量子力学、电磁学、引力、声学、热力学、某些随机过程等的系统——最好的数学模型采用一组偏微分方程的形式。 对于结构工程、气候学、生物学或许多其他领域中出现的复杂现代应用，所得到的偏微分方程只能使用快速计算机来近似求解。 当我们处理新的、更复杂的应用时，开发精确的算法来在计算机上近似求解这些方程仍然是一个巨大的挑战。 上个世纪，随着有限元方法的发展，计算机求解偏微分方程取得了最伟大的进步之一，有限元方法已成为模拟科学和工程中出现的各种现象不可或缺的工具。 有限元的巨大财富在于它们不仅提供了开发模拟数值算法的方法，而且还提供了评估计算解决方案准确性的理论框架。 该项目旨在开发新的算法来扩展有限元方法的适用性，并开发新的工具来更好地理解有限元算法的性能，特别是允许精确验证其准确性。 特别强调的是弹性偏微分方程，它描述了固体在重力、载荷和风等力作用下的变形和可能的断裂，以及电磁学偏微分方程，它用于涉及电力传输、光传输、无线电波和磁的各种建模情况。 但所考虑的框架将非常通用，并且技术将扩展到许多其他应用领域。许多工业和工程应用都需要稳健可靠的方法来求解弹性方程，特别是在最具挑战性的设计应用中，例如飞机、先进建筑和桥梁以及海上石油平台。 最近的设计失败（其中一些是灾难性的）可归因于弹性数值算法的不足。 同样可靠的求解电磁方程的方法是许多现代技术的基础。 因此，该项目有潜力为国家竞争力和公共安全做出贡献。