Applications and development of finite element exterior calculus

涉及固体材料变形、流体运动、电磁和其他现象的物理系统的计算机模拟每天都以无数种方式应用于地球物理、医学和土木工程等各个领域。一旦一个物理系统被一个数学方程系统建模，成功的模拟不仅取决于强大的计算机硬件，还取决于数学算法，这些算法可以利用计算机的高速来获得模型方程的准确解。虽然这样的算法存在于许多重要的物理系统中--而且已经通过数学分析验证，以便我们可以对结果有信心--但仍有很大的改进空间，具有巨大的潜在回报。更重要的是，需要为尚不存在的重要应用程序开发准确、快速和可验证的算法。该项目专注于开发和分析模拟计算算法的新方法，近年来，该方法在涉及从汽车车身到骨骼的固体材料变形的模拟方面取得了巨大成功。该项目的一个主要目标是增加能够准确和自信地模拟的系统的范围。其中一个重点将是将固体和流体结合在一起的复杂材料，例如人脑中的组织，或者地下水流动所在的饱和地下土壤和沙子。第二个重点将是天体物理尺度上的重力模拟，这是一类新的天文观测的核心。首席研究人员将设计、改进和验证由偏微分方程组模拟的复杂物理现象的计算机模拟算法。要开发和研究的算法是有限元方法，它是模拟科学和工程中各种现象的不可或缺的工具，其巨大的资产是，它们不仅提供了一种开发用于模拟的数值算法的方法，而且提供了评估计算解的准确性的理论框架，从而有可能开发出经过验证的方法。这项工作将基于一种称为有限元外微积分的理论，该理论是由首席研究员发起并在过去十年中发展起来的，它极大地提高了我们对有限元方法的理解，扩大了有效有限元方法可以用于的问题的范围。研究的一个主要方向是将新发现的线性弹性材料的方法扩展到更复杂的材料，如非线性、粘弹性和孔弹性材料。第二个主要方向是发展适用于爱因斯坦数值相对论方程的有限元方法。在第三个方向上，首席研究人员将作为计算科学家参加一个数学/物理理论/计算小组，试图阐明无序介质中本征函数局部化这一重要但鲜为人知的现象。虽然这种现象被称为安德森局部化，50年前在诺贝尔奖获得者的研究中发现了这种现象，但我们仍然缺乏准确预测和控制它所需的认识。

Computer simulation of physical systems involving the deformation of solid materials, the motion of fluids, electromagnetism, and other phenomena is applied in countless ways every day in areas as varied as geophysics, medicine, and civil engineering. Once a physical system has been modeled by a system of mathematical equations, successful simulation depends not only on powerful computer hardware but also on mathematical algorithms that can harness the computer's high speed to obtain accurate solutions of the model's equations. While such algorithms exist for many important physical systems -- and moreover have been certified by mathematical analysis so that we can have confidence in the results -- there remains substantial room for improvement, with large potential payoffs. Even more important is the need to develop accurate, fast, and certifiable algorithms for important applications for which they do not yet exist. This project focuses on a new approach to the development and analysis of computational algorithms for simulation that has in recent years achieved great success for simulations involving the deformation of solid materials ranging from auto bodies to bones. A primary goal of the project is to increase the range of systems that can be simulated accurately and confidently. One emphasis will be on complex materials that combine solid and fluid aspects together, such as the tissue in the human brain, or the saturated subsurface soil and sand in which groundwater flows. A second emphasis will be on the simulation of gravity on an astrophysical scale, which is at the heart of a new class of astronomical observatories.The Principal Investigator will devise, improve, and validate algorithms for the computer simulation of complex physical phenomena modeled by partial differential equations. The algorithms to be developed and studied are finite element methods, which are an indispensable tool for simulation of a wide variety of phenomena in science and engineering, with the tremendous asset 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 computed solutions, and thus the possibility to develop validated methods. The present work will be based on a theory called finite element exterior calculus, initiated by the Principal Investigator and developed over the past decade, which has greatly enhanced our understanding of finite element methods and extended the range of problems for which validated finite element methods can be used. A major direction of the research will be the extension of newly discovered methods for linearly elastic materials to more complex materials such as nonlinear, viscoelastic, and poroelastic materials. A second major direction will be the development of finite element methods suited to the Einstein equations of numerical relativity. In a third direction the Principal Investigator will participate as the computational scientist in a mathematics/physics theoretical/computational team seeking to elucidate the important but poorly understood phenomenon of localization of eigenfunctions in disordered media. Although this phenomenon, known as Anderson localization, was discovered 50 years ago in Nobel prize winning research, we still lack the understanding needed to accurately predict and control it.