Applications and development of finite element exterior calculus

物理系统的计算机模拟，包括固体材料的变形、流体的运动、电磁学和其他现象，每天都以无数的方式应用于物理学、医学和土木工程等领域。 一旦一个物理系统被数学方程系统建模，成功的仿真不仅依赖于强大的计算机硬件，还依赖于数学算法，这些算法可以利用计算机的高速来获得模型方程的精确解。 虽然这种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进行了验证，因此我们可以对结果充满信心，但仍然有很大的改进空间，具有很大的潜在回报。 更重要的是，需要为尚未存在的重要应用程序开发准确，快速和可验证的算法。 该项目的重点是开发和分析用于模拟的计算算法的新方法，近年来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。 该项目的主要目标是增加可以准确和自信地模拟的系统范围。 其中一个重点将是结合联合收割机固体和流体方面在一起的复杂材料，如人脑中的组织，或地下水流动的饱和地下土壤和沙子。 第二个重点是在天体物理尺度上模拟重力，这是一类新的天文观测站的核心。首席研究员将设计，改进和验证计算机模拟偏微分方程模拟复杂物理现象的算法。要开发和研究的算法是有限元方法，这是一个不可或缺的工具，用于模拟各种各样的现象，在科学和工程，具有巨大的资产，他们不仅提供了一种方法来开发数值模拟算法，而且还提供了一个理论框架，在其中评估计算的解决方案的准确性，从而有可能开发验证的方法。目前的工作将是基于一个理论，称为有限元外微积分，发起的主要研究者和开发在过去的十年中，这大大提高了我们的理解有限元方法和扩展的范围内的问题验证有限元方法可以使用。 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，粘弹性和多孔弹性材料。第二个主要方向将是发展适合于爱因斯坦数值相对论方程的有限元方法。在第三个方向，首席研究员将作为计算科学家参加数学/物理理论/计算团队，以阐明无序介质中本征函数局部化的重要但知之甚少的现象。尽管这种现象被称为安德森本地化，在50年前的诺贝尔奖获得者研究中被发现，但我们仍然缺乏准确预测和控制它所需的理解。

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.