Mathematical Sciences: Numerical Solution of Differential Equations in Mechanics

9500672阿诺德这个项目将涉及设计、改进和分析由偏微分方程式模拟的复杂物理现象的数值模拟方法。主要重点将放在薄结构(板壳)的建模上。具体地说，将分析一组计算工程师最近提出的一种创新的板弯曲有限元方法，并探索研究人员与Brezzi合作开发和分析用于模拟弹性壳变形的新有限元方法的新方法。后一种方法产生了迄今为止最好的理论结果，避免了困扰壳体计算的膜锁定现象。除了研究数值方法及其引入的误差外，研究者还将考虑薄壁结构的数学模型引入的误差。具体地说，将使用一种新的分析技术来研究薄板二维模型的收敛。这一技术提供了获得全局和远离侧边界(那里的收敛可能更高)的板块模拟误差的精确收敛估计的可能性。人们希望这项研究将阐明在什么情况下Reissner-Mindlin板块模型比经典的Kirchhoff模型提供更好的近似，这是一个几十年来一直未能解决的重要问题。第二个重点领域将是开发、实施、验证和传播模拟黑洞碰撞的有限元代码。这个问题涉及到在四维时空中求解无界区域上的爱因斯坦场方程组，并且在计算上要求非常高。至少在最初阶段，工作的重点将是解决黑洞碰撞的初始数据问题，即确定物理上完全一致和有意义的初始数据。初始数据问题是一个在无界三维区域上提出的非线性椭圆组。对于这个问题，将产生的计算机代码将涉及各种复杂的现代技术，这些技术尚未在计算相对论中应用，包括自适应四面体网格生成和多重网格预处理。研究人员的工作与几个关键的技术问题直接相关。实践者普遍认为，目前炮弹数值建模的技术水平是不够的，这与体育场屋顶和飞机机身等工程结构的故障有关。研究人员在板和壳的有限元方法的发展和分析以及板的建模误差方面的工作都是为了克服这些问题。模拟黑洞碰撞的工作直接受到正在建设的LIGO引力波观测站的推动。LIGO有望在下个十年初探测到黑洞碰撞产生的辐射。需要大量的数值模拟来确定宇宙事件，这些事件是探测到的辐射的来源。很明显，这些模拟突破了现有硬件和软件技术的极限。研究人员的工作旨在通过采用和适应数值算法的最新进展来提高这些模拟的速度和精度。由此产生的计算机代码将适用于并行计算机体系结构，并将为在研究人员指导下工作的研究生和博士后研究员提供学习高性能计算技术的绝佳机会。

9500672 Arnold This project will involve devising, improving, and analyzing methods for the numerical simulation of complex physical phenomena modeled by partial differential equations. A major emphasis will be on the modeling of thin structures (plates and shells). Specifically, an innovative new finite element method recently proposed by a team of computational engineers for plate bending will be analyzed and a new approach the investigator introduced in collaboration with Brezzi to develop and analyze new finite element methods for modeling the deformation of elastic shells will be pursued. This latter approach has produced the best theoretical results to date for avoiding the phenomenon known as membrane locking which has plagued shell computations. In addition to studying numerical methods and the errors they introduce, the investigator will consider the errors introduced by mathematical models of thin structures. Specifically, a new technique of analysis to study the convergence of two-dimensional models of thin plates will be used. This technique gives the possibility of obtaining sharp convergence estimates for plate modeling errors both globally and away from the lateral boundary (where convergence may be higher). It is hoped that this research will clarify in what situations the Reissner-Mindlin plate model affords better approximation than the classical Kirchhoff model, an important question that eluded resolution for decades. A second area of emphasis will be on the development, implementation, validation, and dissemination of a finite element code to simulate the collision of black holes. This problem involves the solution of the Einstein field equations on an unbounded domain in four dimensional space-time and is extremely demanding computationally. Work will focus, at least initially, on the solution of the initial data problem for black hole collisions, i.e., the determination of physical ly consistent and meaningful initial data. The initial data problem is a nonlinear elliptic system posed on an unbounded three-dimensional domain. For this problem the computer code to be produced will involve a variety of sophisticated modern techniques that have not yet been applied in computational relativity, including adaptive tetrahedral mesh generation and multigrid preconditioning. The investigator's work is directly related to several critical technological issues. It is generally agreed by practitioners that the current state of the art of numerical modeling of shells is inadequate, and this has been implicated in the failure of engineering structures such as stadium roofs and airplane fuselages. The investigator's work in both the area of the development and analysis of finite element methods for plates and shells, and on plate modeling errors is directed at overcoming these problems. The work on the simulation of black hole collisions is directly motivated by the ongoing construction of the LIGO gravitational wave observatory. LIGO is expected to be able to detect radiation from black hole collisions early in the next decade. Massive numerical simulations are needed to identify the cosmological events which are the sources for the detected radiation. It is clear that these simulations push the limits of available hardware and software technology. The investigator's work is directed at improving the speed and accuracy of these simulations by employing and adapting recent advances in numerical algorithms. The resulting computer codes will be adapted to parallel computer architectures and will provide an excellent opportunity for the graduate students and postdoctoral fellows working under the supervision of the investigator to learn high performance computing techniques.