Mathematical Sciences: Numerical Solution of Differential Equations in Mechanics

小行星9500672 该项目将涉及设计，改进和分析方法， 复杂物理现象的数值模拟， 偏微分方程 一个主要的重点将是对建模 薄结构（板和壳）。具体而言，一种创新的有限 最近由一组计算工程师提出的单元法， 将分析钢板弯曲，研究者将采用一种新的方法 与Brezzi合作推出，开发和分析新的有限 本文介绍了模拟弹性壳体变形的单元法， 追求。 后一种方法产生了最好的理论结果， 日期，以避免现象称为膜锁定， 令人困扰的外壳计算。 除了研究数值方法和 他们引入的错误，研究者将考虑引入的错误 薄结构的数学模型。 具体来说，一种新的技术， 薄板二维模型收敛性分析 将用于 这项技术使获得尖锐的可能性 板块模拟误差的全局和远场收敛估计 从横向边界（其中收敛可能更高）。 希望 这项研究将澄清在什么情况下， 模型提供了更好的近似比经典基尔霍夫模型， 几十年来一直未能解决的重要问题。 的第二区域 重点将放在开发、实施、验证和 传播有限元代码，以模拟黑色碰撞 孔. 这个问题涉及爱因斯坦场方程的解 四维时空中的一个无界域， 计算上的要求。 至少在最初阶段，工作将侧重于 解决黑洞碰撞的初始数据问题，即，的 确定物理上一致和有意义的初始数据。的 初值问题是一个非线性椭圆型方程组， 三维域 对于这个问题，计算机代码是 生产将涉及各种复杂的现代技术， 尚未应用于计算相对论，包括自适应 四面体网格生成和多重网格预处理。 调查员的工作与几项关键技术直接相关。 问题. 实践者普遍认为，目前的技术水平 壳的数值模拟是不够的，这已经被牵连 在体育场屋顶和飞机等工程结构的破坏中， 机身 调查员在开发领域和 板壳及板上有限元法分析 建模误差旨在克服这些问题。 的工作 黑洞碰撞的模拟直接受到正在进行的 LIGO引力波天文台的建设。 LIGO预计 能够探测到黑洞碰撞产生的辐射 十年 需要大量的数值模拟来确定 宇宙学事件是探测到的辐射源。是 很明显，这些模拟推动了可用硬件的极限， 软件技术。 调查员的工作旨在改善 这些模拟的速度和准确性，采用和适应最近 数值算法的进展。 由此产生的计算机代码将是 适用于并行计算机体系结构，并将提供一个优秀的 研究生和博士后研究员的工作机会 在调查员的监督下学习高性能 计算机技术。

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.