Finite Element Methods for Problems in Solid Mechanics

题目：固体力学问题的有限元方法P.I: Richard S. Falk摘要：本课题的第一个主要研究领域是板壳模型的有限元逼近。一个目标是开发避免“锁定”问题的方法，这种问题在标准近似中导致薄壳和薄板的差结果。项目的第二部分是设计和分析二阶椭圆型边值问题的最小二乘和混合有限元近似的有效多网格预调节器。使用计算和分析技术的第三个一般研究领域是理解与应力驱动的可再结晶棒和膜的不稳定性以及曲率驱动的表面扩散有关的数学模型的预测。项目的最后一部分将尝试开发一阶线性双曲系统的有限元方法以及爱因斯坦方程的简化近似。固体力学中的许多问题都可以用数学模型来研究。这些模型提供了一种经济有效的方法来定量预测机械系统在各种条件下的变化，比如外力的应用。特别是，它们为使用昂贵或困难的实验提供了另一种选择。通常，当现实的数学模型被公式化时，它们是用方程来表示的，这些方程的解代表了工程师和科学家感兴趣的物理量，而这些解是无法解析地确定的，也就是说，用一种简单的形式可以很容易地写下来。然而，通过采用数值方法，仍然可以找到数学模型所描述的物理量的良好近似值。通常，这些数值方法使用高速计算机来进行大量的计算。这个项目是关于设计和分析数值逼近方案的一些重要的数学模型用于力学。考虑的数学模型包括薄壳和薄膜的数学模型。

Proposal NO: DMS-9704556 Title: Finite Element Methods for Problems in Solid Mechanics P.I.: Richard S. Falk Abstract: The first main area of study of this project is the finite element approximation of plate and shell models. One objective is to develop methods which avoid the ``locking'' problem which leads, for standard approximations, to poor results for thin shells and plates. A second part of the project is to design and analyze effective multigrid preconditioners for least squares and mixed finite element approximations to second order elliptic boundary value problems. A third general area of study, using both computational and analytical techniques, is to understand the predictions of mathematical models which are concerned with stress driven instability of recrystallizable rods and films and with curvature driven surface diffusion. The last part of the project will attempt to develop finite element methods for first order linear hyperbolic systems and also for the approximation of simplified versions of the Einstein equations. Many problems in solid mechanics can be studied by using mathematical models. These models offer a cost-effective way to make quantitative predictions about how mechanical systems will change under various conditions, such as the application of external forces. In particular, they offer an alternative to the use of costly or difficult experiments. Typically, when realistic mathematical models are formulated, they are in terms of equations whose solutions, which represent physical quantities of interest to engineers and scientists, are not able to be determined analytically, i.e., in a simple form one can easily write down. However, by employing numerical methods, good approximations to the physical quantities which are described by the mathematical models may still be found. Typically, these numerical methods use high-speed computers to do the large number of calculations involved. This project is concerned with the design and analysis of numerical approximation schemes for a number of important mathematical models used in mechanics. Among the mathematical models considered are those for thin shells and films.