Finite Element Methods for Problems in Solid Mechanics

提案编号：dms-9704556标题：用于解决固体力学问题的有限元方法P.I.：Richard S.Falk摘要：本项目的第一个主要研究领域是板壳模型的有限元近似。一个目标是开发避免“锁定”问题的方法，“锁定”问题对于标准近似会导致薄壳和薄板的结果不佳。该项目的第二部分是设计和分析有效的多重网格预条件，用于二阶椭圆边值问题的最小二乘和混合有限元逼近。使用计算和分析技术的第三个一般研究领域是理解数学模型的预测，该数学模型涉及可再结晶棒和薄膜的应力驱动的不稳定性和曲率驱动的表面扩散。该项目的最后部分将尝试开发一阶线性双曲组的有限元方法，以及简化版本的爱因斯坦方程的近似。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不同条件下的变化，例如外力的作用。特别是，它们提供了一种替代昂贵或困难的实验的方法。通常，当制定现实的数学模型时，它们是以方程的形式表示的，其解代表了工程师和科学家感兴趣的物理量，不能以解析的方式确定，即，以一种简单的形式很容易地写下来。然而，通过使用数值方法，仍然可以找到对由数学模型描述的物理量的很好的近似。通常，这些数值方法使用高速计算机来进行大量的计算。这个项目涉及设计和分析力学中使用的一些重要数学模型的数值逼近方案。所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.