Finite Element Methods for Problems in Solid Mechanics

提案编号：DMS-9704556标题：固体力学问题的有限元方法P.I.： Richard S.福尔克 摘要： 该项目的第一个主要研究领域是板壳模型的有限元近似。 一个目标是开发方法，避免“锁定”的问题，导致标准的近似，薄壳和板的结果差。 该项目的第二部分是设计和分析有效的多重网格预处理最小二乘和混合有限元逼近二阶椭圆边值问题。 第三个一般的研究领域，使用计算和分析技术，是了解数学模型的预测，这是有关应力驱动的不稳定性的再结晶棒和膜，并与曲率驱动的表面扩散。 该项目的最后一部分将试图开发一阶线性双曲系统的有限元方法，也为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具有成本效益的方法，可以定量预测机械系统在各种条件下（例如施加外力）将如何变化。特别是，它们为使用昂贵或困难的实验提供了一种替代方法。 典型地，当实际的数学模型被公式化时，它们是根据其解（其表示工程师和科学家感兴趣的物理量）不能被解析地确定的方程，即，以一种简单的形式，人们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的数值逼近方案的一些重要的数学模型中使用的力学。 其中考虑的数学模型是那些薄壳和薄膜。

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.