Finite Element Methods for Problems in Solid Mechanics

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.