Finite Element Approximation of Problems in Solid Mechanics

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Dear Jong-Shi: I was very pleased to receive your email saying that you plan to recommend funding of my NSF proposal as a 36 month standard award of $150,710. As you requested, here is an abstract of the project. Please let me know if this is acceptable. Regards, Rick Falk Abstract: Finite Element Approximation of Problems in Solid Mechanics The finite element approximation of mathematical models of thin plates and shells is studied. For the Reissner-Mindlin plate model, there are many proven "locking-free" methods (i.e., no accuracy loss for smaller thickness) using triangular and rectangular elements. It is proposed to analyze quadrilateral elements for this problem and study important but unresolved issues about these elements in more general contexts. In addition to the shear locking which causes problems in the approximation of plate models, shells also suffer from the problem of membrane locking. The goal is to improve on the shell elements so far proposed and provide a rigorous analysis of convergence. Variational methods used previously for the derivation and analysis of plate models will be extended to the derivation and analysis of shell models. Discontinuous Galerkin finite element methods are promising candidates for a robust approximation method for both convection-dominated and diffusion-dominated convection-diffusion problems. Further analysis is proposed to demonstrate their effectiveness more conclusively. Computational and analytical techniques are proposed to understand the predictions of 2-D mathematical models concerned with stress driven instability, expanding on previous grant work done on a simpler 1-D model. This proposal is concerned with the use of mathematical models to study several problems in solid mechanics. The use of mathematical models offers a cost-effective way to make quantitative predictions about how mechanical systems will change when external forces are applied and serves as 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, high performance computing is needed 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. These include models of elastic plates and shells (used for example to design the roof of a building to avoid collapse) and models of nano-scale solid crystals (which can be used to study instabilities in certain materials).