Finite Element Approximation of Problems in Solid Mechanics

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Dear Jong-Shi:I was very pleased to receive your email saying that you plan to recommendfunding 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 knowif this is acceptable.Regards,Rick FalkAbstract: Finite Element Approximation of Problems in Solid MechanicsThe finite element approximation of mathematical models of thin plates andshells is studied. For the Reissner-Mindlin plate model, there are manyproven "locking-free" methods (i.e., no accuracy loss for smaller thickness)using triangular and rectangular elements. It is proposed to analyzequadrilateral elements for this problem and study important but unresolvedissues about these elements in more general contexts. In addition to theshear locking which causes problems in the approximation of plate models,shells also suffer from the problem of membrane locking. The goal is toimprove on the shell elements so far proposed and provide a rigorous analysisof convergence. Variational methods used previously for the derivation andanalysis of plate models will be extended to the derivation and analysis ofshell models. Discontinuous Galerkin finite element methods are promisingcandidates for a robust approximation method for both convection-dominated anddiffusion-dominated convection-diffusion problems. Further analysis isproposed to demonstrate their effectiveness more conclusively. Computationaland analytical techniques are proposed to understand the predictions of 2-Dmathematical models concerned with stress driven instability, expanding onprevious grant work done on a simpler 1-D model.This proposal is concerned with the use of mathematical models to studyseveral problems in solid mechanics. The use of mathematical models offers acost-effective way to make quantitative predictions about how mechanicalsystems will change when external forces are applied and serves as analternative to the use of costly or difficult experiments. Typically, whenrealistic mathematical models are formulated, they are in terms of equationswhose solutions, which represent physical quantities of interest to engineersand scientists, are not able to be determined analytically, i.e., in a simpleform one can easily write down. However, by employing numerical methods, goodapproximations to the physical quantities which are described by themathematical models may still be found. Typically, high performance computingis needed to do the large number of calculations involved. This project isconcerned with the design and analysis of numerical approximation schemes fora number of important mathematical models used in mechanics. These includemodels of elastic plates and shells (used for example to design the roof of abuilding to avoid collapse) and models of nano-scale solid crystals (whichcan be used to study instabilities in certain materials).