Finite Element Approximation of Problems in Solid Mechanics

亲爱的Jong-Shi：我很高兴收到您的邮件，说您计划为我的NSF提案推荐一个36个月的标准奖金150,710美元。按你的要求，这是项目摘要。请告诉我这是否可以接受。摘要：固体力学问题的有限元逼近研究了薄板壳数学模型的有限元逼近。对于Reissner-Mindlin板模型，有许多经过验证的“无锁定”方法（即，对于较小厚度没有精度损失）使用三角形和矩形单元。我们建议分析这个问题的等边元素，并在更一般的背景下研究这些元素的重要但尚未解决的问题。除了导致板模型近似问题的剪切锁定外，壳体还遭受膜锁定问题。目标是改进目前提出的壳单元，并提供一个严格的收敛分析。以前用于板模型推导和分析的变分方法将扩展到壳模型的推导和分析。不连续Galerkin有限元方法对于对流主导和扩散主导的对流扩散问题都是一种有希望的鲁棒逼近方法。提出进一步的分析，以更确切地证明其有效性。提出了计算和分析技术来理解与应力驱动不稳定性有关的二维数学模型的预测，扩展了以前在更简单的一维模型上完成的资助工作。这个建议是关于使用数学模型来研究固体力学中的几个问题。数学模型的使用提供了一种经济有效的方法来定量预测当施加外力时机械系统将如何变化，并作为使用昂贵或困难的实验的替代方法。通常，当现实的数学模型被公式化时，它们是用方程来表示的，这些方程的解代表了工程师和科学家感兴趣的物理量，而这些解是无法解析地确定的，也就是说，用一种简单的形式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，需要高性能计算来完成大量的计算。这个项目是关于设计和分析数值逼近方案的一些重要的数学模型用于力学。这些模型包括弹性板和壳的模型（例如用于设计建筑物的屋顶以避免倒塌）和纳米级固体晶体的模型（可用于研究某些材料的不稳定性）。

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).