Finite Element Approximation of Problems in Solid Mechanics

亲爱的Jong-Shi：我很高兴收到你的电子邮件，说你计划将我的NSF提案的资金作为36个月的标准奖励150,710美元。 请让我知道，如果这是可以接受的。问候，里克福克摘要：有限元逼近问题的固体力学研究的有限元逼近的薄板和薄壳的数学模型。 对于Reissner-Mindlin板模型，有许多经过验证的"无锁定"方法（即，对于较小的厚度没有精度损失）使用三角形和矩形元件。建议分析这个问题的四边形元素，并在更一般的背景下研究这些元素的重要但未解决的问题。 除了在板模型的近似中引起问题的剪切锁定之外，壳体还遭受膜锁定的问题。 目的是改进目前提出的壳单元，并提供严格的收敛性分析。将以前用于板模型推导和分析的变分方法推广到壳模型的推导和分析。 间断Galerkin有限元方法是求解对流占优和扩散占优的对流扩散问题的一种强有力的逼近方法。 进一步的分析，建议证明他们的有效性更有说服力。计算和分析技术被提出来理解与应力驱动的不稳定性有关的2-D数学模型的预测，扩展了以前在一个简单的1-D模型上所做的资助工作。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实验的替代方案。 通常，当制定现实的数学模型时，它们是根据其解的方程，其解代表工程师和科学家感兴趣的物理量，不能通过分析确定，即，在一个简单的形式，一个人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，需要高性能计算来进行大量的计算。 本项目涉及力学中一些重要数学模型的数值逼近方案的设计和分析。 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).