Mathematical Sciences: Finite Element Methods for Problems in Solid Mechanics

小行星9403552 该项目的第一个重点领域是二维板模型的数值和分析研究，通常用于研究薄的三维弹性体。最近提出的Reissner-Mindlin板模型的有限元近似方案进行分析，以确定它是否避免了“锁定”的问题，导致大多数方法给薄的近似。本文还将对该方案和文献中提出的其他方案进行数值研究，以比较它们的有效性。该项目的一个相关部分是研究正式的“高阶”板模型，提供一个系统的推导和数学上严格的误差估计的应力和位移。目的是确定这些模型中的任何一个何时可以被严格地证明比最简单的双调和模型更好地近似全三维方程，从而提供模型的理由和决定哪些模型在各种情况下是最合适的方法。该项目的第二个重点领域是研究力学中两个问题的近似时空有限元方法。 第一个问题涉及一个系统的非线性偏微分方程模型的平面运动的一类不可扩展的弹性杆。该系统有一个能量是保守的，该项目的目标是开发一个家庭的任意阶能量守恒有限元方案，这个问题，以获得更准确的近似与更少的计算工作量比可能与有限差分法以前开发的PI。 第二个问题与表面扩散有关。 具体来说，我们的目标是严格建立一个家庭的空间-时间有限元方法的PI开发的偏微分方程系统的近似模型的形状的变化引起的各向同性和均匀的固体的恒定密度下的质量扩散的影响下，体内的稳定性和收敛性。提供 这些方法的严格基础使人们相信，数值近似法能够准确地预测模型的行为 该项目的第一个重点领域是二维板模型的数值和分析研究，工程师通常使用该模型来预测薄三维弹性体在施加各种力时的位移和应力。对于这些问题，使用计算机来获得近似解是必要的，因为除了非常特殊的情况，精确解是未知的。因此，这个项目的一个重要方面是计算算法的分析和比较。由于许多不同的二维板模型出现在文献中，该项目的相关部分是确定何时这些模型中的任何一个可以被严格地显示为比目前使用的最简单的模型更好地近似完整的三维方程，从而提供了模型的理由和决定哪些模型在各种情况下是最合适的方法。该项目的第二个重点领域是开发两个力学问题近似的有效计算算法。第一个问题涉及一个数学模型， 一类不可伸长的弹性杆的平面运动。不可伸展杆的动力学对于从柔性空间结构到长链分子（如聚合物或DNA）的动力学建模的应用范围是重要的。第二个要研究的问题是关于由表面张力驱动的物体的形状变化。待研究的特定数学模型模拟在恒定密度的各向同性和均匀固体中在体内质量扩散的影响下引起的形状变化。其目的是为了开发新的近似方案，并提供严格的误差分析。后者是重要的，在提供信心，获得的数值近似给出准确的预测模型的行为。

9403552 Falk The first area of focus of the project is the numerical and analytical study of two dimensional plate models, commonly used to study thin three dimensional elastic bodies. A recently proposed finite element approximation scheme for the Reissner-Mindlin plate model will be analyzed to determine whether it avoids the "locking" problem which causes most methods to give poor approximations for thin plates. A numerical study of this and other schemes proposed in the literature will also be done to compare their effectiveness. A related portion of the project is to study formally "higher order" plate models, providing both a systematic derivation and mathematically rigorous error estimates for the stresses and displacements. The aim is to determine when any of these models can be rigorously shown to give better approximations to the full three dimensional equations than the simplest biharmonic model, thus providing a justification of the models and a way of deciding which models are the most appropriate in various circumstances. The second area of focus of the project is the study of space-time finite element methods for the approximation of two problems in Mechanics. The first problem involves a system of nonlinear partial differential equations which models the planar motion of a class of inextensible elastic rods. The system has an energy which is conserved and the goal of the project is to develop a family of arbitrary order energy conserving finite element schemes for this problem, in order to obtain more accurate approximations with less computational effort than are possible with a finite difference method previously developed by the PI. The second problem is concerned with surface diffusion. Specifically, the goal is to rigorously establish the stability and convergence of a family of space-time finite element methods developed by the PI for the approximation of a system of partial differential equations that model the changes of shape in duced in an isotropic and homogeneous solid body of constant density under the influence of mass diffusion within the body. Providing a rigorous foundation for these methods gives confidence that the numerical approximations provide accurate predictions of the behavior of the model The first area of focus of the project is the numerical and analytical study of two dimensional plate models, commonly used by engineers to predict displacements and stresses of thin three dimensional elastic bodies when various forces are applied. The use of the computer to obtain approximate solutions is necessary for these problems, since except in very special cases, exact solutions are not known. Hence, one important aspect of this project is the analysis and comparison of computational algorithms. Since many different two dimensional plate models appear in the literature, a related portion of the project is to determine when any of these models can be rigorously shown to give better approximations to the full three dimensional equations than the simplest model currently used, thus providing a justification of the models and a way of deciding which models are the most appropriate in various circumstances. The second area of focus of the project is the development of efficient computational algorithms for the approximation of two problems in mechanics. The first problem involves a mathematical model for the planar motion of a class of inextensible elastic rods. The dynamics of inextensible rods is important for applications ranging from flexible space structures to the modelling of the dynamics of long chain molecules such as polymers or DNA. The second problem to be investigated is concerned with shape changes of a body driven by surface tension. The particular mathematical model to be studied models the changes of shape induced in an isotropic and homogeneous solid body of constant density under the influence of mass diffusion within the body. The aim is t o both develop new approximation schemes and provide a rigorous analysis of the errors. The latter is important in providing confidence that the numerical approximations obtained are giving accurate predictions of the behavior of the models.