Mathematical Sciences: Finite Element Methods for Problems in Solid Mechanics

9403552 Falk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，该模型通常用于研究薄的三维弹性体。对最近提出的Reissner-Mindlin板模型的有限元近似格式进行分析，以确定它是否避免了“锁定”问题，该问题导致大多数方法对薄板的近似较差。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正式研究“高阶”板块模型，为应力和位移提供系统的推导和严格的数学误差估计。其目的是确定何时可以严格地证明这些模型中的任何一个能够比最简单的双调和模型更好地逼近完整的三维方程，从而为这些模型提供理由并确定哪些模型在各种情况下是最合适的。该项目的第二个重点领域是研究用于逼近力学中两个问题的时空有限元方法。第一个问题涉及一个非线性偏微分方程组，它模拟了一类不可伸缩弹性杆的平面运动。该系统的能量是守恒的，本项目的目标是为这个问题发展一族任意阶的能量守恒有限元格式，以便以比PI以前发展的有限差分法更少的计算量获得更精确的近似。第二个问题与表面扩散有关。具体地说，目标是严格地建立由PI发展的一族时空有限元方法，用于近似偏微分方程组，该偏微分方程组模拟了在物体内质量扩散的影响下，在各向同性和均匀的固体中诱导的形状变化。为这些方法提供了严格的基础，使人们相信数值近似提供了对模型行为的准确预测。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，工程师通常使用这种模型来预测薄壁三维弹性体在各种力作用下的位移和应力。对于这些问题，使用计算机来获得近似解是必要的，因为除非在非常特殊的情况下，否则精确解是未知的。因此，该项目的一个重要方面是对计算算法的分析和比较。由于文献中出现了许多不同的二维板模型，本项目的一个相关部分是确定何时可以严格地显示这些模型中的任何一个，以便比目前使用的最简单的模型更好地逼近完整的三维方程，从而提供模型的合理性和确定哪些模型在各种情况下最合适的方法。该项目的第二个重点领域是开发有效的计算算法来近似处理力学中的两个问题。第一个问题涉及一类不可伸缩弹性杆的平面运动的数学模型。不可伸缩棒的动力学对于从柔性空间结构到聚合物或DNA等长链分子动力学建模的应用非常重要。第二个要研究的问题是由表面张力驱动的物体的形状变化。所要研究的特殊数学模型模拟了在物体内质量扩散的影响下，在各向同性、均匀的等密度固体体内引起的形状变化。其目的是开发新的近似格式并提供对误差的严格分析。后者在提供所获得的数值近似给出模型行为的准确预测这一信任度方面很重要。

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.