Mathematical Sciences: Finite Element Methods for Problems in Solid Mechanics

数学科学：固体力学问题的有限元方法

• 批准号: 9403552
• 负责人: Richard Falk
• 金额: $ 6.9万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403552&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Finite; Element; Methods

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.

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。本文将分析最近提出的Reissner-Mindlin板模型的有限元近似方案，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给出较差的近似。还将对这一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”板块模型，为应力和位移提供系统的推导和数学上严格的误差估计。目的是确定这些模型中的任何一个在什么时候可以严格地显示出比最简单的双调和模型更好地近似于完整的三维方程，从而为模型提供理由，并确定哪种模型在各种情况下最合适。该项目的第二个重点领域是研究用于力学中两个问题近似的时空有限元方法。第一个问题涉及一个非线性偏微分方程组，该方程组模拟了一类不可伸缩弹性杆的平面运动。该系统具有守恒的能量，该项目的目标是为该问题开发一组任意阶节能有限元格式，以便以比PI先前开发的有限差分方法更少的计算工作量获得更精确的近似。第二个问题与表面扩散有关。具体来说，目标是严格建立由PI开发的一组时空有限元方法的稳定性和收敛性，这些方法用于逼近一个偏微分方程系统，该系统模拟恒定密度的各向同性均匀固体在体内质量扩散影响下的形状变化。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，通常被工程师用来预测在施加各种力时薄的三维弹性体的位移和应力。对于这些问题，使用计算机来求得近似解是必要的，因为除非在非常特殊的情况下，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。由于文献中出现了许多不同的二维平板模型，因此项目的一个相关部分是确定何时这些模型中的任何一个可以严格地显示出比目前使用的最简单模型更好地近似于完整的三维方程，从而为模型提供理由，并确定哪种模型在各种情况下最合适。该项目的第二个重点领域是为力学中两个问题的逼近开发有效的计算算法。第一个问题涉及到一类不可伸缩弹性杆的平面运动的数学模型。不可扩展棒的动力学对于从柔性空间结构到长链分子（如聚合物或DNA）动力学建模的应用非常重要。第二个要研究的问题是关于物体在表面张力作用下的形状变化。所研究的特殊数学模型模拟了恒定密度的各向同性均匀固体在体内质量扩散的影响下所引起的形状变化。目的是开发新的近似方案，并提供对误差的严格分析。后者在提供信心方面很重要，即所获得的数值近似值能够准确地预测模型的行为。