Finite Element Discretizations for Linear Elasticity
Finite Element Discretizations for Linear Elasticity
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线性弹性的有限元离散
DOI：
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发表时间：
2018
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通讯作者：
Emma Cinatl
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作者：
Emma Cinatl
弹性方程描述了弹性材料在力作用下如何移动。弹性材料是一种在力被解除后恢复到其原始形状的材料。弹性建模在悬索和钉子弯曲等制造应用以及骨骼和肌腱上的重量等生物应用中非常有用[1]。在本论文中，我们研究了近不可压缩线弹性材料的建模。几乎不可压缩的材料是在压力下不会发生太大变化的材料。线弹性材料在力作用下表现出较小的变形。当使用最简单形式的线性弹性方程（纯位移形式）时，标准有限元方法不适用于几乎不可压缩的材料。相反，它们表现出锁定；换句话说，它们在粗网格上产生过小的位移。我们研究了解决此问题的几种方法。一种方法是使用更类似于斯托克斯公式（位移压力公式）的不同形式的线性弹性方程。这种方法有理论支持，但计算成本较高。另一种方法是对方程的一部分使用降积分。这种方法的理论支持较少，但通过正确的设置，该方法比标准方法更便宜、更准确，并且解决了锁定问题。
The elasticity equations describe how an elastic material moves under a force. An elastic material is one that returns to its original shape after the force is lifted. Modeling elasticity is useful in manufacturing applications such as suspension cables and nail bending, and biological applications such as weight on bones and tendons [1]. In this thesis we study the modeling of nearly incompressible linearly elastic materials. A nearly incompressible material is one that does not change much under pressure. Linearly elastic materials exhibit small deformations under a force. Standard finite element methods do not work well on nearly incompressible materials when using the simplest form of the linear elasticity equations (the pure displacement form). Instead, they exhibit locking ; in other words, they produce excessively small displacements on a coarse mesh. We examine several methods that fix this problem. One method is to use a different form of the linear elasticity equations that more closely resembles a Stokes’ formulation (the displacement-pressure formulation). This approach has theoretical support, but is somewhat computationally expensive. Another method is to use reduced integration for part of our equation. This approach has less theoretical support, but with the correct setups, this method is both cheaper and more accurate than the standard method, and solves the locking problem.