论文标题
新的不连续的Galerkin算法和对称应力张量线性弹性的分析
New Discontinuous Galerkin Algorithms and Analysis for Linear Elasticity with Symmetric Stress Tensor
论文作者
论文摘要
本文提出了一种新的和统一的方法，用于对许多现有的衍生和分析以及新的不连续的Galerkin方法，用于线性弹性问题。该分析基于由四个离散变量组成的线性弹性问题的统一离散公式：强烈的对称应力张量$ \ dSig $和每个元素内部的位移$ \ du $，以及这两个变量$ \ hsig $ and $ \ hu $的修改。通过文献中许多相关方法的动机，该公式可用于得出有关线性弹性问题的大多数现有，不合格和符合的Galerkin方法，尤其是开发许多新的不连续的Galerkin方法。事实证明，这种四场配方的许多特殊情况被证明是可杂交的，可以通过消除四个领域中的一个或两个，可以简化为一些已知的杂交不连续的Galerkin，弱Galerkin和局部不连续的Galerkin方法。由于某些稳定参数趋于零，因此这种四场公式被证明会融合到一些线性弹性问题的某些符合和不合格的混合方法。事实证明，两个基于$ h^1 $的inf-sup条件的家族，一个被称为$ h^1 $的家庭，另一个被称为$ h（{\ rm div}）$，被证明在离散空间和参数的不同选择方面是统一有效的。这些INF-SUP条件保证了新提出的方法的适当性，还为文献中许多现有方法作为副产品提供了新的统一分析。提供了一些数值示例来验证理论分析，包括新提出的方法的最佳收敛。
This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.
