The simplest mixed finite element method for linear elasticity in the symmetric formulation on $n$-rectangular grids
The simplest mixed finite element method for linear elasticity in the symmetric formulation on $n$-rectangular grids
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发表时间：
2013-04
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通讯作者：
Jun Hu;Hongying Man;Shangyou Zhang
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作者：
Jun Hu;Hongying Man;Shangyou Zhang
提出了一族混合有限元来求解任意空间维度的一阶线弹性方程组，其中应力场由对称有限元张量近似。该系列单元的应力分量和位移之间具有完美的匹配。矩形网格上法向应力 $\sigma_{ii}$、剪应力 $\sigma_{ij}$ 和位移 $u_i$ 的离散空间分别为 $\operatorname{span}\{1,x_i\}$、$\operatorname{span}\{1,x_i,x_j\}$ 和 $\operatorname{span}\{1\}$。特别是，该定义对于所有空间维度都保持相同。由于这些选择，理论分析也独立于空间维度。在一维中，该元素就是一维拉维阿特-托马斯元素，它是该族中唯一的一致元素。在二维和更高维度中，它们是新元素，但自由度最小。每个单元的总自由度在 1D 中为 2 加 1，在 2D 中为 7 加 2，在 3D 中为 15 加 3。先前记录的最小自由度是，在矩形网格上，2D 中为 13 加 4，3D 中为 54 加 12。这些元素是任何空间维度中最简单的元素。对于纯位移和牵引问题，证明了有限元族的适定性条件和最优先验误差估计。 2D 和 3D 的数值测试显示了新元素相对于其他元素的优越性，令人惊讶地表现出超收敛性。
A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has a perfect matching between the stress components and the displacement. The discrete spaces for the normal stress $\sigma_{ii}$, the shear stress $\sigma_{ij}$ and the displacement $u_i$ are $\operatorname{span}\{1,x_i\}$, $\operatorname{span}\{1,x_i,x_j\}$ and $\operatorname{span}\{1\}$, respectively, on rectangular grids. In particular, the definition remains the same for all space dimensions. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. In 1D, this element is nothing else but the 1D Raviart-Thomas element, which is the only conforming element in this family. In 2D and higher dimensions, they are new elements but of the minimal degrees of freedom. The total degrees of freedom per element is 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D. The previous record of the least degrees of freedom is, 13 plus 4 in 2D, and 54 plus 12 in 3D, on the rectangular grid. These elements are the simplest element for any space dimension. The well-posedness condition and the optimal a priori error estimate of the family of finite elements are proved for both pure displacement and traction problems. Numerical tests in 2D and 3D are presented to show a superiority of the new element over others, as a superconvergence is surprisingly exhibited.