A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension
A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension
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DOI：
10.1007/s10915-013-9736-6
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发表时间：
2013-06
影响因子：
2.5
通讯作者：
Jun-Jue Hu;Hongying Man;Shangyou Zhang
中科院分区：
文献类型：
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作者：
Jun-Jue Hu;Hongying Man;Shangyou Zhang
本文构造了一类任意空间维的低阶矩形协调混合元。该方法中，正应力用二次多项式近似，剪应力用双线性多项式近似，位移用线性多项式近似。每个单元的总自由度（dof）在2D中为10加4，在3D中为21加6，而协调单元的最小dof的先前记录在2D中为17加4，在3D中为72加12。这类单元的特点是，除了简单之外，应力和位移的形函数空间与空间维数无关。由于这些选择，理论分析也独立于空间维度。证明了其适定性条件和最优先验误差估计。数值试验表明了新单元的稳定性和有效性。
We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials, the shear stress by bilinear polynomials, and the displacement by linear polynomials. The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension. As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.