Mixed finite elements for elasticity
Mixed finite elements for elasticity
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DOI：
10.1007/s002110100348
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发表时间：
2002-09-01
影响因子：
2.1
通讯作者：
Winther, R
中科院分区：
文献类型：
--
作者：
Arnold, DN;Winther, R
近四十年来，人们一直致力于发展稳定的混合有限元，用于平面弹性体系的应力-位移方程。这需要发展一对相容的有限元空间，一个离散化求应力场的对称张量空间，一个离散化求位移场的矢量场空间。虽然有许多著名的混合有限元对来解决涉及矢量场和标量场的类似问题，但应力场的对称性是一个很大的额外困难，这里给出的单元是第一个使用多项式形函数的单元，众所周知是稳定的。我们给出了一族这样的有限元空间，每个多项式次数一个，应力从二次开始，位移从一次开始，并证明了稳定性和最优阶逼近。我们还分析了这种有限元空间构造的一些障碍，这些障碍是造成可用单元稀缺的原因。
There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.