Nonconforming mixed finite elements for linear elasticity on simplicial grids

Abstract

This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking-free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case. KEYWORDSlinear elasticity, mixed finite element, nonconforming simplicial element

New Discontinuous Galerkin Algorithms and Analysis for Linear Elasticity with Symmetric Stress Tensor

“…Another framework to construct stable weakly symmetric mixed finite elements was presented in [11], where two approaches were particularly proposed with the first one based on Stokes's problems and the second one based on interpolation operators. To keep the symmetry of discrete stresses, a second way is to relax the continuity of the normal components of discrete stresses across the internal edges or faces of grids, which leads to nonconforming mixed FEMs with strong symmetric stress tensor [4,9,10,25,32,33,37,[47][48][49]. In 2002, based on the elasticity complexes, the first family of symmetric conforming mixed elements with polynomial shape functions was proposed for two-dimensional cases in [8], which was extended to threedimensional cases in [2].…”

High order mixed finite elements with mass lumping for elasticity on triangular grids

“…These stability constraints make it challengeable to construct stable finite element pairs with symmetric stresses. In this field, we refer to [1-5, 7, 12, 21] for some conforming mixed methods and to [6,19,22,25,30] for some nonconforming methods. In [23,24] Hu and Zhang designed a family of conforming symmetric mixed finite elements with optimal convergence orders for linear elasticity on triangular and tetrahedral grids.…”