Some equilibrium finite element methods for two-dimensional elasticity problems

Abstract

Summary. We consider some equilibrium finite element methods for twodimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establish Lz-estimates of order O(h 2) for both stresses and displacements.

Conforming equilibrium finite element methods for some elliptic plane problems

“…Some non-conforming methods, where the equilibrium équation or the symmetry of the stress tensor is satisfied only approximately, are described in [15,25,26,27]. …”

“…Finite element spaces of fonctions, whose divergence exists in the sensé of distributions, and various degrees of freedom (parameters) of these spaces are given in [6,7,15,17,24], However, ifwe add the equilibrium condition div a = 0 we get a constraint among the parameters of each element, ie., the equilibrium fmite element method then consists in minimizing some quadratic functional with linear constraints. These constraints can be removed e.g.…”

Untitled

“…The natural discretization of the latter space is evident-piecewise polynomial of some degree without interelement continuity constraints-but the development of an appropriate finite element subspace of H(div, Ω; S) to use with these is a long-standing and challenging problem. For plane elasticity, the known stable mixed finite element methods have mostly involved composite elements for the stress [6,15,16,21]. To avoid these, other authors have modified the standard mixed variational formulation of elasticity to a formulation that uses general, rather than symmetric, tensors for the stress, with the symmetry imposed weakly; see [2,5,7,17,18,19,20,8].…”

A mixed nonconforming finite element for linear elasticity

“…There have been many efforts during past four decades to develop stable mixed finite element methods for the system of planar linear elasticity (see [1][2][3][4][5][6][7][8][9][10][11]). Unlike mixed methods for second-order scalar elliptic problems, stress-displacement finite elements are extremely difficult to construct.…”