Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

Abstract

Abstract. In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

Mixed formulations for a class of variational inequalities

“…Observe that the conforming condition on V h ensures that the bilinear form a is uniformly coercive on V h . Theorem 2.3 and Lemma 2.2 apply in the discrete context too, and readily give that problem (18) has one and only one solution uniformly bounded in h. For formulations where K = X and a is coercive on all of X, the well-posedness of the discrete problem and the convergence of its solution are obtained simply through the usual Brezzi's condition (20) with W h = X h [2,10,18]. Theorem 2.6.…”

“…A similar problem related to a frictionless unilateral contact problem of two elastic bodies has been considered in [10,18]. In all these works, the stresses are removed from the formulation.…”

“…To establish (10), it is sufficient to remark that for 0 < t < 1, (1 − t)u + t(v/t) ∈ K and to let t → 0.…”

“…This is the frame of the estimates given in [2,10,18]. Unfortunately, such a property does not hold in a mixed formulation involving both the displacements and the stresses and explicitly enforcing the equilibrium conditions.…”

A modified perturbed Lagrangian formulation for contact problems

“…In recent years segment-to-segment formulations like the mortar method [8] have been successfully applied to solving a wide variety of contact problems in 2D [35,27,55] and 3D [39,38], with linear and quadratic elements [28,53], in large and small deformations including Coulomb friction [40,41,17,18,42,50,20] and dynamic problems [24]. The theoretical basis of the mortar method is well known [15,28,32,30,31]. The compatibility of the displacement field and the contact stresses allows the Brezzi-Babuska-InfSup condition to be fulfilled, so the optimal convergence rate of the finite element solution can be achieved.…”