Mixed finite element methods for two-body contact problems

Abstract

a b s t r a c tThis paper presents mixed finite element methods of higher-order for two-body contact problems of linear elasticity. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The main focus is on the convergence of the scheme and on a priori estimates for the h-and the p-method. For this purpose, a discrete inf-sup condition is proven which guarantees the stability of the mixed method. Numerical results confirm the theoretical findings.

Semi-smooth Newton methods for mixed FEM discretizations of higher-order for frictional, elasto-plastic two-body contact problems

“…Following the work [9] we transfer the presented active-set strategies for Mortar methods to mixed finite elements, which were introduced by Haslinger [28] for low order. We generalize the solving methods to higher-order discretizations that are proposed in [29] and general friction laws depending on the contact forces.…”

“…However the splitting of the normal and tangential parts is clear. Stability of the described discretization is proven for the elastic case and balanced h, H, p, and q in [29]. It is shown that if the inequality…”

“…In this section we introduce a higher-order discretization for the described problem (13)- (14) that was mentioned in [29]. Denote by T h,m and B H triangulations of Ω m , m = 1, 2 and Γ 1 C , respectively, with corresponding affine transformations…”

“…For constant ansatz functions we define C 0 := {0 d−1 }, whereas for q = 1 the set C 1 consists of the corners of the reference element [−1, 1] d−1 . The non-conforming ansatz was proposed in [36] and convergence is shown for an elastic two-body problem in [29]. In the case of lower polynomial degrees q = 0, 1, this ansatz becomes conforming, since the choice of C q leads to a uniform sign condition on the elements E ∈ B H .…”

“…Our approach has compared to Mortar methods the advantage that the higher-order basis functions are given by simple Lagrange basis functions. We transfer the approach of active-set strategies that have been developed for Mortar methods [9,10] to mixed finite elements introduced by Haslinger [28] and higher-order discretizations presented in [29,30]. Lagrange multipliers capture the geometrical contact and frictional constraints.…”