A duality-based path-following semismooth Newton method for elasto-plastic contact problems

Abstract

A Fenchel dualization scheme for the one-step time-discretized contact problem of quasi-static elasto-plasticity with combined kinematic-isotropic hardening is considered. The associated path is induced by a coupled Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the optimal displacementstress-strain triple of the original elasto-plastic contact problem in the space-continuous setting. This property relies on the density of the intersection of certain convex sets which is shown as well. It is also argued that the mappings associated with the resulting problems are Newton-or slantly differentiable. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence. For efficiency purposes, an inexact path-following approach is proposed and a numerical validation of the theoretical results is given.Keywords: elasto-plastic contact, variational inequality of the 2nd kind, Fenchel duality, Moreau-Yosida/Tichonov regularization, path-following, semismooth Newton 2010 MSC: 49M15, 49M29, 74C05, 74S05

Density of convex intersections and applications

“…An elasto-plastic contact problem arises if the movement of the material is additionally restricted by the presence of a rigid obstacle. From a mathematical point of view, the problem can be equivalently reformulated in terms of the normal stress z * at the (sufficiently smooth) contact boundary Γ c ⊂∂ Ω , and a variable q that is related to the deviatoric part of the material stress; for details we refer to [ 24 , p.154]: Here, d := N ( N +1)/2−1 and G is a strongly convex, continuous and coercive functional that models the elasto-plastic material behaviour subject to given external loads. Furthermore, a contact constraint on the normal component of the displacement is imposed by a function ψ , which lies in the trace space H 1/2 ( Γ c ).…”

“…As a consequence, the Newton iterates are superlinearly convergent, and the convergence rate is mesh-independent upon discretization. For details, see [ 24 , section 5]. In order to prove the stability of ( 6.2 ) with regard to the limit problem ( 6.1 ) in the sense of theorem 2.1 , we show that the problems ( 6.2 ) define a quasi-monotone perturbation of with respect to the dense subspace H 1 ( Γ c )× H 1 ( Ω ) d ⊂ H 1/2 ( Γ c )*× L 2 ( Ω ) d (cf.…”

On the density of classes of closed convex sets with pointwise constraints in Sobolev spaces

“…We briefly mention several motivating applications where (1) emerges from Fenchel dualization (Ekeland and Temam, 1976) and semismooth Newton solvers (Hintermüller et al, 2003). In a rather abstract setting, regularized total variation type image restoration (Hintermüller and Stadler, 2006), energies related to Bingham fluids (Hintermüller and de los Reyes, 2011), simplified friction problems or elastoplastic problems in material science (Duvaut and Lions, 1976;Johnson, 1976;Carstensen, 1997;Stadler, 2004;Hintermüller and Rösel, 2013) can be associated with the following problem:…”

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

“…Linear and quadratic finite elements have been regarded in [15]. Semi-smooth Newton methods for elasto-plastic problems are analyzed involving frictionless contact in [16,17] and frictional contact in [18][19][20]. Besides Mortar methods there exist a series of strategies for the numerical treatment of frictional contact problems.…”