Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

Abstract

Abstract. The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinitedimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.Mathematics Subject Classification. 49M05, 49M29, 74M10, 74M15, 74B05.

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

“…Further, mesh independence results [25,29] are available. We emphasize that the semismooth Newton method has found considerable attention throughout the last decade as it has proved to be a remarkably efficient method, notably for the solution of various problems in PDE-constrained optimization [30,26,27] and variational inequalities [15,28,39], to mention only a few. We further rely on the following calculus rules related to the Newton differentiability of several nonsmooth functions which can be found in [30] and [28].…”

“…While some attention has been paid to infinite-dimensional methods in linear elasticity with (frictional) contact [39,45], elasto-plastic problems are still less researched. Among the few available references we mention [8] for domain decomposition methods leading to a linear rate of convergence.…”

Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity

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A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

“…In the case of unilateral constraints, the resulting system is only piecewise continuously differentiable and Newton's method can be extended to this class of nonsmooth systems [28]. Newton's method for unilateral contact problems has been used for instance in [1,22]. In particular, it has been applied to the problem of unilateral contact with cohesive forces in [25].…”