A Nonsmooth Optimization Approach for Hemivariational Inequalities with Applications to Contact Mechanics

Abstract

In this paper we introduce an abstract nonsmooth optimization problem and prove existence and uniqueness of its solution. We present a numerical scheme to approximate this solution. The theory is later applied to a sample static contact problem describing an elastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on normal and tangential components of displacement. Finally, computational simulations are performed to illustrate obtained results.

A survey of numerical methods for hemivariational inequalities with applications to Contact Mechanics

“…Other nonsmooth optimization algorithms such as proximal bundle method could also be adapted (see [3]). Direct optimization method in a more complex setting (with function h τ dependent on u ν (x)) using Uzawa algorithm is presented in [10].…”

“…Detailed arguments can be found in [10] and are omitted here. We only mention main steps of the proof.…”

“…We remark that functional J defined by (3.10)- (3.11) under assumptions H(j ν ), H(j τ ) and H(h) satisfies H(J) (cf. [10,Lemma 4]).…”

“…In this paper we consider a nonmonotone friction law that depends on both normal and tangential components of displacement and its weak formulation which leads to hemivariational inequality. We consider similar mechanical model to the one described in [10], but with more general A operator as potential operator. Next, we compare three popular methods of solving introduced problem: direct optimization method, augmented Lagrangian method and primal-dual active set strategy.…”

“…Nevertheless, usually in papers containing Contact Mechanics simulations one method is chosen and presented. Early ideas about optimization in Contact Mechanics were presented in [20] and further details can be found in [10,20]. Augmented Lagrangian method was reviewed or applied in [1,5,21,22].…”