## Necas center for Mathematical Modelling

## Memorial Seminar

"Jindrich Necas and Contact problems"## Monday, December 1, 2008

Seminar room K1

2nd floor, Sokolovska 83, Praha 8 - Karlin

Chairman : Jaroslav Haslinger14:00 Part I - Opening 14:15 Prof. Dr. Barbara Wohlmuth

(University of Stuttgart, Germany)Stabilized semi-smooth Newton algorithms for variational inequalities

Numerical simulation for partial differential equations plays an important role in many fields. Although efficient solution strategies are well known for simple settings, more complex models involving non-smooth nonlinearities are still quite challenging. In this talk we present some applications from structural mechanics and finance where variational inequalities play an important role. Surprisingly frictional contact problems with plasticity fit into the same mathematical framework as American options. We propose fast and robust numerical approaches based on the concept of domain decomposition, adaptivity and stabilized time integration.15:15 Coffee Chairman : Jiri Jarusek15:40 Part II - Opening 15:50 Prof. Dr. Christof Eck

(University of Stuttgart, Germany)Existence results for contact problems with friction

Contact problems in elasticity belong to the classical topics of applied mathematics, but there are still many questions unsolved, in particular if friction is taken into account. The main difficulty is the combination of the unilateral contact condition with the Coulomb friction law that leads to a non-monotone and non-compact formulation. This requires special approaches to prove the existence of solutions. The most successful approach to analyze contact problems with Coulomb friction was established by Jindrich Necas together with Jiri Jarusek and Jaroslav Haslinger in a seminal paper "On the solution of the variational inequality to the Signorini problem with small friction." (Boll. Un. Mat. Ital. B 5(17), pp. 943-958). It consists in the approximation of the problem by some convex problem, the proof of some addition regularity of the solution by a certain translation technique and the application of a fixed point argument. This approach was first applied to static contact problems with Coulomb friction. It was later extended to many other types of frictional contact problems, as e.g. quasistatic contact problems and dynamic problems for viscoelastic and viscoplastic materials. Some results also include the transport of heat generated by friction. In the lecture we present this approach and give a survey on the available existence results for static, quasistatic and dynamic frictional contact problems.16:50 Prof. RNDr. Igor Bock, PhD

(Slovak University of Technology Bratislava, Slovakia).Dynamic Contact Problems for von Karman Plates

We deal with systems consisting of a nonlinear hyperbolic variational inequality for a deflection and a nonlinear elliptic equation for the Airy stress function. The systems describe moderately large deflections of thin elastic and viscoelastic plates with an inner obstacle and in a dynamic action. The dynamic contact problems are not frequently solved in the framework of variational inequalities. Mainly for the elastic problems there is only limited amount of results available. The aim of the presentation is to extend these results to von K\'arm\'an plates. We neglect the rotational inertia member in the elastic case. We will consider the short memory and the long memory material in the case of a viscoelastic plate. The fist one is expressed by a pseudo-hyperbolic unilateral problem. The long memory material will be considered as an integro-differential variational inequality with a singular kernel.17:15 Wine and Cheese

In the name of the organizers: M. Feistauer, J. Haslinger, J. Jarušek, J. Málek, Š. Nečasová, M. Rokyta, T. Roubíček

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