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Recent New Developments in Contact Mechanics

During the last years considerable effort was devoted to better numerical treatment of contact problems. This fact is due to the growing computing power which lead to more and more sophistication and detailed technical models within engineering analysis. Due to the more precise modelling within the associated discretization process often unilateral constraints have to be considered. Hence better discretization techniques, especially for finite deformations, are needed to solve problems with contact constraints in an efficient and robust way. In this paper we will discuss some recently developed discretization schemes and algorithms for the treatment of contact constraints. The presentation is split into two parts. The first one is devoted to discretization techniques for contact problems which fulfill the BB-condition needed for a stable contact discretization scheme. This leads to a discussion of weak enforcement of the contact constraint conditions which results in so-called mortar methods for linear and nonlinear problems. Here also special remarks are made with regard to efficient solution schemes which are based on a total gap vector at the contact interface. The second part of the presentation is related to adaptive finite element methods for large deformation thermo-mechanical contact problems. Here a special staggered scheme is developed in which different finite element meshes are combined to solve the thermo-mechanical contact problem. Based of the methodology of the Zienkiewicz, Zhu error indicators based on superconvergent patch recovery special error indicators are developed for the the mechanical and thermal part of the problem including the contact constraints. Furthermore an error indication in time is derived for the thermal heat conductance equation based on a time-space discretization which uses a continuous Galerkin scheme for the time integration. Using such integration algorithm one can derive again a error indicator by assuming superconvergent time points. This method is applied to solve an example with known analytical solution which allows the computation of efficiency indices. Here it can be shown that the developed adaptive time stepping scheme results in very good efficiency of the method. For all parts, the basic theoretical basis is derived, algorithmic implications are discussed and explanatory examples are presented to show the properties of formulations when compared to existing ones.