Mortar contact formulation using dual lagrange multiplier

Abstract

Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretized? In recent years, the mortar method has become popular in developing new contact algorithms. The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach, the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers. In this thesis, we present an approach for two dimensional frictionless multibody contact based on a mortar formulation and using a primal-dual active set strategy for contact constraint enforcement. So called dual Lagrange multipliers are introduced for the contact pressure but can be eliminated from the global system of equations by static condensation, thus avoiding an increase in system size. For this type of contact formulation, we provide a full linearization of both contact forces and normal (non-penetration) and tangential (frictionless sliding) contact constraints in the finite deformation frame. Furthermore, the discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master element, respectively. By further interpreting the active set search as a semismooth Newton method, contact nonlinearity and geometrical nonlinearity can be solved within one single iterative scheme. This yields a robust and highly efficient algorithm for frictionless finite deformation, contact problems. Three numerical examples and also one patch test demonstrate the efficiency of our method with the high quality of results.