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Continua a l'article "A contact domain method for large deformation frictional contact problems. Part 2: Numerical aspects" publicat a la revista "Computer methods in applied mechanics and engineering", 2009, vol. 198, p. 2607-2631. In the first part of this work, the theoretical basis of a frictional contact domain method for two-dimensional large deformation problems is presented. Most of the existing contact formulations impose the contact constraints on the boundary of one of the contacting bodies, which necessitates the projection of certain quantities from one contacting surface onto the other. In this work, the contact constraints are formulated on a so-called contact domain, which has the same dimension as the contacting bodies. This contact domain can be interpreted as a fictive intermediate region connecting the potential contact surfaces of the deformable bodies. The introduced contact domain is subdivided into a non-overlapping set of patches and is endowed with a displacement field, interpolated from the displacements at the contact surfaces. This leads to a contact formulation that is based on dimensionless, strain-like measures for the normal and tangential gaps and that exactly passes the contact patch test. In addition, the contact constraints are enforced using a stabilized Lagrange multiplier formulation based on an interior penalty method (Nitsche method). This allows the condensation of the introduced Lagrange multipliers and leads to a purely displacement driven problem. An active set strategy, based on the concept of effective gaps as entities suitable for smooth extrapolation, is used for determining the active normal stick and slip patches of the contact domain.

This work focuses on the assessment of the relative performance of the so-called contact domain method, using either the Lagrange multiplier or the penalty strategies. The mathematical formulation of the contact domain method and the imposition of the contact constraints using a stabilized Lagrange multiplier method are taken from the seminal work (as cited later), whereas the penalty based implementation is firstly described here. Although both methods result into equivalent formulations, except for the difference in the constraint imposition strategy, in the Lagrange multiplier method the constraints are enforced using a stabilized formulation based on an interior penalty method, which results into a different estimation of the contact forces compared to the penalty method. Several numerical examples are solved to assess certain numerical intricacies of the two implementations. The results show that both methods perform similarly as one increases the value of the penalty parameter or decreases the value of the stabilization factor (in case of the Lagrange multiplier method). However there seems to exist a clear advantage in using the Lagrange multiplier based strategy in a few critical situations, where the penalty method fails to produce convincing results due to excessive penetration. The Spanish Ministry of Science and Innovation and the Catalan Government Research Department are gratefully acknowledged for their financial support under Grants BIA2008-00411 and 2009 SGR 1510, respectively.

Continuació de l'article "A contact domain method for large deformation frictional contact problems. Part 1: Theoretical basis" publicat a la revista Computer Methods in Applied Mechanics and Engineering, Vol. 198, #33-36, July 2009, p. 2591-2606 This second part of the work describes the numerical aspects of the developed contact domain method for large deformation frictional contact problems. The theoretical basis of this contact method is detailed in the first part of this work. Starting from this, the present contribution focuses on describing important algorithmic details that go along with the finite element implementation for two-dimensional problems. Important aspects are the construction of the contact domain mesh, via a constraint Delaunay triangulation, the linearization of the discretized contact contributions and some important technical aspects about the extrapolation procedure used for the predictive active set strategy. Finally a set of numerical examples is presented to demonstrate the performance of the developed contact strategy. Demanding static and dynamic contact problems in the context of large deformations, including frictional effects as well as self contact, show the wide applicability and the robustness of the proposed method.