Keywords: contact, Lagrange multiplier, mortar, large deformation, finite element, convergence.

In recent years, the mortar finite element method has been successfully applied to solve contact problems using non-conforming discretizations of the bodies in contact. Unlike the classical node-to-segment approach, the mortar method prevents locking or over-constraint and it allows the optimal convergence rate of the error to be obtained [1,2,3,4].

In this work an implementation of the mortar method is proposed to solve three-dimensional contact problems in the context of large deformations. The contact constraints and non-linear equilibrium equations are derived from the augmented Lagrange method which is a C¹ continuous functional. This method was first used by Pietrzak and Curnier [5] for node-to-segment contact formulations. Newton's method is used to solve the non-linear system of equations. As the proposed formulation is based on the mortar method, the constraints are imposed in a weak integral sense. In order to compute the contact integrals, we use an inexact numerical quadrature that facilitates the linearization of the contact variables (gap, normal, etc.) and reduces the computational cost.

In the numerical examples we show that, although the contact integrals are not exact, the optimal convergence rate of the discretization error is still preserved. We have performed the full linearization of the equilibrium equations and good quadratic convergence is obtained in the numerical examples.

1

M.A. Puso, T. Laursen, "A mortar segment-to-segment frictional contact method for large deformations", Computer Methods in Applied Mechanics and Engineering, 193, 4891-4913, 2004. doi:10.1016/j.cma.2004.06.001

2

B. Yang, T. Laursen, X. Meng, "Two dimensional mortar contact methods for large deformation frictional sliding", International Journal for Numerical Methods in Engineering, 62(9), 1183-1225, 2005. doi:10.1002/nme.1222

3

M. Tur, F.J. Fuenmayor, P. Wriggers, "A mortar-based frictional contact formulation for large deformations using Lagrange multipliers", Computer Methods in Applied Mechanics and Engineering, 198, 2860-2873, 2009. doi:10.1016/j.cma.2009.04.007

4

A. Popp, M. Gitterle, M.W. Gee, W.A. Wall, "A dual mortar approach for 3D finite deformation contact with consistent linearization", International Journal for Numerical Methods in Engineering, 2010. doi:10.1002/nme.2866

5

G. Pietrzak, A. Curnier, "Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment", Computer Methods in Applied Mechanics and Engineering, 177, 351-381, 1999. doi:10.1016/S0045-7825(98)00388-0

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