A dual mortar approach for 3D finite deformation contact with consistent linearization

Abstract

SUMMARYIn this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal-dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis.

Isogeometric dual mortar methods for computational contact mechanics

“…Using these methods based on a standard Lagrange multiplier interpolation, a system of increased size containing both displacement and Lagrange multiplier degrees of freedom has to be solved. In this work, we follow a different approach using dual shape functions for the Lagrange multiplier, which were initially introduced in domain decomposition problems [19,45] and extended to contact problems in [20][21][22][23][24][25][26][27] and recently reviewed in [29]. While dual mortar methods are meanwhile well-established in finite elements, the present work, to the authors knowledge, is the first application of dual basis functions in the context of IGA for both domain decomposition and finite deformation frictional contact.…”

“…Here, the use of dual shape functions yields a diagonal mortar matrix D, which allows for an easy condensation of the Lagrange multiplier degrees of freedom from the global system of equations, see e.g. [19,[22][23][24][25].…”

“…A consistently linearized system of both the discrete balance equation and the complementarity function can be solved with a non-smooth variant of Newton's method [52], where the NCP functions serve as the basis for a primal dual active set strategy [53]. Since the algorithm and linearization details are the same as for Lagrangian finite elements, we refer to [23][24][25] and omit the details here.…”

“…Proposing an alternative approximation of the Lagrange multiplier, so-called dual mortar methods were originally introduced in the context of domain decomposition [19] and later extended to small deformation frictionless contact in 2D [20] and 3D [21], small deformation frictional contact [22], finite deformation frictionless contact [23,24] and finite deformation frictional contact [25]. Since NURBS are naturally associated with higher order approximations, the extension of the dual mortar method to second order Lagrange finite elements in [26,27] including optimal a priori error estimates is also worth mentioning.…”

A Higher Order Phase-Field Approach to Fracture for Finite-Deformation Contact Problems

“…like the node-to-surface method, the variational consistent Mortar method has been wellestablished (see e.g. [31,15,30,36]). In a recent publication (see [7]) we applied the Mortar method for thermo-mechanical frictional contact problems.…”