Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization

Abstract

SUMMARYA two-dimensional, finite deformation frictional contact formulation with Coulomb's law is presented. The approach considers multibody contact and is based on a mortar formulation. The enforcement of contact constraints is realized with dual Lagrange multipliers. These alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase. Friction kinematic variables are formulated in an objective way and enter non-smooth complementarity functions for expressing the contact constraints. An active set strategy is derived by applying a semi-smooth Newton method, which treats contact nonlinearities, material and geometrical nonlinearities in one single iterative scheme. By further carrying out a consistent linearization for both normal and frictional contact forces and constraints, a robust and highly efficient algorithm for linear and higher-order (quadratic) interpolation is achieved. Efficiency of the proposed method and quality of results are demonstrated in several examples.

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.…”

2d Frictional B-Spline Smoothed Mortar Contact Problems Part I: Matching Phase

“…Discretizing with mortar approach, part of the STS method family, is stable and passes the patch test but its implementation is difficult and requires a lot of technical expertise. This technique has been successfully applied for normal contact problems [3,8,9] and for contact problems with friction [2,11,13,53,[54][55][56][57].…”