Improved robustness and consistency of 3D contact algorithms based on a dual mortar approach

Abstract Mortar finite element methods have been successfully applied as discretization scheme to a wide range of contact and impact problems in recent years. The 3D finite deformation contact formulation taken up and enhanced in this paper is based on a mortar approach using so-called dual Lagrange multipliers, which substantially facilitate the treatment of interface constraints as compared with standard mortar techniques. Despite being quite well-established in the meantime, dual mortar methods may lack robustness or even consistency in certain situations, e.g., when large curvatures occur in the contact zone or when one body slides off another at a sharp edge. However, since such scenarios are regularly appearing in engineering practice, the present contribution provides several new extensions that completely resolve these issues and thus significantly improve the applicability of dual mortar formulations for challenging contact problems. The proposed extensions include a consistent biorthogonalization procedure to obtain feasible dual Lagrange multiplier shape functions close to boundaries of the contact surfaces, an improved conditioning of the global linear system of equations by nodal scaling and a novel approach to unify the advantages of standard and dual mortar methods via a Petrov–Galerkin type of Lagrange multiplier interpolation. Several numerical examples demonstrate the achievable improvements in terms of consistency and robustness for 3D contact analysis including finite deformations.

[1]  P. Alart,et al.  A mixed formulation for frictional contact problems prone to Newton like solution methods , 1991 .

[2]  Peter Wriggers,et al.  Frictionless 2D Contact formulations for finite deformations based on the mortar method , 2005 .

[3]  Peter Betsch,et al.  A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems , 2009 .

[4]  Tod A. Laursen,et al.  A mortar segment-to-segment frictional contact method for large deformations , 2003 .

[5]  Zdenek Dostl.  Optimal Quadratic Programming Algorithms: With Applications to Variational Inequalities , 2009 .

[6]  Barbara Wohlmuth,et al.  Variationally consistent discretization schemes and numerical algorithms for contact problems* , 2011, Acta Numerica.

[7]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[8]  Barbara I. Wohlmuth,et al.  Dual Quadratic Mortar Finite Element Methods for 3D Finite Deformation Contact , 2012, SIAM J. Sci. Comput..

[9]  Ekkehard Ramm,et al.  Unilateral non‐linear dynamic contact of thin‐walled structures using a primal‐dual active set strategy , 2007 .

[10]  M. Puso,et al.  A mortar segment-to-segment contact method for large deformation solid mechanics , 2004 .

[11]  Peter Wriggers,et al.  Computational Contact Mechanics , 2002 .

[12]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[13]  T. Laursen.  Computational Contact and Impact Mechanics , 2003 .

[14]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[15]  Barbara Wohlmuth,et al.  An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements , 2012 .

[16]  Peter Betsch,et al.  Transient three-dimensional contact problems: mortar method. Mixed methods and conserving integration , 2011 .

[17]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[18]  Michael A. Puso,et al.  A 3D mortar method for solid mechanics , 2004 .

[19]  Barbara I. Wohlmuth,et al.  Higher Order Mortar Finite Element Methods in 3D with Dual Lagrange Multiplier Bases , 2005, Numerische Mathematik.

[20]  Barbara I. Wohlmuth,et al.  A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier , 2000, SIAM J. Numer. Anal..

[21]  Barbara Wohlmuth,et al.  Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D , 2007 .

[22]  Barbara I. Wohlmuth,et al.  A Primal-Dual Active Set Algorithm for Three-Dimensional Contact Problems with Coulomb Friction , 2008, SIAM J. Sci. Comput..

[23]  P. W. Christensen,et al.  Formulation and comparison of algorithms for frictional contact problems , 1998 .

[24]  Wolfgang A. Wall,et al.  A dual mortar approach for 3D finite deformation contact with consistent linearization , 2010 .

[25]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[26]  Wolfgang A. Wall,et al.  Finite deformation frictional mortar contact using a semi‐smooth Newton method with consistent linearization , 2010 .

[27]  P. Wriggers,et al.  A mortar-based frictional contact formulation for large deformations using Lagrange multipliers , 2009 .

[28]  Wolfgang A. Wall,et al.  A finite deformation mortar contact formulation using a primal–dual active set strategy , 2009 .

[29]  Tod A. Laursen,et al.  A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations , 2007 .

[30]  Barbara Wohlmuth,et al.  A primal–dual active set strategy for non-linear multibody contact problems , 2005 .

[31]  Barbara I. Wohlmuth,et al.  Discretization Methods and Iterative Solvers Based on Domain Decomposition , 2001, Lecture Notes in Computational Science and Engineering.

[32]  Tod A. Laursen,et al.  A segment-to-segment mortar contact method for quadratic elements and large deformations , 2008 .

[33]  Rolf Krause,et al.  Efficient simulation of multi‐body contact problems on complex geometries: A flexible decomposition approach using constrained minimization , 2009 .

[34]  M. Bischoff,et al.  Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers , 2011 .