A time discretization scheme based on Rothe’s method for dynamical contact problems with friction

Abstract We present a new dissipative and contact-stabilized time discretization scheme for dynamic frictional contact problems, which is based on Rothe’s method. Especially for the case of Coulomb friction the stability of the contact stresses is of crucial importance as they directly influence the frictional behavior. In our approach, we obtain an accurate description of the frictional stresses by a time-discretized friction law allowing for an implicit treatment of the contact forces in the framework of the Newmark scheme. Moreover, undesirable oscillations at the contact interface are removed by employing an additional L 2 -projection within the predictor step. Since the implicit treatment of the material behavior and the frictional response requires a (quasi-)variational inequality to be solved in each time step, we derive a non-smooth multiscale method, which allows for the efficient and robust solution of these highly non-linear problems. The convergence of this multiscale method is proven for the case of Tresca friction. For the case of Coulomb friction, an inexact fixed point iteration in the normal stresses is used. We furthermore, consider the case of two-body contact. Here, the information transfer at the contact interface is realized by means of mortar methods, which provide a stable discretization of the relative displacements and the stresses at the contact boundary. Numerical results for the resulting fully discrete scheme in 3D are presented, showing the high accuracy of the proposed method.

[1]  Erich Rothe,et al.  Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben , 1930 .

[2]  Jaroslav Haslinger,et al.  Approximation of the signorini problem with friction, obeying the coulomb law , 1983 .

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

[4]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[5]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[6]  Tod A. Laursen,et al.  Energy consistent algorithms for frictional contact problems , 1998 .

[7]  Tod A. Laursen,et al.  Improved implicit integrators for transient impact problems––dynamic frictional dissipation within an admissible conserving framework , 2003 .

[8]  Peter Deuflhard,et al.  A contact‐stabilized Newmark method for dynamical contact problems , 2008 .

[9]  Ralf Kornhuber,et al.  On constrained Newton linearization and multigrid for variational inequalities , 2002, Numerische Mathematik.

[10]  Rolf Krause,et al.  Monotone Multigrid Methods on Nonmatching Grids for Nonlinear Multibody Contact Problems , 2003, SIAM J. Sci. Comput..

[11]  ROLF KRAUSE,et al.  A Nonsmooth Multiscale Method for Solving Frictional Two-Body Contact Problems in 2D and 3D with Multigrid Efficiency , 2008, SIAM J. Sci. Comput..

[12]  Patrick Laborde,et al.  On the discretization of contact problems in elastodynamics , 2006 .

[13]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[14]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

[15]  P. G. Ciarlet,et al.  Three-dimensional elasticity , 1988 .

[16]  Rolf Krause,et al.  Monotone Multigrid Methods for Signorini's Problem with Friction , 2001 .

[17]  Michel Raous,et al.  Remarks on a Numerical Method for Unilateral Contact Including Friction , 1991 .

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

[19]  T. Laursen,et al.  DESIGN OF ENERGY CONSERVING ALGORITHMS FOR FRICTIONLESS DYNAMIC CONTACT PROBLEMS , 1997 .

[20]  P. Laborde,et al.  A energy conserving approximation for elastodynamic contact problems , 2006 .

[21]  R. Kornhuber,et al.  Adaptive multigrid methods for Signorini’s problem in linear elasticity , 2001 .

[22]  E. A. Repetto,et al.  Finite element analysis of nonsmooth contact , 1999 .

[23]  Patrick Laborde,et al.  Mass redistribution method for finite element contact problems in elastodynamics , 2008 .

[24]  R. Glowinski,et al.  Numerical Methods for Nonlinear Variational Problems , 1985 .

[25]  T. Laursen,et al.  Improved implicit integrators for transient impact problems—geometric admissibility within the conserving framework , 2002, International Journal for Numerical Methods in Engineering.

[26]  Ralf Kornhuber,et al.  A monotone multigrid solver for two body contact problems in biomechanics , 2007 .