A fast solver for finite deformation contact problems

We present a new solver for large-scale two-body contact problems in nonlinear elasticity. It is based on an SQP-trust-region approach. This guarantees global convergence to a first-order critical point of the energy functional. The linearized contact conditions are discretized using mortar elements. A special basis transformation known from linear contact problems allows to use a monotone multigrid solver for the inner quadratic programs. They can thus be solved with multigrid complexity. Our algorithm does not contain any regularization or penalization parameters, and can be used for all hyperelastic material models.

[1]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[2]  R. Kornhuber.  Monotone multigrid methods for elliptic variational inequalities I , 1994 .

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

[4]  Serge Gratton,et al.  Recursive Trust-Region Methods for Multiscale Nonlinear Optimization , 2008, SIAM J. Optim..

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

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

[7]  J. Schöberl.  Efficient contact solvers based on domain decomposition techniques , 2001 .

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

[9]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[10]  Francisco J. Prieto,et al.  A Sequential Quadratic Programming Algorithm Using an Incomplete Solution of the Subproblem , 1995, SIAM J. Optim..

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

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

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