A multiscale large time increment/FAS algorithm with time‐space model reduction for frictional contact problems

SUMMARY A multiscale strategy using model reduction for frictional contact computation is presented. This new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies. This strategy is based on a combination between the LATIN (LArge Time INcrement) method and the FAS multigrid solver. The LATIN method is an iterative solver operating on the whole time-space domain. Applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned. Time-space vectors forming the so-called reduced basis depict particular scales of the problem. It becomes easy to make analogies with multigrid method to take full advantage of multiscale information. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Joachim Schöberl,et al.  Minimizing Quadratic Functions Subject to Bound Constraints with the Rate of Convergence and Finite Termination , 2005, Comput. Optim. Appl..

[2]  J. Haslinger,et al.  Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique , 2002 .

[3]  Pierre Ladevèze,et al.  A 3D shock computational strategy for real assembly and shock attenuator , 2002 .

[4]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[5]  F. Jourdan,et al.  A Gauss-Seidel like algorithm to solve frictional contact problems , 1998 .

[6]  Anthony Gravouil,et al.  A global model reduction approach for 3D fatigue crack growth with confined plasticity , 2011 .

[7]  Daniel Marceau,et al.  The adapted augmented Lagrangian method: a new method for the resolution of the mechanical frictional contact problem , 2012 .

[8]  T. Laursen,et al.  An algorithm for the matrix-free solution of quasistatic frictional contact problems , 1999 .

[9]  J. C. Simo,et al.  An augmented lagrangian treatment of contact problems involving friction , 1992 .

[10]  C. Farhat,et al.  A numerically scalable domain decomposition method for the solution of frictionless contact problems , 2001 .

[11]  Gianluigi Rozza,et al.  Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity , 2009, J. Comput. Phys..

[12]  A. Curnier,et al.  Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment , 1999 .

[13]  F. Chinesta,et al.  A Short Review in Model Order Reduction Based on Proper Generalized Decomposition , 2018 .

[14]  Siamak Niroomandi,et al.  Real-time deformable models of non-linear tissues by model reduction techniques , 2008, Comput. Methods Programs Biomed..

[15]  A. Lubrecht,et al.  MultiLevel Methods in Lubrication , 2013 .

[16]  Pierre Alart,et al.  Conjugate gradient type algorithms for frictional multi-contact problems: applications to granular materials , 2005 .

[17]  Charbel Farhat,et al.  Nonlinear model order reduction based on local reduced‐order bases , 2012 .

[18]  P. Alart,et al.  A generalized Newton method for contact problems with friction , 1988 .

[19]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[20]  Masao Fukushima,et al.  Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems , 2000, Comput. Optim. Appl..

[21]  P. Ladevèze,et al.  A modular approach to 3-D impact computation with frictional contact , 2000 .

[22]  P. Wriggers.  Finite element algorithms for contact problems , 1995 .

[23]  Pierre Alart,et al.  A domain decomposition strategy for nonclassical frictional multi-contact problems , 2001 .

[24]  Florent Galland,et al.  An adaptive model reduction approach for 3D fatigue crack growth in small scale yielding conditions , 2011 .

[25]  Pierre-Alain Boucard,et al.  A multiparametric strategy for the two step optimization of structural assemblies , 2013 .

[26]  Yves Renard,et al.  Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity , 2013 .

[27]  Tod A. Laursen,et al.  Formulation and treatment of frictional contact problems using finite elements , 1992 .

[28]  Pierre Kerfriden,et al.  Local/global model order reduction strategy for the simulation of quasi‐brittle fracture , 2011, 1108.3167.

[29]  L. Champaney,et al.  Large scale applications on parallel computers of a mixed domain decomposition method , 1997 .

[30]  P. Alart,et al.  Solution of frictional contact problems using ILU and coarse/fine preconditioners , 1995 .

[31]  C. Coulomb.  Théorie des machines simples, en ayant égard au frottement de leurs parties et a la roideur des cordages , 1968 .

[32]  Anthony Gravouil,et al.  A new fatigue frictional contact crack propagation model with the coupled X-FEM/LATIN method , 2007 .

[33]  Pierre Ladevèze,et al.  MODULAR ANALYSIS OF ASSEMBLAGES OF THREE-DIMENSIONAL STRUCTURES WITH UNILATERAL CONTACT CONDITIONS , 1999 .

[34]  Laurent Champaney,et al.  Une nouvelle approche modulaire pour l'analyse d'assemblages de structures tridimensionnelles , 1996 .

[35]  P. Ladevèze,et al.  The LATIN multiscale computational method and the Proper Generalized Decomposition , 2010 .

[36]  Pierre-Alain Boucard,et al.  A suitable computational strategy for the parametric analysis of problems with multiple contact , 2003 .

[37]  Marie-Christine Baietto,et al.  A two-scale extended finite element method for modelling 3D crack growth with interfacial contact , 2010 .

[38]  P. Ladevèze.  Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation , 1998 .

[39]  Frédéric Lebon,et al.  Multigrid Methods for Unilateral Contact Problems with Friction , 2007 .

[40]  Pierre-Alain Boucard,et al.  A parallel, multiscale domain decomposition method for the transient dynamic analysis of assemblies with friction , 2010 .

[41]  Pierre Alart,et al.  A scalable multiscale LATIN method adapted to nonsmooth discrete media , 2008 .

[42]  A. Ammar,et al.  Space–time proper generalized decompositions for the resolution of transient elastodynamic models , 2013 .

[43]  Pierre Alart,et al.  Méthode de Newton généralisée en mécanique du contact , 1997 .

[44]  Gabriella Bolzon,et al.  An effective computational tool for parametric studies and identification problems in materials mechanics , 2011 .

[45]  P Kerfriden,et al.  Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. , 2011, Computer methods in applied mechanics and engineering.

[46]  Marie-Christine Baietto,et al.  Optimization of a stabilized X-FEM formulation for frictional cracks , 2012 .

[47]  Francisco Chinesta,et al.  Recent Advances and New Challenges in the Use of the Proper Generalized Decomposition for Solving Multidimensional Models , 2010 .

[48]  Alain Combescure,et al.  Three dimensional automatic refinement method for transient small strain elastoplastic finite element computations , 2012 .

[49]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[50]  Pierre Ladevèze,et al.  Nonlinear Computational Structural Mechanics , 1999 .

[51]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[52]  Alain Combescure,et al.  Efficient FEM calculation with predefined precision through automatic grid refinement , 2005 .

[53]  M. Anitescu,et al.  Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems , 1997 .

[54]  David Néron,et al.  Multiparametric analysis within the proper generalized decomposition framework , 2012 .

[55]  A. Brandt,et al.  Multiscale solvers and systematic upscaling in computational physics , 2005, Comput. Phys. Commun..

[56]  Michel Saint Jean,et al.  The non-smooth contact dynamics method , 1999 .