Explicit Verlet time-integration for a Nitsche-based approximation of elastodynamic contact problems

The aim of the present paper is to study theoretically and numerically the Verlet scheme for the explicit time-integration of elastodynamic problems with a contact condition approximated by Nitsche’s method. This is a continuation of papers (Chouly et al. ESAIM Math Model Numer Anal 49(2), 481–502, 2015; Chouly et al. ESAIM Math Model Numer Anal 49(2), 503–528, 2015) where some implicit schemes (theta-scheme, Newmark and a new hybrid scheme) were proposed and proved to be well-posed and stable under appropriate conditions. A theoretical study of stability is carried out and then illustrated with both numerical experiments and numerical comparison to other existing discretizations of contact problems.

[1]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[2]  T. Laursen,et al.  Contact—impact modeling in explicit transient dynamics , 2000 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  Franz Chouly,et al.  A Nitsche-Based Method for Unilateral Contact Problems: Numerical Analysis , 2013, SIAM J. Numer. Anal..

[5]  Alexandre Ern,et al.  Analysis of the Modified Mass Method for the Dynamic Signorini Problem with Coulomb Friction , 2011, SIAM J. Numer. Anal..

[6]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[7]  Gary Cohen,et al.  Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations , 2016 .

[8]  Erik Burman,et al.  A Nitsche-based formulation for fluid-structure interactions with contact , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[9]  M. Schatzman,et al.  Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: La corde vibrante avec obstacle ponctuel , 1980 .

[10]  Jérôme Pousin,et al.  An overview of recent results on Nitsche's method for contact problems , 2016 .

[11]  Laetitia Paoli,et al.  Numerical simulation of the dynamics of an impacting bar , 2007 .

[12]  G. Burton.  Sobolev Spaces , 2013 .

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

[14]  Barbara Wohlmuth,et al.  A stable energy‐conserving approach for frictional contact problems based on quadrature formulas , 2008 .

[15]  L. Paoli,et al.  Approximation et existence en vibro-impact , 1999 .

[16]  Laetitia Paoli,et al.  A Numerical Scheme for Impact Problems I: The One-Dimensional Case , 2002, SIAM J. Numer. Anal..

[17]  M. Schatzman.  A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle , 1980 .

[18]  Jérôme Pousin,et al.  A weighted finite element mass redistribution method for dynamic contact problems , 2019, J. Comput. Appl. Math..

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

[20]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[21]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[22]  J. Pousin,et al.  A robust finite element redistribution approach for elastodynamic contact problems , 2016 .

[23]  J. J.,et al.  Numerical aspects of the sweeping process , 1999 .

[24]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[25]  David E. Stewart,et al.  Existence of Solutions for a Class of Impact Problems Without Viscosity , 2006, SIAM J. Math. Anal..

[26]  Nathan M. Newmark,et al.  A Method of Computation for Structural Dynamics , 1959 .

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

[28]  C. Mariotti,et al.  An explicit energy-momentum conserving time-integration scheme for Hamiltonian dynamics , 2017 .

[29]  F. Armero,et al.  Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems , 1998 .

[30]  Mark O. Neal,et al.  Contact‐impact by the pinball algorithm with penalty and Lagrangian methods , 1991 .

[31]  Franz Chouly,et al.  An adaptation of Nitscheʼs method to the Tresca friction problem , 2014 .

[32]  Yves Renard,et al.  The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems , 2010, J. Comput. Appl. Math..

[33]  Adrian J. Lew,et al.  An explicit asynchronous contact algorithm for elastic body–rigid wall interaction , 2012 .

[34]  Matthew West,et al.  Decomposition contact response (DCR) for explicit finite element dynamics , 2005, International Journal for Numerical Methods in Engineering.

[35]  Laetitia Paoli,et al.  A Numerical Scheme for Impact Problems II: The Multidimensional Case , 2002, SIAM J. Numer. Anal..

[36]  M. Raous,et al.  Friction and instability of steady sliding: squeal of a rubber/glass contact , 1999 .

[37]  Franz Chouly,et al.  On convergence of the penalty method for unilateral contact problems , 2012, 1204.4136.

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

[39]  Alexandre Ern,et al.  Time-Integration Schemes for the Finite Element Dynamic Signorini Problem , 2011, SIAM J. Sci. Comput..

[40]  Michel Salaün,et al.  Vibro-impact of a plate on rigid obstacles: existence theorem, convergence of a scheme and numerical simulations , 2013 .

[41]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[42]  D. Flanagan,et al.  PRONTO 3D: A three-dimensional transient solid dynamics program , 1989 .

[43]  Arne S. Gullerud,et al.  Solution verification for explicit transient dynamics problems in the presence of hourglass and contact forces. , 2004 .

[44]  Franz Chouly,et al.  A Nitsche finite element method for dynamic contact : 1. Semi-discrete problem analysis and time-marching schemes , 2014 .

[45]  Alexandre Ern,et al.  Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem , 2009 .

[46]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[47]  Franz Chouly,et al.  Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments , 2014, Math. Comput..

[48]  Michelle Schatzman,et al.  A wave problem in a half-space with a unilateral constraint at the boundary , 1984 .

[49]  P. Tallec,et al.  Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact , 2006 .

[50]  Laetitia Paoli,et al.  Vibrations avec contraintes unilatérales et perte d'énergie aux impacts, en dimension finie , 1993 .

[51]  J. U. Kim,et al.  A boundary thin obstacle problem for a wave equation , 1989 .

[52]  Vincent Acary,et al.  Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: Definition and outlook , 2014, Math. Comput. Simul..

[53]  Jérôme Pousin,et al.  Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary , 2014 .

[54]  Sebastian Wolff,et al.  Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints , 2013, International journal for numerical methods in engineering.

[55]  J. Moreau,et al.  Unilateral Contact and Dry Friction in Finite Freedom Dynamics , 1988 .

[56]  Franz Chouly,et al.  A Nitsche finite element method for dynamic contact : 1. Semi-discrete problem analysis and time-marching schemes , 2014 .

[57]  A. Ern,et al.  Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models , 2013 .