An explicit time-integrator with singular mass for non-smooth dynamics

This article addresses the simulation of non-smooth dynamics problems with unilateral contact constraints between rigid and deformable bodies. It proposes a modified CD-Lagrange scheme with a singular mass matrix. This scheme is explicit, and based on a contact condition on velocity. The formulation is designed for a 1D impact problem between deformable and rigid body with unilateral constraint. The singular mass matrix allows to get a more accurate energy balance on the discrete system, especially during non-smooth events. An extension is presented then for the 3D cases. Its implementation is easy, and fully compatible with large deformations or non-linear materials. Indeed it consists only in adding a numerical parameter for each contact node. The energy balance for the singular 3D formulation is improved compared to the consistent one.

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

[2]  R. Taylor,et al.  Lagrange constraints for transient finite element surface contact , 1991 .

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

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

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

[6]  Anthony Gravouil,et al.  A new heterogeneous asynchronous explicit–implicit time integrator for nonsmooth dynamics , 2017 .

[7]  A. Cardona,et al.  Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-α scheme , 2014 .

[8]  M. Bischoff,et al.  Hybrid‐mixed discretization of elasto‐dynamic contact problems using consistent singular mass matrices , 2013 .

[9]  J. Moreau.  Numerical aspects of the sweeping process , 1999 .

[10]  Dinesh K. Pai,et al.  Geometric Numerical Integration of Inequality Constrained, Nonsmooth Hamiltonian Systems , 2010, SIAM J. Sci. Comput..

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

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

[13]  J. C. Simo,et al.  Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics , 1992 .

[14]  Jerrold E. Marsden,et al.  Nonsmooth Lagrangian Mechanics and Variational Collision Integrators , 2003, SIAM J. Appl. Dyn. Syst..

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

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

[17]  Patrice Hauret,et al.  Mixed interpretation and extensions of the equivalent mass matrix approach for elastodynamics with contact , 2010 .

[18]  David Dureisseix,et al.  Benchmark cases for robust explicit time integrators in non-smooth transient dynamics , 2019, Advanced Modeling and Simulation in Engineering Sciences.

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

[20]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[21]  P. Panagiotopoulos,et al.  New developments in contact problems , 1999 .

[22]  Rolf Krause,et al.  Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme , 2012 .

[23]  Thomas J. R. Hughes,et al.  Improved numerical dissipation for time integration algorithms in structural dynamics , 1977 .

[24]  V. Acary,et al.  A nonsmooth generalized‐ α scheme for flexible multibody systems with unilateral constraints , 2013 .

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

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

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

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

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

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

[31]  Vincent Acary,et al.  Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts , 2012 .

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

[33]  J. Marsden,et al.  Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems , 2000 .

[34]  Steen Krenk,et al.  Energy conservation in Newmark based time integration algorithms , 2006 .

[35]  J. Marsden,et al.  Discrete mechanics and variational integrators , 2001, Acta Numerica.

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

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