THE SINGULAR DYNAMIC METHOD FOR DYNAMIC CONTACT OF THIN ELASTIC STRUCTURES

This paper adresses the approximation of the dynamic impact of thin elastic structures. The principle of the presented method is the use of a singular mass matrix obtained by different discretizations of the deflection and velocity. The obtained semi-discretized problem is proved to be well-posed and energy conserving. The method is applied on some membrane, beam and plate models and associated numerical experiments are discussed.

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

[2]  Laetitia Paoli,et al.  Time discretization of vibro‐impact , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

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

[5]  Houari Boumediène Khenous.  Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique. Etude mathématique et résolution numérique. , 2005 .

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

[7]  Robert L. Taylor,et al.  On a finite element method for dynamic contact/impact problems , 1993 .

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

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

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

[11]  J. Moreau.  Liaisons unilatérales sans frottement et chocs inélastiques , 1983 .

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

[13]  Yves Renard,et al.  Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers , 2006 .

[14]  P. Alart,et al.  A mixed formulation for frictional contact problems prone to Newton like solution methods , 1991 .

[15]  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.

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

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

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

[19]  Laetitia Paoli,et al.  VIBRATIONS OF A BEAM BETWEEN OBSTACLES. CONVERGENCE OF A FULLY DISCRETIZED APPROXIMATION , 2006 .

[20]  A. Curnier,et al.  A finite element method for a class of contact-impact problems , 1976 .

[21]  K. Deimling.  Multivalued Differential Equations , 1992 .

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

[23]  Thomas J. R. Hughes,et al.  A reduction scheme for problems of structural dynamics , 1976 .

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