Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary

This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.

[1]  J. Moreau,et al.  Nonsmooth Mechanics and Applications , 1989 .

[2]  The Fundamental Theorem of Calculus for Lebesgue Integral , 2000 .

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

[4]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

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

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

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

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

[9]  P. Wriggers.  Computational contact mechanics , 2012 .

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

[11]  M. Schatzman,et al.  Numerical approximation of a wave equation with unilateral constraints , 1989 .

[12]  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.

[13]  R. Dautray,et al.  Analyse mathématique et calcul numérique pour les sciences et les techniques , 1984 .

[14]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

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

[16]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[17]  W. Rudin.  Real and complex analysis , 1968 .

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

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

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

[21]  J. Ball.  Strongly continuous semigroups, weak solutions, and the variation of constants formula , 1977 .

[22]  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.

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

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

[25]  Jerry J. Koliha,et al.  A Fundamental Theorem of Calculus for Lebesgue Integration , 2006, Am. Math. Mon..

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

[27]  J. Aubin.  Approximation of Elliptic Boundary-Value Problems , 1980 .

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

[29]  Laetitia Paoli,et al.  A numerical scheme for impact problems , 1999 .

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

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

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

[33]  C. HAGER,et al.  Analysis of a Space-Time Discretization for Dynamic Elasticity Problems Based on Mass-Free Surface Elements , 2009, SIAM J. Numer. Anal..

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

[35]  M. Crouzeix,et al.  Analyse Numérique des équations Différentielles , 1987 .

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