Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-02T11:49:03.809Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  08 July 2014

Farshid Dabaghi ,
Adrien Petrov ,
Jérôme Pousin  and
Yves Renard
Farshid Dabaghi
Affiliation:
Universitéde Lyon, CNRS, INSA-Lyon, Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, 69621 Villeurbanne, France. . farshid.dabaghi@insa-lyon.fr; apetrov@math.univ-lyon1.fr jerome.pousin@insa-lyon.fr; Yves.Renard@insa-lyon.fr
Adrien Petrov
Affiliation:
Universitéde Lyon, CNRS, INSA-Lyon, Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, 69621 Villeurbanne, France. . farshid.dabaghi@insa-lyon.fr; apetrov@math.univ-lyon1.fr jerome.pousin@insa-lyon.fr; Yves.Renard@insa-lyon.fr
Jérôme Pousin
Affiliation:
Universitéde Lyon, CNRS, INSA-Lyon, Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, 69621 Villeurbanne, France. . farshid.dabaghi@insa-lyon.fr; apetrov@math.univ-lyon1.fr jerome.pousin@insa-lyon.fr; Yves.Renard@insa-lyon.fr
Yves Renard
Affiliation:
Universitéde Lyon, CNRS, INSA-Lyon, Institut Camille Jordan UMR 5208, 20 Avenue A. Einstein, 69621 Villeurbanne, France. . farshid.dabaghi@insa-lyon.fr; apetrov@math.univ-lyon1.fr jerome.pousin@insa-lyon.fr; Yves.Renard@insa-lyon.fr
Get access

Abstract

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.

Keywords

Existenceuniquenessconvergencemass redistribution methodvariational inequalityunilateral contact
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 1147 - 1169
Copyright
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alart, P. and Curnier, A., A generalized Newton method for contact problems with friction. J. Mech. Theor. Appl. 7 (1988) 67–82. Google Scholar
Armero, F. and Petocz, E., Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput. Methods Appl. Mech. Engrg. 158 (1998) 269–300. Google Scholar
J.P. Aubin, Approximation of elliptic boundary-value problems. Pure and Applied Mathematics, Vol. XXVI. Wiley-Interscience (1972).
Ball, J.M., Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 63 (1977) 370–373. Google Scholar
Bárcenas, D., The fundamental theorem of calculus for Lebesgue integral. Divulg. Mat. 8 (2000) 75–85. Google Scholar
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam (1973).
M. Crouzeix and A.L. Mignot, Analyse numérique des équations différentielles. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1984).
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 8. INSTN: Collection Enseignement. Masson, Paris (1988).
K. Deimling, Multivalued differential equations. Vol. 1 of de Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter & Co., Berlin (1992).
Doyen, D. and Ern, A., Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem. Commun. Math. Sci. 7 (2009) 1063–1072. Google Scholar
D. Doyen, A. Ern and S. Piperno, Time-integration schemes for the finite element dynamic Signorini problem. SIAM J. Sci. Comput. (2011) 223–249.
A. Ern and J.L. Guermond, Theory and practice of finite elements. Appl. Math. Sci., vol. 159. Springer-Verlag, New York (2004).
Hager, C. and Wohlmuth, B.I., Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements. SIAM J. Num. Anal. 47 (2009) 1863–1885. Google Scholar
Hauret, P., Mixed interpretation and extensions of the equivalent mass matrix approach for elastodynamics with contact. Comput. Methods Appl. Mech. Engrg. 199 (2010) 2941–2957. Google Scholar
Hugues, T.J.R., Taylor, R.L., Sackman, J.L., A. Curnier and W. Kano Knukulchai, A finite method for a class of contact-impact problems. Comput. Methods Appl. Mech. Engrg. 8 (1976) 249–276. Google Scholar
Khenous, H.B., Laborde, P. and Renard, Y., Mass redistribution method for finite element contact problems in elastodynamics. Eur. J. Mech. A Solids 27 (2008) 918–932. Google Scholar
Krenk, S., Energy conservation in Newmark based time integration algorithms. Comput. Methods Appl. Mech. Engrg. 195 (2006) 6110–6124. Google Scholar
N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies Appl. Math. SIAM, Philadelphia, Pa (1988).
Kim, J.U., A boundary thin obstacle problem for a wave equation. Commun. Partial Differential Eqs. 14 (1989) 1011–1026. Google Scholar
Laursen, T.A. and Chawla, V., Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Engrg. 40 (1997) 863–886. Google Scholar
Laursen, T.A. and Love, G.R., Improved implicit integrators for transient impact problems-geometric admissibility within the conserving framework. Int. J. Numer. Methods Engrg. 53 (2002) 245–274. Google Scholar
Lebeau, G. and Schatzman, M., A wave problem in a half-space with a unilateral constraint at the boundary. J. Differ. Eqs. 53 (1984) 309–361. Google Scholar
Moreau, J.-J., Liaisons unilatérales sans frottement et chocs inélastiques. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 296 (1983) 1473–1476. Google Scholar
J.-J. Moreau and P.D. Panagiotopoulos, Nonsmooth mechanics and applications. Vol. 302 of CISM Courses Lect. Springer-Verlag, Vienna (1988).
Paoli, L., Time discretization of vibro-impact. R. Soc. London Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001) 2405–2428. Google Scholar
Paoli, L. and Schatzman, M., A numerical scheme for impact problem I. SIAM J. Numer. Anal. 40 (2002) 702–733. Google Scholar
Paoli, L. and Schatzman, M., Approximation et existence en vibro-impact. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 1003–1007. Google Scholar
Renard, Y., Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comput. Meth. Appl. Mech. Engng. 256 (2013) 38–55. Google Scholar
Y. Renard and J. Pommier, Getfem++. An Open Source generic C++ library for finite element methods. http://home.gna.org/getfem.
W. Rudin, Real and complex analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York, 2nd edn (1974).
Schatzman, M., A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle. J. Math. Anal. Appl. 73 (1980) 138–191. Google Scholar
Schatzman, M. and Bercovier, M., Numerical approximation of a wave equation with unilateral constraints. Math. Comput. 53 (1989) 55–79. Google Scholar
K. Schweizerhof, J.O. Hallquist and D. Stillman, Efficiency refinements of contact strategies and algorithms in explicit finite element programming. Compt. Plasticity. Edited by Owen, Onate, Hinton, Pineridge (1992) 457–482.
Simon, J., Compact sets in the space Lp(0,T;B). Ann. Mat. Pura Appl. 146 (1987) 65–96. Google Scholar
P. Wriggers, Computational contact mechanics. John Wiley and Sons Ltd. (2002).