Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T10:25:22.735Z Has data issue: false hasContentIssue false
  • English
  • Français

A frictionless contact algorithm for deformable bodies*

Published online by Cambridge University Press:  02 August 2010

Olivier Pantz
Olivier Pantz*
Affiliation:
CMAP, École Polytechnique, 91128 Palaiseau Cedex, France. olivier.pantz@polytechnique.org
Get access

Abstract

This article is devoted to the presentation of a new contact algorithm for bodies undergoing finite deformations. We only address the kinematic aspect of the contact problem, that is the numerical treatment of the non-intersection constraint. In consequence, mechanical aspects like friction, adhesion or wear are not investigated and we restrict our analysis to the simplest frictionless case. On the other hand, our method allows us to treat contacts and self-contacts, thin or non-thin structures in a single setting.

Keywords

Contactfrictionlessself-contactelasticityfinite deformations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 2 , March 2011 , pp. 235 - 254
Copyright
© EDP Sciences, SMAI, 2010

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

Astorino, M., Gerbeau, J.-F., Pantz, O. and Traoré, K.-F., Fluid-structure interaction and multi-body contact: Application to aortic valves. Comput. Methods Appl. Mech. Eng. 198 (2009) 3603–3612. CrossRef
Ball, J.M., Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 315–328. CrossRef
D. Baraff, Analytical methods for dynamic simulation of non-penetrating rigid bodies, in SIGGRAPH '89: Proceedings of the 16th annual conference on computer graphics and interactive techniques, ACM Press, New York, USA (1989) 223–232.
D. Baraff, Fast contact force computation for nonpenetrating rigid bodies, in SIGGRAPH '94: Proceedings of the 21st annual conference on computer graphics and interactive techniques, ACM Press, New York, USA (1994) 23–34.
D. Baraff and A. Witkin, Dynamic simulation of non-penetrating flexible bodies, in SIGGRAPH '92: Proceedings of the 19th annual conference on computer graphics and interactive techniques, ACM Press, New York (1992) 303–308.
D. Baraff and A. Witkin, Large steps in cloth simulation, in SIGGRAPH '98: Proceedings of the 25th annual conference on computer graphics and interactive techniques, ACM Press, New York (1998) 43–54.
Ciarlet, P.G. and Nečas, J., Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171–188. CrossRef
Giaquinta, M., Modica, G. and Souček, J., Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106 (1989) 97–159. CrossRef
M. Giaquinta, G. Modica and J. Souček, Erratum and addendum to: “Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity” [Arch. Rational Mech. Anal. 106 (1989) 97–159; MR 90c:58044]. Arch. Rational Mech. Anal. 109 (1990) 385–392.
Gonzalez, O., Maddocks, J.H., Schuricht, F. and von der Mosel, H., Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differ. Equ. 14 (2002) 29–68. CrossRef
Hallquist, J.O., Goudreau, G.L. and Benson, D.J., Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comput. Methods Appl. Mech. Eng. 51 (1985) 107–137. CrossRef
M. Heinstein, S. Attaway, J. Swegle and F. Mello, A general-purpose contact detection algorithm for nonlinear structural analysis code. Sandia Report SAND92-2141, Sandia National Laboratories, Alburquerque (1993).
Hirota, G., Fisher, S. and State, A., An improved finite-element contact model for anatomical simulations. Vis. Comput. 19 (2003) 291–309. CrossRef
Jean, M., The non-smooth contact dynamics method. Computational modeling of contact and friction. Comput. Methods Appl. Mech. Eng. 177 (1999) 235–257. CrossRef
Jean, M., Acary, V. and Monerie, Y., Non-smooth contact dynamics approach of cohesive materials. Non-smooth mechanics. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 359 (2001) 2497–2518. CrossRef
N. Kikuchi and J.T. Oden, Contact problems in elasticity: a study of variational inequalities and finite element methods, SIAM Studies in Applied Mathematics 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1988).
Klarbring, A., Large displacement frictional contact: a continuum framework for finite element discretization. Eur. J. Mech. A Solids 14 (1995) 237–253.
T.A. Laursen, Formulation and treatment of frictional contact problems using finite elements. SUDAM Report 92 (1992).
T.A. Laursen, Computational contact and impact mechanics, Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer-Verlag, Berlin (2002).
Laursen, T.A. and Simo, J.C., A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int. J. Numer. Methods Eng. 36 (1993) 3451–3485. CrossRef
V.J. Milenkovic and H. Schmidl, Optimization-based animation, in SIGGRAPH '01: Proceedings of the 28th annual conference on computer graphics and interactive techniques, ACM Press, New York (2001) 37–46.
B.S. Mordukhovich, Variational analysis and generalized differentiation. I Basic theory, Grundlehren der Mathematischen Wissenschaften 330. Springer-Verlag, Berlin (2006).
B.S. Mordukhovich, Variational analysis and generalized differentiation. II Applications, Grundlehren der Mathematischen Wissenschaften 331. Springer-Verlag, Berlin (2006).
Moreau, J.J., An introduction to unilateral dynamics. Lect. Notes Appl. Comput. Mech. 14 (2004) 1–26. CrossRef
O. Pantz, The modeling of deformable bodies with frictionless (self-)contacts. Rapport Interne 585, CMAP, École Polytechnique, Palaiseau (2005).
O. Pantz, Contacts en dimension 2 : Une méthode de pénalisation. Rapport Interne 597, CMAP, École Polytechnique, Palaiseau (2006).
Pantz, O., The modeling of deformable bodies with frictionless (self-)contacts. Arch. Rational Mech. Anal. 188 (2008) 183–212. CrossRef
Puso, M.A. and Laursen, T.A., A 3D contact smoothing method using Gregory patches. Int. J. Numer. Methods Eng. 54 (2002) 1161–1194. CrossRef
Schuricht, F., A variational approach to obstacle problems for shearable nonlinearly elastic rods. Arch. Rational Mech. Anal. 140 (1997) 103–159. CrossRef
Schuricht, F., Regularity for shearable nonlinearly elastic rods in obstacle problems. Arch. Rational Mech. Anal. 145 (1998) 23–49. CrossRef
Schuricht, F., Variational approach to contact problems in nonlinear elasticity. Calc. Var. Partial Differ. Equ. 15 (2002) 433–449. CrossRef
Schuricht, F. and von der Mosel, H., Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Rational Mech. Anal. 168 (2003) 35–82. CrossRef
Tang, Q., Almost-everywhere injectivity in nonlinear elasticity. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988) 79–95.
Wriggers, P., Finite element algorithms for contact problems. Arch. Comput. Methods Eng. 2 (1995) 1–49. CrossRef
P. Wriggers, Computational Contact Mechanics. Springer, New York (2006).
Yang, B., Laursen, T.A. and Meng, X., Two dimensional mortar contact methods for large deformation frictional sliding. Int. J. Numer. Methods Eng. 62 (2005) 1183–1225. CrossRef