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.
@article{M2AN_2011__45_2_235_0, author = {Pantz, Olivier}, title = {A frictionless contact algorithm for deformable bodies}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {45}, year = {2011}, pages = {235-254}, doi = {10.1051/m2an/2010041}, mrnumber = {2804638}, zbl = {1267.74016}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2011__45_2_235_0} }
Pantz, Olivier. A frictionless contact algorithm for deformable bodies. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 45 (2011) pp. 235-254. doi : 10.1051/m2an/2010041. http://gdmltest.u-ga.fr/item/M2AN_2011__45_2_235_0/
[1] Fluid-structure interaction and multi-body contact: Application to aortic valves. Comput. Methods Appl. Mech. Eng. 198 (2009) 3603-3612. | MR 2571824 | Zbl 1229.74095
, , and ,[2] Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 315-328. | MR 616782 | Zbl 0478.46032
,[3] 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.
,[4] 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.
,[5] 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.
and ,[6] 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.
and ,[7] Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987) 171-188. | MR 862546 | Zbl 0628.73043
and ,[8] Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 106 (1989) 97-159. | MR 980756 | Zbl 0677.73014
, and ,[9] 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. | MR 980756 | Zbl 0712.73009
, and ,[10] Global curvature and self-contact of nonlinearly elastic curves and rods. Calc. Var. Partial Differ. Equ. 14 (2002) 29-68. | MR 1883599 | Zbl 1006.49001
, , and ,[11] Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comput. Methods Appl. Mech. Eng. 51 (1985) 107-137. | MR 822741 | Zbl 0567.73120
, and ,[12] A general-purpose contact detection algorithm for nonlinear structural analysis code. Sandia Report SAND92-2141, Sandia National Laboratories, Alburquerque (1993).
, , and ,[13] An improved finite-element contact model for anatomical simulations. Vis. Comput. 19 (2003) 291-309.
, and ,[14] The non-smooth contact dynamics method. Computational modeling of contact and friction. Comput. Methods Appl. Mech. Eng. 177 (1999) 235-257. | MR 1710453 | Zbl 0959.74046
,[15] 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. | MR 1884312 | Zbl 1011.74051
, and ,[16] 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). | MR 961258 | Zbl 0685.73002
and ,[17] Large displacement frictional contact: a continuum framework for finite element discretization. Eur. J. Mech. A Solids 14 (1995) 237-253. | MR 1331237 | Zbl 0824.73061
,[18] Formulation and treatment of frictional contact problems using finite elements. SUDAM Report 92 (1992).
,[19] Computational contact and impact mechanics, Fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer-Verlag, Berlin (2002). | MR 1902698 | Zbl 0996.74003
,[20] 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. | MR 1242927 | Zbl 0833.73057
and ,[21] 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.
and ,[22] Variational analysis and generalized differentiation. I Basic theory, Grundlehren der Mathematischen Wissenschaften 330. Springer-Verlag, Berlin (2006). | MR 2191744 | Zbl 1100.49002
,[23] Variational analysis and generalized differentiation. II Applications, Grundlehren der Mathematischen Wissenschaften 331. Springer-Verlag, Berlin (2006). | MR 2191745 | Zbl 1100.49002
,[24] An introduction to unilateral dynamics. Lect. Notes Appl. Comput. Mech. 14 (2004) 1-26. | Zbl 1123.70014
,[25] The modeling of deformable bodies with frictionless (self-)contacts. Rapport Interne 585, CMAP, École Polytechnique, Palaiseau (2005). | Zbl 1138.74038
,[26] Contacts en dimension 2 : Une méthode de pénalisation. Rapport Interne 597, CMAP, École Polytechnique, Palaiseau (2006).
,[27] The modeling of deformable bodies with frictionless (self-)contacts. Arch. Rational Mech. Anal. 188 (2008) 183-212. | MR 2385740 | Zbl 1138.74038
,[28] A 3D contact smoothing method using Gregory patches. Int. J. Numer. Methods Eng. 54 (2002) 1161-1194. | MR 1911054 | Zbl 1098.74711
and ,[29] A variational approach to obstacle problems for shearable nonlinearly elastic rods. Arch. Rational Mech. Anal. 140 (1997) 103-159. | MR 1482930 | Zbl 0898.73080
,[30] Regularity for shearable nonlinearly elastic rods in obstacle problems. Arch. Rational Mech. Anal. 145 (1998) 23-49. | MR 1656476 | Zbl 0915.73078
,[31] Variational approach to contact problems in nonlinear elasticity. Calc. Var. Partial Differ. Equ. 15 (2002) 433-449. | MR 1942127 | Zbl 1050.35015
,[32] Euler-Lagrange equations for nonlinearly elastic rods with self-contact. Arch. Rational Mech. Anal. 168 (2003) 35-82. | MR 2029004 | Zbl 1030.74029
and ,[33] Almost-everywhere injectivity in nonlinear elasticity. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988) 79-95. | MR 952330 | Zbl 0656.73010
,[34] Finite element algorithms for contact problems. Arch. Comput. Methods Eng. 2 (1995) 1-49. | MR 1367147 | Zbl 0840.73067
,[35] Computational Contact Mechanics. Springer, New York (2006).
,[36] Two dimensional mortar contact methods for large deformation frictional sliding. Int. J. Numer. Methods Eng. 62 (2005) 1183-1225. | MR 2120292 | Zbl 1161.74497
, and ,