Stable IMEX schemes for a Nitsche-based approximation of elastodynamic contact problems. Selective mass scaling interpretation

Bretin, Élie; Renard, Yves

The SMAI Journal of computational mathematics, Tome 6 (2020), pp. 159-185.

We introduce some IMEX schemes (implicit-explicit schemes with an implicit term being linear) for approximating elastodynamic contact problems when the contact condition is taken into account with a Nitsche method. We develop a theoretical and numerical study of the properties of the schemes, especially in terms of stability, provide some numerical comparisons with standard explicit and implicit scheme and propose some improvements to obtain a more reliable approximation of motion for large time steps. We also show how selective mass scaling techniques can be interpreted as IMEX schemes.

Classification : 74H15, 65N30, 74M15
Mots clés : unilateral contact, elastodynamics, Nitsche’s method, IMEX schemes, stability, finite element method, selective mass scaling

[1] Adams, R.-A. Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | Zbl

[2] Ahn, J.; Stewart, D. E. Existence of solutions for a class of impact problems without viscosity, SIAM J. Math. Anal., Volume 38 (2006) no. 1, pp. 37-63 | DOI | MR | Zbl

[3] Alart, P.; Curnier, A. A generalized Newton method for contact problems with friction, J. Méc. Théor. Appl., Volume 7 (1988) no. 1, pp. 67-82 | MR | Zbl

[4] Annavarapu, C.; Hautefeuille, M.; Dolbow, J. E. A Nitsche stabilized finite element method for frictional sliding on embedded interfaces. Part I: Single interface, Comput. Methods Appl. Mech. Eng., Volume 268 (2014), pp. 417-436 | DOI | MR | Zbl

[5] Armero, F.; Petöcz, E. Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems, Comput. Methods Appl. Mech. Eng., Volume 158 (1998) no. 3-4, pp. 269-300 | DOI | MR | Zbl

[6] Belhytschko, T.; Neal, M. O. Contact-impact by the pinball algorithm with penaly and Lagrangian methods, Int. J. Numer. Meth. Engng., Volume 31 (1991), pp. 547-572 | DOI

[7] Boiveau, T.; Burman, E. A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity, IMA J. Numer. Anal., Volume 36 (2016) no. 2, pp. 770-795 | DOI | MR | Zbl

[8] Chouly, F.; Fabre, M.; Hild, P.; Mlika, R.; Pousin, J.; Renard, Y. An overview of recent results on Nitsche’s method for contact problems, Geometrically unfitted finite element methods and applications (Lecture Notes in Computational Science and Engineering), Volume 121, Springer, 2018, pp. 93-141 | DOI | Zbl

[9] Chouly, F.; Hild, P. A Nitsche-based method for unilateral contact problems: numerical analysis, SIAM J. Numer. Anal., Volume 51 (2013) no. 2, pp. 1295-1307 | DOI | MR | Zbl

[10] Chouly, F.; Hild, P.; Renard, Y. A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes, ESAIM, Math. Model. Numer. Anal., Volume 49 (2015) no. 2, pp. 481-502 | DOI | MR | Zbl

[11] Chouly, F.; Hild, P.; Renard, Y. A Nitsche finite element method for dynamic contact: 2. Stability of the schemes and numerical experiments, ESAIM, Math. Model. Numer. Anal., Volume 49 (2015) no. 2, pp. 503-528 | DOI | MR | Zbl

[12] Chouly, F.; Hild, P.; Renard, Y. Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments, Math. Comput., Volume 84 (2015) no. 293, pp. 1089-1112 | DOI | MR | Zbl

[13] Chouly, F.; Renard, Y. Explicit Verlet time-integration for a Nitsche-based approximation of elastodynamic contact problems, Adv. Model. Simul. Eng. Sci., Volume 5 (2019), pp. 93-141

[14] The finite element method for elliptic problems (Ciarlet, P.-G.; Lions, J. L., eds.), Handbook of Numerical Analysis, 2, North-Holland, 1991

[15] Cocchetti, G.; Pagani, M.; Perego, U. Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements, Computers & Structures, Volume 127 (2013), pp. 39-52 | DOI

[16] Dabaghi, F.; Petrov, A.; Pousin, J.; Renard, Y. Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary, ESAIM, Math. Model. Numer. Anal., Volume 48 (2014) no. 4, pp. 1147-1169 | DOI | Numdam | MR | Zbl

[17] Doyen, D.; Ern, A.; Piperno, S. Time-integration schemes for the finite element dynamic Signorini problem, SIAM J. Sci. Comput., Volume 33 (2011) no. 1, pp. 223-249 | DOI | MR | Zbl

[18] Eck, C.; Jarušek, J.; Krbec, M. Unilateral contact problems. Variational methods and existence theorems, Pure and Applied Mathematics (Boca Raton), 270, Chapman & Hall/CRC, 2005, x+398 pages | MR | Zbl

[19] Eyre, D. J Unconditionally gradient stable time marching the Cahn-Hilliard equation, MRS Online Proceedings Library Archive, Volume 529 (1998) | DOI | MR

[20] Glasner, K.; Orizaga, S. Improving the accuracy of convexity splitting methods for gradient flow equations, J. Comput. Phys., Volume 315 (2016), pp. 52-64 | DOI | MR | Zbl

[21] Hager, C.; Hüeber, S.; Wohlmuth, B. I. A stable energy-conserving approach for frictional contact problems based on quadrature formulas, Int. J. Numer. Meth. Engng., Volume 5 (2008) no. 27, pp. 918-932 | Zbl

[22] Hauret, P.; Le Tallec, P. Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 37-40, pp. 4890-4916 | DOI | MR | Zbl

[23] Heinstein, M. W.; Mello, F. J.; Attaway, S. W.; Laursen, T. A. Contact-impact modeling in explicit transient dynamics, Comput. Methods Appl. Mech. Eng., Volume 187 (2000), pp. 621-640 | DOI | Zbl

[24] Khenous, H. B. Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique. Etude mathématique et résolution numérique., Ph. D. Thesis, INSA de Toulouse (2005)

[25] Khenous, H. B.; Laborde, P.; Renard, Y. Mass redistribution method for finite element contact problems in elastodynamics, Eur. J. Mech., A, Solids, Volume 27 (2008) no. 5, pp. 918-932 | DOI | MR | Zbl

[26] Kikuchi, N.; Oden, J. T. 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, 1988, xiv+495 pages | MR | Zbl

[27] Kim, J. U. A boundary thin obstacle problem for a wave equation, Commun. Partial Differ. Equations, Volume 14 (1989) no. 8-9, pp. 1011-1026 | DOI | MR

[28] Laursen, T. A.; Chawla, V. Design of energy conserving algorithms for frictionless dynamic contact problems, Int. J. Numer. Meth. Engng., Volume 40 (1997) no. 5, pp. 863-886 | DOI | MR | Zbl

[29] Lebeau, G.; Schatzman, M. A wave problem in a half-space with a unilateral constraint at the boundary, J. Differ. Equations, Volume 53 (1984) no. 3, pp. 309-361 | DOI | MR | Zbl

[30] Moreau, J. J. Unilateral contact and dry friction in finite freedom dynamics, Springer (1988), pp. 1-82 | Zbl

[31] Moreau, J. J. Numerical aspects of the sweeping process, Comput. Methods Appl. Mech. Eng., Volume 177 (1999), pp. 329-349 | DOI | MR | Zbl

[32] Newmark, N. M. A method of computation for structural dynamics, J. Eng. Mech. Div., Volume 85 (1959) no. 3, pp. 67-94

[33] Olovsson, L.; Simonsson, K.; Unosson, M. Selective mass scaling for explicit finite element analyses, Int. J. Numer. Meth. Engng., Volume 63 (2005) no. 10, pp. 1436-1445 | DOI | Zbl

[34] Olovsson, L.; Unosson, M.; Simonsson, K. Selective mass scaling for thin walled structures modeled with tri-linear solid elements, Comput. Mech., Volume 34 (2004) no. 2, pp. 134-136 | DOI | Zbl

[35] Paoli, L.; Schatzman, M. A numerical scheme for impact problems. I. The one-dimensional case, SIAM J. Numer. Anal., Volume 40 (2002) no. 2, pp. 702-733 | DOI | MR | Zbl

[36] Paoli, L.; Schatzman, M. A numerical scheme for impact problems. II. The multidimensional case, SIAM J. Numer. Anal., Volume 40 (2002) no. 2, pp. 734-768 | DOI | MR | Zbl

[37] Pozzolini, C.; Renard, Y.; Salaün, M. Vibro-impact of a plate on rigid obstacles: existence theorem, convergence of a scheme and numerical simulations, IMA J. Numer. Anal., Volume 33 (2013) no. 1, pp. 261-294 | DOI | MR | Zbl

[38] Renard, Y.; Poulios, K. Automated FE modeling of multiphysics problems based on a generic weak form language (2020) (Submitted)

[39] Rosales, R. R; Seibold, B.; Shirokoff, D.; Zhou, D. Unconditional Stability for Multistep ImEx Schemes: Theory, SIAM J. Numer. Anal., Volume 55 (2017) no. 5, pp. 2336-2360 | DOI | MR | Zbl

[40] Schatzman, M. A hyperbolic problem of second order with unilateral constraints: the vibrating string with a concave obstacle, J. Math. Anal. Appl., Volume 73 (1980) no. 1, pp. 138-191 | DOI | MR | Zbl

[41] Schatzman, M. Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: la corde vibrante avec obstacle ponctuel, J. Differ. Equations, Volume 36 (1980) no. 2, pp. 295-334 | DOI | MR | Zbl

[42] Schindler, T.; Acary, V. Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook, Math. Comput. Simulation, Volume 95 (2014), pp. 180-199 | DOI | MR

[43] Seitz, A. Computational Methods for Thermo-Elasto-Plastic Contact, Ph. D. Thesis, Technische Universität München (2019)

[44] Seitz, A.; Wall, Wolfgang A.; Popp, A. Nitsche’s method for finite deformation thermomechanical contact problems, Comput. Mech. (2018), pp. 1-20 | Zbl

[45] Stern, A.; Grinspun, E. Implicit-explicit variational integration of highly oscillatory problems, Multiscale Model. Simul., Volume 7 (2009) no. 4, pp. 1779-1794 | DOI | MR | Zbl

[46] Taylor, L. M.; Flanagan, D. P. PRONTO3D: A three-dimensionnal transient solid dynamics program, SAND89-1912, Sandia National Laboratories, 1989

[47] Tkachuk, A.; Bischoff, M. Local and global strategies for optimal selective mass scaling, Comput. Mech., Volume 53 (2014) no. 6, pp. 1197-1207 | DOI | MR | Zbl

[48] Vola, D.; Raous, M.; Martins, J. A. C. Friction and instability of steady sliding: squeal of a rubber/glass contact, Int. J. Numer. Meth. Engng., Volume 46 (1999) no. 10, pp. 1699-1720 | DOI | MR | Zbl

[49] Wohlmuth, B. I. Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numer., Volume 20 (2011), pp. 569-734 | DOI | MR | Zbl