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An energy-preserving Discrete Element Method for elastodynamics∗

Published online by Cambridge University Press:  13 June 2012

Laurent Monasse  and
Christian Mariotti
Laurent Monasse
Affiliation:
Université Paris-Est, CERMICS 6 et 8 avenue Blaise Pascal, Cité Descartes – Champs-sur-Marne 77455 Marne-la-Vallée Cedex 2, France. [email protected] CEA DAM DIF, 91297 Arpajon, France; [email protected]; [email protected]
Christian Mariotti
Affiliation:
CEA DAM DIF, 91297 Arpajon, France; [email protected]; [email protected]
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Abstract

We develop a Discrete Element Method (DEM) for elastodynamics using polyhedral elements. We show that for a given choice of forces and torques, we recover the equations of linear elastodynamics in small deformations. Furthermore, the torques and forces derive from a potential energy, and thus the global equation is an Hamiltonian dynamics. The use of an explicit symplectic time integration scheme allows us to recover conservation of energy, and thus stability over long time simulations. These theoretical results are illustrated by numerical simulations of test cases involving large displacements.

Keywords

Solidselasticitydiscrete element methodHamiltonianexplicit time integration
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1527 - 1553
Copyright
© EDP Sciences, SMAI, 2012

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References

Andersen, H.C., RATTLE : A “velocity” version of the SHAKE algorithm for molecular dynamics calculations. J. Comput. Phys. 52 (1983) 24–34. Google Scholar
Antoci, C., Gallati, M. and Sibilla, S., Numerical simulation of fluid-structure interaction by SPH. 4th MIT Conference on Computational Fluid and Solid Mechanics. Comput. Struct. 85 (2007) 879–890. Google Scholar
Bonet, J. and Lok, T.S.L., Variational and momentum preservation aspects of Smooth particle hydrodynamic formulations. Comput. Meth. Appl. Mech. Eng. 180 (1999) 97–115. Google Scholar
Cundall, P.A. and Strack, O.D.L., A discrete numerical model for granular assemblies. Geotech. 29 (1979) 47–65. Google Scholar
D’Addetta, G.A., Kun, F. and Ramm, E., On the application of a discrete model to the fracture process of cohesive granular materials. Granul. Matter 4 (2002) 77–90. Google Scholar
De Hoop, A.T., A modification of Cagniard’s method for solving seismic pulse problem. Appl. Sci. Res. B 8 (1960) 349–356. Google Scholar
A.C. Eringen, Theory of micropolar elasticity, in Fracture, edited by H. Liebowitz. Academic Press, New York 2 (1968) 621–729.
Fahrenthold, E.P. and Horban, B.A., An improved hybrid particle-element method for hypervelocity impact simulation. Symposium on Hypervelocity Impact, Galveston. Texas (2000). Int. J. Impact Eng. 26 (2001) 169–178. Google Scholar
Fahrenthold, E.P. and Shivarama, R., Extension and validation of a hybrid particle-finite element method for hypervelocity impact simulation. Hypervelocity Impact Symposium. Int. J. Impact Eng. 29 (2003) 237–246. Google Scholar
Feng, Y.T., Han, K., Li, C.F. and Owen, D.R.J., Discrete thermal element modelling of heat conduction in particle systems : Basic formulations. J. Comput. Phys. 227 (2008) 5072–5089. Google Scholar
Forest, S., Pradel, F. and Sab, K., Asymptotic analysis of heterogeneous Cosserat media. Int. J. Solids Struct. 38 (2001) 4585–4608. Google Scholar
Gingold, R.A. and Monaghan, J.J., smoothed particle hydrodynamics : Theory and application to nonspherical stars. Mon. Not. R. Astron. Soc. 181 (1977) 375–389. Google Scholar
Gonzalez, O., Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Meth. Appl. Mech. Eng. 190 (2000) 1763–1783. Google Scholar
Hairer, E. and Vilmart, G., Preprocessed discrete Moser-Veselov algorithm for the full dynamics of a rigid body. J. Phys. A 39 (2006) 13225–13235. Google Scholar
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration : Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition. Springer Series in Comput. Math. 31 (2006).
Han, K., Feng, Y.T. and Owen, D.R.J., Coupled lattice Boltzmann and discrete element modelling of fluid-particle interaction problems, in 4th MIT Conference on Computational Fluid and Solid Mechanics. Comput. Struct. 85 (2007) 1080–1088. Google Scholar
Hauret, P. and Le Tallec, P., Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Meth. Appl. Mech. Eng. 195 (2006) 4890–4916. Google Scholar
Hicks, D.L., Swegle, J.W. and Attaway, S.W., Conservative smoothing stabilizes discrete-numerical instabilities in SPH material dynamics computations. Appl. Math. Comput. 85 (1997) 209–226. Google Scholar
W.G. Hoover, Smooth Particle Applied Mechanics : The State of the Art (World Scientific). Adv. Ser. Nonlinear Dyn. 25 (2006).
Hoover, W.G., Arhurst, W.T. and Olness, R.J., Two-dimensional studies of crystal stability and fluid viscosity. J. Chem. Phys. 60 (1974) 4043–4047. Google Scholar
Ibrahimbegovic, A. and Delaplace, A., Microscale and mesoscale discrete models for dynamic fracture of structures built of brittle material. Comput. Struct. 81 (2003) 1255–1265. Google Scholar
Koo, J.C. and Fahrenthold, E.P., Discrete Hamilton’s equations for arbitrary Lagrangian-Eulerian dynamics of viscous compressible flow. Comput. Meth. Appl. Mech. Eng. 189 (2000) 875–900. Google Scholar
Koshizuka, S. and Oka, Y., Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123 (1996) 421–434. Google Scholar
Koshizuka, S., Nobe, A. and Oka, Y., Numerical analysis of breaking waves using the moving particle semi-implicit method. Int. J. Numer. Meth. Fluids 26 (1998) 751–769. Google Scholar
S. Koshizuka, M.S. Song and Y. Oka, A particle method for three-dimensional elastic analysis, in Proc. of 6th World Cong. Computational Mechanics (WCCM VI). Beijing (2004).
Kun, F. and Herrmann, H., A study of fragmentation processes using a discrete element method. Comput. Meth. Appl. Mech. Eng. 138 (1996) 3–18. Google Scholar
Lamb, H., On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. Lond. A 203 (1904) 1–42. Google Scholar
Laursen, T.A. and Meng, X.N., A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics. Comput. Meth. Appl. Mech. Eng. 190 (2001) 6309–6322. Google Scholar
Lee, C.J.K., Noguchi, H. and Koshizuka, S., Fluid-shell structure interaction analysis by coupled particle and finite element method, in 4th MIT Conference on Computational Fluid and Solid Mechanics. Comput. Struct. 85 (2007) 688–697. Google Scholar
Leimkuhler, B.J. and Skeel, R.D., Symplectic numerical integrators in constrained Hamiltonian systems. J. Comput. Phys. 112 (1994) 117–125. Google Scholar
Lew, A., Marsden, J.E., Ortiz, M. and West, M., Variational time integrators. Int. J. Numer. Meth. Eng. 60 (2004) 153–212. Google Scholar
Libersky, L.D., Petschek, A.G., Carney, T.C., Hipp, J.R. and Allahdadi, F.A., High strain Lagrangian hydrodynamics : A three-dimensional SPH code for dynamic material response. J. Comput. Phys. 109 (1993) 76–83. Google Scholar
Lucy, L.B., A numerical approach to the testing of the fission hypothesis. Astron. J. 82 (1977) 1013–1024. Google Scholar
Mariotti, C., Lamb’s problem with the lattice model Mka3D. Geophys. J. Int. 171 (2007) 857–864. Google Scholar
Monaghan, J.J., Simulating free surface flows with SPH. J. Comput. Phys. 110 (1994) 399–406. Google Scholar
Potyondy, D.O. and Cundall, P.A., A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci. 41 (2004) 1329–1364. Google Scholar
Ries, A., Wolf, D.E. and Unger, T., Shear zones in granular media : Three-dimensional contact dynamics simulation. Phys. Rev. E 76 (2007) 051301. Google ScholarPubMed
Simo, J.C., Tarnow, N. and Wong, K.K., Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Meth. Appl. Mech. Eng. 100 (1992) 63–116. Google Scholar
Suzuki, Y. and Koshizuka, S., A Hamiltonian particle method for non-linear elastodynamics. Int. J. Numer. Meth. Eng. 74 (2008) 1344–1373. Google Scholar
Swegle, J.W., Hicks, D.L. and Attaway, S.W., smoothed particle hydrodynamics stability analysis. J. Comput. Phys. 116 (1995) 123–134. Google Scholar
Sze, K.Y., Liu, X.H. and Lo, S.H., Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des. 40 (2004) 1551–1569. Google Scholar
Yserentant, H., A new class of particle methods. Numer. Math. 76 (1997) 87–109. Google Scholar