# An energy-preserving Discrete Element Method for elastodynamics∗

Laurent Monasse; Christian Mariotti

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 6, page 1527-1553
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMonasse, Laurent, and Mariotti, Christian. "An energy-preserving Discrete Element Method for elastodynamics∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1527-1553. <http://eudml.org/doc/276378>.

@article{Monasse2012,

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.},

author = {Monasse, Laurent, Mariotti, Christian},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Solids; elasticity; discrete element method; Hamiltonian; explicit time integration; solids},

language = {eng},

month = {6},

number = {6},

pages = {1527-1553},

publisher = {EDP Sciences},

title = {An energy-preserving Discrete Element Method for elastodynamics∗},

url = {http://eudml.org/doc/276378},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Monasse, Laurent

AU - Mariotti, Christian

TI - An energy-preserving Discrete Element Method for elastodynamics∗

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/6//

PB - EDP Sciences

VL - 46

IS - 6

SP - 1527

EP - 1553

AB - 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.

LA - eng

KW - Solids; elasticity; discrete element method; Hamiltonian; explicit time integration; solids

UR - http://eudml.org/doc/276378

ER -

## References

top- H.C. Andersen, RATTLE : A “velocity” version of the SHAKE algorithm for molecular dynamics calculations. J. Comput. Phys.52 (1983) 24–34. Zbl0513.65052
- C. Antoci, M. Gallati and S. Sibilla, Numerical simulation of fluid-structure interaction by SPH. 4th MIT Conference on Computational Fluid and Solid Mechanics. Comput. Struct.85 (2007) 879–890.
- J. Bonet and T.S.L. Lok, Variational and momentum preservation aspects of Smooth particle hydrodynamic formulations. Comput. Meth. Appl. Mech. Eng.180 (1999) 97–115. Zbl0962.76075
- P.A. Cundall and O.D.L. Strack, A discrete numerical model for granular assemblies. Geotech.29 (1979) 47–65.
- G.A. D’Addetta, F. Kun and E. Ramm, On the application of a discrete model to the fracture process of cohesive granular materials. Granul. Matter4 (2002) 77–90. Zbl1074.74633
- A.T. De Hoop, A modification of Cagniard’s method for solving seismic pulse problem. Appl. Sci. Res. B8 (1960) 349–356. Zbl0100.44208
- A.C. Eringen, Theory of micropolar elasticity, in Fracture, edited by H. Liebowitz. Academic Press, New York 2 (1968) 621–729.
- E.P. Fahrenthold and B.A. Horban, 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.
- E.P. Fahrenthold and R. Shivarama, 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.
- Y.T. Feng, K. Han, C.F. Li and D.R.J. Owen, Discrete thermal element modelling of heat conduction in particle systems : Basic formulations. J. Comput. Phys.227 (2008) 5072–5089. Zbl1144.80012
- S. Forest, F. Pradel and K. Sab, Asymptotic analysis of heterogeneous Cosserat media. Int. J. Solids Struct.38 (2001) 4585–4608. Zbl1033.74038
- R.A. Gingold and J.J. Monaghan, smoothed particle hydrodynamics : Theory and application to nonspherical stars. Mon. Not. R. Astron. Soc.181 (1977) 375–389. Zbl0421.76032
- O. Gonzalez, Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Meth. Appl. Mech. Eng.190 (2000) 1763–1783. Zbl1005.74075
- E. Hairer and G. Vilmart, Preprocessed discrete Moser-Veselov algorithm for the full dynamics of a rigid body. J. Phys. A39 (2006) 13225–13235. Zbl1127.70002
- 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). Zbl1094.65125
- K. Han, Y.T. Feng and D.R.J. Owen, 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.
- P. Hauret and P. Le Tallec, Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Meth. Appl. Mech. Eng.195 (2006) 4890–4916. Zbl1177.74379
- D.L. Hicks, J.W. Swegle and S.W. Attaway, Conservative smoothing stabilizes discrete-numerical instabilities in SPH material dynamics computations. Appl. Math. Comput.85 (1997) 209–226. Zbl0882.76064
- W.G. Hoover, Smooth Particle Applied Mechanics : The State of the Art (World Scientific). Adv. Ser. Nonlinear Dyn.25 (2006). Zbl1141.76001
- W.G. Hoover, W.T. Arhurst and R.J. Olness, Two-dimensional studies of crystal stability and fluid viscosity. J. Chem. Phys.60 (1974) 4043–4047.
- A. Ibrahimbegovic and A. Delaplace, Microscale and mesoscale discrete models for dynamic fracture of structures built of brittle material. Comput. Struct.81 (2003) 1255–1265.
- J.C. Koo and E.P. Fahrenthold, Discrete Hamilton’s equations for arbitrary Lagrangian-Eulerian dynamics of viscous compressible flow. Comput. Meth. Appl. Mech. Eng.189 (2000) 875–900. Zbl0991.76045
- S. Koshizuka and Y. Oka, Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng.123 (1996) 421–434.
- S. Koshizuka, A. Nobe and Y. Oka, Numerical analysis of breaking waves using the moving particle semi-implicit method. Int. J. Numer. Meth. Fluids26 (1998) 751–769. Zbl0928.76086
- 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).
- F. Kun and H. Herrmann, A study of fragmentation processes using a discrete element method. Comput. Meth. Appl. Mech. Eng.138 (1996) 3–18. Zbl0881.73106
- H. Lamb, On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. Lond. A203 (1904) 1–42. Zbl34.0859.02
- T.A. Laursen and X.N. Meng, 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. Zbl1022.74013
- C.J.K. Lee, H. Noguchi and S. Koshizuka, 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.
- B.J. Leimkuhler and R.D. Skeel, Symplectic numerical integrators in constrained Hamiltonian systems. J. Comput. Phys.112 (1994) 117–125. Zbl0817.65057
- A. Lew, J.E. Marsden, M. Ortiz and M. West, Variational time integrators. Int. J. Numer. Meth. Eng.60 (2004) 153–212. Zbl1060.70500
- L.D. Libersky, A.G. Petschek, T.C. Carney, J.R. Hipp and F.A. Allahdadi, High strain Lagrangian hydrodynamics : A three-dimensional SPH code for dynamic material response. J. Comput. Phys.109 (1993) 76–83. Zbl0791.76065
- L.B. Lucy, A numerical approach to the testing of the fission hypothesis. Astron. J.82 (1977) 1013–1024.
- C. Mariotti, Lamb’s problem with the lattice model Mka3D. Geophys. J. Int.171 (2007) 857–864.
- J.J. Monaghan, Simulating free surface flows with SPH. J. Comput. Phys.110 (1994) 399–406. Zbl0794.76073
- D.O. Potyondy and P.A. Cundall, A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci.41 (2004) 1329–1364.
- A. Ries, D.E. Wolf and T. Unger, Shear zones in granular media : Three-dimensional contact dynamics simulation. Phys. Rev. E76 (2007) 051301.
- J.C. Simo, N. Tarnow and K.K. Wong, Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Meth. Appl. Mech. Eng.100 (1992) 63–116. Zbl0764.73096
- Y. Suzuki and S. Koshizuka, A Hamiltonian particle method for non-linear elastodynamics. Int. J. Numer. Meth. Eng.74 (2008) 1344–1373. Zbl1159.74459
- J.W. Swegle, D.L. Hicks and S.W. Attaway, smoothed particle hydrodynamics stability analysis. J. Comput. Phys.116 (1995) 123–134. Zbl0818.76071
- K.Y. Sze, X.H. Liu and S.H. Lo, Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des.40 (2004) 1551–1569.
- H. Yserentant, A new class of particle methods. Numer. Math.76 (1997) 87–109. Zbl0876.76067

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.