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.
@article{M2AN_2012__46_6_1527_0,
author = {Monasse, Laurent and Mariotti, Christian},
title = {An energy-preserving Discrete Element Method for elastodynamics},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {1527-1553},
doi = {10.1051/m2an/2012015},
mrnumber = {2996339},
zbl = {1267.74114},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_6_1527_0}
}
Monasse, Laurent; Mariotti, Christian. An energy-preserving Discrete Element Method for elastodynamics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 1527-1553. doi : 10.1051/m2an/2012015. http://gdmltest.u-ga.fr/item/M2AN_2012__46_6_1527_0/
[1] , RATTLE : A “velocity” version of the SHAKE algorithm for molecular dynamics calculations. J. Comput. Phys. 52 (1983) 24-34. | Zbl 0513.65052
[2] , and , Numerical simulation of fluid-structure interaction by SPH. 4th MIT Conference on Computational Fluid and Solid Mechanics. Comput. Struct. 85 (2007) 879-890.
[3] and , Variational and momentum preservation aspects of Smooth particle hydrodynamic formulations. Comput. Meth. Appl. Mech. Eng. 180 (1999) 97-115. | MR 1735279 | Zbl 0962.76075
[4] and , A discrete numerical model for granular assemblies. Geotech. 29 (1979) 47-65.
[5] , and , On the application of a discrete model to the fracture process of cohesive granular materials. Granul. Matter 4 (2002) 77-90. | Zbl 1074.74633
[6] , A modification of Cagniard's method for solving seismic pulse problem. Appl. Sci. Res. B 8 (1960) 349-356. | Zbl 0100.44208
[7] , Theory of micropolar elasticity, in Fracture, edited by H. Liebowitz. Academic Press, New York 2 (1968) 621-729. | Zbl 0266.73004
[8] and , 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.
[9] and , 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.
[10] , , and , Discrete thermal element modelling of heat conduction in particle systems : Basic formulations. J. Comput. Phys. 227 (2008) 5072-5089. | Zbl 1144.80012
[11] , and , Asymptotic analysis of heterogeneous Cosserat media. Int. J. Solids Struct. 38 (2001) 4585-4608. | MR 1837063 | Zbl 1033.74038
[12] 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. | Zbl 0421.76032
[13] , Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Meth. Appl. Mech. Eng. 190 (2000) 1763-1783. | MR 1807477 | Zbl 1005.74075
[14] and , Preprocessed discrete Moser-Veselov algorithm for the full dynamics of a rigid body. J. Phys. A 39 (2006) 13225-13235. | MR 2266054 | Zbl 1127.70002
[15] , and , Geometric Numerical Integration : Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition. Springer Series in Comput. Math. 31 (2006). | MR 2221614 | Zbl 1094.65125
[16] , and , 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.
[17] and , Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Meth. Appl. Mech. Eng. 195 (2006) 4890-4916. | MR 2240585 | Zbl 1177.74379
[18] , and , Conservative smoothing stabilizes discrete-numerical instabilities in SPH material dynamics computations. Appl. Math. Comput. 85 (1997) 209-226. | MR 1468047 | Zbl 0882.76064
[19] , Smooth Particle Applied Mechanics : The State of the Art (World Scientific). Adv. Ser. Nonlinear Dyn. 25 (2006). | MR 2309657 | Zbl 1141.76001
[20] , and , Two-dimensional studies of crystal stability and fluid viscosity. J. Chem. Phys. 60 (1974) 4043-4047.
[21] and , Microscale and mesoscale discrete models for dynamic fracture of structures built of brittle material. Comput. Struct. 81 (2003) 1255-1265.
[22] and , Discrete Hamilton's equations for arbitrary Lagrangian-Eulerian dynamics of viscous compressible flow. Comput. Meth. Appl. Mech. Eng. 189 (2000) 875-900. | Zbl 0991.76045
[23] and , Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123 (1996) 421-434.
[24] , and , Numerical analysis of breaking waves using the moving particle semi-implicit method. Int. J. Numer. Meth. Fluids 26 (1998) 751-769. | Zbl 0928.76086
[25] , and , A particle method for three-dimensional elastic analysis, in Proc. of 6th World Cong. Computational Mechanics (WCCM VI). Beijing (2004).
[26] and , A study of fragmentation processes using a discrete element method. Comput. Meth. Appl. Mech. Eng. 138 (1996) 3-18. | Zbl 0881.73106
[27] , On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. Lond. A 203 (1904) 1-42. | JFM 34.0859.02
[28] and , 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. | MR 1857698 | Zbl 1022.74013
[29] , and , 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.
[30] and , Symplectic numerical integrators in constrained Hamiltonian systems. J. Comput. Phys. 112 (1994) 117-125. | MR 1277499 | Zbl 0817.65057
[31] , , and , Variational time integrators. Int. J. Numer. Meth. Eng. 60 (2004) 153-212. | MR 2073073 | Zbl 1060.70500
[32] , , , and , High strain Lagrangian hydrodynamics : A three-dimensional SPH code for dynamic material response. J. Comput. Phys. 109 (1993) 76-83. | Zbl 0791.76065
[33] , A numerical approach to the testing of the fission hypothesis. Astron. J. 82 (1977) 1013-1024.
[34] , Lamb's problem with the lattice model Mka3D. Geophys. J. Int. 171 (2007) 857-864.
[35] , Simulating free surface flows with SPH. J. Comput. Phys. 110 (1994) 399-406. | Zbl 0794.76073
[36] and , A bonded-particle model for rock. Int. J. Rock Mech. Min. Sci. 41 (2004) 1329-1364.
[37] , and , Shear zones in granular media : Three-dimensional contact dynamics simulation. Phys. Rev. E 76 (2007) 051301.
[38] , and , Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Meth. Appl. Mech. Eng. 100 (1992) 63-116. | MR 1187632 | Zbl 0764.73096
[39] and , A Hamiltonian particle method for non-linear elastodynamics. Int. J. Numer. Meth. Eng. 74 (2008) 1344-1373. | MR 2410182 | Zbl 1159.74459
[40] J.W. Swegle, D.L. Hicks and S.W. Attaway, smoothed particle hydrodynamics stability analysis. J. Comput. Phys. 116 (1995) 123-134. | MR 1315212 | Zbl 0818.76071
[41] , and , Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des. 40 (2004) 1551-1569.
[42] , A new class of particle methods. Numer. Math. 76 (1997) 87-109. | MR 1438683 | Zbl 0876.76067