Ghost forces and spurious effects in atomic-to-continuum coupling methods by the Arlequin approach
Ludovic Chamoin
LMT, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, Cachan Cedex 94235, France
Search for more papers by this authorSerge Prudhomme
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.
Search for more papers by this authorH. Ben Dhia
Laboratoire de Mécanique des Sols, Structures et Matériaux, Ecole Centrale de Paris, 1 Grande Voie des Vignes, Chatenay-Malabry Cedex 92295, France
Search for more papers by this authorCorresponding Author
Tinsley Oden
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.Search for more papers by this authorLudovic Chamoin
LMT, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, Cachan Cedex 94235, France
Search for more papers by this authorSerge Prudhomme
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.
Search for more papers by this authorH. Ben Dhia
Laboratoire de Mécanique des Sols, Structures et Matériaux, Ecole Centrale de Paris, 1 Grande Voie des Vignes, Chatenay-Malabry Cedex 92295, France
Search for more papers by this authorCorresponding Author
Tinsley Oden
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.
Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, U.S.A.Search for more papers by this authorAbstract
The main objective of this work is to identify spurious effects and propose a corrective method when coupling a particle model involving long–range interaction potentials with a first gradient homogenized model using the Arlequin framework. The most significant spurious effects are generally created by the so-called ghost forces that arise in coupling methods based on the minimization of a global energy functional. They depend as well on the coupling formulation itself, on the notion of representative volume element, and on the discretization of the continuum model. The proposed corrective technique is based on post-processing of the approximate solution by introducing dead forces that can be systematically evaluated and consistently inserted within the Arlequin formulation. Efficiency of the corrective procedure is demonstrated on 1D and 2D examples. Copyright © 2010 John Wiley & Sons, Ltd.
REFERENCES
- 1 Liu WK, Karpov EG, Zhang S, Park HS. An introduction to computational nanomechanics and materials. Computer Methods in Applied Mechanics and Engineering 2004; 193: 1529–1578.
- 2 Fish J. Bridging the scales in nano engineering and science. Journal of Nanoparticle Research 2006; 8(6): 577–594.
- 3 Bauman PT, Oden JT, Prudhomme S. Adaptive multiscale modeling of polymeric materials: Arlequin coupling and goals algorithm. Computer Methods in Applied Mechanics and Engineering 2009; 198: 799–818.
- 4 Fish J, Nuggehally MA, Shephard MS, Picu CR, Badia S, Parks ML, Gunzburger M. Concurrent AtC coupling based on a blend of the continuum stress and the atomic force. Computer Methods in Applied Mechanics and Engineering 2007; 196: 4548–4560.
- 5 Broughton JQ, Abraham FF, Bernstein N, Kaxiras E. Concurrent coupling of length scales: methodology and application. Physical Review B 1999; 60(4): 2391–2403.
- 6 Wagner GJ, Liu WK. Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics 2003; 190(1): 249–274.
- 7 Xiao SP, Belytschko T. A bridging domain method for coupling continua with molecular dynamics. Computer Methods in Applied Mechanics and Engineering 2004; 193: 1645–1669.
- 8 Ben Dhia H. Multiscale mechanical problems: the Arlequin method. Comptes Rendus de l'Académie des Sciences, Paris, Série II B 1998; 326(12): 899–904.
- 9 Tadmor EB, Ortiz M, Phillips R. Quasicontinuum analysis of defects in solids. Philosophical Magazine 1996; A73: 1529–1563.
10.1080/01418619608243000 Google Scholar
- 10 Shilkrot LE, Miller RE, Curtin WA. Coupled atomistic and discrete dislocation plasticity. Physical Review Letters 2002; 89:025501.
- 11 Bauman PT, Ben Dhia H, Elkhodja N, Oden JT, Prudhomme S. On the application of the Arlequin method to the coupling of particle and continuum models. Computational Mechanics 2008; 42: 511–530.
- 12 Ben Dhia H, Elkhodja N. Coupling of atomistic and continuum models in the Arlequin framework. Proceedings of Congrès de Mécanique, El Jadida, Maroc, 2007; 133–135.
- 13 Prudhomme S, Chamoin L, Ben Dhia H, Bauman P. An adaptive strategy for the control of modeling error in two-dimensional atomic-to-continuum coupling simulations. Computer Methods in Applied Mechanics and Engineering 2009; 198: 1887–1901.
- 14 Chamoin L, Oden JT, Prudhomme S. A stochastic coupling method for atomistic-to-continuum Monte-Carlo simulations. Computer Methods in Applied Mechanics and Engineering 2008; 197(43–44): 3530–3546.
- 15 Ben Dhia H. Further insights by theoretical investigations of the multiscale Arlequin method. Multiscale Computational Engineering 2008; 6(3): 215–232.
- 16 Xu M, Gracie R, Belytschko T. A continuum-to-atomistic bridging domain method for composite lattices. International Journal for Numerical Methods in Engineering 2009; 81(13): 1635–1658.
- 17 Shenoy VB, Miller RE, Tadmor EB, Rodney D, Phillips R, Ortiz M. An adaptive finite element approach to atomic-scale mechanics: the quasicontinuum method. Journal of the Mechanics and Physics of Solids 1999; 47: 611–642.
- 18 Knap J, Ortiz M. An analysis of the quasicontinuum method. Journal of the Mechanics and Physics of Solids 2001; 49: 1899–1923.
- 19 Miller RE, Tadmor EB. The quasicontinuum method: overview, applications and current directions. Journal of Computer-Aided Materials Design 2002; 9: 203–239.
- 20 Curtin WA, Miller RE. Atomistic/continuum coupling in computational material science. Modeling and Simulation in Materials Science and Engineering 2003; 11: R33–R68.
- 21 Badia S, Bochev P, Fish J, Gunzburger M, Lehoucq R, Nuggehally M, Parks M. A force-based blending model for atomistic-to-continuum coupling. International Journal for Multiscale Computational Engineering 2007; 5: 387–406.
- 22 Badia S, Parks M, Bochev P, Gunzburger M, Lehoucq R. On atomistic-to-continuum (atc) coupling by blending. SIAM Journal on Multiscale Modeling and Simulation 2008; 7(1): 381–406.
- 23 Ben Dhia H, Rateau G. The Arlequin method as a flexible engineering design tool. International Journal for Numerical Methods in Engineering 2005; 62(11): 1442–1462.
- 24 Ben Dhia H, Rateau G. Analyse mathématique de la méthode Arlequin mixte. Comptes Rendus de l'Académie des Sciences, Paris, Série I 2001; 332: 649–654.
- 25 Guidault PA, Belytschko T. On the L2 and the H1 couplings for an overlapping domain decomposition method using Lagrange multipliers. International Journal for Numerical Methods in Engineering 2007; 70(3): 322–350.
- 26 Dobson M, Luskin M. An analysis of the effect of ghost force oscillation on quasicontinuum error. M2AN Mathematical Modeling and Numerical Analysis 2009; 43(3): 591–604.
- 27 Zhang S, Khare R, Lu Q, Belytschko T. A bridging domain and strain computation method for coupled atomistic-continuum modelling of solids. International Journal for Numerical Methods in Engineering 2007; 70: 913–933.
