no. 6
The role of the patch test in 2D atomistic-to-continuum coupling methods
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1275-1319.

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy-Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.

DOI : 10.1051/m2an/2012005
Classification : 65N12, 65N15, 70C20
Mots clés : atomistic models, atomistic-to-continuum coupling, quasicontinuum method, coarse graining, ghost forces, patch test, consistency 
@article{M2AN_2012__46_6_1275_0,
     author = {Ortner, Christoph},
     title = {The role of the patch test in {2D} atomistic-to-continuum coupling methods},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1275--1319},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     doi = {10.1051/m2an/2012005},
     mrnumber = {2996328},
     zbl = {1269.82063},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2012005/}
}
TY  - JOUR
AU  - Ortner, Christoph
TI  - The role of the patch test in 2D atomistic-to-continuum coupling methods
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 1275
EP  - 1319
VL  - 46
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2012005/
DO  - 10.1051/m2an/2012005
LA  - en
ID  - M2AN_2012__46_6_1275_0
ER  - 
%0 Journal Article
%A Ortner, Christoph
%T The role of the patch test in 2D atomistic-to-continuum coupling methods
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 1275-1319
%V 46
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2012005/
%R 10.1051/m2an/2012005
%G en
%F M2AN_2012__46_6_1275_0
Ortner, Christoph. The role of the patch test in 2D atomistic-to-continuum coupling methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1275-1319. doi : 10.1051/m2an/2012005. http://archive.numdam.org/articles/10.1051/m2an/2012005/

[1] A. Abdulle, P. Lin and A. Shapeev, Homogenization-based analysis of quasicontinuum method for complex crystals. arXiv:1006.0378.

[2] N.C. Admal and E.B. Tadmor, A unified interpretation of stress in molecular systems. J. Elasticity 100 (2010) 63-143. | MR | Zbl

[3] R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1-37 (electronic). | MR | Zbl

[4] D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26 (1989) 1276-1290. | MR | Zbl

[5] S. Badia, M. Parks, P. Bochev, M. Gunzburger and R. Lehoucq, On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7 (2008) 381-406. | MR | Zbl

[6] G.P. Bazeley, Y.K. Cheung, B.M. Irons and O.C. Zienkiewicz, Triangle elements in plate bending : conforming and nonconforming solutions, in Proc. Conf. Matrix Meth. Struc. Mech. Wright Patterson AFB, Ohio (1966).

[7] T. Belytschko, W.K. Liu and B. Moran, Nonlinear finite elements for continua and structures. John Wiley & Sons Ltd., Chichester (2000). | MR | Zbl

[8] P.G. Ciarlet, The finite element method for elliptic problems. Classics in Appl. Math. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA 40 (2002). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. | MR | Zbl

[9] M. Dobson, There is no pointwise consistent quasicontinuum energy. arXiv:1109.1897. | MR

[10] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM : M2AN 42 (2008) 113-139. | Numdam | MR | Zbl

[11] M. Dobson and M. Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM : M2AN 43 (2009) 591-604. | Numdam | MR | Zbl

[12] M. Dobson and M. Luskin, An optimal order error analysis of the one-dimensional quasicontinuum approximation. SIAM J. Numer. Anal. 47 (2009) 2455-2475. | MR | Zbl

[13] M. Dobson, R. Elliot, M. Luskin and E. Tadmor, A multilattice quasicontinuum for phase transforming materials : cascading cauchy born kinematics. J. Computer-Aided Mater. Design 14 (2007) 219-237.

[14] M. Dobson, M. Luskin and C. Ortner, Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58 (2010) 1741-1757. | MR | Zbl

[15] M. Dobson, M. Luskin and C. Ortner, Stability, instability, and error of the force-based quasicontinuum approximation. Arch. Rational Mech. Anal. 197 (2010) 179-202. | MR | Zbl

[16] W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and prospects of contemporary applied mathematics. Ser. Contemp. Appl. Math. CAM 6 (2005) 18-32. | MR | Zbl

[17] W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids : static problems. Arch. Rational Mech. Anal. 183 (2007) 241-297. | MR | Zbl

[18] W. E, J. Lu and J.Z. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115.

[19] B. Eidel and A. Stukowski, A variational formulation of the quasicontinuum method based on energy sampling in clusters. J. Mech. Phys. Solids 57 (2009) 87-108. | MR | Zbl

[20] M. Finnis, Interatomic Forces in Condensed Matter. Oxford Series on Materials Modelling 1 (2003).

[21] J. Fish, M.A. Nuggehally, M.S. Shephard, C.R. Picu, S. Badia, M.L. Parks, and M. Gunzburger, Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force. Comput. Methods Appl. Mech. Eng. 196 (2007) 4548-4560. | MR | Zbl

[22] M. Gunzburger and Y. Zhang, A quadrature-rule type approximation to the quasi-continuum method. Multiscale Model. Simul. 8 (2009/2010) 571-590. | MR | Zbl

[23] M. Iyer and V. Gavini, A field theoretical approach to the quasi-continuum method. J. Mech. Phys. Solids 59 (2011) 1506-1535. | MR | Zbl

[24] P.A. Klein and J.A. Zimmerman, Coupled atomistic-continuum simulations using arbitrary overlapping domains. J. Comput. Phys. 213 (2006) 86-116. | MR | Zbl

[25] J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl

[26] S. Kohlhoff and S. Schmauder, A new method for coupled elastic-atomistic modelling, in Atomistic Simulation of Materials : Beyond Pair Potentials, edited by V. Vitek and D.J. Srolovitz. Plenum Press, New York (1989) 411-418.

[27] X.H. Li and M. Luskin, An analysis of the quasi-nonlocal quasicontinuum approximation of the embedded atom model. To appear in Int. J. Multiscale Comput. Eng., arXiv:1008.3628.

[28] X.H. Li and M. Luskin, A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction. To appear in IMA J. Numer. Anal., arXiv:1007.2336. | MR | Zbl

[29] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675. | MR | Zbl

[30] P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects. SIAM J. Numer. Anal. 45 (2007) 313-332 (electronic). | MR | Zbl

[31] J. Lu and P. Ming, Convergence of a force-based hybrid method for atomistic and continuum models in three dimension. arXiv:1102.2523.

[32] M. Luskin and C. Ortner, An analysis of node-based cluster summation rules in the quasicontinuum method. SIAM J. Numer. Anal. 47 (2009) 3070-3086. | MR | Zbl

[33] C. Makridakis, C. Ortner and E. Süli, A priori error analysis of two force-based atomistic/continuum models of a periodic chain. Numer. Math. 119 (2011) 83-121. | MR | Zbl

[34] C. Makridakis, C. Ortner and E. Süli, Stress-based atomistic/continuum coupling : a new variant of the quasicontinuum approximation. Int. J. Multiscale Comput. Eng. forthcoming.

[35] R.E. Miller and E.B. Tadmor, The quasicontinuum method : overview, applications and current directions. J. Computer-Aided Mater. Design 9 (2003) 203-239.

[36] R.E. Miller and E.B. Tadmor, A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17 (2009).

[37] P. Ming and J.Z. Yang, Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul. 7 (2009) 1838-1875. | MR | Zbl

[38] M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.

[39] C. Ortner, Analysis of the Quasicontinuum Method. Ph.D. thesis, University of Oxford (2006).

[40] C. Ortner, A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comp. 80 (2011) 1265-1285. | MR

[41] C. Ortner and A.V. Shapeev, Analysis of an energy-based atomistic/continuum coupling approximation of a vacancy in the 2d triangular lattice. To appear in Math. Comp., arXiv1104.0311. | MR | Zbl

[42] C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM : M2AN 42 (2008) 57-91. | Numdam | MR | Zbl

[43] C. Ortner and H. Wang, A priori error estimates for energy-based quasicontinuum approximations of a periodic chain. Math. Models Methods Appl. Sci. 21 (2011) 2491-2521. | MR | Zbl

[44] C. Ortner and L. Zhang, work in progress.

[45] C. Ortner and L. Zhang, Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces : a 2d model problem. arXiv:1110.0168. | MR | Zbl

[46] M.L. Parks, P.B. Bochev and R.B. Lehoucq, Connecting atomistic-to-continuum coupling and domain decomposition. Multiscale Model. Simul. 7 (2008) 362-380. | MR | Zbl

[47] D. Pettifor, Bonding and structure of molecules and solids. Oxford University Press (1995).

[48] K. Polthier and E. Preuß, Identifying vector field singularities using a discrete Hodge decomposition, in Visualization and mathematics III, Math. Vis. Springer, Berlin (2003) 113-134. | MR | Zbl

[49] A.V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions. Multiscale Model. Simul. 9 (2011) 905-932. | MR | Zbl

[50] V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids 47 (1999) 611-642. | MR | Zbl

[51] L.E. Shilkrot, R.E. Miller and W.A. Curtin, Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89 (2002) 025501.

[52] T. Shimokawa, J.J. Mortensen, J. Schiotz and K.W. Jacobsen, Matching conditions in the quasicontinuum method : removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B 69 (2004) 214104.

[53] G. Strang and G. Fix, An Analysis of the Finite Element Method. Wellesley-Cambridge Press (2008). | MR | Zbl

[54] B. Van Koten and M. Luskin, Development and analysis of blended quasicontinuum approximations. To appear in SIAM J. Numer. Anal., arXiv:1008.2138. | MR

[55] B. Van Koten, Z.H. Li, M. Luskin and C. Ortner, A computational and theoretical investigation of the accuracy of quasicontinuum methods, in Numerical Analysis of Multiscale Problems, edited by I. Graham, T. Hou, O. Lakkis and R. Scheichl. Springer Lect. Notes Comput. Sci. Eng. 83 (2012). | MR | Zbl

[56] S.P. Xiao and T. Belytschko, A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193 (2004) 1645-1669. | MR | Zbl

Cité par Sources :