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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {1275-1319},
doi = {10.1051/m2an/2012005},
mrnumber = {2996328},
zbl = {1269.82063},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 1275-1319. doi : 10.1051/m2an/2012005. http://gdmltest.u-ga.fr/item/M2AN_2012__46_6_1275_0/
[1] , and , Homogenization-based analysis of quasicontinuum method for complex crystals. arXiv:1006.0378.
[2] and , A unified interpretation of stress in molecular systems. J. Elasticity 100 (2010) 63-143. | MR 2653867 | Zbl 1260.74005
[3] and , A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36 (2004) 1-37 (electronic). | MR 2083851 | Zbl 1070.49009
[4] and , A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26 (1989) 1276-1290. | MR 1025088 | Zbl 0696.73040
[5] , , , and , On atomistic-to-continuum coupling by blending. Multiscale Model. Simul. 7 (2008) 381-406. | MR 2399551 | Zbl 1160.65338
[6] , , and , Triangle elements in plate bending : conforming and nonconforming solutions, in Proc. Conf. Matrix Meth. Struc. Mech. Wright Patterson AFB, Ohio (1966).
[7] , and , Nonlinear finite elements for continua and structures. John Wiley & Sons Ltd., Chichester (2000). | MR 1897447 | Zbl 1279.74002
[8] , 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 520174 | Zbl 0999.65129
[9] , There is no pointwise consistent quasicontinuum energy. arXiv:1109.1897. | MR 3269435 | Zbl pre06370506
[10] and , Analysis of a force-based quasicontinuum approximation. ESAIM : M2AN 42 (2008) 113-139. | Numdam | MR 2387424 | Zbl 1140.74006
[11] and , An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM : M2AN 43 (2009) 591-604. | Numdam | MR 2536250 | Zbl 1165.81414
[12] and , An optimal order error analysis of the one-dimensional quasicontinuum approximation. SIAM J. Numer. Anal. 47 (2009) 2455-2475. | MR 2525607 | Zbl 1203.82089
[13] , , and , A multilattice quasicontinuum for phase transforming materials : cascading cauchy born kinematics. J. Computer-Aided Mater. Design 14 (2007) 219-237.
[14] , and , Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids 58 (2010) 1741-1757. | MR 2742030 | Zbl 1200.74005
[15] , and , Stability, instability, and error of the force-based quasicontinuum approximation. Arch. Rational Mech. Anal. 197 (2010) 179-202. | MR 2646818 | Zbl 1268.74006
[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 2249291 | Zbl 1188.74019
[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 2278407 | Zbl 1106.74019
[18] W. E, J. Lu and J.Z. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214115.
[19] and , A variational formulation of the quasicontinuum method based on energy sampling in clusters. J. Mech. Phys. Solids 57 (2009) 87-108. | MR 2479021 | Zbl 1298.74011
[20] , Interatomic Forces in Condensed Matter. Oxford Series on Materials Modelling 1 (2003).
[21] , , , , , , and , 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 2354453 | Zbl 1173.74303
[22] and , A quadrature-rule type approximation to the quasi-continuum method. Multiscale Model. Simul. 8 (2009/2010) 571-590. | MR 2581034 | Zbl 1188.70046
[23] and , A field theoretical approach to the quasi-continuum method. J. Mech. Phys. Solids 59 (2011) 1506-1535. | MR 2848058 | Zbl 1270.74006
[24] and , Coupled atomistic-continuum simulations using arbitrary overlapping domains. J. Comput. Phys. 213 (2006) 86-116. | MR 2203436 | Zbl 1137.74367
[25] and , An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49 (2001) 1899-1923. | Zbl 1002.74008
[26] and , 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] and , 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] and , A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction. To appear in IMA J. Numer. Anal., arXiv:1007.2336. | MR 2911393 | Zbl 1241.82078
[29] , Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675. | MR 1954960 | Zbl 1010.74003
[30] , Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects. SIAM J. Numer. Anal. 45 (2007) 313-332 (electronic). | MR 2285857 | Zbl 1220.74010
[31] and , Convergence of a force-based hybrid method for atomistic and continuum models in three dimension. arXiv:1102.2523.
[32] and , An analysis of node-based cluster summation rules in the quasicontinuum method. SIAM J. Numer. Anal. 47 (2009) 3070-3086. | MR 2551158 | Zbl 1196.82122
[33] , and , A priori error analysis of two force-based atomistic/continuum models of a periodic chain. Numer. Math. 119 (2011) 83-121. | MR 2824856 | Zbl 1225.82070
[34] , and , Stress-based atomistic/continuum coupling : a new variant of the quasicontinuum approximation. Int. J. Multiscale Comput. Eng. forthcoming.
[35] and , The quasicontinuum method : overview, applications and current directions. J. Computer-Aided Mater. Design 9 (2003) 203-239.
[36] and , A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model. Simul. Mater. Sci. Eng. 17 (2009).
[37] and , Analysis of a one-dimensional nonlocal quasi-continuum method. Multiscale Model. Simul. 7 (2009) 1838-1875. | MR 2539201 | Zbl 1177.74169
[38] , and , Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.
[39] , Analysis of the Quasicontinuum Method. Ph.D. thesis, University of Oxford (2006).
[40] , A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comp. 80 (2011) 1265-1285. | MR 2785458 | Zbl pre05918690
[41] and , 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 3073196 | Zbl 1276.82013
[42] and , Analysis of a quasicontinuum method in one dimension. ESAIM : M2AN 42 (2008) 57-91. | Numdam | MR 2387422 | Zbl 1139.74004
[43] and , A priori error estimates for energy-based quasicontinuum approximations of a periodic chain. Math. Models Methods Appl. Sci. 21 (2011) 2491-2521. | MR 2864639 | Zbl 1242.74213
[44] C. Ortner and L. Zhang, work in progress.
[45] and , Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces : a 2d model problem. arXiv:1110.0168. | MR 3253459 | Zbl 1269.82019
[46] , and , Connecting atomistic-to-continuum coupling and domain decomposition. Multiscale Model. Simul. 7 (2008) 362-380. | MR 2399550 | Zbl 1160.65343
[47] , Bonding and structure of molecules and solids. Oxford University Press (1995).
[48] and , Identifying vector field singularities using a discrete Hodge decomposition, in Visualization and mathematics III, Math. Vis. Springer, Berlin (2003) 113-134. | MR 2047004 | Zbl 1065.37018
[49] , Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions. Multiscale Model. Simul. 9 (2011) 905-932. | MR 2831585 | Zbl 1246.74065
[50] , , , , and , An adaptive finite element approach to atomic-scale mechanics - the quasicontinuum method. J. Mech. Phys. Solids 47 (1999) 611-642. | MR 1675219 | Zbl 0982.74071
[51] , and , Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett. 89 (2002) 025501.
[52] , , and , 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] and , An Analysis of the Finite Element Method. Wellesley-Cambridge Press (2008). | MR 2743037 | Zbl 1171.65081
[54] and , Development and analysis of blended quasicontinuum approximations. To appear in SIAM J. Numer. Anal., arXiv:1008.2138. | MR 2861715
[55] , , and , 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 3050911 | Zbl 1245.74102
[56] and , A bridging domain method for coupling continua with molecular dynamics. Comput. Methods Appl. Mech. Eng. 193 (2004) 1645-1669. | MR 2069430 | Zbl 1079.74509