Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice

Ortner, C.; Shapeev, A.

doi:10.1090/S0025-5718-2013-02687-7

Analysis of an energy-based atomistic/continuum approximation of a vacancy in the 2D triangular lattice
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by C. Ortner and A. V. Shapeev PDF

Math. Comp. 82 (2013), 2191-2236 Request permission

Abstract:

We present an a priori error analysis of a practical energy based atomistic/continuum coupling method (A. V. Shapeev, Multiscale Model. Simul., 9(3):905–932, 2011) in two dimensions, for finite-range pair-potential interactions, in the presence of vacancy defects.

We establish first-order consistency and stability of the method, from which we obtain a priori error estimates in the $\mathrm {H}^1$-norm and the energy in terms of the mesh size and the “smoothness” of the atomistic solution in the continuum region. From these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh, as well as convergence rates in terms of the number of degrees of freedom. Our analytical predictions are supported by extensive numerical tests.

References

Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
L. Demkowicz, Ph. Devloo, and J. T. Oden, On an $h$-type mesh-refinement strategy based on minimization of interpolation errors, Comput. Methods Appl. Mech. Engrg. 53 (1985), no. 1, 67–89. MR 812590, DOI 10.1016/0045-7825(85)90076-3
A. Demlow, D. Leykekhman, A. H. Schatz, and L. B. Wahlbin, Best approximation property in the $W^{1}_{\infty }$ norm for finite element methods on graded meshes, Math. Comp. 81 (2012), no. 278, 743–764. MR 2869035, DOI 10.1090/S0025-5718-2011-02546-9
Matthew Dobson and Mitchell Luskin, An optimal order error analysis of the one-dimensional quasicontinuum approximation, SIAM J. Numer. Anal. 47 (2009), no. 4, 2455–2475. MR 2525607, DOI 10.1137/08073723X
M. Dobson, M. Luskin, and C. Ortner, Accuracy of quasicontinuum approximations near instabilities, J. Mech. Phys. Solids 58 (2010), no. 10, 1741–1757. MR 2742030, DOI 10.1016/j.jmps.2010.06.011
Matthew Dobson, Mitchell Luskin, and Christoph Ortner, Stability, instability, and error of the force-based quasicontinuum approximation, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 179–202. MR 2646818, DOI 10.1007/s00205-009-0276-z
W. E. J. Lu, and J. Z. Yang, Uniform accuracy of the quasicontinuum method, Phys. Rev. B 74(21) (2006), 214115.
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
F. C. Frank and J. H. van der Merwe, One-dimensional dislocations. I. static theory, Proc. R. Soc. London A198 (1949), 205–216.
Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
Thomas Hudson and Christoph Ortner, On the stability of Bravais lattices and their Cauchy-Born approximations, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 81–110. MR 2846368, DOI 10.1051/m2an/2011014
Mrinal Iyer and Vikram Gavini, A field theoretical approach to the quasi-continuum method, J. Mech. Phys. Solids 59 (2011), no. 8, 1506–1535. MR 2848058, DOI 10.1016/j.jmps.2010.12.002
P. A. Klein and J. A. Zimmerman, Coupled atomistic-continuum simulations using arbitrary overlapping domains, J. Comput. Phys. 213 (2006), no. 1, 86–116. MR 2203436, DOI 10.1016/j.jcp.2005.08.014
S. Kohlhoff and S. Schmauder, A new method for coupled elastic-atomistic modelling, Atomistic Simulation of Materials: Beyond Pair Potentials (V. Vitek and D. J. Srolovitz, eds.), Plenum Press, New York, 1989, pp. 411–418.
Xingjie Helen Li and Mitchell Luskin, A generalized quasinonlocal atomistic-to-continuum coupling method with finite-range interaction, IMA J. Numer. Anal. 32 (2012), no. 2, 373–393. MR 2911393, DOI 10.1093/imanum/drq049
Xingjie Helen Li, Mitchell Luskin, and Christoph Ortner, Positive definiteness of the blended force-based quasicontinuum method, Multiscale Model. Simul. 10 (2012), no. 3, 1023–1045. MR 3022030, DOI 10.1137/110859270
J. Lu and P. Ming, Convergence of a force-based hybrid method for atomistic and continuum models in three dimension, arXiv:1102.2523.
Charalambos Makridakis, Christoph Ortner, and Endre Süli, A priori error analysis of two force-based atomistic/continuum models of a periodic chain, Numer. Math. 119 (2011), no. 1, 83–121. MR 2824856, DOI 10.1007/s00211-011-0380-5
R. Miller and E. Tadmor, A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods, Modelling Simul. Mater. Sci. Eng. 17 (2009).
Pingbing Ming and Jerry Zhijian Yang, Analysis of a one-dimensional nonlocal quasi-continuum method, Multiscale Model. Simul. 7 (2009), no. 4, 1838–1875. MR 2539201, DOI 10.1137/080725842
M. Ortiz, R. Phillips, and E. B. Tadmor, Quasicontinuum analysis of defects in solids, Philosophical Magazine A 73(6) (1996), 1529–1563.
Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 6, 1275–1319. MR 2996328, DOI 10.1051/m2an/2012005
Christoph Ortner, A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D, Math. Comp. 80 (2011), no. 275, 1265–1285. MR 2785458, DOI 10.1090/S0025-5718-2010-02453-6
Christoph Ortner and Dirk Praetorius, On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems, SIAM J. Numer. Anal. 49 (2011), no. 1, 346–367. MR 2783229, DOI 10.1137/090781073
C. Ortner and A. V. Shapeev, Analysis of an energy-based atomistic/continuum coupling approximation of a vacancy in the 2D triangular lattice, arXiv:1104.0311v1.
Christoph Ortner and Endre Süli, Analysis of a quasicontinuum method in one dimension, M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, 57–91. MR 2387422, DOI 10.1051/m2an:2007057
C. Ortner and H. Wang, A priori error estimates for energy-based quasicontinuum approximations of a periodic chain, Math. Models Methods Appl. Sc. 21 (2011), 2491–2521.
Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437–445. MR 645661, DOI 10.1090/S0025-5718-1982-0645661-4
Alexander V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in three dimensions, SIAM J. Sci. Comput. 34 (2012), no. 3, B335–B360. MR 2970282, DOI 10.1137/110844544
Alexander V. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potentials in one and two dimensions, Multiscale Model. Simul. 9 (2011), no. 3, 905–932. MR 2831585, DOI 10.1137/100792421
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), no. 3, 611–642. MR 1675219, DOI 10.1016/S0022-5096(98)00051-9
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(21) (2004), 214104.
B. Van Koten, X. H. Li, M. Luskin, and C. Ortner, A computational and theoretical investigation of the accuracy of quasicontinuum methods, Numerical Analysis of Multiscale Problems (Ivan Graham, Tom Hou, Omar Lakkis, and Rob Scheichl, eds.), Springer Lecture Notes in Computational Science and Engineering, vol. 83, Springer, 2012.
S. P. Xiao and T. Belytschko, A bridging domain method for coupling continua with molecular dynamics, Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 17-20, 1645–1669. MR 2069430, DOI 10.1016/j.cma.2003.12.053