The role of the patch test in 2D atomistic-to-continuum coupling methods∗

Christoph Ortner

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1275-1319
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Ortner, Christoph. "The role of the patch test in 2D atomistic-to-continuum coupling methods∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1275-1319. <http://eudml.org/doc/276383>.

@article{Ortner2012,
abstract = {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.},
author = {Ortner, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Atomistic models; atomistic-to-continuum coupling; quasicontinuum method; coarse graining; ghost forces; patch test; consistency; atomistic models},
language = {eng},
month = {3},
number = {6},
pages = {1275-1319},
publisher = {EDP Sciences},
title = {The role of the patch test in 2D atomistic-to-continuum coupling methods∗},
url = {http://eudml.org/doc/276383},
volume = {46},
year = {2012},
}

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
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1275
EP - 1319
AB - 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.
LA - eng
KW - Atomistic models; atomistic-to-continuum coupling; quasicontinuum method; coarse graining; ghost forces; patch test; consistency; atomistic models
UR - http://eudml.org/doc/276383
ER -

References

top
  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. Elasticity100 (2010) 63–143.  Zbl1260.74005
  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).  Zbl1070.49009
  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.  Zbl0696.73040
  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.  Zbl1160.65338
  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).  Zbl0959.74001
  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)].  
  9. M. Dobson, There is no pointwise consistent quasicontinuum energy. arXiv:1109.1897.  Zbl1305.82044
  10. M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM : M2AN42 (2008) 113–139.  Zbl1140.74006
  11. M. Dobson and M. Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error. ESAIM : M2AN43 (2009) 591–604.  Zbl1165.81414
  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.  Zbl1203.82089
  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. Design14 (2007) 219–237.  
  14. M. Dobson, M. Luskin and C. Ortner, Accuracy of quasicontinuum approximations near instabilities. J. Mech. Phys. Solids58 (2010) 1741–1757.  Zbl1200.74005
  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.  Zbl1268.74006
  16. W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and prospects of contemporary applied mathematics. Ser. Contemp. Appl. Math. CAM6 (2005) 18–32.  
  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.  Zbl1106.74019
  18. W. E, J. Lu and J.Z. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B74 (2006) 214115.  
  19. B. Eidel and A. Stukowski, A variational formulation of the quasicontinuum method based on energy sampling in clusters. J. Mech. Phys. Solids57 (2009) 87–108.  Zbl1298.74011
  20. M. Finnis, Interatomic Forces in Condensed Matter. Oxford Series on Materials Modelling1 (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.  Zbl1173.74303
  22. M. Gunzburger and Y. Zhang, A quadrature-rule type approximation to the quasi-continuum method. Multiscale Model. Simul.8 (2009/2010) 571–590.  Zbl1188.70046
  23. M. Iyer and V. Gavini, A field theoretical approach to the quasi-continuum method. J. Mech. Phys. Solids59 (2011) 1506–1535.  Zbl1270.74006
  24. P.A. Klein and J.A. Zimmerman, Coupled atomistic-continuum simulations using arbitrary overlapping domains. J. Comput. Phys.213 (2006) 86–116.  Zbl1137.74367
  25. J. Knap and M. Ortiz, An analysis of the quasicontinuum method. J. Mech. Phys. Solids49 (2001) 1899–1923.  Zbl1002.74008
  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.  
  29. P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp.72 (2003) 657–675.  Zbl1010.74003
  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).  Zbl1220.74010
  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.  Zbl1196.82122
  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.  Zbl1225.82070
  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.  Zbl1225.82070
  35. R.E. Miller and E.B. Tadmor, The quasicontinuum method : overview, applications and current directions. J. Computer-Aided Mater. Design9 (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.  Zbl1177.74169
  38. M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A73 (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.  Zbl05918690
  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.  Zbl1276.82013
  42. C. Ortner and E. Süli, Analysis of a quasicontinuum method in one dimension. ESAIM : M2AN42 (2008) 57–91.  Zbl1139.74004
  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.  Zbl1242.74213
  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.  Zbl1269.82019
  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.  Zbl1160.65343
  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.  Zbl1065.37018
  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.  Zbl1246.74065
  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. Solids47 (1999) 611–642.  Zbl0982.74071
  51. L.E. Shilkrot, R.E. Miller and W.A. Curtin, Coupled atomistic and discrete dislocation plasticity. Phys. Rev. Lett.89 (2002) 025501.  Zbl1049.74015
  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. B69 (2004) 214104.  
  53. G. Strang and G. Fix, An Analysis of the Finite Element Method. Wellesley-Cambridge Press (2008).  Zbl1171.65081
  54. B. Van Koten and M. Luskin, Development and analysis of blended quasicontinuum approximations. To appear in SIAM J. Numer. Anal., arXiv:1008.2138.  Zbl1241.82026
  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).  Zbl1245.74102
  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.  Zbl1079.74509

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.