Geodesic
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

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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
Keywords: atomistic models, atomistic-to-continuum coupling, quasicontinuum method, coarse graining, ghost forces, patch test, consistency 
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     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://geodesic.mathdoc.fr/articles/10.1051/m2an/2012005/}
}
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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

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