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The role of the patch test in 2D atomistic-to-continuum coupling methods. (English) Zbl 1269.82063
Summary: For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a \(P{_1}\)-finite element discretisation of Cauchy-Born nonlinear elasticity, this paper addresses 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 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.

82D20 Statistical mechanics of solids
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
70C20 Statics
74S05 Finite element methods applied to problems in solid mechanics
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