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Analysis of energy-based blended quasi-continuum approximations. (English) Zbl 1241.82026

Quasi-continuum energy methods have been devised for a computer-assisted analysis of multidimensional crystalline solids modeled by many-body potentials. Their core lies in coupling localised strain-field areas (where a mesoscopic accuracy is required) with smoothened larger regions, where an efficient approximation is provided by coarse-grained continuum finite element models. Orginal two- and three-dimensional quasi-continuuum approximations proved not to be patch test consistent.
In the present paper, an error analysis is performed for the blended quasi-continuum approximation of a periodic one-dimensional chain of atoms with next-nearest neighbor interactions. The analysis includes a proper choice and then the optimization of the blending function. The reduction of the strain error is achieved for an optimal blending function, by a factor \(k^{3/2}\), where \(k\) is the number of atoms in the blending region. The error in the critical strain can be reduced by a factor \(k^2\), thus demonstrating that energy-based blended quasi-continuum approximations have the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.

MSC:

82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
70C20 Statics
74S20 Finite difference methods applied to problems in solid mechanics
65G40 General methods in interval analysis
82B80 Numerical methods in equilibrium statistical mechanics (MSC2010)
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
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