An analysis of the boundary layer in the 1D surface Cauchy-Born model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 109-123
The surface Cauchy-Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” - the stiffness of the interaction potential - with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.
DOI : https://doi.org/10.1051/m2an/2012021
Classification:  70C20,  70-08,  65N12,  65N30
Keywords: surface-dominated materials, surface Cauchy-Born rule, coarse-graining
@article{M2AN_2013__47_1_109_0,
     author = {Jayawardana, Kavinda and Mordacq, Christelle and Ortner, Christoph and Park, Harold S.},
     title = {An analysis of the boundary layer in the 1D surface Cauchy-Born model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {109-123},
     doi = {10.1051/m2an/2012021},
     zbl = {1273.74004},
     mrnumber = {2968697},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_1_109_0}
}
Jayawardana, Kavinda; Mordacq, Christelle; Ortner, Christoph; Park, Harold S. An analysis of the boundary layer in the 1D surface Cauchy-Born model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 109-123. doi : 10.1051/m2an/2012021. http://www.numdam.org/item/M2AN_2013__47_1_109_0/

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