An Analysis of Surface Relaxation in the Surface Cauchy--Born Model
Provides theoretical error bounds for a multi-scale method used in materials science, but the analysis is limited to a linearized 1D model.
The paper analyzes the Surface Cauchy-Born method for surface-dominated crystalline materials, showing that the error is O(1) in mesh size but exponentially small in the stiffness of the interaction potential for mean strain. It proposes an improvement by enforcing atomistic mesh spacing at the free boundary.
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 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.