Analysis of Energy-Based Blended Quasicontinuum Approximations

Abstract

The development of patch test consistent quasicontinuum energies for multi-dimensional crystalline solids modeled by many-body potentials remains a challenge. The original quasicontinuum energy (QCE) has been implemented for many-body potentials in two and three space dimensions, but it is not patch test consistent. We propose that by blending the atomistic and corresponding Cauchy-Born continuum models of QCE in an interfacial region with thickness of a small number $k$ of blended atoms, a general quasicontinuum energy (BQCE) can be developed with the potential to significantly improve the accuracy of QCE near lattice instabilities such as dislocation formation and motion. In this paper, we give an error analysis of the blended quasicontinuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions. Our analysis includes the optimization of the blending function for an improved convergence rate. We show that the ${\ell }^2$ strain error for the non-blended QCE energy (QCE), which has low order $\text{O}({\epsilon }^{1/2})$ where $\epsilon$ is the atomistic length scale, can be reduced by a factor of $k^{3/2}$ for an optimized blending function where $k$ is the number of atoms in the blending region. The QCE energy has been further shown to suffer from a O$(1)$ error in the critical strain at which the lattice loses stability. We prove that the error in the critical strain of BQCE can be reduced by a factor of $k^2$ for an optimized blending function, thus demonstrating that the BQCE energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.

26 pages, 1 figure

References in corpus (7)

• Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in One and Two Dimensions
• Accuracy of Quasicontinuum Approximations Near Instabilities
• A Priori and A Posteriori Analysis of the Quasi-Nonlocal Quasicontinuum Method in 1D
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Cited by in corpus (6)

• Consistent Energy-based Atomistic/Continuum Coupling for Two-body Potentials in Three Dimensions
• A posteriori error control for a quasicontinuum approximation of a periodic chain
• A Computational and Theoretical Investigation of the Accuracy of Quasicontinuum Methods
• The role of the patch test in 2D atomistic-to-continuum coupling methods
• Linear Stationary Iterative Methods for the Force-based Quasicontinuum Approximation
• Analysis of the quasi-nonlocal approximation of linear and circular chains in the plane