Skip to content..
Carnegie Mellon Center for Nonlinear Analysis
Energy-Based Blended Quasicontinuum Approximations


Mitchell Luskin
University of Minnesota
School of Mathematics

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 of blended atoms, a general quasicontinuum energy (BQCE) can be developed with the potential to significantly improve the accuracy of QCE near lattice stabilities such as dislocation formation and motion.

We give an error analysis of the blended quasicontinuum energy (BQCE) for a periodic one-dimensional chain of atoms with next-nearest neighbor interactions which allows the blending function to be optimized for an improved convergence rate. We show that the l2 strain error for the non-blended QCE energy (QCE), which has low order ${\rm O}(\varepsilon^{1/2})$ where $\varepsilon$ is the scaled atomistic length scale, can be reduced by a factor of $k^{3/2}$ 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 k2 where k is the number of atoms in the blended interface region, thus demonstrating that the BQCE energy has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth.

Joint with Brian Van Koten.