*Energy-Based Blended Quasicontinuum Approximations*

**Mitchell Luskin**

University of Minnesota

School of Mathematics

**luskin@umn.edu**

**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 strain error for the
non-blended QCE energy (QCE), which has low order
where is the scaled atomistic
length scale, can be reduced by a factor of where 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.