Analysis of energy-based blended quasi-continuum approximations
B. van Koten and M. Luskin
SIAM Journal of Numerical Analysis, 49, 2182-2209 (2011).
ABSTRACT
The development of patch test consistent quasi-continuum energies for
multidimensional crystalline solids modeled by many-body potentials remains a
challenge. The original quasi-continuum energy (QCE) [R. Miller and E. Tadmor,
Model. Simul. Mater. Sci. Eng., 17 (2009), 053001] 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 blended quasi-continuum 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 quasi-continuum 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 l(2) strain error for the
nonblended QCE energy, which has low order O(epsilon(1/2)), where e is the
atomistic length scale [M. Dobson and M. Luskin, SIAM J. Numer. Anal., 47
(2009), pp. 2455-2475, P. Ming and J. Z. Yang, Multiscale Model. Simul., 7
(2009), pp. 1838-1875], 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 [M. Dobson, M. Luskin, and C. Ortner, J.
Mech. Phys. Solids, 58 (2010), pp. 1741-1757]. 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.