Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation
M. Dobson and M. Luskin and C. Ortner
Multiscale Modeling and Simulation, 8, 782-802 (2010).
ABSTRACT
The accuracy of atomistic-to-continuum hybrid methods can be guaranteed only for
deformations where the lattice configuration is stable for both the atomistic
energy and the hybrid energy. For this reason, a sharp stability analysis of
atomistic-to-continuum coupling methods is essential for evaluating their
capabilities for predicting the formation of lattice defects. We formulate a
simple one-dimensional model problem and give a detailed analysis of the linear
stability of the force-based quasicontinuum (QCF) method at homogeneous
deformations. The focus of the analysis is the question of whether the QCF
method is able to predict a critical load at which fracture occurs. Numerical
experiments show that the spectrum of a linearized QCF operator is identical to
the spectrum of a linearized energy-based quasi-nonlocal quasicontinuum (QNL)
operator, which we know from our previous analyses to be positive below the
critical load. However, the QCF operator is nonnormal, and it turns out that it
is not generally positive definite, even when all of its eigenvalues are
positive. Using a combination of rigorous analysis and numerical experiments, we
investigate in detail for which choices of "function spaces" the QCF operator is
stable, uniformly in the size of the atomistic system.