  A sharp stability analysis of atomistic-to-continuum coupling methods is
essential for evaluating their capabilities for predicting the formation and
motion of lattice defects. We formulate a simple one-dimensional model problem
and give a detailed analysis of the stability of the force-based quasicontinuum
(QCF) method. The focus of the analysis is the question 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
operator (QNL), which we know from our previous analyses to be positive below
the critical load. However, the QCF operator is non-normal 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.
Force-based multi-physics coupling methods are popular techniques to circumvent
the difficulties faced in formulating consistent energy-based coupling
pproaches. Even though the QCF method is possibly the simplest coupling method
of this kind, we anticipate that many of our observations apply more generally.
