Symmetries of 2-lattices and second order accuracy of the Cauchy-Born model

We show that the Cauchy-Born model of a single-species 2-lattice is second order if the atomistic and continuum kinematics are connected in a novel way. Our proof uses a generalization to 2-lattices of the point symmetry of Bravais lattices. Moreover, by identifying similar symmetries in multispecies pair interaction models, we construct a new stored energy density, using shift gradients but not strain gradients, that is also second order accurate. These results can be used to develop highly accurate continuum models and atomistic/continuum coupling methods for materials such as graphene, hcp metals, and shape memory alloys

A priori and a posteriori W1,∞W^{1,\infty} error analysis of a QC method for complex lattices

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a $W^{1,\infty}$ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the error bounds (both, a priori and a posteriori). We illustrate our error estimates with numerical experiments.Comment: 23 page

Analysis of the quasi-nonlocal approximation of linear and circular chains in the plane

We give an analysis of the stability and displacement error for linear and circular atomistic chains in the plane when the atomistic energy is approximated by the Cauchy-Born continuum energy and by the quasi-nonlocal atomistic-to-continuum coupling energy. We consider atomistic energies that include Lennard-Jones type nearest neighbor and next nearest neighbor pair-potential interactions. Previous analyses for linear chains have shown that the Cauchy-Born and quasi-nonlocal approximations reproduce (up to the order of the lattice spacing) the atomistic lattice stability for perturbations that are constrained to the line of the chain. However, we show that the Cauchy-Born and quasi-nonlocal approximations give a finite increase for the lattice stability of a linear or circular chain under compression when general perturbations in the plane are allowed. We also analyze the increase of the lattice stability under compression when pair-potential energies are augmented by bond-angle energies. Our estimates of the largest strain for lattice stability (the critical strain) are sharp (exact up to the order of the lattice scale). We then use these stability estimates and modeling error estimates for the linearized Cauchy-Born and quasi-nonlocal energies to give an optimal order (in the lattice scale) {\em a priori} error analysis for the approximation of the atomistic strain in $\ell^2_\epsilon$ due to an external force.Comment: 27 pages, 0 figure