## Section: New Results

### Atomistic to continuum methods

Participants : Matthew Dobson, Claude Le Bris, Frédéric Legoll.

The project-team has continued their theoretical and numerical efforts on the general topic of "passage from the atomistic to the continuum". This concerns theoretical issues arising in this passage but also the development and the improvement of numerical simulations coupling the two scales.

The quasicontinuum method couples an atomistic model to a continuum approximation in order to compute deformed states of a crystalline lattice at a reduced computational cost compared to a full atomistic simulation. In collaboration with M. Luskin (University of Minnesota) and C. Ortner (Warwick), M. Dobson analyzed the use of numerical solvers for approximating solutions to the equilibrium equations of the force-based quasicontinuum method [37] . In particular, it was shown that a previously-proposed modified conjugate gradient algorithm has unstable modes since the linearized operator is generally not positive-definite. Based on observed properties of the spectrum, convergence rates are given for the GMRES method applied to the operator.

Still in the framework of quasicontinuum methods, several consistent couplings have been proposed in the literature in the past years. M. Dobson showed the impossibility of constructing higher-order consistent couplings for quasicontinuum energies [36] . The analysis is performed in the one-dimensional situation, and is based on the fact that the truncation error gives lower bounds on the global error on the deformation gradient. A consequence of this result is that the so-called quasi-nonlocal energy (which is one of the coupling schemes proposed earlier) has asymptotically optimal error bounds.

The team has also addressed questions related to the *finite temperature*
modeling of atomistic systems and
derivation of coarse-grained descriptions, such as canonical averages of
observables depending only on a few variables. In the one-dimensional
setting, an efficient strategy that bypasses the simulation of the whole
system had been proposed in 2010. In collaboration with X. Blanc (CEA),
F. Legoll has extended this strategy to the so-called membrane
setting [20] : the system is composed of atoms that
lie on a two-dimensional lattice, and have a unique degree of
freedom, representing their height. The strategy
can also be used to derive the stress-strain relation for
one-dimensional chains of atoms, e.g. the relation between the
elongation of the chain and the stress, at any given
temperature [63] .