Homogenization of hexagonal lattices

International audienceWe characterize the macroscopic e ffective behavior of a graphene sheet modeled by a hexagonal lattice of elastic bars, using Gamma-convergence

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

Minimising movements for the motion of discrete screw dislocations along glide directions

In [3] a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in [8] and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on $\Gamma$-convergence techniques

On the variational limits of lattice energies on prestrained elastic bodies

We study the asymptotic behaviour of the discrete elastic energies in presence of the prestrain metric $G$, assigned on the continuum reference configuration $\Omega$. When the mesh size of the discrete lattice in $\Omega$ goes to zero, we obtain the variational bounds on the limiting (in the sense of $\Gamma$-limit) energy. In case of the nearest-neighbour and next-to-nearest-neibghour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to $G$

Joint density for the local times of continuous-time Markov chains

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\mathbb{Z}^d$ tending to $\mathbb{Z}^d$ as time diverges. We finally discuss the relation of our density formula to the Ray--Knight theorem for continuous-time simple random walk on $\mathbb{Z}$, which is analogous to the well-known Ray--Knight description of Brownian local times.Comment: Published at http://dx.doi.org/10.1214/009171906000001024 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Overall properties of a discrete membrane with randomly distributed defects

A prototype for variational percolation problems with surface energies is considered: the description of the macroscopic properties of a (two-dimensional) discrete membrane with randomly distributed defects in the spirit of the weak membrane model of Blake and Zisserman (quadratic springs that may break at a critical length of the elongation). After introducing energies depending on suitable independent identically distributed random variables, this is done by exhibiting an equivalent continuum energy computed using Delta-convergence, geometric measure theory, and percolation arguments. We show that below a percolation threshold the effect of the defects is negligible and the continuum description is given by the Dirichlet integral, while above that threshold an additional (Griffith) fracture term appears in the energy, which depends only on the defect probability through the chemical distance on the "weak cluster of defects"