Computational modeling of microstructure

Many materials such as martensitic or ferromagnetic crystals are observed to be in metastable states exhibiting a fine-scale, structured spatial oscillation called microstructure; and hysteresis is observed as the temperature, boundary forces, or external magnetic field changes. We have developed a numerical analysis of microstructure and used this theory to construct numerical methods that have been used to compute approximations to the deformation of crystals with microstructure

An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error

The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the optimal rate O($h$) in the discrete $\ell^\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for $1 \leq p < \infty.$ where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays away from the interface to O($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$) in the continuum region. E, Ming, and Yang previously gave a counterexample to convergence in the $w^{1,\infty}$ norm for a harmonic interatomic potential. Our work gives an explicit and simplified form for the decay of the effect of the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and $w^{1,p}$ norms.Comment: 14 pages, 1 figur

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

On solutions of Maxwell's equations with dipole sources over a thin conducting film

We derive and interpret solutions of time-harmonic Maxwell's equations with a vertical and a horizontal electric dipole near a planar, thin conducting film, e.g. graphene sheet, lying between two unbounded isotropic and non-magnetic media. Exact expressions for all field components are extracted in terms of rapidly convergent series of known transcendental functions when the ambient media have equal permittivities and both the dipole and observation point lie on the plane of the film. These solutions are simplified for all distances from the source when the film surface resistivity is large in magnitude compared to the intrinsic impedance of the ambient space. The formulas reveal the analytical structure of two types of waves that can possibly be excited by the dipoles and propagate on the film. One of these waves is intimately related to the surface plasmon-polariton of transverse-magnetic (TM) polarization of plane waves.Comment: 48 pages, 4 figure

Numerical Analysis of Parallel Replica Dynamics

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.Comment: 37 pages, 4 figures, revised and new estimates from the previous versio