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

Cephalometric variability among siblings

OBJECTIVE: To determine whether multiple siblings in a family resemble one another in terms of their craniofacial characteristics.  METHODS: This pilot study was conducted retrospectively using the Forsyth Twin sample. 32 families were included, each with at least 4 siblings who had lateral cephalometric radiographs taken after skeletal maturity was documented, for a total of 142 subjects. Headfilms were digitized and skeletal landmarks located to allow measurement of 6 parameters indicating sagittal jaw relationships and vertical status.  Dixon’s Q test was applied to identify any outliers in a family for a given parameter. Manhattan distance quantified similarity among siblings per parameter. Scatter plots visually displayed subject’s measure relative to the mean and standard deviation of each parameter to assess clinical relevance.  RESULTS: 11 families (34.4%) had no outliers on any of the 6 parameters, 13 families (40.6%) had outliers on only 1 parameter, and 8 families (25%) had outliers on at least 2 parameters. Our analyses identified 29 individuals with at least one outlying measure (20.4%). Of those, only 2 individuals (1.4%) were significantly different from their siblings for more than 1 measurement.  Although the majority of the families did not demonstrate a statistical outlier for any given measurement, the ranges were clinically relevant as they might lead to differing orthodontic treatment plans. CONCLUSIONS: Although families are generally not statistically dissimilar in their craniofacial characteristics as measured on cephalometric radiographs, measurements from siblings cannot be used to predict the measurements of another sibling in a clinically meaningful way

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

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