arXiv subjects

Anthony Roberts

Publications and source records attributed to Anthony Roberts.

8 recordsLinked to original sources

arXiv2012

Approximation of the random inertial manifold of singularly perturbed stochastic wave equations

By applying Rohlin's result on the classification of homomorphisms of Lebesgue space, the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter. This approximation can be seen as the Smolukowski--Kramers approximation as time goes to infinity. However, as time goes infinity, the approximation changes with the small parameter, which is different from the approximation on a finite interval.

math.AP↗

arXiv2012

Diffusion approximation for self-similarity of stochastic advection in Burgers' equation

Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic Burgers' equation. The analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self-similar variables. Furthermore, by a diffusion approximation, the long time convergence to the self-similar solution is proved in the sense of distribution.

math.AP↗

arXiv2000

Elastic moduli of model random three-dimensional closed-cell cellular solids

Most cellular solids are random materials, while practically all theoretical results are for periodic models. To be able to generate theoretical results for random models, the finite element method (FEM) was used to study the elastic properties of solids with a closed-cell cellular structure. We have computed the density ($ρ$) and microstructure dependence of the Young's modulus ($E$) and Poisson's ratio (PR) for several different isotropic random models based on Voronoi tessellations and level-cut Gaussian random fields. The effect of partially open cells is also considered. The results, which are best described by a power law $E\proptoρ^n$ ($1 < n <2$), show the influence of randomness and isotropy on the properties of closed-cell cellular materials, and are found to be in good agreement with experimental data.

cond-mat.mtrl-sci↗

arXiv2000

Transport and Elastic Properties of Fractal Media

We investigate the influence of fractal structure on material properties. We calculate the statistical correlation functions of fractal media defined by level-cut Gaussian random fields. This allows the modeling of both surface fractal and mass fractal materials. Variational bounds on the conductivity, diffusivity and elastic moduli of the materials are evaluated. We find that a fractally rough interface has a relatively strong influence on the properties of composites. In contrast a fractal volume (mass) has little effect on material properties.

cond-mat.mtrl-sci↗

arXiv2000

Transport properties of heterogeneous materials derived from Gaussian random fields: Bounds and Simulation

We investigate the effective conductivity ($σ_e$) of a class of amorphous media defined by the level-cut of a Gaussian random field. The three point solid-solid correlation function is derived and utilised in the evaluation of the Beran-Milton bounds. Simulations are used to calculate $σ_e$ for a variety of fields and volume fractions at several different conductivity contrasts. Relatively large differences in $σ_e$ are observed between the Gaussian media and the identical overlapping sphere model used previously as a `model' amorphous medium. In contrast $σ_e$ shows little variability between different Gaussian media.

cond-mat.mtrl-sci↗

arXiv1999

Structure-Property Correlations in Model Composite Materials

We investigate the effective properties (conductivity, diffusivity and elastic moduli) of model random composite media derived from Gaussian random fields and overlapping hollow spheres. The morphologies generated in the models exhibit low percolation thresholds and give a realistic representation of the complex microstructure observed in many classes of composites. The statistical correlation functions of the models are derived and used to evaluate rigorous bounds on each property. Simulation of the effective conductivity is used to demonstrate the applicability of the bounds. The key morphological features which effect composite properties are discussed.

cond-mat.dis-nn↗

arXiv1999

Statistical reconstruction of three-dimensional porous media from two-dimensional images

A method of modelling the three-dimensional microstructure of random isotropic two-phase materials is proposed. The information required to implement the technique can be obtained from two-dimensional images of the microstructure. The reconstructed models share two-point correlation and chord-distribution functions with the original composite. The method is designed to produce models for computationally and theoretically predicting the effective macroscopic properties of random materials (such as electrical and thermal conductivity, permeability and elastic moduli). To test the method we reconstruct the morphology and predict the conductivity of the well known overlapping sphere model. The results are in very good agreement with data for the original model.

cond-mat.dis-nn↗

arXiv1999

Chord distribution functions of three-dimensional random media: Approximate first-passage times of Gaussian processes

The main result of this paper is a semi-analytic approximation for the chord distribution functions of three-dimensional models of microstructure derived from Gaussian random fields. In the simplest case the chord functions are equivalent to a standard first-passage time problem, i.e., the probability density governing the time taken by a Gaussian random process to first exceed a threshold. We obtain an approximation based on the assumption that successive chords are independent. The result is a generalization of the independent interval approximation recently used to determine the exponent of persistence time decay in coarsening. The approximation is easily extended to more general models based on the intersection and union sets of models generated from the iso-surfaces of random fields. The chord distribution functions play an important role in the characterization of random composite and porous materials. Our results are compared with experimental data obtained from a three-dimensional image of a porous Fontainebleau sandstone and a two-dimensional image of a tungsten-silver composite alloy.

cond-mat.dis-nn↗