Fast computation of effective diffusivities using a semi-analytical solution of the homogenization boundary value problem for block locally-isotropic heterogeneous media

Direct numerical simulation of diffusion through heterogeneous media can be difficult due to the computational cost of resolving fine-scale heterogeneities. One method to overcome this difficulty is to homogenize the model by replacing the spatially-varying fine-scale diffusivity with an effective diffusivity calculated from the solution of an appropriate boundary value problem. In this paper, we present a new semi-analytical method for solving this boundary value problem and computing the effective diffusivity for pixellated, locally-isotropic, heterogeneous media. We compare our new solution method to a standard finite volume method and show that equivalent accuracy can be achieved in less computational time for several standard test cases. We also demonstrate how the new solution method can be applied to complex heterogeneous geometries represented by a grid of blocks. These results indicate that our new semi-analytical method has the potential to significantly speed up simulations of diffusion in heterogeneous media.Comment: 29 pages, 4 figures, 5 table

Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450-464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work.Comment: 24 pages, 8 figures, accepted version of paper published in Applied Mathematics and Computatio

The Search for Perfection: Lake Forest and the Progressive Era

Lake Forest, Illinois in the Progressive Era was a highly exclusive safe haven for the elite of Chicago. Lake Forest, however, was more than a high priced suburb for successful businessmen and the children of entrepreneurs; it was a community where wealthy individuals would attempt to create the perfect environment. This paper will explore how the residents attempted to create a perfect community, from the European culture they chose to imitate to the architecture they chose for their estates, as well as the social world they created for themselves

An investigation into grid patching techniques

In the past decade significant advances were made using flow field methods in the calculation of external transonic flows over aerodynamic configurations. It is now possible to calculate inviscid transonic flow over three dimensional configurations by solving the potential equation. However, with the exception of the transonic small disturbance methods which have the advantage of a simple cartesian grid, the configurations over which it is possible to calculate such flows are relatively simple. The major reason for this is the difficulty of producing compatibility between grid generation and flow equation solutions. The main programs in use, use essentially analytic transformations for prescribed configurations and, as such, are not easy to extend. While there is work in progress to extend this type of system to a limited extent, the long term effort is directed towards a more general approach. This approach should not be restricted to producing grid systems in isolation but rather a consideration of the overall problem of flow field solution

Holes in the walls: primordial black holes as a solution to the cosmological domain wall problem

We propose a scenario in which the cosmological domain wall and monopole problems are solved without any fine tuning of the initial conditions or parameters in the Lagrangian of an underlying filed theory. In this scenario domain walls sweep out (unwind) the monopoles from the early universe, then the fast primordial black holes perforate the domain walls, change their topology and destroy them. We find further that the (old vacuum) energy density released from the domain walls could alleviate but not solve the cosmological flatness problem.Comment: References added; Published in Phys. Rev.

An Empirical Analysis of Community Center Rents

This article is the winner of the Retail Real Estate manuscript prize (sponsored by the International Council of Shopping Centers) presented at the 2001 American Real Estate Society Annual Meeting. This study empirically models the determinants of community center rent. It employs a two-stage model that estimates center vacancy in the first stage and then includes predicted vacancy in a second stage demand model investigating endogenous and exogenous determinants of community center rent. The data includes information on maximum and minimum square foot rent for 118 community centers in Atlanta, Georgia. Maximum community center rent is highly correlated with a center’s predicted vacancy rate and location within the Atlanta area. Additionally, rent at both maximum and minimum levels is influenced by trade area purchasing power, property age and to a lesser extent by proximity to a regional mall, center design and neighborhood factors.