Meshless Methods for the Neutron Transport Equation

Mesh-based methods for the numerical solution of partial differential equations (PDEs) require the division of the problem domain into non-overlapping, contiguous subdomains that conform to the problem geometry. The mesh constrains the placement and connectivity of the solution nodes over which the PDE is solved. In meshless methods, the solution nodes are independent of the problem geometry and do not require a mesh to determine connectivity. This allows the solution of PDEs on geometries that would be difficult to represent using even unstructured meshes. The ability to represent difficult geometries and place solution nodes independent of a mesh motivates the use of meshless methods for the neutron transport equation, which often includes spatially-dependent PDE coefficients and strong localized gradients. The meshless local Petrov-Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. The MLPG method uses weighted residuals of the transport equation to solve for basis function expansion coefficients of the neutron angular flux. Connectivity of the solution nodes is determined by the shared support domain of overlapping meshless functions, such as radial basis functions (RBFs) and moving least squares (MLS) functions. To prevent oscillations in the neutron flux, the MLPG transport equation is stabilized by the streamline upwind Petrov-Galerkin (SUPG) method, which adds numerical diffusion to the streaming term. Global neutron conservation is enforced by using MLS basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. Two integration schemes for the basis and weight functions in the MLPG method are presented, including a background mesh integration and a fully meshless integration approach. The method of manufactured solutions (MMS) is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations. Finally, meshless heat transfer equations are derived using a similar MLPG approach and verified using the MMS. These heat equation are coupled to the MLPG neutron transport equations, and results for a pincell are compared to values from a commercial pressurized water reactor.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145796/1/brbass_1.pd

An assessment of a mesh-based method for slab-geometry transport in stochastic materials

There exists a variety of methods to solve the radiation transport equation in stochastic materials, including the Levermore-Pomraning method and the atomic mix method, each of which is designed to replicate the results given by benchmarks with materials rearranged in a randomized fashion. While the atomic mix method is very simple to implement, the results given are less than ideal due to the assumption that the materials are mixed at the atomic level, precluding the account of larger individual pieces of a material. The Levermore-Pomraning method is often used to approximate the benchmark solutions for exponentially-distributed materials, but includes a coupling term in its differential equation that is more difficult to implement and is significantly inaccurate in optically thick and highly scattering materials. By assuming that the materials will not change within a cell, as happens in the benchmark solutions if the realizations are generated on a constant mesh, a set of differential equations can be derived that are coupled only at the cell boundaries. The analytic and then discretized equations for this method and each of the standard and benchmark methods are derived herein. Two separate methods of coupling for the mesh-based equations are shown, and their results using a variety of material parameters are compared with each of the standard and benchmark methods

Wealth and sexual behaviour among men in Cameroon

BACKGROUND: The 2004 Demographic and Health Survey (DHS) in Cameroon revealed a higher prevalence of HIV in richest and most educated people than their poorest and least educated compatriots. It is not certain whether the higher prevalence results partly or wholly from wealthier people adopting more unsafe sexual behaviours, surviving longer due to greater access to treatment and care, or being exposed to unsafe injections or other HIV risk factors. As unsafe sex is currently believed to be the main driver of the HIV epidemic in sub-Saharan Africa, we designed this study to examine the association between wealth and sexual behaviour in Cameroon. METHODS: We analysed data from 4409 sexually active men aged 15–59 years who participated in the Cameroon DHS using logistic regression models, and have reported odds ratios (OR) with confidence intervals (CI). RESULTS: When we controlled for the potential confounding effects of marital status, place of residence, religion and age, men in the richest third of the population were less likely to have used a condom in the last sex with a non-spousal non-cohabiting partner (OR 0.43, 95% CI 0.32–0.56) and more likely to have had at least two concurrent sex partners in the last 12 months (OR 1.38, 95% CI 1.12–1.19) and more than five lifetime sex partners (OR 1.97, 95% CI 1.60–2.43). However, there was no difference between the richest and poorest men in the purchase of sexual services. Regarding education, men with secondary or higher education were less likely to have used a condom in the last sex with a non-spousal non-cohabiting partner (OR 0.24, 95% CI 0.16–0.38) and more likely to have started sexual activity at age 17 years or less (OR 2.73, 95% CI 2.10–3.56) and had more than five lifetime sexual partners (OR 2.59, 95% CI 2.02–3.31). There was no significant association between education and multiple concurrent sexual partnerships in the last 12 months or purchase of sexual services. CONCLUSION: Wealthy men in Cameroon are more likely to start sexual activity early and have both multiple concurrent and lifetime sex partners, and are less likely to (consistently) use a condom in sex with a non-spousal non-cohabiting partner. These unsafe sexual behaviours may explain the higher HIV prevalence among wealthier men in the country. While these findings do not suggest a redirection of HIV prevention efforts from the poor to the wealthy, they do call for efforts to ensure that HIV prevention messages get across all strata of society

Paraventricular thalamic nucleus: Subcortical connections and innervation by serotonin, orexin, and corticotropin-releasing hormone in macaque monkeys

The present study examines subcortical connections of paraventricular thalamic nucleus (Pa) following small anterograde and retrograde tracer injections in cynomolgus monkeys ( Macaca fascicularis ). An anterograde tracer injection into the dorsal midline thalamus revealed strong projections to the accumbens nucleus, basal amygdala, lateral septum, and hypothalamus. Retrograde tracer injections into these areas labeled neurons specifically in Pa. Following a retrograde tracer injection into Pa, labeled neurons were found in the hypothalamus, dorsal raphe, and periaqueductal gray. Pa contained a remarkably high density of axons and axonal varicosities immunoreactive for serotonin (5-HT) and orexin/hypocretin (ORX), as well as a moderate density of fibers immunoreactive for corticotropin-releasing hormone (CRH). A retrograde tracer injection into Pa combined with immunohistochemistry demonstrated that ORX and 5-HT axons originate from neurons in the hypothalamus and midbrain. Pa-projecting neurons were localized in the same nuclei of the hypothalamus, amygdala, and midbrain as CRH neurons, although no double labeling was found. The connections of Pa and its innervation by 5-HT, ORX, and CRH suggest that it may relay stress signals between the midbrain and hypothalamus with the accumbens nucleus, basal amygdala, and subgenual cortex as part of a circuit that manages stress and possibly stress-related psychopathologies. J. Comp. Neurol. 512:825–848, 2009. © 2008 Wiley-Liss, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61435/1/21934_ftp.pd

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

A recent publication provides the network graph for a neocortical microcircuit comprising 8 million connections between 31,000 neurons (H. Markram, et al., Reconstruction and simulation of neocortical microcircuitry, Cell, 163 (2015) no. 2, 456-492). Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of such a biological network, we explored whether methods from algebraic topology could provide a new perspective on its structural and functional organization. Structural topological analysis revealed that directed graphs representing connectivity among neurons in the microcircuit deviated significantly from different varieties of randomized graph. In particular, the directed graphs contained in the order of $10^7$ simplices {\DH} groups of neurons with all-to-all directed connectivity. Some of these simplices contained up to 8 neurons, making them the most extreme neuronal clustering motif ever reported. Functional topological analysis of simulated neuronal activity in the microcircuit revealed novel spatio-temporal metrics that provide an effective classification of functional responses to qualitatively different stimuli. This study represents the first algebraic topological analysis of structural connectomics and connectomics-based spatio-temporal activity in a biologically realistic neural microcircuit. The methods used in the study show promise for more general applications in network science