Acceleration of the Surface Test Integral Using Vertex Functions

In recent years, many papers have reported on the efficient and accurate evaluation of the double surface integrals that arise in the Method of Moments. Most have focused on the careful evaluation of the inner integral and assumed that the outer integral is sufficiently smooth to be easily evaluated numerically. More recently, several papers have appeared where the double integral is treated as a whole using the divergence theorem. These papers show promising results, though their implementation may imply changes to the integration paradigm for the associated codes. Here, instead, we investigate a technique that improves the numerical evaluation of the test integral without affecting the treatment of the source integral. From the integrand of the outer integral, we subtract pairs of quasi-static, so-called vertex functions defined on the source triangle. The approach is compared to standard Gauss-triangle schemes to demonstrate its effectiveness

Evaluation of Static Potential Integrals on Triangular Domains

Static potential integrals for constant and linear sources on triangles are derived in a straightforward way. The new representations, as presented, are robust with respect to machine evaluation in important limiting cases. The potential integrals comprise up to six functions, each dependent on the relative position and orientation (with respect to an observation point) of a vertex and edge, respectively, of the source triangle. Gradients of the potentials are derived by differentiation, thus preserving relations between the representations. Each such vertex function reveals any anomalous functional behavior near its associated vertex or edge, which is useful information for devising test integral schemes. Potential plots in the source plane of an equilateral triangle illustrate such behavior, as do similar plots for each vertex function and gradient components near their associated edge and vertex

6-D MoM Reaction Integrals Evaluated via the Divergence Theorem

In this contribution we propose an accurate and efficient numerical evaluation of 6-D reaction integrals that appear in the Method of Moment (MoM) discretization of Volume Integral Equations (VIEs)

Reducing the Dimensionality of 6-D MoM Integrals Applying Twice the Divergence Theorem

In this paper we propose a scheme for evaluating the 6-D interaction integrals appearing in volume integral equation solved with the Method of Moments and tetrahedral elements. We treat as a whole the double volume integral, applying the divergence theorem first on the source domain and then on the test domain. With the proper variable transformation and reordering, the 6-D integrals are expressed as two radial integrals plus four linear integrals over the source and observation domain planes

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller–Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated