We consider the simulation of electromagnetic scattering by single and multiple isotropic homogeneous dielectric particles using boundary integral equations. Galerkin discretizations of the classical Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) boundary integral equation formulation provide accurate solutions for complex particle geometries, but are well-known to lead to ill-conditioned linear systems. In this paper we carry out an experimental investigation into the performance of Calderón preconditioning techniques for single and multiple absorbing obstacles, which involve a squaring of the PMCHWT operator to produce a well-conditioned second-kind formulation. For single-particle scattering configurations we find that Calderón preconditioning is actually often outperformed by simple "mass-matrix" preconditioning, i.e. working with the strong form of the discretized PMCHWT operator. In the case of scattering by multiple particles we find that a significant saving in computational cost can be obtained by performing block-diagonal Calderón preconditioning in which only the self-interaction blocks are preconditioned. Using the boundary element software library Bempp (www.bempp.com) the numerical performance of the different methods is compared for a range of wavenumbers, particle geometries and complex refractive indices relevant to the scattering of light by atmospheric ice crystals.

No abstract

We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve significant reductions in computational cost by (i) using "reduced" Calderón preconditioners obtained by discarding constituent boundary integral operators that are not essential for regularisation, and (ii) adopting a "bi-parametric" approach in which we use a lower quality (cheaper) H-matrix assembly routine for the preconditioner than for the original operator, including a novel approach of discarding far-field interactions in the preconditioner. Using the boundary element software Bempp (www.bempp.com), we compare the performance of different combinations of these techniques in the context of scattering by multiple dielectric particles. Applying our accelerated implementation to 3D electromagnetic scattering by an aggregate consisting of 8 monomer ice crystals of overall diameter 1cm at 664GHz leads to a 99% reduction in memory cost and at least a 75% reduction in total computation time compared to a non-accelerated implementation.

We study the evolution of flow and temperature of a fluid moving upstream in a long, thin vertical pipe when a boiler element is involved. The main goal of this work is to understand how the size and position of the boiler will affect the flow and temperature in the pipe over time, as current literature considers cases where the heater or boiler covers the whole length of the pipe, or when already boiling fluid enters a pipe without a boiler. Therefore, we shall allow for a boiling element which covers only a fraction of the pipe when devising out mathematical model. The boiling process results in a transition to different multiphase flow regimes, and we therefore consider a two-phase flow model. From this model, we obtained a simplified one-dimensional model, since we are concerned with a long, thin pipe, under reasonable assumptions and reductions which still preserve the desired physics. We performed a stability analysis for the boiling boundary denoting the phase change in this model. We then obtained numerical simulations for the steady and transient solutions. The numerical results suggest that both the size and position of the boiler strongly affect the flow regime. In particular, depending on the size of the boiler, transition to other phases might not always occur, and depending on its position along the pipe, the fluid coming out at the top of the pipe might not have the desired thermal profile. As such, one may tailor the position and size of the boiler element in order to obtain a useful thermal profile for particular applications. Such results are of possible relevance in industrial applications where heating or boiling of fluid is required.