Novel boundary element based solvers for light scattering from complex ice crystals

Lead Research Organisation: University College London
Department Name: Mathematics

Abstract

The Met Office operational global model and its its configurations over and under predict the cloudy short-wave radiative effect by ~30 Wm-2 to - 40 Wm-2 when compared against the latest observations. As part of ongoing significant investments by the Met Office and space agencies in understanding and reducing these errors an accurate understanding of the scattering and absorption properties of ice crystals is required. In the limit of very large particles, at solar wavelengths, ray tracing has been successfully used for simulating light-scattering from ice crystals. However, in the practically very important mid-size and small ice range, ray tracing is not applicable and direct simulation methods are required that solve the underlying Maxwell equation. Here, standard T-Matrix techniques are popular tools in the athmospheric research community, which are restricted to rather simple homogeneous shapes.

An alternative are boundary element methods (BEM), a type of direct solution techniques that do not suffer from the geometric restrictions of standard T-Matrix approaches. However, due to their significant higher implementational complexity these have been so far little used in the climate research community. Since 2011, a novel open-source BEM code (BEM++, www.bempp.org) has been in development at UCL, funded by EPSRC grants EP/I030042/1 and EP/K03829X/1.

BEM++ was previously used for a first feasibility study for the simulation of light-scattering from complex ice crystal configurations that are out of reach of currently publicly available T-Matrix methods (ice crystals with varying internal air cavities and bullet-rosettes). We are planning to build on this very promising initial study to develop a powerful framework for the simulation of light-scattering from ice crystals, including:

Fast High-Frequency solvers for Maxwell. In recent years, fast directional FMM methods for the oscillatory kernels arising in Maxwell BEM have been developed that scale almost optimally with respect to the wavenumber. Together with operator preconditioning for the Maxwell equations these lead to fast solvers across a wide range of frequencies. Implementation of such methods in BEM++ is ongoing and together with the CASE Student we will apply and investigate the scalability of these methods for complex ice crystal configurations at medium to high frequencies. This will bridge the gap between geometric optics and electromagnetism.[Time: 1 year]

Simulation of electromagnetic scattering from ice crystals with internal inhomogeneities. We apply the method to the microwave and submillimetre regions of the spectrum to take advantage of new instrumentation developed by the Met Office. At the larger particle sizes of relevance to these incident frequencies, the particle volume is likely to consist of mixtures of ice, water and air. To simulate random inhomogeneities in ice crystals we will couple finite element methods (FEM) with a BEM for the exterior scattering problem. [Time: 1.5 year]

Complex multiple scattering configurations. We plan to extend the BEM based solution capabilities to arrays of ice crystals with random orientation. In work by Ganesh, Hesthaven and Stamm a reduced basis approach was proposed in which individual scattering computations of electromagnetic particles are used to accelerate the computation of the scattered field of an array of particles. Based on the BEM++ solver capabilities we will build on this approach and apply it to arrays of ice crystals. [Time: 1.5 years]

Publications

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Kleanthous A (2019) Calderón preconditioning of PMCHWT boundary integral equations for scattering by multiple absorbing dielectric particles in Journal of Quantitative Spectroscopy and Radiative Transfer

 
Description Discretisation of boundary integral equation formulations results into linear systems of the form Ax = b, which are then solved numerically via direct or iterative solvers such as GMRES.

In the problem of interest here, electromagnetic scattering by single and multiple absorbing dielectric bodies we use the PMCHWT formulation.
The PMCHWT formulation suffers from ill-conditioning upon discretisation resulting in a large number of iterations required by GMRES to converge. To remedy this, the Calderòn identities are used in the form of a preconditioner (P).

For single-particle problems the preconditioner P is the same as the operator A leading to a system G^{-1}PG^{-1}Ax = G^{-1}PG^{-1}b, where G^{-1} is the mass matrix. In numerical experiments published recently, we found that a simple mass-matrix preconditioning often outperforms the Calderòn preconditioner (in terms of overall matrix-vector multiplications).

For multi-particle problems, one can consider two types of Calderòn preconditioners: a full Calderòn preconditioner where P=A as in the single-particle case, and a block diagonal Calderòn preconditioner where P = diag(A). As in the single-particle case, we can also consider mass-matrix preconditioning for the multi-particle case. In numerical experiments published in the same article as above, we found that the block-diagonal preconditioner results in a reduced number of matrix-vector multiplications as well as a reduced number of GMRES iterations.
Exploitation Route NA
Sectors Environment

URL https://www.sciencedirect.com/science/article/pii/S0022407318306320
 
Description The QJMAM Fund for Applied Mathematics
Amount £720 (GBP)
Organisation Institute of Mathematics and its Applications 
Sector Learned Society
Country United Kingdom
Start 06/2018 
End 06/2018