project_brief.tex

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\begin{document}

\title{Determining size and geometry of the particles from the
polarisation change of the light scattered from the particles}

\author{Aman Abhishek Tiwari}
\author{Dr. Pankaj Jain(Supervisor)}
\affiliation{Indian Institute of Technology Kanpur, Department of Physics}

\date{\today}

\begin{abstract}
An important problem in atmospheric physics is to characterize the ambient aerosol
distribution. While a majority of current laser-based detectors can measure the size
spectrum of the scattering particles, they do not give information about the geometry
of the scatterers. We aim to compute the effect of the scatterers on the polarization
of the incoming radiation and to use the measured radiation to infer the size as well
as the geometry. In order to do so, we will write a code to solve Maxwell’s equations
for arbitrary geometries using the Discontinuous Galerkin method and then use this
code to explore the effect of scatterer geometry on the incoming radiation. In this
article we try to find the time evolution of the $1D$ and $2D$ wave equations using
the discontinuous Galerkin method.\end{abstract}

\maketitle

\section{Introduction}
Light based aerosol counters are used to determine the size distribution of
particles in the air. An optical aerosol counter \cite{wiki:particle_counter}
determines the size distribution
by analysing the light scattered by the particles. It works by illuminating the
sample with a LASER beam, which gets scattered by the particles. A detector is
used to detect the scattered radiation. By analysing the scattered radiation, it
determines the size distribution of the particle.

The problem with these aerosol counters is that they assume spherically shaped
particles. They also discard information about the polarisation changes in the
scattered light which can be helpful in finding more information about the shape
and size of the particles. In this research, we assume ellipsoidal shaped particles
and use the polarisation change of the scattered radiation to determine the
parameters describing the ellipsoid.

To find the shape and size distribution of the particles from the scattered
radiation, we plan to numerically calculate the scattering solution for particles
of different shape and size parameters present in the medium. By doing this
iteratively for different number of particles, with varying shape and size
parameters, we plan to get the given scattering solution.

A Maxwell's equation solver needs to be developed to determine the scattering
solution. To find the scattering solution we are developing a Maxwell's equation
solver using the discontinuous Galerkin method. In this article we try to find the
solution of the $1D$ and $2D$ wave equation using the discontinuous Galerkin method.
Developing these solvers are the primary steps towards the development of the full
Maxwell's equation solver.

\subsection{Earlier Work}
This project is a continuation of the project I took in the last semester. By the
end of the last semester we managed to develop an Advection equation solver for
1D domain. Prototype code for 2D Advection solver was also written and tested.

\pagebreak

\section{Objectives}
\begin{itemize}
\item Modify the Advection equation solver to solve the Maxwell's equations with
reflective boundary conditions for both 1D and 2D case.
\item Enhance the 2D Advection equation solver to work for Arbitrary shaped meshes.
\item Parallelize the Advection equation solver.
\item Obtain the scattering solutions numerically using the Maxwell's equations
solver and compare it with the analytically calculated scattering solution.
\end{itemize}

\section{Methods}
We use the Nodal Discontinuous Galerkin method to solve the Advection equations.
We divide our domain of interest in a mesh. We use $2^{nd}$ order quadrangular mesh
for solving the Advection equation in 2D.

\section{Plan}
I have planned to finish the listed tasks by the given dates.
\begin{enumerate}
\item {\bf January 30, 2018:} Get numerical solutions for Maxwell's equations being
solved inside a metallic cavity (using both 1D and 2D solver).

\item {\bf February 10, 2018:} Modify the 2D Advection solver to work for arbitrary
shaped mesh made of $\mathrm{2}^{\mathrm{nd}}$ order quadranular elements.

\item {\bf February 18:} Code cleanup and refactoring.

\item {\bf March 1, 2018:} Parallizing the 1D and 2D Advection equation solver.

\item {\bf Before End-semester:} Modify the code to obtain the scattering solution
for particles of arbitrary geometry and test the scattering solution against the
analytically obtained solution.
\end{enumerate}


\section{Comments}
One of the hurdles is that the solver is very slow even when running on GPU.
The code has to be parallelized so that numerical solution can be obtained to
increase the simulation speed.

\end{document}