CoCalc Shared FilesPM_2_5 / DG_Maxwell_reports / thesis / aman / project_brief / project_brief.tex
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24\begin{document}
25
26\title{Determining size and geometry of the particles from the
27polarisation change of the light scattered from the particles}
28
29\author{Aman Abhishek Tiwari}
30\author{Dr. Pankaj Jain(Supervisor)}
31\affiliation{Indian Institute of Technology Kanpur, Department of Physics}
32
33\date{\today}
34
35\begin{abstract}
36An important problem in atmospheric physics is to characterize the
37ambient aerosol distribution. While a majority of current laser-based
38detectors can measure the size spectrum of the scattering particles,
39they do not give information about the geometry of the scatterers. We
40aim to compute the effect of the scatterers on the polarization of the
41incoming radiation and to use the measured radiation to infer the size
42as well as the geometry. In order to do so, we will write a code to
43solve Maxwell's equations for arbitrary geometries using the
44Discontinuous Galerkin method and then use this code to explore the
45effect of scatterer geometry on the incoming radiation.
46\end{abstract}
47
48\maketitle
49
50\section{Introduction}
51Current optical aerosol counters used to characterize the particulate
52matter suspended in air are only capable of measuring the size spectrum
53of the particles, but they assume that the particles are spherical in
54shape.
56the particles as an ellipsoid and calculate the size spectrum $N(r)$ where
57$r$ defines the size of the particles, eccentricity spectrum  $N(e)$ and
58where $e$ is the eccentricity of the particles.
59
60\subsection{Experimental Setup}
61A typical aerosol counter setup has a LASER light source, a sample
62chamber in which air sample to be investigated is held and an array of
63photosensitive detectors which detects the scattered light, it will
64provide us our scattering data. Figure.\ref{fig:aerosol_counter} shows a
65schematic of a typical aerosol counter.
66
67\begin{figure}[h!]
68\centering
69\includegraphics[scale=1.0]{aerosol_counter.png}
70\caption{\label{fig:aerosol_counter}Schematic of a typical aerosol
71                counter}
72\end{figure}
73
74\pagebreak
75
76\section{Simulation}
77
78\subsubsection{Simulation Domain}
79We find the scattering solution for an arbitrarily shaped
80particle inside a rectangular domain by solving the Maxwell's equations.
81A schematic of the domain is shown in the figure
82\ref{fig:domainOfInterest}.
83
84\begin{figure}[h!]
85\centering
86\includegraphics[scale=0.6]{domain_of_interest_2.png}
87\caption{\label{fig:domainOfInterest}Figure showing an arbitrarily shaped
88particle inside a rectangular domain in which we will solve Maxwell's equations.}
89\end{figure}
90
91\subsubsection{Maxwell's Equations}
92
93The Maxwell's Equations are given by(ref \cite{book:electrodynamics_DJ}):
94
95\begin{align} \label{eq:Maxwell_eq}
96    \nabla \cdot  \vec{D}  & = \rho \\
97    \nabla \cdot  \vec{B}  & = 0 \\
98    \nabla \times \vec{E} & = -\frac{\partial \vec{B}}{\partial t} \\
99    \nabla \times \vec{H} & = \vec{J_f}
100                            + \frac{\partial \vec{D}}{\partial t}
101\end{align}
102
103here,
104
105$\vec{E}$ is the electric field.
106
107$\vec{B}$ is the Magnetic field.
108
109$\vec{H}$ is the magnetic field strength.
110
111$\vec{D}$ is the electric displacement field.
112
113$\vec{J_f}$ is the free current density.
114
115
116\subsubsection{Tools and Methods to be used for simulation}
117To solve Maxwell's equations for the domain shown in the figure
118\ref{fig:domainOfInterest} we are divide the domain into second order
119quadrangular elements. We will then use the Discontinuous Galerkin method
120to solve Maxwell's equations over the domain.
121Figure \ref{fig:meshingOverDomain} shows an example mesh made
122over the domain shown in figure \ref{fig:domainOfInterest}.
123Our mesh for solving Maxwell's equations will similar to this.
124
125\begin{figure}[h!]
126\centering
127\includegraphics[scale=0.6]{meshingOverDomain.png}
128\caption{\label{fig:meshingOverDomain}$2^{nd}$ order Quadrangular meshing
129done over the domain shown in the figure \ref{fig:domainOfInterest}.}
130\end{figure}
131
132
133\subsection{Code Verification}
134To test our code for solving Maxwell's equations in the
135domain shown in figure \ref{fig:domainOfInterest}, we plan to compare the
136numerical scattering solution against analytic as well
137as semi-analytic solutions.
138
139We also plan to test the code by solving the scattering solution for a
140homogeneous sphere. The analytic scattering solutions for scattering by
141a homogeneous sphere is given by the Mie scattering solutions
142\cite{wiki:mie_scattering}. We will compare our numerical solutions
143against the Mie scatt    ering solutions.
144
145\subsection{Extracting particle characteristics}
146The domain of our final version of our simulation will be a rectangular
147domain similar to the one shown in figure \ref{fig:domainOfInterest}
148but with much more number of particles inside it.
149
150We aim to find the relationships $N(r)$ vs $r$ and $N(e)$ vs $e$ from the
151simulation for a given input scattering data. We plan to do this by
152iteratively varying the total number of particles $N_0$, and $r$, $e$, and
153$\epsilon$(dielectric constant) for each of the particle inside the domain
154until we get a scattering solution same as the input scattering
155data.
156
157\section{Conclusion}
158Through this project we aim to devise an algorithm to extract more
159information from the scattering data than is commonly available.
160
161\medskip
162
163\bibliographystyle{unsrt}
164\bibliography{references.bib}
165
166% \begin{thebibliography}{5}
167
168% \bibitem{WHO_aq_guide}Ambient (outdoor) air quality and health, WHO
169% \url{<http://www.who.int/mediacentre/factsheets/fs313/en/>}
170
171% \bibitem{mie_scattering}Mie scattering, Wikipedia
172% \url{<https://en.wikipedia.org/wiki/Mie_scattering>}
173
174% \end{thebibliography}
175
176\end{document}
177