CoCalc Shared FilesPM_2_5 / DG_Maxwell_reports / thesis / aman / project_brief / project_brief.texOpen in CoCalc with one click!

\documentclass[prb,11pt]{revtex4-1}12% set font encoding for PDFLaTeX or XeLaTeX3\usepackage{ifxetex}4\ifxetex5\usepackage{fontspec}6\else7\usepackage[T1]{fontenc}8\usepackage[utf8]{inputenc}9\usepackage{lmodern}10\fi1112\usepackage{amsmath} % need for subequations13\usepackage{graphicx} % need for figures14\usepackage{verbatim} % useful for program listings15\usepackage{color} % use if color is used in text16\usepackage{subfigure} % use for side-by-side figures17\usepackage{hyperref} % use for hypertext links, including those to external documents and URLs18% \usepackage{url}19\raggedbottom % don't add extra vertical space2021\graphicspath{{./images/}}2223\begin{document}2425\title{Determining size and geometry of the particles from the26polarisation change of the light scattered from the particles}2728\author{Aman Abhishek Tiwari}29\author{Dr. Pankaj Jain(Supervisor)}30\affiliation{Indian Institute of Technology Kanpur, Department of Physics}3132\date{\today}3334\begin{abstract}35An important problem in atmospheric physics is to characterize the36ambient aerosol distribution. While a majority of current laser-based37detectors can measure the size spectrum of the scattering particles,38they do not give information about the geometry of the scatterers. We39aim to compute the effect of the scatterers on the polarization of the40incoming radiation and to use the measured radiation to infer the size41as well as the geometry. In order to do so, we will write a code to42solve Maxwell's equations for arbitrary geometries using the43Discontinuous Galerkin method and then use this code to explore the44effect of scatterer geometry on the incoming radiation.45\end{abstract}4647\maketitle4849\section{Introduction}50Current optical aerosol counters used to characterize the particulate51matter suspended in air are only capable of measuring the size spectrum52of the particles, but they assume that the particles are spherical in53shape.54We aim to recover additional information about the particles by modeling55the particles as an ellipsoid and calculate the size spectrum $N(r)$ where56$r$ defines the size of the particles, eccentricity spectrum $N(e)$ and57where $e$ is the eccentricity of the particles.5859\subsection{Experimental Setup}60A typical aerosol counter setup has a LASER light source, a sample61chamber in which air sample to be investigated is held and an array of62photosensitive detectors which detects the scattered light, it will63provide us our scattering data. Figure.\ref{fig:aerosol_counter} shows a64schematic of a typical aerosol counter.6566\begin{figure}[h!]67\centering68\includegraphics[scale=1.0]{aerosol_counter.png}69\caption{\label{fig:aerosol_counter}Schematic of a typical aerosol70counter}71\end{figure}7273\pagebreak7475\section{Simulation}7677\subsubsection{Simulation Domain}78We find the scattering solution for an arbitrarily shaped79particle inside a rectangular domain by solving the Maxwell's equations.80A schematic of the domain is shown in the figure81\ref{fig:domainOfInterest}.8283\begin{figure}[h!]84\centering85\includegraphics[scale=0.6]{domain_of_interest_2.png}86\caption{\label{fig:domainOfInterest}Figure showing an arbitrarily shaped87particle inside a rectangular domain in which we will solve Maxwell's equations.}88\end{figure}8990\subsubsection{Maxwell's Equations}9192The Maxwell's Equations are given by(ref \cite{book:electrodynamics_DJ}):9394\begin{align} \label{eq:Maxwell_eq}95\nabla \cdot \vec{D} & = \rho \\96\nabla \cdot \vec{B} & = 0 \\97\nabla \times \vec{E} & = -\frac{\partial \vec{B}}{\partial t} \\98\nabla \times \vec{H} & = \vec{J_f}99+ \frac{\partial \vec{D}}{\partial t}100\end{align}101102here,103104$\vec{E}$ is the electric field.105106$\vec{B}$ is the Magnetic field.107108$\vec{H}$ is the magnetic field strength.109110$\vec{D}$ is the electric displacement field.111112$\vec{J_f}$ is the free current density.113114115\subsubsection{Tools and Methods to be used for simulation}116To solve Maxwell's equations for the domain shown in the figure117\ref{fig:domainOfInterest} we are divide the domain into second order118quadrangular elements. We will then use the Discontinuous Galerkin method119to solve Maxwell's equations over the domain.120Figure \ref{fig:meshingOverDomain} shows an example mesh made121over the domain shown in figure \ref{fig:domainOfInterest}.122Our mesh for solving Maxwell's equations will similar to this.123124\begin{figure}[h!]125\centering126\includegraphics[scale=0.6]{meshingOverDomain.png}127\caption{\label{fig:meshingOverDomain}$2^{nd}$ order Quadrangular meshing128done over the domain shown in the figure \ref{fig:domainOfInterest}.}129\end{figure}130131132\subsection{Code Verification}133To test our code for solving Maxwell's equations in the134domain shown in figure \ref{fig:domainOfInterest}, we plan to compare the135numerical scattering solution against analytic as well136as semi-analytic solutions.137138We also plan to test the code by solving the scattering solution for a139homogeneous sphere. The analytic scattering solutions for scattering by140a homogeneous sphere is given by the Mie scattering solutions141\cite{wiki:mie_scattering}. We will compare our numerical solutions142against the Mie scatt ering solutions.143144\subsection{Extracting particle characteristics}145The domain of our final version of our simulation will be a rectangular146domain similar to the one shown in figure \ref{fig:domainOfInterest}147but with much more number of particles inside it.148149We aim to find the relationships $N(r)$ vs $r$ and $N(e)$ vs $e$ from the150simulation for a given input scattering data. We plan to do this by151iteratively varying the total number of particles $N_0$, and $r$, $e$, and152$\epsilon$(dielectric constant) for each of the particle inside the domain153until we get a scattering solution same as the input scattering154data.155156\section{Conclusion}157Through this project we aim to devise an algorithm to extract more158information from the scattering data than is commonly available.159160\medskip161162\bibliographystyle{unsrt}163\bibliography{references.bib}164165% \begin{thebibliography}{5}166167% \bibitem{WHO_aq_guide}Ambient (outdoor) air quality and health, WHO168% \url{<http://www.who.int/mediacentre/factsheets/fs313/en/>}169170% \bibitem{mie_scattering}Mie scattering, Wikipedia171% \url{<https://en.wikipedia.org/wiki/Mie_scattering>}172173% \end{thebibliography}174175\end{document}176177