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Cory Balaton 2023-10-24 21:03:15 +02:00
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18
latex/sections/methods.tex
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@ -33,7 +33,7 @@ First, we consider Newton's second law \eqref{eq:newton_second}, to determine th
m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i} m \ddot{\mathbf{r}} = \sum_{i} \mathbf{F}_{i}
\end{equation} % \end{equation} %
% %
In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force acting the particle. In addition, we introduce the Lorentz force \eqref{eq:lorentz_force}, which describes the force acting on the particle.
\begin{equation}\label{eq:lorentz_force} \begin{equation}\label{eq:lorentz_force}
\mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B}, \mathbf{F} = q \mathbf{E} + q \mathbf{v} \times \mathbf{B},
\end{equation} % \end{equation} %
@ -122,7 +122,7 @@ We will also consider the numerical simulation of multiple particles, confined i
\end{equation} % \end{equation} %
The Coulomb constant $k_{e}$ value can be found in table \ref{tab:constants}. The Coulomb constant $k_{e}$ value can be found in table \ref{tab:constants}.
For multiple particles we have to modify the equations of motion, by adding a term for the force a given particle experience at a given point. When we scale eq. \eqref{eq:electric_field_interaction} by charge an mass, we get a new set of equations of motion For multiple particles we have to modify the equations of motion, by adding a term for the force a given particle experience at a given point. When we scale eq. \eqref{eq:electric_field_interaction} by charge and mass, we get a new set of equations of motion
\begin{align} \begin{align}
\label{eq:coulomb_motion_x} \label{eq:coulomb_motion_x}
\ddot{x}_{i} - \omega_{0,i} \dot{y}_{i} - \frac{1}{2} \omega_{z,i}^{2} x_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{x_{i} - x_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, \\ \ddot{x}_{i} - \omega_{0,i} \dot{y}_{i} - \frac{1}{2} \omega_{z,i}^{2} x_{i} - k_{e} \frac{q_{i}}{m_{i}}\sum_{j \neq i} q_j \frac{x_{i} - x_{j}}{|\mathbf{r}_{i} - \mathbf{r}_{j}|^{3}} &= 0, \\
@ -173,7 +173,7 @@ We first implemented the forward Euler method, using the expression for a couple
\end{algorithm} \end{algorithm}
\end{figure} \end{figure}
We also implemented the 4th order Runge-Kutta (RK4) method, using the expressin given in appendix \ref{sec:algo_rk4}. We define the RK4 algorithm in \ref{algo:rk4}. $\mathbf{F}$ does not take any arguments, however, the total force acting on the particle is calculated using the value of position and velocity within $particles$. We also implemented the 4th order Runge-Kutta (RK4) method, using the expression given in appendix \ref{sec:algo_rk4}. We define the RK4 algorithm in \ref{algo:rk4}. $\mathbf{F}$ does not take any arguments, however, the total force acting on the particle is calculated using the value of position and velocity within $particles$.
\begin{figure} \begin{figure}
\begin{algorithm}[H] \begin{algorithm}[H]
\caption{RK4 method} \caption{RK4 method}
@ -225,12 +225,12 @@ We also implemented the 4th order Runge-Kutta (RK4) method, using the expressin
\State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$ \State $\vb{k}_{\vb{v},4,i} \leftarrow \frac{\vb{F}}{m}$
\State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6} \State $tmp\_p_i.\vb{r} \leftarrow orig\_p_i.\vb{r} + \frac{dt}{6}
\cdot \left( \vb{k}_{\vb{r},1,i} + \vb{k}_{\vb{r},2,i} \cdot \left( \vb{k}_{\vb{r},1,i} + 2 \vb{k}_{\vb{r},2,i}
+ \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$ + 2 \vb{k}_{\vb{r},3,i} + \vb{k}_{\vb{r},4,i} \right)$
\State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6} \State $tmp\_p_i.\vb{v} \leftarrow orig\_p_i.\vb{v} + \frac{dt}{6}
\cdot \left( \vb{k}_{\vb{v},1,i} + \vb{k}_{\vb{v},2,i} \cdot \left( \vb{k}_{\vb{v},1,i} + 2 \vb{k}_{\vb{v},2,i}
+ \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$ + 2 \vb{k}_{\vb{v},3,i} + \vb{k}_{\vb{v},4,i} \right)$
\EndFor \EndFor
@ -253,5 +253,5 @@ The maximum error of a simulation $k$ with a step size $h_{k}$ is given by
where $\mathbf{r}_{i, \text{exact}}$ is the analytical solution, and $\mathbf{r}_{i}$ is the computed result. where $\mathbf{r}_{i, \text{exact}}$ is the analytical solution, and $\mathbf{r}_{i}$ is the computed result.
\subsection{Tools} \subsection{Tools}
The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler Scalasca and score-p instrumentation We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. The numerical methods are implemented in C++, and parallelized using \verb|OpenMP| \cite{openmp:2018}. In addition, we used profiler \verb|Scalasca| and \verb|score-p| instrumentation. We used the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures.
\end{document} \end{document}

16
latex/sections/results.tex
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@ -98,7 +98,7 @@ In figure \ref{fig:two_particles_radial_interaction} we see the movement in radi
Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. % Something Finally, by subjecting the system to a time-dependent field, making the replacement in \ref{eq:pertubation}, we study the fraction of particles left at different amplitudes $f$. We can see how the different amplitudes lead to loss of particles, at different angular frequencies $\omega_{V}$ in \ref{fig:wide_sweep}. % Something
When we study ... closer, in figure \ref{fig:} , we observe ... Certain angular frequencies are more effective in pushing particles out of the Penning trap. When we study the angular frequency $\omega_{V} \in [1.1, 1.7]$ closer, in figure \ref{fig:narrow_sweep}, we observe that there are a few spots where more particles will escape. The most prominent one is where $\omega_V \in [1.1, 1.7]$, and when looking closer to the range, it seems like there's a resonating frequency at around $1.4MHz$ where All the particles will escape no matter the amplitude. When looking at the different angular frequencies with particle interaction like in \ref{fig:narrow_sweep_interactions}, we see that the amount of particles left is roughly the same as when there are no particle interactions, but that it's less predictable.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf} \includegraphics[width=\linewidth]{images/particles_left_wide_sweep.pdf}
@ -106,4 +106,16 @@ When we study ... closer, in figure \ref{fig:} , we observe ... Certain angular
\label{fig:wide_sweep} \label{fig:wide_sweep}
\end{figure} \end{figure}
\end{document} \begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{images/particles_left_narrow_sweep.pdf}
\caption{Exploring different angular frequencies more closely}
\label{fig:narrow_sweep}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{images/particles_left_narrow_sweep_interactions.pdf}
\caption{Exploring particles}
\label{fig:wide_sweep_interactions}
\end{figure}
\end{document}