Poisson's equation - Revision history

E6WikiAdmin at 17:02, 22 January 2024

2024-01-22T17:02:46Z

E6WikiAdmin at 17:02, 4 December 2022

2022-12-04T17:02:49Z

E6WikiAdmin at 15:27, 10 November 2022

2022-11-10T15:27:47Z

E6WikiAdmin at 19:44, 22 October 2022

2022-10-22T19:44:41Z

E6WikiAdmin at 19:44, 22 October 2022

2022-10-22T19:44:10Z

E6WikiAdmin: /* One vs two domain formulation */

2022-10-22T15:08:54Z

‎One vs two domain formulation

E6WikiAdmin: /* One vs two domain formulation */

2022-10-22T15:02:11Z

‎One vs two domain formulation

E6WikiAdmin: /* One vs two domain formulation */

2022-10-22T14:57:53Z

‎One vs two domain formulation

E6WikiAdmin: /* Higher dimensional space */

2022-10-22T14:56:07Z

‎Higher dimensional space

E6WikiAdmin: /* Convergence rate */

2021-04-29T12:15:25Z

‎Convergence rate

@@ Line 1: / Line 1: @@
 {{Box-round|title= Related papers |
-[https://e6.ijs.si/ParallelAndDistributedSystems/publications/52715011.pdf M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y]
+[https://e6.ijs.si/ParallelAndDistributedSystems/publications/52715011.pdf M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y]]
 [https://e6.ijs.si/ParallelAndDistributedSystems/publications/69777155.pdf J. Slak, G. Kosec; Medusa : A C++ library for solving PDEs using strong form mesh-free methods, ACM transactions on mathematical software, vol. 47, 2021 [DOI: 10.1145/3450966]]

← Older revision		Revision as of 17:02, 4 December 2022
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	}}		}}

		+	__TOC__

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

@@ Line 1: / Line 1: @@
 {{Box-round|title= Related papers |
 [https://e6.ijs.si/ParallelAndDistributedSystems/publications/52715011.pdf M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y]
 }}

← Older revision		Revision as of 19:44, 22 October 2022
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	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].
−
−	~~__FORCETOC__~~

	== Dirichlet boundary conditions ==		== Dirichlet boundary conditions ==

← Older revision		Revision as of 19:44, 22 October 2022
Line 1:		Line 1:
		+	{{Box-round\|title= Related papers \|
		+	[https://e6.ijs.si/ParallelAndDistributedSystems/publications/52715011.pdf M. Jančič, J. Slak, G. Kosec; Monomial augmentation guidelines for RBF-FD from accuracy versus computational time perspective, Journal of scientific computing, vol. 87, 2021 [DOI: 10.1007/s10915-020-01401-y]
		+	}}
		+
		+
	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

← Older revision		Revision as of 15:08, 22 October 2022
Line 293:		Line 293:

	[[File:4Dsolution.png\|800px]]		[[File:4Dsolution.png\|800px]]
−
−	~~== One vs two domain formulation ==~~
−	~~Consider a composite material made of two concentric circles with radii~~
−	~~$R_1$ and $R_2$, $0 < R_1 < R_2$, having thermal diffusivities $\lambda_1$~~
−	~~and $\lambda_2$. The internal circle is heated with a volumetric~~
−	~~heat density $q(r) = q_0 r$. The external circle's boundary is kept at~~
−	~~constant temperature 0.~~
−
−	~~One domain formulation:~~
−	~~\begin{align}~~
−	~~\lambda(r) \triangle u(r) &= q(r), \text{ for } 0 \leq r < R_2, \\~~
−	~~u(R_2) &= 0, \\~~
−	~~q(r) &= \begin{cases} q_0r; & 0 \leq r < R_1 \\ 0; & R_1 \leq r < R_2 \end{cases}, \\~~
−	~~\lambda(r) &= \begin{cases} \lambda_1; & 0 \leq r < R_1 \\ \lambda_2; & R_1 \leq r < R_2 \end{cases}.~~
−	~~\end{align}~~
−
−	~~Two domain formulation:~~
−	~~\begin{align}~~
−	~~\lambda_1 \triangle u_1(r) &= q0 r, \text{ for } 0 \leq r < R_1, \\~~
−	~~u_1(R_2) &= u_2(R_1), \\~~
−	~~\lambda_1\frac{\partial u_1}{\partial \vec{n}} (R_1) &= -\lambda_2 \frac{\partial u_2}{\partial \vec{n}}(R_1), \\~~
−	~~\lambda_2 \triangle u_2(r) &= 0, \text{ for } R_1 \leq r < R_2, \\~~
−	~~u_2(R_2) &= 0, \\~~
−	~~\end{align}~~
−	~~and finally,~~
−	~~\[ u(r) = \begin{cases} u_1(r); & 0 \leq r < R_1 \\ u_2(r); & R_1 \leq r \leq R_2 \end{cases}. \]~~
−
−	~~The solution to both problems is~~
−	~~\begin{equation}~~
−	~~u(r) = \begin{cases}~~
−	~~\frac{q_0}{4 \lambda _1 \lambda _2} \left(\lambda _2 \left(R_1^2-r^2\right)+2 \lambda _1 R_1^2 \log~~
−	~~\left(\frac{R_2}{R_1}\right)\right) & 0\leq~~
−	~~r<R_1 \\~~
−	~~\frac{q_0 R_1^2}{2 \lambda _2} \log \left(\frac{R_2}{r}\right) &~~
−	~~R_1\leq r\leq R_2~~
−	~~\end{cases}.~~
−	~~\end{equation}~~
−
−	~~The solution for three regimes is presented in Figure~\ref{fig:sol}.~~
−	~~\begin{figure}[h]~~
−	~~\centering~~
−	~~\includegraphics[width=0.32\textwidth]{sol15.pdf}~~
−	~~\includegraphics[width=0.32\textwidth]{sol22.pdf}~~
−	~~\includegraphics[width=0.32\textwidth]{sol51.pdf}~~
−	~~\caption{Plot of $u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}~~
−	~~\label{fig:sol}~~
−	~~\end{figure}~~
−
−	~~\section{Numerical results}~~
−	~~We have tested numerical approach of two and one domain formulation on the~~
−	~~three test cases illustrated in figure~\ref{fig:sol}.~~
−	~~The results are be presented in terms of convergence and cross section~~
−	~~plots in figure~\ref{fig:res}.~~
−
−	~~\begin{figure}[h]~~
−	~~\centering~~
−	~~\includegraphics[width=0.49\textwidth]{conv_22.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_22.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{conv_51.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_51.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{conv_15.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_15.pdf}~~
−	~~\caption{Plot of numerically obtained~~
−	~~$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}~~
−	~~\label{fig:res}~~
−	~~\end{figure}~~
−
−	~~And two more cases in figure~\ref{fig:res2}~~
−	~~\begin{figure}[h]~~
−	~~\centering~~
−	~~\includegraphics[width=0.49\textwidth]{conv_21.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_21.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{conv_12.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_12.pdf}~~
−	~~\caption{Plot of numerically obtained~~
−	~~$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}~~
−	~~\label{fig:res2}~~
−	~~\end{figure}~~
−
−	~~\section{Effect of smoothing}~~
−	~~In one domain formulation, grave inaccuracies are caused by discontinuous~~
−	~~$\lambda$, which has no derivative at $r = R_1$. This can be remedied by~~
−	~~smoothing $\lambda$ from $\lambda_1$ to $\lambda_2$ linearly on interval~~
−	~~$R_1-\varepsilon$ to $R_1+\varepsilon$. Choosing $\varepsilon \approx dx$,~~
−	~~we obtain a smoothed solution as seen in figure~\ref{fig:smooth}. The~~
−	~~displayed section is from the solution with the least error.~~
−
−	~~\begin{figure}[h]~~
−	~~\centering~~
−	~~\includegraphics[width=0.49\textwidth]{conv_51_smooth.pdf}~~
−	~~\includegraphics[width=0.49\textwidth]{sec_51_smooth.pdf}~~
−	~~\caption{Plot of numerically obtained $u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}~~
−	~~\label{fig:smooth}~~
−	~~\end{figure}~~
−
−
−	~~\section{Execution time}~~
−	~~Results are presented in fig.~\ref{fig:time}.~~
−
−	~~\begin{figure}[h]~~
−	~~\centering~~
−	~~\includegraphics[width=0.75\textwidth]{time_22.pdf}~~
−	~~\includegraphics[width=0.75\textwidth]{time_51.pdf}~~
−	~~\includegraphics[width=0.75\textwidth]{time_15.pdf}~~
−	~~\caption{Plot of numerically obtained~~
−	~~$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}~~
−	~~\label{fig:time}~~
−	~~\end{figure}~~


	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

@@ Line 295: / Line 295: @@
 == One vs two domain formulation ==
 Consider a composite material made of two concentric circles with radii
 $R_1$ and $R_2$, $0 < R_1 < R_2$, having thermal diffusivities $\lambda_1$
@@ Line 316: / Line 313: @@
    \lambda_1 \triangle u_1(r) &= q0 r,  \text{ for } 0 \leq r < R_1, \\
    u_1(R_2) &= u_2(R_1),   \\
-   \lambda_1\dpar{u_1}{\vec{n}} (R_1) &= -\lambda_2 \dpar{u_2}{\vec{n}}(R_1),   \\
+   \lambda_1\frac{\partial u_1}{\partial \vec{n}} (R_1) &= -\lambda_2 \frac{\partial u_2}{\partial \vec{n}}(R_1),   \\
    \lambda_2 \triangle u_2(r) &= 0,  \text{ for } R_1 \leq r < R_2, \\
    u_2(R_2) &= 0, \\

← Older revision		Revision as of 14:57, 22 October 2022
Line 295:		Line 295:

	== One vs two domain formulation ==		== One vs two domain formulation ==
		+
		+	\section{Test case}
		+
		+	Consider a composite material made of two concentric circles with radii
		+	$R_1$ and $R_2$, $0 < R_1 < R_2$, having thermal diffusivities $\lambda_1$
		+	and $\lambda_2$. The internal circle is heated with a volumetric
		+	heat density $q(r) = q_0 r$. The external circle's boundary is kept at
		+	constant temperature 0.
		+
		+	One domain formulation:
		+	\begin{align}
		+	\lambda(r) \triangle u(r) &= q(r), \text{ for } 0 \leq r < R_2, \\
		+	u(R_2) &= 0, \\
		+	q(r) &= \begin{cases} q_0r; & 0 \leq r < R_1 \\ 0; & R_1 \leq r < R_2 \end{cases}, \\
		+	\lambda(r) &= \begin{cases} \lambda_1; & 0 \leq r < R_1 \\ \lambda_2; & R_1 \leq r < R_2 \end{cases}.
		+	\end{align}
		+
		+	Two domain formulation:
		+	\begin{align}
		+	\lambda_1 \triangle u_1(r) &= q0 r, \text{ for } 0 \leq r < R_1, \\
		+	u_1(R_2) &= u_2(R_1), \\
		+	\lambda_1\dpar{u_1}{\vec{n}} (R_1) &= -\lambda_2 \dpar{u_2}{\vec{n}}(R_1), \\
		+	\lambda_2 \triangle u_2(r) &= 0, \text{ for } R_1 \leq r < R_2, \\
		+	u_2(R_2) &= 0, \\
		+	\end{align}
		+	and finally,
		+	\[ u(r) = \begin{cases} u_1(r); & 0 \leq r < R_1 \\ u_2(r); & R_1 \leq r \leq R_2 \end{cases}. \]
		+
		+	The solution to both problems is
		+	\begin{equation}
		+	u(r) = \begin{cases}
		+	\frac{q_0}{4 \lambda _1 \lambda _2} \left(\lambda _2 \left(R_1^2-r^2\right)+2 \lambda _1 R_1^2 \log
		+	\left(\frac{R_2}{R_1}\right)\right) & 0\leq
		+	r<R_1 \\
		+	\frac{q_0 R_1^2}{2 \lambda _2} \log \left(\frac{R_2}{r}\right) &
		+	R_1\leq r\leq R_2
		+	\end{cases}.
		+	\end{equation}
		+
		+	The solution for three regimes is presented in Figure~\ref{fig:sol}.
		+	\begin{figure}[h]
		+	\centering
		+	\includegraphics[width=0.32\textwidth]{sol15.pdf}
		+	\includegraphics[width=0.32\textwidth]{sol22.pdf}
		+	\includegraphics[width=0.32\textwidth]{sol51.pdf}
		+	\caption{Plot of $u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}
		+	\label{fig:sol}
		+	\end{figure}
		+
		+	\section{Numerical results}
		+	We have tested numerical approach of two and one domain formulation on the
		+	three test cases illustrated in figure~\ref{fig:sol}.
		+	The results are be presented in terms of convergence and cross section
		+	plots in figure~\ref{fig:res}.
		+
		+	\begin{figure}[h]
		+	\centering
		+	\includegraphics[width=0.49\textwidth]{conv_22.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_22.pdf}
		+	\includegraphics[width=0.49\textwidth]{conv_51.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_51.pdf}
		+	\includegraphics[width=0.49\textwidth]{conv_15.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_15.pdf}
		+	\caption{Plot of numerically obtained
		+	$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}
		+	\label{fig:res}
		+	\end{figure}
		+
		+	And two more cases in figure~\ref{fig:res2}
		+	\begin{figure}[h]
		+	\centering
		+	\includegraphics[width=0.49\textwidth]{conv_21.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_21.pdf}
		+	\includegraphics[width=0.49\textwidth]{conv_12.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_12.pdf}
		+	\caption{Plot of numerically obtained
		+	$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}
		+	\label{fig:res2}
		+	\end{figure}
		+
		+	\section{Effect of smoothing}
		+	In one domain formulation, grave inaccuracies are caused by discontinuous
		+	$\lambda$, which has no derivative at $r = R_1$. This can be remedied by
		+	smoothing $\lambda$ from $\lambda_1$ to $\lambda_2$ linearly on interval
		+	$R_1-\varepsilon$ to $R_1+\varepsilon$. Choosing $\varepsilon \approx dx$,
		+	we obtain a smoothed solution as seen in figure~\ref{fig:smooth}. The
		+	displayed section is from the solution with the least error.
		+
		+	\begin{figure}[h]
		+	\centering
		+	\includegraphics[width=0.49\textwidth]{conv_51_smooth.pdf}
		+	\includegraphics[width=0.49\textwidth]{sec_51_smooth.pdf}
		+	\caption{Plot of numerically obtained $u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}
		+	\label{fig:smooth}
		+	\end{figure}
		+
		+
		+	\section{Execution time}
		+	Results are presented in fig.~\ref{fig:time}.
		+
		+	\begin{figure}[h]
		+	\centering
		+	\includegraphics[width=0.75\textwidth]{time_22.pdf}
		+	\includegraphics[width=0.75\textwidth]{time_51.pdf}
		+	\includegraphics[width=0.75\textwidth]{time_15.pdf}
		+	\caption{Plot of numerically obtained
		+	$u$ for $q_0 = 10$, $R_1 = 1$ and $R_2 = 2$.}
		+	\label{fig:time}
		+	\end{figure}


	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

← Older revision		Revision as of 14:56, 22 October 2022
Line 293:		Line 293:

	[[File:4Dsolution.png\|800px]]		[[File:4Dsolution.png\|800px]]
		+
		+	== One vs two domain formulation ==
		+

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

@@ Line 215: / Line 215: @@
 [[File:triceratops_example.png|800px]]
-== Convergence rate ==
+== Convergence rate - including high order solution ==
 Assume a $d$-dimensional Poisson problem with mixed boundary conditions.