Ghost nodes - Revision history

Zigavaupotic at 21:05, 5 November 2023

2023-11-05T21:05:00Z

← Older revision		Revision as of 21:05, 5 November 2023
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	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

	See the [[Ghost nodes (theory)]] page for what ghost nodes and how they are used. We will use them in this example to reliably ~~solver~~ a 3D mixed Dirichlet and Neumann problem on an irregular domain.		See the [[Ghost nodes (theory)]] page for what ghost nodes and how they are used. We will use them in this example to reliably solve a 3D mixed Dirichlet and Neumann problem on an irregular domain.

	We will solve the problem		We will solve the problem

2019-09-09T12:33:14Z

← Older revision		Revision as of 12:33, 9 September 2019
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	The code below constructs ghost nodes and remembers the corresponding ghost node for each boundary node.		The code below constructs ghost nodes and remembers the corresponding ghost node for each boundary node.
	<syntaxhighlight lang="cpp">		<syntaxhighlight lang="cpp">
	Range<int> gh = domain.addGhostNodes(dx, neu);		Range<int> gh = domain.addGhostNodes(dx, 0, neu);
	</syntaxhighlight>		</syntaxhighlight>
	It is equivalent to:		It is equivalent to:

2019-09-09T12:31:48Z

← Older revision		Revision as of 12:31, 9 September 2019
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	The code below constructs ghost nodes and remembers the corresponding ghost node for each boundary node.		The code below constructs ghost nodes and remembers the corresponding ghost node for each boundary node.
			<syntaxhighlight lang="cpp">
			Range<int> gh = domain.addGhostNodes(dx, neu);
			</syntaxhighlight>
			It is equivalent to:
	<syntaxhighlight lang="cpp">		<syntaxhighlight lang="cpp">
	Range<int> gh(domain.size(), -1); // Maps boundary nodes to their corresponding ghost nodes.		Range<int> gh(domain.size(), -1); // Maps boundary nodes to their corresponding ghost nodes.
	for (int i : neu) { // Add ghost nodes for each Neumann boundary node		for (int i : neu) { // Add ghost nodes for each Neumann boundary node
	gh[i] = domain.~~addInternalNode~~(domain.pos(i) + dx*domain.normal(i), 2);		gh[i] = domain.addNode(domain.pos(i) + dx*domain.normal(i), 0); // add node with type 0
	}		}
	</syntaxhighlight>		</syntaxhighlight>

2019-05-20T11:10:24Z

← Older revision		Revision as of 11:10, 20 May 2019
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	[[File:solution_ghost_example.png\|600px]]		[[File:solution_ghost_example.png\|600px]]

			The code is available as [https://gitlab.com/e62Lab/medusa/blob/dev/examples/ghost_nodes/poisson_mixed_3D_ghost.cpp poisson_mixed_3D_ghost.cpp] in the Medusa examples.

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

2019-05-18T09:38:50Z

← Older revision		Revision as of 09:38, 18 May 2019
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	}		}
	</syntaxhighlight>		</syntaxhighlight>

	~~The matrix of the system can be seen below~~

	~~[[File:matrix-ghost-labelled.png\|400px]]~~

	The system is then solved as usual. The solution (without the ghost nodes) is shown below:		The system is then solved as usual. The solution (without the ghost nodes) is shown below:

2019-05-18T09:37:55Z

← Older revision		Revision as of 09:37, 18 May 2019
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	The matrix of the system can be seen below		The matrix of the system can be seen below

	[[File:matrix-ghost-labelled.png\|~~600px~~]]		[[File:matrix-ghost-labelled.png\|400px]]

	The system is then solved as usual. The solution (without the ghost nodes) is shown below:		The system is then solved as usual. The solution (without the ghost nodes) is shown below:

2019-05-18T09:37:38Z

← Older revision		Revision as of 09:37, 18 May 2019
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	}		}
	</syntaxhighlight>		</syntaxhighlight>

			The matrix of the system can be seen below

			[[File:matrix-ghost-labelled.png\|600px]]

	The system is then solved as usual. The solution (without the ghost nodes) is shown below:		The system is then solved as usual. The solution (without the ghost nodes) is shown below:

2019-05-18T09:01:00Z

← Older revision		Revision as of 09:01, 18 May 2019
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	</syntaxhighlight>		</syntaxhighlight>

	When assembling the system,		When assembling the system, we write two equations for each Neumann node. One is the Neumann BC and is written in the row corresponding to the boundary node.
			The other is the fact that the PDE also holds at the boundary node, but we write that equation in the row, corresponding to the ghost node.

	<syntaxhighlight lang="cpp">		<syntaxhighlight lang="cpp">
Line 33:		Line 34:
	</syntaxhighlight>		</syntaxhighlight>

			The system is then solved as usual. The solution (without the ghost nodes) is shown below:

	The solution is shown below:

	[[File:solution_ghost_example.png\|600px]]		[[File:solution_ghost_example.png\|600px]]

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

2019-05-18T08:58:41Z

@@ Line 9: / Line 9: @@
 where $\Omega = B(\boldsymbol{0}, 1) - B(\boldsymbol{1}, 1.5)$, $\partial \Omega_{+}$ is the part of the boundary with the nonnegative $x$ coordinate and $\partial \Omega_{-}$ the part of the boundary with negative $x$ coordinate.
 <syntaxhighlight lang="cpp">
-int main () {
+Range<int> gh(domain.size(), -1);  // Maps boundary nodes to their corresponding ghost nodes.
+}
 </syntaxhighlight>

2019-05-18T08:55:49Z

← Older revision		Revision as of 08:55, 18 May 2019
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	where $\Omega = B(\boldsymbol{0}, 1) - B(\boldsymbol{1}, 1.5)$, $\partial \Omega_{+}$ is the part of the boundary with the nonnegative $x$ coordinate and $\partial \Omega_{-}$ the part of the boundary with negative $x$ coordinate.		where $\Omega = B(\boldsymbol{0}, 1) - B(\boldsymbol{1}, 1.5)$, $\partial \Omega_{+}$ is the part of the boundary with the nonnegative $x$ coordinate and $\partial \Omega_{-}$ the part of the boundary with negative $x$ coordinate.

			<syntaxhighlight lang="cpp">
	~~TODO code, explanation, matrix.~~		int main () {
			}
			</syntaxhighlight>