Cahn-Hilliard equation - Revision history

MihaR at 18:08, 12 October 2020

2020-10-12T18:08:24Z

← Older revision		Revision as of 18:08, 12 October 2020
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	}		}
	</syntaxhighlight>		</syntaxhighlight>

			<figure id="fig:timeEvolutionStatic">
			[[File:cahnHilliardTimeEvolution.png\|thumb\|upright=1.5\|<caption>Time evolution of the concentration. Initially mixed fluid separates into domains that grow with time.
			</caption>]]
			</figure>

	=== Mixed solution ===		=== Mixed solution ===

MihaR at 14:10, 11 February 2020

2020-02-11T14:10:53Z

← Older revision		Revision as of 14:10, 11 February 2020
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	</syntaxhighlight>		</syntaxhighlight>

			The complete example is avalible on [https://gitlab.com/e62Lab/medusa/blob/dev/examples/cahnHilliard_equation git].

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

MihaR at 13:38, 11 February 2020

2020-02-11T13:38:02Z

Show changes

MihaR at 00:51, 11 February 2020

2020-02-11T00:51:42Z

← Older revision		Revision as of 00:51, 11 February 2020
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	== Periodic boundary conditions ==		== Periodic boundary conditions ==
	The simplest method of implementing a periodic boundary is adding a band of periodic nodes that mirror values from nodes that lie close to the opposite boundary. We have to create periodic nodes based on locations of interior nodes and construct a mapping that will allow us to update their values based on their interior counterparts. Width of the periodic band should be chosen with support size in mind. Periodic band should be wide enough to include all nodes required to calculate interior values.

	[[File:~~periodicBands~~.png\|~~center~~\|~~300px]]~~		<figure id="fig:periodicBands">
	~~TODO:~~ node ~~visualization~~ with same ~~colors for~~ periodic nodes, support in ~~corner visualization~~		[[File:PeriodicBands.png\|thumb\|upright=2\|<caption>Periodic band display.

			Left subplot shows an example of filled domain with $border = 10$ and $dx = 0.2$. Interior nodes and their periodic counterparts share colors to illustrate periodicity.

			Right subplot shows the neighbourhood of bottom left corner node $(-border, -border)$, with its support nodes highlighted in red. This example uses same fill parameters as the left figure and a support size of 100.
			</caption>]]
			</figure>

			The simplest method of implementing a periodic boundary is adding a band of periodic nodes that mirror values from nodes that lie close to the opposite boundary. We have to create periodic nodes based on locations of interior nodes and construct a mapping that will allow us to update their values based on their interior counterparts. Width of the periodic band should be chosen with support size in mind. Periodic band should be wide enough to include all nodes required to calculate interior values. An example that can help us determine the required width of the periodic band is shown on the right subplot of <xr id="fig:periodicBands"/> and shows that we need a periodic band width of $6 dx$ for a support size of 100.

	We start by creating a square domain centered on 0 and uniformly filling it with nodes. $border$ parameter determines the square size while $dx$, the typical distance between nodes, determines the node density.		We start by creating a square domain centered on 0 and uniformly filling it with nodes. $border$ parameter determines the square size while $dx$, the typical distance between nodes, determines the node density.

MihaR at 00:17, 11 February 2020

2020-02-11T00:17:48Z

← Older revision		Revision as of 00:17, 11 February 2020
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	The simplest method of implementing a periodic boundary is adding a band of periodic nodes that mirror values from nodes that lie close to the opposite boundary. We have to create periodic nodes based on locations of interior nodes and construct a mapping that will allow us to update their values based on their interior counterparts. Width of the periodic band should be chosen with support size in mind. Periodic band should be wide enough to include all nodes required to calculate interior values.		The simplest method of implementing a periodic boundary is adding a band of periodic nodes that mirror values from nodes that lie close to the opposite boundary. We have to create periodic nodes based on locations of interior nodes and construct a mapping that will allow us to update their values based on their interior counterparts. Width of the periodic band should be chosen with support size in mind. Periodic band should be wide enough to include all nodes required to calculate interior values.

			[[File:periodicBands.png\|center\|300px]]
	TODO: node visualization with same colors for periodic nodes, support in corner visualization		TODO: node visualization with same colors for periodic nodes, support in corner visualization

MihaR at 15:40, 7 February 2020

2020-02-07T15:40:23Z

← Older revision		Revision as of 15:40, 7 February 2020
Line 14:		Line 14:

	== Periodic boundary conditions ==		== Periodic boundary conditions ==
			The simplest method of implementing a periodic boundary is adding a band of periodic nodes that mirror values from nodes that lie close to the opposite boundary. We have to create periodic nodes based on locations of interior nodes and construct a mapping that will allow us to update their values based on their interior counterparts. Width of the periodic band should be chosen with support size in mind. Periodic band should be wide enough to include all nodes required to calculate interior values.

			TODO: node visualization with same colors for periodic nodes, support in corner visualization

			We start by creating a square domain centered on 0 and uniformly filling it with nodes. $border$ parameter determines the square size while $dx$, the typical distance between nodes, determines the node density.

	<syntaxhighlight lang="cpp">		<syntaxhighlight lang="cpp">
	~~/// Create the square domain and fill it with nodes.~~
	~~typedef mm::Vec<double, dim> vec_t;~~
	mm::BoxShape<vec_t> box(-border, border);		mm::BoxShape<vec_t> box(-border, border);
	mm::DomainDiscretization<vec_t> domain = box.discretizeBoundaryWithStep(dx);		mm::DomainDiscretization<vec_t> domain = box.discretizeBoundaryWithStep(dx);

MihaR at 13:52, 7 February 2020

2020-02-07T13:52:21Z

@@ Line 10: / Line 10: @@
 \]
 where $D$ is the diffusion constant and $\sqrt{\gamma}$ is the characteristic transition length between domains.
 Go back to [[Medusa#Examples|Examples]].

MihaR at 02:09, 7 February 2020

2020-02-07T02:09:07Z

← Older revision		Revision as of 02:09, 7 February 2020
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	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

	In this example we will show how to solve the [https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation Cahn-Hilliard] equation in Medusa. The solution provides an implementation of periodic boundary conditions and shows how to combine implicit and explicit operators. This example uses a custom biharmonic operator, so we recommend firstly reading tutorial on [[Customization]].		In this example we will show how to solve the [https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation Cahn-Hilliard] equation in Medusa. The solution provides an implementation of periodic boundary conditions and shows how to combine implicit and explicit operators. This example uses a custom biharmonic operator, so we recommend firstly reading the tutorial on [[Customization]].

	== Cahn-Hilliard equation ==		== Cahn-Hilliard equation ==

MihaR: MihaR moved page CahnHilliard equation to Cahn-Hilliard equation

2020-02-07T02:07:55Z

MihaR moved page CahnHilliard equation to Cahn-Hilliard equation

← Older revision	Revision as of 02:07, 7 February 2020
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MihaR: Created page with "Go back to Examples. In this example we will show how to solve the [https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation Cahn-Hilliard] equation i..."

2020-02-07T02:05:51Z

Created page with "Go back to Examples. In this example we will show how to solve the [https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation Cahn-Hilliard] equation i..."

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

In this example we will show how to solve the [https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation Cahn-Hilliard] equation in Medusa. The solution provides an implementation of periodic boundary conditions and shows how to combine implicit and explicit operators. This example uses a custom biharmonic operator, so we recommend firstly reading tutorial on [[Customization]].

== Cahn-Hilliard equation ==
The Cahn-Hilliard equation describes the separation of binary fluid into pure domains. We define concentration, a scalar field $c(x, y) \in [-1, 1]$ to describe the fluid, with values $c \pm 1$ representing pure phases. In this case the equation can be written as

\[
\frac{\partial c}{\partial t} = D \nabla^2 (c^3 - c - \gamma \nabla^2 c),
\]
where $D$ is the diffusion constant and $\sqrt{\gamma}$ is the characteristic transition length between domains.

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