Cahn-Hilliard equation - Revision history

MihaR at 18:08, 12 October 2020

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MihaR at 14:10, 11 February 2020

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MihaR at 13:38, 11 February 2020

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MihaR at 00:51, 11 February 2020

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MihaR at 00:17, 11 February 2020

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MihaR at 15:40, 7 February 2020

2020-02-07T15:40:23Z

MihaR at 13:52, 7 February 2020

2020-02-07T13:52:21Z

MihaR at 02:09, 7 February 2020

2020-02-07T02:09:07Z

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

2020-02-07T02:07:55Z

MihaR moved page CahnHilliard equation to Cahn-Hilliard equation

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..."

New page

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]].

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 === Mixed solution ===

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		+	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]].

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 fill.seed(randomSeed);
 domain.fill(fill, dx);
 </syntaxhighlight>
 == Solution ==
 We need to handle two uncommon terms to solve the equation. Biharmonic ($\nabla^4$) operator can be implemented with custom operators as shown in [[Customization]] example. Nonlinear ($\nabla^2 c^3$) term proves to be more problematic, as it can not be solved implicitly. We either need to solve the entire equation with explicit time stepping or include the nonlinear term in the right hand side of our system of equations.
 === Explicit solution ===
 <syntaxhighlight lang="cpp">
 </syntaxhighlight>
 === Mixed solution ===
 <syntaxhighlight lang="cpp">
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 == Periodic boundary conditions ==
-[[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.
 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.

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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

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

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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]].
+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 ==

← Older revision	Revision as of 02:07, 7 February 2020
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