Level Set

Topics
	Advection
	Advection methods for level sets.
	Closest Point
	The Closest Point (CP) toolbox.
	Geometry
	Geometrical methods for level sets.
	Reinitialization
	Reinitialization methods.

Detailed Description

Definition of level sets

Level set (LS) methods are used for capturing a geometrical separation - ie. interfaces - between two or more domains. A quick overview of LS methods is given in presentation [1] and article [2].

Let \( \Omega_1 \) and \( \Omega_2 \) be two domains, then the interface \( \Gamma_{12} \) between them two is defined as \( \Gamma_{12} = \Omega_1 \cap \Omega_2 \).

For each domain, we can define a scalar fields \( \phi_i(\boldsymbol{x}) \) such that:

\[ \begin{cases} \phi_{i}(\boldsymbol{x})<0 & \text{for }\boldsymbol{x}\in\Omega_{i}\\ \phi_{i}(\boldsymbol{x})>0 & \text{for }\boldsymbol{x}\notin\Omega_{i}\\ \phi_{i}(\boldsymbol{x})=0 & \text{for }\boldsymbol{x}\in\partial\Omega_{i} \end{cases} \]

Those functions are called level sets. They are mostly used to capture implicitely the interfaces between two phases, ie. where \( \phi_1(\boldsymbol{x}) = \phi_2(\boldsymbol{x}) = 0 \).

Note

Usually, for practical reasons, level sets are considered (or desired) to be signed distance functions, ie. \( \phi(x) = \pm d(\boldsymbol{x}, \Gamma ) \), the Euclidian distance to the interface, negative inside the domain and positive outside.

When considering two domains, only 1 level set can be used to capture the interface as \( \phi_1(\boldsymbol{x}) = -\phi_2(\boldsymbol{x}) \). This simplification is commonly used in two phase flows problems.

References

[1] The Level Set Method; Per-Olof Persson; MIT classes. http://math.mit.edu/classes/18.086/2007/levelsetpres.pdf

[2] A review of level-set methods and some recent applications; Frederic Gibou, Ronald Fedkiw, Stanley Osher; JCP #353