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.

Modules
module	variables_levelset_cp
	Variables associated to the level set representationMainly default values for LS parameters.
module	mod_lsm_notus
	Interface between the LSM module and Notus. This is where we deal with the translation from a phase to a level set representation. It is mainly used to impose Notus' boundary conditions on VF for LS.
module	type_levelset_reinit_cp_parameters
	Level set CP parameters as a structureAlso define default values.
module	type_levelset
	The level set type.
module	type_levelset_cp_parameters
	Level set CP parameters as a structureAlso define default values.
module	type_levelset_parameters
	Level set parameters as a structureAlso define default values.
module	variables_levelset
	Variables associated to the level set representationMainly default values for LS parameters.

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