Level Sets

The levelsets package tracks a material interface with a level-set function \(\phi\), whose zero contour \(\phi = 0\) marks the interface. The field is advected with the flow and periodically reinitialized to remain a signed distance function near the interface. It provides a sharp interface representation complementing the diffuse volume-fraction description of Chapter Hydrodynamics.

Warning

Experimental: The level-set package currently supports a single level set and requires hydrodynamics to be enabled.

Governing Equations

Advection

The level-set function is advected by the hydrodynamic velocity using the same flux-conservative machinery (reconstruction and Riemann solver) as the other advected fields,

\[\frac{\partial \phi}{\partial t} + \nabla\!\cdot\!\left(\phi\vec{v}\right) = 0 .\]

By convention \(\phi > 0\) in the designated sharp material (sharp_mat) and \(\phi < 0\) elsewhere, so the interface is the \(\phi = 0\) contour.

Reinitialization

Advection distorts \(\phi\) away from a signed distance function (\(|\nabla\phi| = 1\)). Every reinit_modcyc cycles the package restores that property near the interface by evolving a hyperbolic Eikonal equation in a pseudo-time \(\tau\) (Sussman & Fatemi, SIAM J. Sci. Comput. 20, 1165, 1999),

\[\frac{\partial\phi}{\partial\tau} = \operatorname{sign}(\phi_0)\left(1 - |\nabla\phi|\right),\]

where \(\phi_0\) is the level set at the start of reinitialization and \(\operatorname{sign}(\phi_0)\) is a smoothed sign function. The gradient magnitude is formed with upwind differences selected by the sign of \(\phi_0\) (forward differences where \(\phi_0 > 0\), backward where \(\phi_0 < 0\)) using second-order minmod-limited reconstruction. Reinitialization is applied only within a band of half-width reinit_width cells around the interface; outside the band \(\phi\) is held at a constant magnitude. The pseudo-time update runs for reinit_nstep steps, each propagating the correction roughly one cell.

Input Parameters

Level sets are enabled with the levelsets toggle in the <physics> block (Section Enabling Physics: the <physics> Block); hydro must also be enabled. The remaining controls live in the <levelsets> block.

Parameters in the `<levelsets>` block.
Parameter	Type	Default	Description
sharp_mat	int	`0`	Index of the material on the \(\phi > 0\) side of the interface.
reinit_modcyc	int	`25`	Reinitialize every this many cycles (\(1\) = every cycle).
reinit_width	int	`4`	Half-width (in cells) of the band kept as a signed distance function.
reinit_nstep	int	`15`	Number of pseudo-time steps per reinitialization (ideally \(>\) `reinit_width`).

Registered Fields

The package registers the single advected level-set field in the table below, plus two single-copy scratch fields used only during reinitialization (\(\phi_0\) storage and the pseudo-time right-hand side).

Fields registered by the level-set package.
Field	Symbol	Components	Metadata / description
levelset	\(\phi\)	1	Cell, Independent, Intensive, FillGhost, Advected, WithFluxes; the level-set function.
levelset0	\(\phi_0\)	1	Cell, Independent, Intensive, OneCopy; level set stored at the start of reinitialization.
dudt_reinitialize	\(\partial_\tau\phi\)	1	Cell, Independent, Intensive, OneCopy; reinitialization right-hand side.

Example

Track the interface of material 0, reinitializing every 25 cycles:

riot.input("physics", hydro=True, levelsets=True)
riot.input("levelsets", sharp_mat=0, reinit_modcyc=25,
           reinit_width=4, reinit_nstep=15)