README.md

Level Set Solver

The Level Set Solver is designed to model dynamically evolving fronts, such as interfaces or phase boundaries, on unstructured meshes. It evolves the level set function $\phi$, where $\phi=0$ represents the front.

Governing Equation

The level set evolution equation is:

$$ \frac{\partial \phi}{\partial t} + H(\nabla \phi, \mathbf{x}) = 0 $$

where the Hamiltonian $H(\nabla \phi, \mathbf{x}) = v_n ||\nabla \phi||$ and $v_n$ is the normal velocity magnitude.

Numerical Methods

Discretization

A finite difference approach is used on unstructured meshes. The upwind value of $H(\nabla \phi, \mathbf{x})$ is calculated using corner weights $\gamma_{hp}$:

$$ H_{+h} = \frac{\sum \limits_{hp \in h} (\gamma_{hp} \mathbf{v}_{hp} \cdot \nabla \phi_p)}{\sum \limits_{hp \in h} \gamma_{hp}} $$

where the subscript $+$ denotes the upwind direction relative to element $h$.

Nodal Gradient

Nodal gradients are calculated using the central difference method:

$$ \nabla \phi_p = -\frac{1}{V_p} \sum \limits_{hp \in p} \mathbf{N}_{hp} \phi_{g(h)} $$

where $\mathbf{N}_{hp}$ is the corner normal.

Corner Weights

Corner weights $\gamma_{hp}$ are calculated based on the dot product of the corner velocity and the element corner unit normal:

$$ \gamma_{hp} = \max\left(0, -\frac{\mathbf{v}_{hp}}{||\mathbf{v}_{hp}||} \cdot \mathbf{n}_{hp}\right) $$

This ensures stable upwinding.

Time Integration

The level set equation is integrated forward in time using a multi-stage Runge-Kutta method (e.g., two-stage).

Results

Example: Four interfaces propagating radially outward and colliding.

Implementation Details

• Element Type: Unstructured hexahedral meshes.
• Field Location: Level set field at element centers (Gauss points), gradients at nodes.
• Material Handling: Evolves level set field within regions containing specified materials.