This projects starts with a
newly developed conservative level-set method
for multiphase flows. This method combines ideas from the level-set and
the volume of fluid schemes into a monolithic model. The model contains a consistent term
that regularizes the Jacobian of the non-linear equation and that penalizes deviations
from the signed distance function.

The result is a conservative level-set model that does not require reconstruction of the interface and that produces an approximation of the signed distance function (to the fluids interface).

In the current project we aim to improve the method in the following three fronts:

- The current form of the method does not require extra stabilization of the advective term since it depends upon the penalization term. This however, implies that one can't reduce the influence of the penalization or instabilities might start to appear. We want to introduce extra and independent stabilization to the advective term.

The model is a conservation law for a regularized Heaviside function. The reason for this is that integration of discontinuous functions requires non-standard methodologies within the context of finite elements.

We plan to use state of the art integration techniques that would allow us to use exact Heaviside functions improving the conservation properties and overall quality of the solution.

- The model contains a user defined parameter. To obtain qualitatively good results one might need to select this parameter depending on the problem. This dependency can be mitigated via optimal control theory that would allow the algorithm to automatically select an optimal parameter for any given problem.

We plan to test each modification to the method via a set of well established benchmarks in the area of multiphase flows.

Implementation
and testing of the proposed algorithm

Applied
mathematics and computational science