Numerical methods for computational evolving manifolds in industrial applications

In this talk, a cell-centered finite volume method is used to solve the G-equation on polyhedral meshes in three-dimensional space, that is, a general type of the level-set equation including advective, normal, and mean curvature flow motions. Numerical experiments quantitatively show that the size of time step proportional to an average size of computational cells is enough to obtain the second-order convergence in space and time for smooth solutions of the general level set equation. A qualitative comparison is presented for a nontrivial example to compare numerical results obtained with hexahedral and polyhedral meshes. Furthermore, we propose to use a nonlinear boundary condition when advective or normal flow equations in the level-set formulation are numerically solved on polyhedral meshes. Since a common choice of the initial condition in the level set method is a signed distance function, the eikonal equation on the boundary is compatibly correct at the starting point. Enforcing the eikonal equation on the boundary along time can effectively eliminate an inflow boundary condition which is typically necessary in the transport equation. For the chosen examples, the numerical results confirm that the eikonal boundary condition provides comparable accuracy and robustness in the evolution of the surface using the exact Dirichlet boundary condition.

This project No. 2140/01/01 has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 945478.