Front tracking with a twist: the eXtreme mesh deformation approach (X-MESH)

Nicolas Moës

Ecole Centrale de Nantes, France

The arbitrary Eulerian Lagrangian (ALE) formulation is a common approach to tracking fronts in finite element simulations. It is, however, tricky to track fronts over long distances, as the mesh density generally becomes too low on one side of the front (increasingly large elements). Moreover, traditional ALE front tracking cannot cope with changes in front topology. To remedy the above problems (at least the first one), remeshing is required from time to time to maintain correct mesh approximation capability on both sides of the front. This remeshing requires projection of the field and updating of the database in the simulation, which is detrimental to the speed of the code and the accuracy of the solution.

We introduce a new approach in which the set of nodes located on the front evolves over time. This allows the front to migrate through the mesh without breaking the approximation capability of the mesh. Topological changes are also easily taken into account. For example, a small front can form, propagate and merge with other fronts as it propagates. The small front may be represented initially by three or four nodes, then by hundreds of nodes as it lengthens.

For the new approach to work properly, we have to accept that some elements become very small and possibly of zero measure. This means that the elements can deform in extreme ways, hence the acronym X-MESH. Surprisingly, as we shall show, this situation does not prevent simulations from being carried out.

In short, X-MESH simply uses node movements to propagate fronts over long distances, even in the event of topological changes. The mesh topology remains unchanged during simulation. The size and sparsity of the finite element matrices are therefore fixed throughout the simulation, and no field projection is required. As the simulation progresses, nodes arrive and depart from the front.

X-MESH's capability will be demonstrated for several important applications in mechanics and physics, such as front tracking in the Stefan phase change model or the simulation of immiscible two-phase flows.

The work is funded by a European Research Council (ERC) Synergy Grant whose co-pI is J.F. Remacle.

References:

- N. Moës, J.-F. Remacle, J. Lambrechts, B. Lé and N. Chevaugeon. The eXtreme Mesh Deformation Approach (X-MESH) for the Stefan Phase Change Model. Journal of Computational Physics 2023, 477, 111878. https://doi.org/10.1016/j.jcp.2022.111878.

- A. Quiriny, J. Lambrechts, N. Moës and J.-F. Remacle, X-Mesh: A new approach for the simulation of two-phase flow with sharp interface, Journal of Computational Physics 2024, 112775. https://doi.org/10.1016/j.jcp.2024.112775.