Body-fitted tracking of 2d open curves with a level set based mesh evolution method

This article describes a novel numerical algorithm for tracking the motion of a collection of open curves in two space dimensions.

The proposed strategy combines two complementary representations of these curves at each iteration of the evolution process: on the one hand, they are meshed explicitly, as a sub-collection of the entities of a mesh of the total computational domain.

Concurrently, using a variant of the Level Set Method, they are captured implicitly as algebraic combinations of the negative, zero and positive subsets of two auxiliary scalar functions, defined on the whole ambient space.

This coupling of representations allows to perform accurate geometric or mechanical computations on these curves, while leaving room for large evolution of their shape.

After the description of its main numerical ingredients, several applications examples of this methodology are proposed, where it is used to simulate the motion of open physical discontinuities, such as a vortex sheet roll-up, and to optimize the shape of open-ended curves, e.g. with respect to their anisotropic length, with the aim to improve the trajectory of a laser acting on a powder bed in the context of additive manufacturing, or to reconstruct fracture sets.