Body-fitted tracking within a surface via a level set based mesh evolution method

This article introduces a robust numerical strategy for tracking the arbitrarily large motion of a region $G(t)$ within a three-dimensional surface $S$ under the effect of a complex velocity field $V(t, x)$. Following our earlier work about evolving domains in the Euclidean space $\mathbb{R}^d$, two complementary representations of the region $G(t) \subset S$ are combined at each stage of the iterative process. On the one hand, $G(t)$ is meshed exactly, which allows for precise geometric and finite element computations, such as those required by the evaluation of $V(t, x)$. On the other hand, $G(t)$ is represented implicity, via the level set method-a format under which dramatic deformations of this region can be captured, including changes in its topology. Efficient numerical algorithms make it possible to switch consistently from one of these representations to the other, depending on its relevance with respect to the ongoing operation. After numerical validation, this strategy is applied to address two concrete physical problems, namely the simulation of the evolution of a fire front within a complex landscape, and the optimization of the shape of regions supporting the boundary conditions of a mechanical boundary value problem.