A linear MARS method for three-dimensional interface tracking

For explicit interface tracking in three dimensions, we propose a linear MARS method that (a) represents the interface by a partially ordered set of glued surfaces and approximates each glued surface with a triangular mesh, (b) maintains an $(r,h,\theta)$-regularity on each triangular mesh so that the distance between any pair of adjacent markers is within the range $[rh,h]$ and no angle in any triangle is less than $\theta$, (c) applies to three-dimensional continua with arbitrarily complex topology and geometry, (d) preserves topological structures and geometric features of moving phases under diffeomorphic and isometric flow maps, and (e) achieves second-order and third-order accuracy in terms of the Lagrangian and Eulerian length scales, respectively. Results of classic benchmark tests verify the effectiveness of the novel mesh adjustment algorithms in enforcing the $(r,h,\theta)$-regularity and demonstrate the high accuracy and efficiency of the proposed linear MARS method.