# Highlight on Moment-of-Fluid method

Moment-of-Fluid (MOF) is a semi-Lagrangian method to advect interfaces in multiphase flow simulation. This method is the apex of the VOF methods using Piecewise Linear Interface Calculation. Indeed, MOF can manage the reconstruction of a large number of interfaces with *automatic ordering* using only a *one-cell stencil* (compared to the 9 (27 in 3D) point stencil of the PLIC method).

These performances are achieved through the exploitation of the centroids in addition to the usual volume fraction. MoF consists in finding a polygonal approximation of a reference subset ω* (see following figure) for which volume is conserved and distance of the relative centroïd to the reference one is minimized.

*Figure 1: Reference subset (left) and reconstructed subset (right)*

This cell-wise minimization process generates fairly substantial computational costs. We have reduced these costs by finding an analytical solution to this minimization process on Cartesian grids which also increases robustness. Consider a rectangular cell of dimension (Cx, Cy) such as represented in the center of Fig. 2. The locus of the centroids for a given reference volume is a closed convex curve. In Fig. 2, we have represented the locus of the centroids for various reference volumes such that V ≤ 0.5CxCy . We observe 8 different configurations. 4 configurations where the reconstructed polygon is a triangle (odd numbers on the figure) and 4 configurations where the reconstructed polygon is a quadrangle (even numbers on the figure). Only the first two configurations can be considered since any other configuration can be transformed into the first two by symmetry and/or inverting the role of Cx and Cy . We have proven that when the reconstructed polygon is a triangle, the locus is a hyperbola H and when the reconstructed polygon is a quadrangle, the locus is a parabola P. Then it is possible to find the minimal distance from the reference centroid to the various parts of the curve to find the global minimum of the minimization problem. The normal (nx,ny) is thus obtained and the volume can be reconstruted by a flood algorithm.

*Figure 2: Representation of the different configurations of the locus of the centroids (cyan) in a rectangular cell of dimension (Cx, Cy) for various reference volumes V such that V ≤ 0.5CxCy . Depending on the reconstructed polygon, the locus is either a hyperbola H or a parabola P;*

If we compare the computational time of the minimization process with a tolerance value that gives an equivalent precision for the centroid defect, we observe that the analytic reconstruction is about 160% faster. With a tolerance value that would be used for a physical simulation (about 10−6 ), we observe that the analytic reconstruction is at least 44% faster, but the centroid defect does not reach the machine error.

When more than two materials are involved (see movie 1 below), the proposed algorithm can be applied only to the reconstruction of the first material, the remaining materials are reconstructed using the original minimization algorithm. Even in this case, we have shown that this algorithm is interesting in conjunction with the minimization algorithm.

**Movie 1: Interface Moment-of-Fluid reconstruction of a 4 material disk in a reverse-shear flow**

**Reference**

A. Lemoine, S. Glockner, J. Breil, Moment-of-Fluid Analytic Reconstruction on 2D Cartesian Grids, *Journal of Computational Physics,* 328 (2017) 131–139 (pdf).