Analytic gradient for the moment-of-fluid method in axisymmetric and on general polyhedrons in any dimension

Abstract

The moment-of-fluid method (MOF) is an interface reconstruction method similar to the volume-of-fluid method with piecewise linear interface reconstruction (VOF-PLIC). In the MOF method, the normal to the interface is found by minimizing the distance between the centroid of the polyhedron below the interface and a reference centroid under a volume constraint. To solve this minimization problem, the gradient of the objective function must be evaluated. Analytic formulas have been proposed by many authors to compute the gradient in 2D on general polygons with a polar parametrization and in 3D on convex polyhedrons with a spherical parametrization. In this short note, we propose a more general formula that covers non-convex polyhedrons in any dimension and axisymmetric coordinates. Furthermore, this formula does not depend on the parametrization of the normal to the interface. We also provide some practical way to use the formula in a code.