Analytical Relations Connecting Linear Interfaces and Volume Fractions in Rectangular Grids

The computational-geometric problems arising when a linear interface cuts a cube are considered. They are of interest in particular for the calculation of volume fractions or interface positions in three-dimensional interface calculations in the Volume of Fluid (VOF) methods. Typically, the normal vector is known. One then wants to compute the volume fraction knowing the interface position, or conversely the interface position knowing the volume fraction. Explicit expressions of general use are given, and the algorithms used to search for solutions are described in detail. Explicit formulas for cubic roots are found to be less than two thirds as time consuming as Newton–Raphson iterations.