Numerical integration of implicit functions for the initialization of the VOF function

•Implicit representation of interfaces, f(x, y, z) = 0.•Initialization of the volume fraction scalar field via numerical integration.•Numerical computation of internal and external limits of integration.•Double Gauss–Legendre integration with a variable number of nodes.•Examples in two and three dimensions to show the method accuracy and robustness.

A numerical method of initializing cell volume fraction demarcated by implicitly defined fluid interfaces is presented. Each cell of the computational domain is examined for the presence of the reference phase. When a cell is not full or empty, limits are found that allow volume fraction to be computed by numerical integration. The method enlists a number of algorithms including root finding and minimum search on an oriented segment, a preconditioned conjugate gradient minimum search on a cell face and a double Gauss–Legendre integration with a variable number of nodes, among others. Practical multi-phase fluid examples in two- and three-dimensions are presented to demonstrate the accuracy and robustness of the method.