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

A numerical method of initializing cell volume fraction demarcated by implicitly de- fined 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.