High-order, refinement-based computation of the volume of an arbitrary polyhedron intersected by an implicitly defined fluid body

Abstract

A high-order accurate, efficient and easy-to-implement method is presented for computing the fluid volume bounded by an arbitrary polyhedron, whether it is convex or non-convex, and an implicitly-defined fluid body. This method is an improved version of a previous one that used a recursive local grid refinement of the polyhedron and linear interpolations to determine the intersections of the interface that delimits the fluid body (or simply fluid-body interface) with the polyhedron boundaries. The proposed method first determines the volume of a polyhedral approximation of the bounded fluid region by using a general clipping-lookup and capping procedure valid for arbitrary polyhedra, where the points of intersection between the polyhedron and the fluid-body interface are obtained using a root-finding method rather than linear interpolations. The approximated polyhedral volume is subsequently corrected by using simple Gaussian quadrature rules over a triangulated approximation of the intersected fluid-body interface to achieve high-order accuracy. Recursive local grid refinement of the polyhedron also enables reductions in fluid volume errors. The proposed method requires no assumption of any particular local parametrization of the fluid-body interface, whether paraboloidal or of any other type, or deriving any complex analytical expressions to compute the volume of fluid contained within the polyhedron, thereby making the method easy to implement and generally applicable to any implicitly-defined function. A detailed assessment shows global fourth-order convergent accuracies on structured and unstructured grids even for complex fluid-body interfaces of a high degree. Speedups of several orders of magnitude with respect to the previous refinement method with linear interpolations are achieved even for relatively coarse grids. Comparisons with other methods are also presented, and the software with the implemented method and tests used for the assessment is freely available.