A CONSISTENT ANALYTICAL FORMULATION FOR VOLUME ESTIMATION OF GEOMETRIES ENCLOSED BY IMPLICITLY DEFINED SURFACES

摘要

Volume estimation of implicitly de fined geometries is of interest for many subjects related to computational science. Previous work in the literature mainly focuses on the development of accurate, robust, and efficient numerical methods for computing the volume of complex geometries, based, e.g., on geometrical reconstruction, smoothed Heaviside function, recursive subdivision, and Gaussian quadrature. In this paper, we have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly de fined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all two-dimensional cases and for elementary three-dimensional cases by which the volume of general three-dimensional cases can be computed. Second, our method addresses the inconsistency issue due to mesh refinement. Compared to existing numerical volume-estimation methods, the proposed analytical method produces negligible consistency errors. Its computational cost is as small as that of the fast numerical volume estimation methods. It is demonstrated by several two-dimensional and three-dimensional cases that this analytical formulation exhibits second-order accuracy. One potential application of this method is the development of a fully adaptive multiphase method where both the interface and flow field are re fined or coarsened during simulations.