A consistent analytical formulation for volume-estimation of geometries enclosed by implicitly defined surfaces

Abstract: We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined 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 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. It is demonstrated by several two-dimensional and three-dimensional cases that this analytical formulation exhibits 2nd-order accuracy.