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We have derived an analytical formulation for estimating the volume ofgeometries enclosed by implicitly defined surfaces. The novelty of this work isdue to two aspects. First we provide a general analytical formulation for alltwo-dimensional cases, and for elementary three three-dimensional cases bywhich the volume of general three-dimensional cases can be computed. Second,our method addresses the inconsistency issue due to mesh refinement. It isdemonstrated by several two-dimensional and three-dimensional cases that thisanalytical formulation exhibits 2nd-order accuracy.

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