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3D Volume Calculation For the Marching Cubes Algorithm in Cartesian Coordinates

From a scalar field defined at the corner of a cube, an isosurface can be extracted using the Marching Cube algorithm. The isosurface separates the cell into two or more partial cells. A similar situation arises when an material interface in the Front Tracking method cuts through the computational cells. A popular method to calculate the volumes of the partial cells is to first partition the cells into tetrahedra and then sum together the volumes of the tetrahedra for the corresponding partial cells. In this paper, the divergence theorem is used to calculate the volumes of the partial cells generated by the Marching Cubes algorithm. This method is both more robust and efficient compared with the tetrahedralization approach.