# A Consistent Analytical Formulation for Volume Estimation of Geometries Enclosed by Implicitly Defined Surfaces

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*The rubble-pile asteroid Itokawa as observed by Hayabusa*, Science, 312 (2006), pp. 1330--1334.

*3-dimensional reconstruction of asteroid $2867$ Steins*, in Proceeding of the Lunar and Planetary Science Conference, Vol. 40, 2009, p. 1545.

*Radar observations and a physical model of Asteroid $4660$ Nereus, a prime space mission target*, Icarus, 201 (2009), pp. 153--166.

*A conservative interface method for compressible flows*, J. Comput. Phys., 219 (2006), pp. 553--578.

*Anti-diffusion method for interface steepening in two-phase incompressible flow*, J. Comput. Phys., 230 (2011), pp. 5155--5177.

*A coupled level set and volume-of-fluid method for computing $3$D and axisymmetric incompressible two-phase flows*, J. Comput. Phys., 162 (2000), pp. 301--337.

*Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows*, J. Comput. Phys., 152 (1999), pp. 423--456.

*Multi-scale modeling of compressible multi-fluid flows with conservative interface method*, in Proceedings of the Summer Program, Center for Turbulence Research, 2010.

*Analytical relations connecting linear interfaces and volume fractions in rectangular grids*, J. Comput. Phys., 164 (2000), pp. 228--237.

*An interface reconstruction method based on analytical formulae for $2$D planar and axisymmetric arbitrary convex cells*, J. Comput. Phys., 275 (2014), pp. 53--64.

*An interface reconstruction method based on an analytical formula for $3$D arbitrary convex cells*, J. Comput. Phys., 305 (2016), pp. 63--74.

*Fetal lung volume: Estimation at MR imaging---initial results*, Radiology, 219 (2001), pp. 236--241.

*Prostate cancer tumor volume: Measurement with endorectal MR and MR spectroscopic Imaging*, Radiology, 223 (2002), pp. 91--97.

*Brain tumor volume measurement: Comparison of manual and semiautomated methods*, Radiology, 212 (1999), pp. 811--816.

*Usefulness of tumor volumetry by magnetic resonance imaging in assessing response to radiation therapy in carcinoma of the uterine cervix*, Internat. J. Radiation Oncol. Biol. Phys., 35 (1996), pp. 915--924.

*Three-dimensional reconstruction volume: A novel method for volume measurement in kidney cancer*, J. Endourology, 28 (2014), pp. 745--750.

*Relationship among total kidney volume, renal function and age*, J. Urology, 187 (2012), pp. 344--349.

*Normal values for renal length and volume as measured by magnetic resonance imaging*, Clinical J. Amer. Soci. Nephrology, 2 (2007), pp. 38--45.

*Ultrasonographic determination of renal mass and renal volume*, J. Ultrasound Medicine, 2 (1983), pp. 151--154.

*A mass-conserving level-set method for modelling of multi-phase flows*, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 339--361.

*Coupling level set/vof/ghost fluid methods: Validation and application to $3$D simulation of the primary break-up of a liquid jet*, Int. J. Multiph. Flow, 33 (2007), pp. 510--524.

*Efficient implementation of THINC scheme: A simple and practical smoothed VOF algorithm*, J. Comput. Phys., 226 (2007), pp. 1985--2002.

*A variational level set approach to multiphase motion*, J. Comput. Phys., 127 (1996), pp. 179--195.

*A level set method for structural topology optimization*, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 227--246.

*A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI*, IEEE Trans. Image Process., 20 (2011), pp. 2007--2016.

*Volume Fractions and Surface Areas for 3-D Grid Cells Cut by an Interface*, Technical Report SUNYSB-AMS-07-01, State University of New York at Stony Brook, 2007.

*Introduction to the mechanics of a continuous Medium*, Prentice-Hall, Upper Saddle River, NJ, 1969.

*3D Volume Calculation for the Marching Cubes Algorithm in Cartesian Coordinates*, preprint, arXiv:13080387, 2013.

*Marching cubes: A high resolution 3D surface construction algorithm*, in Proceedings of SIGGRAPH, ACM, 1987.

*A survey of the marching cubes algorithm*, Comput. & Graphics, 30 (2006), pp. 854--879.

*Area of and volume enclosed by digital and triangulated surfaces*, in Proceeding of Medical Imaging 2002. International Society for Optics and Photonics, 2002, pp. 669--680.

*Geometric integration over irregular domains with application to level-set methods*, J. Comput. Phys., 226 (2007), pp. 1432--1443.

*The middle-cut triangulations of the n-cube*, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 407--419.

*Numerical modelling and investigation of symmetric and asymmetric cavitation bubble dynamics*, Comput & Fluids, 69 (2012), pp. 1--19.

*Higher-order accurate integration of implicit geometries*, Internat. J. Numer. Methods Engrg., 106 (2016), pp. 323--371.

*On gray scale image measurements: II. Surface area and volume*, CVGIP Graphical Models Image Process., 53 (1991), pp. 550--562.

*Multi-dimensional quadrature of singular and discontinuous functions*, BIT, 42 (2002), pp. 644--669.

*Simple multidimensional integration of discontinuous functions with application to level set methods*, Internat. J. Numer. Methods Engrg. 92 (2012), pp. 637--651.

*Numerical integration over implicitly defined domains for higher order unfitted finite element methods*, Lobachevskii J. Math., 37 (2016), pp. 582--596.

*Robust second-order accurate discretizations of the multi-dimensional heaviside and dirac delta functions*, J. Comput. Phys., 227 (2008), pp. 9686--9695.

*High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles*, SIAM J. Sci. Comput., 37 (2015), pp. A993--A1019.

*Highly accurate surface and volume integration on implicit domains by means of moment-fitting*, Internat. J. Numer. Methods Engrg., 96 (2013), pp. 512--528.

*An adaptive level set approach for incompressible two-phase flows*, J. Comput. Phys., 148 (1999), pp. 81--124.

*A second order accurate level set method on non-graded adaptive cartesian grids*, J. Comput. Phys., 225 (2007), pp. 300--321.

*An efficient block parallel AMR method for two phase interfacial flow simulations*, Comput. & Fluids, 44 (2011), pp. 339--357.

*An adaptive Cartesian cut-cell/level-set method to simulate incompressible two-phase flows with embedded moving solid boundaries*, Comput. & Fluids, 71 (2013), pp. 469--486.

*Adaptive multi-resolution method for compressible multi-phase flows with sharp interface model and pyramid data structure*, J. Comput. Phys., 262 (2014), pp. 131--152.

*A conservative sharp interface method for incompressible multiphase flows*, J. Comput. Phys., 284 (2015), pp. 547--565.

*Scale separation for multi-scale modeling of free-surface and two-phase flows with the conservative sharp interface method*, J. Comput. Phys., 280 (2015), pp. 387--403.

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**Submitted**: 19 April 2017

**Accepted**: 20 February 2018

**Published online**: 24 May 2018

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