Methods and Algorithms for Scientific Computing

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


Volume estimation of implicitly defined 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 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-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 refined or coarsened during simulations.


  1. analytical formulation
  2. volumes estimation
  3. consistency
  4. implicitly defined surfaces

MSC codes

  1. 65D30
  2. 65D18

Get full access to this article

View all available purchase options and get full access to this article.


A. Fujiwara, J. Kawaguchi, D. Yeomans, M. Abe, T. Mukai, T. Okada, et al., The rubble-pile asteroid Itokawa as observed by Hayabusa, Science, 312 (2006), pp. 1330--1334.
S. Besse, O. Groussin, L. Jorda, P. Lamy, M. Kaasalainen, G. Gesquiere, et al., 3-dimensional reconstruction of asteroid $2867$ Steins, in Proceeding of the Lunar and Planetary Science Conference, Vol. 40, 2009, p. 1545.
M. Brozovic, S. J. Ostro, L. A. Benner, J. D. Giorgini, R. F. Jurgens, R. Rose, et al., Radar observations and a physical model of Asteroid $4660$ Nereus, a prime space mission target, Icarus, 201 (2009), pp. 153--166.
X. Hu, B. Khoo, N. Adams, and F. Huang, A conservative interface method for compressible flows, J. Comput. Phys., 219 (2006), pp. 553--578.
K. So, X. Hu, and N. Adams, Anti-diffusion method for interface steepening in two-phase incompressible flow, J. Comput. Phys., 230 (2011), pp. 5155--5177.
M. Sussman and E. G. Puckett, 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.
D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, and S. Zaleski, Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys., 152 (1999), pp. 423--456.
X. Hu, N. Adams, M. Herrmann, and G. Iaccarino, Multi-scale modeling of compressible multi-fluid flows with conservative interface method, in Proceedings of the Summer Program, Center for Turbulence Research, 2010.
R. Scardovelli and S. Zaleski, Analytical relations connecting linear interfaces and volume fractions in rectangular grids, J. Comput. Phys., 164 (2000), pp. 228--237.
S. Diot, M. M. François, and E. D. Dendy, 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.
S. Diot and M. M. rançois, An interface reconstruction method based on an analytical formula for $3$D arbitrary convex cells, J. Comput. Phys., 305 (2016), pp. 63--74.
F. Rypens, T. Metens, N. Rocourt, P. Sonigo, F. Brunelle, M. P. Quere, et al., Fetal lung volume: Estimation at MR imaging---initial results, Radiology, 219 (2001), pp. 236--241.
F. V. Coakley, J. Kurhanewicz, Y. Lu, K. D.Jones, M. G. Swanson, and S. D. Chang, et al., Prostate cancer tumor volume: Measurement with endorectal MR and MR spectroscopic Imaging, Radiology, 223 (2002), pp. 91--97.
B. N. Joe, M. B. Fukui, C. C. Meltzer, Q. S. Huang, R. S. Day, and P. J. Greer, et al., Brain tumor volume measurement: Comparison of manual and semiautomated methods, Radiology, 212 (1999), pp. 811--816.
N. A. Mayr, V. A. Magnotta, J. C. Ehrhardt, J. A. Wheeler, J. I. Sorosky, and B. C. Wen, et al., 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.
T. A. Durso, J. Carnell, T. T. Turk, and G. N. Gupta, Three-dimensional reconstruction volume: A novel method for volume measurement in kidney cancer, J. Endourology, 28 (2014), pp. 745--750.
I. H. Gong, J. Hwang, D. K. Choi, S. R. Lee, Y. K. Hong, and J. Y. Hong, et al., Relationship among total kidney volume, renal function and age, J. Urology, 187 (2012), pp. 344--349.
B. Cheong, M. F. R. Raja Muthupillai, and S. D. Flamm, Normal values for renal length and volume as measured by magnetic resonance imaging, Clinical J. Amer. Soci. Nephrology, 2 (2007), pp. 38--45.
T. B. Jones, and L. R. Riddick, M. D. Harpen, R. L. Dubuisson, and D. Samuels, Ultrasonographic determination of renal mass and renal volume, J. Ultrasound Medicine, 2 (1983), pp. 151--154.
S. Van der Pijl, A. Segal, C. Vuik, and P. Wesseling, A mass-conserving level-set method for modelling of multi-phase flows, Internat. J. Numer. Methods Fluids, 47 (2005), pp. 339--361.
T. Ménard, S. Tanguy, and A. Berlemont, 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.
K. Yokoi, Efficient implementation of THINC scheme: A simple and practical smoothed VOF algorithm, J. Comput. Phys., 226 (2007), pp. 1985--2002.
H. K. Zhao, T. Chan, B. Merriman, and S. Osher, A variational level set approach to multiphase motion, J. Comput. Phys., 127 (1996), pp. 179--195.
M. Y. Wang, X. Wang and D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 227--246.
C. Li, R. Huang, Z. Ding, J. C. Gatenby, D. N. Metaxas, and J. C. Gore, 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.
D. Kim, J. Pestieau, and J. Glimm, 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.
L. Lawrence, Introduction to the mechanics of a continuous Medium, Prentice-Hall, Upper Saddle River, NJ, 1969.
S. Wang, 3D Volume Calculation for the Marching Cubes Algorithm in Cartesian Coordinates, preprint, arXiv:13080387, 2013.
W. E. Lorensen and H. E. Cline, Marching cubes: A high resolution 3D surface construction algorithm, in Proceedings of SIGGRAPH, ACM, 1987.
T. S. Newman and H. Yi, A survey of the marching cubes algorithm, Comput. & Graphics, 30 (2006), pp. 854--879.
I. Nystroem, J. K. Udupa, G. J. Grevera, and B. E. Hirsch, 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.
C. Min and F. Gibou, Geometric integration over irregular domains with application to level-set methods, J. Comput. Phys., 226 (2007), pp. 1432--1443.
J. F. Sallee, The middle-cut triangulations of the n-cube, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 407--419.
E. Lauer, X. Hu, S. Hickel, and N. A. Adams, Numerical modelling and investigation of symmetric and asymmetric cavitation bubble dynamics, Comput & Fluids, 69 (2012), pp. 1--19.
T. P. Fries and S. Omerović, Higher-order accurate integration of implicit geometries, Internat. J. Numer. Methods Engrg., 106 (2016), pp. 323--371.
D. Eberly, J. Lancaster, and A. Alyassin, On gray scale image measurements: II. Surface area and volume, CVGIP Graphical Models Image Process., 53 (1991), pp. 550--562.
A. K. Tornberg, Multi-dimensional quadrature of singular and discontinuous functions, BIT, 42 (2002), pp. 644--669.
B. Müller, F. Kummer, M. Oberlack, and Y. Wang, Simple multidimensional integration of discontinuous functions with application to level set methods, Internat. J. Numer. Methods Engrg. 92 (2012), pp. 637--651.
M. A. Olshanskii and D. Safin, Numerical integration over implicitly defined domains for higher order unfitted finite element methods, Lobachevskii J. Math., 37 (2016), pp. 582--596.
C. Min and F. Gibou, Robust second-order accurate discretizations of the multi-dimensional heaviside and dirac delta functions, J. Comput. Phys., 227 (2008), pp. 9686--9695.
R. Saye, High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37 (2015), pp. A993--A1019.
B. Müller, F. Kummer, and M. Oberlack, Highly accurate surface and volume integration on implicit domains by means of moment-fitting, Internat. J. Numer. Methods Engrg., 96 (2013), pp. 512--528.
M. Sussman, A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, and M. L. Welcome, An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys., 148 (1999), pp. 81--124.
C. Min, and F. Gibou, A second order accurate level set method on non-graded adaptive cartesian grids, J. Comput. Phys., 225 (2007), pp. 300--321.
D. Zuzio and J. Estivalezes, An efficient block parallel AMR method for two phase interfacial flow simulations, Comput. & Fluids, 44 (2011), pp. 339--357.
M. H. Chung, 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.
L. Han, X. Hu, and N. Adams, 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.
J. Luo, X. Hu, and N. Adams, A conservative sharp interface method for incompressible multiphase flows, J. Comput. Phys., 284 (2015), pp. 547--565.
L. Han, X. Hu, and N. Adams, 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.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1523 - A1543
ISSN (online): 1095-7197


Submitted: 19 April 2017
Accepted: 20 February 2018
Published online: 24 May 2018


  1. analytical formulation
  2. volumes estimation
  3. consistency
  4. implicitly defined surfaces

MSC codes

  1. 65D30
  2. 65D18



Funding Information

China Scholarship Council : 201306290030

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account