First Moments of a Polyhedron Clipped by a Paraboloid
Abstract.
We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron’s faces with the paraboloid. We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at IRL, step-by-step guide, 8e77b35.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
The GNU Compiler Collection 10.3.0, https://gcc.gnu.org/onlinedocs/gcc-10.3.0.
2.
A. Abedian, J. Parvizian, A. Düster, H. Khademyzadeh, and E. Rank, Performance of different integration schemes in facing discontinuities in the finite cell method, Int. J. Comput. Methods, 10 (2013), 1350002, https://doi.org/10.1142/S0219876213500023.
3.
P. Antolin and T. Hirschler, Quadrature-free Immersed Isogeometric Analysis, Eng. Comput., 38 (2022), pp. 4475–4499.
4.
S. Bnà, S. Manservisi, R. Scardovelli, P. Yecko, and S. Zaleski, Numerical integration of implicit functions for the initialization of the VOF function, Comput. & Fluids, 113 (2015), pp. 42–52, https://doi.org/10.1016/j.compfluid.2014.04.010.
5.
E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing, CutFEM: Discretizing geometry and partial differential equations, Internat. J. Numer. Methods Engrg., 104 (2015), pp. 472–501, https://doi.org/10.1002/nme.4823.
6.
A. Chierici, L. Chirco, V. Le Chenadec, R. Scardovelli, P. Yecko, and S. Zaleski, An optimized Vofi library to initialize the volume fraction field, Comput. Phys. Commun., (2022), 108506, https://doi.org/10.1016/j.cpc.2022.108506.
7.
E. B. Chin and N. Sukumar, An efficient method to integrate polynomials over polytopes and curved solids, Comput. Aided Geom. Design, 82 (2020), 101914, https://doi.org/10.1016/j.cagd.2020.101914.
8.
R. Chiodi and O. Desjardins, General, robust, and efficient polyhedron intersection in the interface reconstruction library, J. Comput. Phys., 449 (2022), 110787, https://doi.org/10.1016/j.jcp.2021.110787.
9.
L. Dagum and R. Menon, OpenMP: An industry standard API for shared-memory programming, IEEE Comput. Sci. Eng., 5 (1998), pp. 46–55, https://doi.org/10.1109/99.660313.
10.
S. C. Divi, C. V. Verhoosel, F. Auricchio, A. Reali, and E. H. van Brummelen, Error-estimate-based adaptive integration for immersed isogeometric analysis, Comput. Math. Appl., 80 (2020), pp. 2481–2516, https://doi.org/10.1016/j.camwa.2020.03.026.
11.
D. A. Dunavant, High degree efficient symmetrical Gaussian quadrature rules for the triangle, Internat. J. Numer. Methods Engrg., 21 (1985), pp. 1129–1148, https://doi.org/10.1002/nme.1620210612.
12.
C. Engwer, S. May, A. Nüßing, and F. Streitbürger, A Stabilized DG Cut Cell Method for Discretizing the Linear Transport Equation, SIAM J. Sci. Comput., 42 (2020), pp. A3677–A3703, https://doi.org/10.1137/19M1268318.
13.
M. Evans and T. Swartz, Approximating Integrals via Monte Carlo and Deterministic Methods, Oxford Statist. Sci. Ser 20, Oxford University Press, Oxford, 2000.
14.
G. E. Farin, From conics to NURBS: A tutorial and survey, IEEE Comput. Graphs Appl., 12 (1992), pp. 78–86, https://doi.org/10.1109/38.156017.
15.
G. E. Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th ed., Morgan Kaufmann Ser. Comput. Graph. Geom. Model., Morgan Kaufmann, San Francisco, CA, 2001.
16.
T. Hahn, Cuba—a library for multidimensional numerical integration, Comput. Phys. Commun., 168 (2005), pp. 78–95, https://doi.org/10.1016/j.cpc.2005.01.010.
17.
E. Hartmann, G2 interpolation and blending on surfaces, Vis. Comput., 12 (1996), pp. 181–192, https://doi.org/10.1007/BF01782321.
18.
T. Hughes, J. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 4135–4195, https://doi.org/10.1016/j.cma.2004.10.008.
19.
B. W. Jones, A. G. Malan, and N. A. Ilangakoon, The initialisation of volume fractions for unstructured grids using implicit surface definitions, Comput. & Fluids, 179 (2019), pp. 194–205, https://doi.org/10.1016/j.compfluid.2018.10.021.
20.
W. Kahan, Pracniques: Further remarks on reducing truncation errors, Commun. ACM, 8 (1965), p. 40, https://doi.org/10.1145/363707.363723.
21.
J. Kromer and D. Bothe, Highly accurate computation of volume fractions using differential geometry, J. Comput. Phys., 396 (2019), pp. 761–784, https://doi.org/10.1016/j.jcp.2019.07.005.
22.
J. Kromer and D. Bothe, Third-order accurate Initialization of VOF volume fractions on unstructured grids with arbitrary polyhedral cells, J. Comput. Phys., 475 (2023), 111840.
23.
L. Kudela, N. Zander, S. Kollmannsberger, and E. Rank, Smart octrees: Accurately integrating discontinuous functions in 3D, Comput. Methods Appl. Mech. Engrg., 306 (2016), pp. 406–426, https://doi.org/10.1016/j.cma.2016.04.006.
24.
S. Lodha and J. Warren, Bézier representation for quadric surface patches, Comput.-Aided Des., 22 (1990), pp. 574–579, https://doi.org/10.1016/0010-4485(90)90042-B.
25.
Y. Renardy and M. Renardy, PROST: A parabolic reconstruction of surface tension for the volume-of-fluid method, J. Comput. Phys., 183 (2002), pp. 400–421, https://doi.org/10.1006/jcph.2002.7190.
26.
R. Sevilla, S. Fernández-Méndez, and A. Huerta, NURBS-enhanced finite element method (NEFEM), Internat. J. Numer. Methods Engrg., 76 (2008), pp. 56–83, https://doi.org/10.1002/nme.2311.
27.
S. J. Teller and C. H. Séquin, Modeling Implicit Quadrics and Free-form Surfaces with Trimmed Rational Quadratic Bezier Patches, Technical report CSD-90-577, Computer Science Division, University of California at Berkeley, Berkeley, CA, 1990.
28.
T. Tolle, D. Gründing, D. Bothe, and T. Marić, triSurfaceImmersion: Computing volume fractions and signed distances from triangulated surfaces immersed in unstructured meshes, Comput. Phys. Commun., 273 (2022), 108249, https://doi.org/10.1016/j.cpc.2021.108249.
29.
G. Turk, The Stanford 3D Scanning Repository, 1994, http://graphics.stanford.edu/data/3Dscanrep/.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A2250 - A2274
DOI: 10.1137/22M1524308
ISSN (online): 1095-7197
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 26 September 2022
Accepted: 5 May 2023
Published online: 21 September 2023
Keywords
MSC codes
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at IRL, step-by-step guide, 8e77b35.
Authors
Funding Information
H2020 Marie Skłodowska-Curie Actions (MSCA): 101026017
Multidisciplinary University Research Initiative (MURI): N00014-16-1-2617
Funding: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement 101026017. Second, third, and fifth author were sponsored by the Office of Naval Research (ONR) as part of the Multidisciplinary University Research Initiatives (MURI) Program, under grant N00014-16-1-2617. The views and conclusions contained herein are those of the authors only and should not be interpreted as representing those of ONR, the U.S. Navy, or the U.S. Government.
Metrics & Citations
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.
Cited By
- Microstructure-based prediction of hydrodynamic forces in stationary particle assembliesInternational Journal of Multiphase Flow, Vol. 175 | 1 May 2024
- Comparison of methods for curvature estimation from volume fractionsInternational Journal of Multiphase Flow, Vol. 174 | 1 Apr 2024
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].