An exact general remeshing scheme applied to physically conservative voxelization
Abstract
We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass, momentum, and energy conservation while applying higher-order polynomial interpolation. We elaborate on applications of our algorithm arising in the analysis of cosmological N-body data, computer graphics, and continuum mechanics problems. We focus on the particular case of remeshing tetrahedral cells onto a Cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh. We refer to this as “physically conservative voxelization.” At the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other. This algorithm is an implementation of the ideas presented abstractly by Sugihara [48], who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output. This makes our implementation robust to geometric degeneracy in the input. We employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain. We also addressmore »
- Authors:
- Publication Date:
- Research Org.:
- SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC); Stanford SGF
- OSTI Identifier:
- 1228391
- Alternate Identifier(s):
- OSTI ID: 1183695
- Report Number(s):
- SLAC-PUB-16295
Journal ID: ISSN 0021-9991; S0021999115003563; PII: S0021999115003563
- Grant/Contract Number:
- AC02-76SF00515; DE-AC02-76SF00515
- Resource Type:
- Published Article
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Name: Journal of Computational Physics Journal Volume: 297 Journal Issue: C; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; ASTRO, COMP, GRQC
Citation Formats
Powell, Devon, and Abel, Tom. An exact general remeshing scheme applied to physically conservative voxelization. United States: N. p., 2015.
Web. doi:10.1016/j.jcp.2015.05.022.
Powell, Devon, & Abel, Tom. An exact general remeshing scheme applied to physically conservative voxelization. United States. https://doi.org/10.1016/j.jcp.2015.05.022
Powell, Devon, and Abel, Tom. Tue .
"An exact general remeshing scheme applied to physically conservative voxelization". United States. https://doi.org/10.1016/j.jcp.2015.05.022.
@article{osti_1228391,
title = {An exact general remeshing scheme applied to physically conservative voxelization},
author = {Powell, Devon and Abel, Tom},
abstractNote = {We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass, momentum, and energy conservation while applying higher-order polynomial interpolation. We elaborate on applications of our algorithm arising in the analysis of cosmological N-body data, computer graphics, and continuum mechanics problems. We focus on the particular case of remeshing tetrahedral cells onto a Cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh. We refer to this as “physically conservative voxelization.” At the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other. This algorithm is an implementation of the ideas presented abstractly by Sugihara [48], who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output. This makes our implementation robust to geometric degeneracy in the input. We employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain. We also address practical issues arising in a software implementation, including numerical stability in geometric calculations, management of cancellation errors, and extension to two dimensions. In a comparison to recent work, we show substantial performance gains. We provide a C implementation intended to be a fast, accurate, and robust tool for geometric calculations on polyhedral mesh elements.},
doi = {10.1016/j.jcp.2015.05.022},
journal = {Journal of Computational Physics},
number = C,
volume = 297,
place = {United States},
year = {Tue Sep 01 00:00:00 EDT 2015},
month = {Tue Sep 01 00:00:00 EDT 2015}
}
https://doi.org/10.1016/j.jcp.2015.05.022
Web of Science
Works referenced in this record:
E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh
journal, January 2010
- Springel, Volker
- Monthly Notices of the Royal Astronomical Society, Vol. 401, Issue 2
Generalizing the Formula for Areas of Polygons to Moments
journal, February 1989
- Bockman, S. F.
- The American Mathematical Monthly, Vol. 96, Issue 2
Conservative voxelization
journal, June 2007
- Zhang, Long; Chen, Wei; Ebert, David S.
- The Visual Computer, Vol. 23, Issue 9-11
Accurate Conservative Remapping (Rezoning) for Arbitrary Lagrangian-Eulerian Computations
journal, May 1987
- Dukowicz, John K.; Kodis, John W.
- SIAM Journal on Scientific and Statistical Computing, Vol. 8, Issue 3
A Novel Approach to Visualizing Dark Matter Simulations
journal, December 2012
- Kaehler, R.; Hahn, O.; Abel, T.
- IEEE Transactions on Visualization and Computer Graphics, Vol. 18, Issue 12
Enzo: an Adaptive mesh Refinement code for Astrophysics
journal, March 2014
- Bryan, Greg L.; Norman, Michael L.; O'Shea, Brian W.
- The Astrophysical Journal Supplement Series, Vol. 211, Issue 2
Two design principles of geometric algorithms in finite-precision arithmetic
journal, January 1989
- Sugihara, Kokichi; Iri, Masao
- Applied Mathematics Letters, Vol. 2, Issue 2
The Universe at extreme scale: Multi-petaflop sky simulation on the BG/Q
conference, November 2012
- Habib, Salman; Morozov, Vitali; Finkel, Hal
- 2012 SC - International Conference for High Performance Computing, Networking, Storage and Analysis, 2012 International Conference for High Performance Computing, Networking, Storage and Analysis
Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids
journal, February 1998
- Liu, Yen; Vinokur, Marcel
- Journal of Computational Physics, Vol. 140, Issue 1
A new approach to simulating collisionless dark matter fluids
journal, July 2013
- Hahn, Oliver; Abel, Tom; Kaehler, Ralf
- Monthly Notices of the Royal Astronomical Society, Vol. 434, Issue 2
Analytical and geometrical tools for 3D volume of fluid methods in general grids
journal, June 2008
- López, J.; Hernández, J.
- Journal of Computational Physics, Vol. 227, Issue 12
Cosmological hydrodynamics with adaptive mesh refinement: A new high resolution code called RAMSES
journal, April 2002
- Teyssier, R.
- Astronomy & Astrophysics, Vol. 385, Issue 1
Second-order sign-preserving conservative interpolation (remapping) on general grids
journal, January 2003
- Margolin, L. G.; Shashkov, Mikhail
- Journal of Computational Physics, Vol. 184, Issue 1
Lagrangian-Eulerian finite element formulation for incompressible viscous flows
journal, December 1981
- Hughes, Thomas J. R.; Liu, Wing Kam; Zimmermann, Thomas K.
- Computer Methods in Applied Mechanics and Engineering, Vol. 29, Issue 3
Exact formulae for areas, volumes and moments of polygons and polyhedra
journal, November 1988
- Liggett, James A.
- Communications in Applied Numerical Methods, Vol. 4, Issue 6
ReALE: A reconnection-based arbitrary-Lagrangian–Eulerian method
journal, June 2010
- Loubère, Raphaël; Maire, Pierre-Henri; Shashkov, Mikhail
- Journal of Computational Physics, Vol. 229, Issue 12
Analytic Anti-Aliasing of Linear Functions on Polytopes
journal, May 2012
- Auzinger, T.; Guthe, M.; Jeschke, S.
- Computer Graphics Forum, Vol. 31, Issue 2pt1
Numerical Simulation of Moving Contact Line Problems Using a Volume-of-Fluid Method
journal, July 2001
- Renardy, Michael; Renardy, Yuriko; Li, Jie
- Journal of Computational Physics, Vol. 171, Issue 1
zobov: a parameter-free void-finding algorithm
journal, June 2008
- Neyrinck, Mark C.
- Monthly Notices of the Royal Astronomical Society, Vol. 386, Issue 4
A polyhedron clipping and capping algorithm and a display system for three dimensional finite element models
journal, September 1975
- Stephenson, Michael B.; Christiansen, Henry N.
- ACM SIGGRAPH Computer Graphics, Vol. 9, Issue 3
A Mnemonic for Areas of Polygons
journal, June 1986
- Stone, M. G.
- The American Mathematical Monthly, Vol. 93, Issue 6
Volume of fluid (VOF) method for the dynamics of free boundaries
journal, January 1981
- Hirt, C. W.; Nichols, B. D.
- Journal of Computational Physics, Vol. 39, Issue 1
A hidden-surface algorithm with anti-aliasing
journal, August 1978
- Catmull, Edwin
- ACM SIGGRAPH Computer Graphics, Vol. 12, Issue 3
Fast and Accurate Computation of Polyhedral Mass Properties
journal, January 1996
- Mirtich, Brian
- Journal of Graphics Tools, Vol. 1, Issue 2
Local Robustness and its Application to Polyhedral Intersection
journal, March 1994
- Stewart, A. James
- International Journal of Computational Geometry & Applications, Vol. 04, Issue 01
2HOT: an improved parallel hashed oct-tree n-body algorithm for cosmological simulation
conference, January 2013
- Warren, Michael S.
- Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis on - SC '13
The cosmological simulation code gadget-2
journal, December 2005
- Springel, Volker
- Monthly Notices of the Royal Astronomical Society, Vol. 364, Issue 4
An efficient linearity and bound preserving conservative interpolation (remapping) on polyhedral meshes
journal, February 2007
- Garimella, Rao; Kucharik, Milan; Shashkov, Mikhail
- Computers & Fluids, Vol. 36, Issue 2
An adaptively refined phase–space element method for cosmological simulations and collisionless dynamics
journal, November 2015
- Hahn, Oliver; Angulo, Raul E.
- Monthly Notices of the Royal Astronomical Society, Vol. 455, Issue 1
Tracing the dark matter sheet in phase space: Tracing the dark matter sheet
journal, October 2012
- Abel, Tom; Hahn, Oliver; Kaehler, Ralf
- Monthly Notices of the Royal Astronomical Society, Vol. 427, Issue 1
A Robust and Consistent Algorithm for Intersecting Convex Polyhedra
journal, August 1994
- Sugihara, Kokichi
- Computer Graphics Forum, Vol. 13, Issue 3
Conservative Remapping and Region Overlays by Intersecting Arbitrary Polyhedra
journal, January 1999
- Grandy, Jeffrey
- Journal of Computational Physics, Vol. 148, Issue 2
Towards noiseless gravitational lensing simulations
journal, September 2014
- Angulo, Raul E.; Chen, Ruizhu; Hilbert, Stefan
- Monthly Notices of the Royal Astronomical Society, Vol. 444, Issue 3
Adaptive Refinement Tree: A New High‐Resolution N ‐Body Code for Cosmological Simulations
journal, July 1997
- Kravtsov, Andrey V.; Klypin, Anatoly A.; Khokhlov, Alexei M.
- The Astrophysical Journal Supplement Series, Vol. 111, Issue 1
Monotonicity in high-order curvilinear finite element arbitrary Lagrangian-Eulerian remap: MONOTONICITY IN HIGH-ORDER CURVILINEAR FINITE ELEMENT ALE REMAP
journal, October 2014
- Anderson, R. W.; Dobrev, V. A.; Kolev, Tz. V.
- International Journal for Numerical Methods in Fluids, Vol. 77, Issue 5
Reentrant polygon clipping
journal, January 1974
- Sutherland, Ivan E.; Hodgman, Gary W.
- Communications of the ACM, Vol. 17, Issue 1
On finite element integration in natural co-ordinates
journal, January 1973
- Eisenberg, Martin A.; Malvern, Lawrence E.
- International Journal for Numerical Methods in Engineering, Vol. 7, Issue 4
Nyx: A MASSIVELY PARALLEL AMR CODE FOR COMPUTATIONAL COSMOLOGY
journal, February 2013
- Almgren, Ann S.; Bell, John B.; Lijewski, Mike J.
- The Astrophysical Journal, Vol. 765, Issue 1