research-article

Open access

Spiral-spectral fluid simulation

Authors:

Timothy Langlois,

Theodore KimAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 40, Issue 6

Article No.: 202, Pages 1 - 16

https://doi.org/10.1145/3478513.3480536

Published: 10 December 2021 Publication History

Abstract

We introduce a fast, expressive method for simulating fluids over radial domains, including discs, spheres, cylinders, ellipses, spheroids, and tori. We do this by generalizing the spectral approach of Laplacian Eigenfunctions, resulting in what we call spiral-spectral fluid simulations. Starting with a set of divergence-free analytical bases for polar and spherical coordinates, we show that their singularities can be removed by introducing a set of carefully selected enrichment functions. Orthogonality is established at minimal cost, viscosity is supported analytically, and we specifically design basis functions that support scalable FFT-based reconstructions. Additionally, we present an efficient way of computing all the necessary advection tensors. Our approach applies to both three-dimensional flows as well as their surface-based, codimensional variants. We establish the completeness of our basis representation, and compare against a variety of existing solvers.

Supplementary Material

ZIP File (a202-cui.zip)

Supplemental files.

Download
10.75 MB

MP4 File (a202-cui.mp4)

Download
410.04 MB

References

[1]

Alexis Angelidis. 2017. Multi-Scale Vorticle Fluids. ACM Trans. Graph. 36, 4, Article 104 (July 2017), 12 pages.

Digital Library

[2]

Alexis Angelidis and Fabrice Neyret. 2005. Simulation of Smoke Based on Vortex Filament Primitives. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 87--96.

Digital Library

[3]

Ferdinand Baer and George W Platzman. 1961. A Procedure for Numerical Integration of the Spectral Vorticity Equation. Journal of Meteorology 18, 3 (Jun 1961), 393--401.

[4]

Jernej Barbič and Doug L. James. 2005. Real-Time Subspace Integration for St. Venant-Kirchhoff Deformable Models. ACM Trans. Graph. 24, 3 (July 2005), 982--990.

Digital Library

[5]

Rubén G Barrera, GA Estevez, and J Giraldo. 1985. Vector spherical harmonics and their application to magnetostatics. European Journal of Physics 6, 4 (1985), 287.

[6]

John P Boyd. 1978. The Choice of Spectral Functions on a Sphere for Boundary and Eigenvalue Problems: A Comparison of Chebyshev, Fourier and Associated Legendre Expansions. Mon. Weather Rev. 106, 8 (1978), 1184 -- 1191.

[7]

John P Boyd. 2001. Chebyshev and Fourier spectral methods. Dover Publications.

[8]

John P Boyd and Fu Yu. 2011. Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions. J. Comput. Phys. 230, 4 (2011), 1408--1438.

Digital Library

[9]

Robert Bridson. 2015. Fluid simulation for computer graphics. CRC Press.

Digital Library

[10]

Keaton J Burns, Geoffrey M Vasil, Jeffrey S Oishi, Daniel Lecoanet, and Benjamin P Brown. 2020. Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Review Research 2, 2 (2020), 023068.

[11]

Qiaodong Cui, Pradeep Sen, and Theodore Kim. 2018. Scalable Laplacian eigenfluids. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1--12.

Digital Library

[12]

Tyler De Witt, Christian Lessig, and Eugene Fiume. 2012. Fluid Simulation Using Laplacian Eigenfunctions. ACM Trans. Graph. 31, 1, Article 10 (2012), 11 pages.

Digital Library

[13]

John B Drake. 2014. Climate modeling for scientists and engineers. SIAM.

Digital Library

[14]

Stefen Fangmeier. 1996. Designing digital tornados. Comput. Graph. World 19, 8 (1996), 58.

[15]

Matteo Frigo and Steven G. Johnson. 2005. The Design and Implementation of FFTW3. Proc. IEEE 93, 2 (2005), 216--231. Special issue on "Program Generation, Optimization, and Platform Adaptation".

[16]

Dan Gerszewski, Ladislav Kavan, Peter-Pike Sloan, and Adam W. Bargteil. 2015. Basis Enrichment and Solid-fluid Coupling for Model-reduced Fluid Simulation. Computer Animation and Virtual Worlds 26 (Mar/Apr 2015).

Digital Library

[17]

Gene H Golub and Charles F Van Loan. 2012. Matrix computations. Vol. 3. JHU Press.

[18]

Denis S Grebenkov and B-T Nguyen. 2013. Geometrical structure of Laplacian eigenfunctions. siam REVIEW 55, 4 (2013), 601--667.

[19]

Michael Griebel, Thomas Dornseifer, and Tilman Neunhoeffer. 1998. Numerical simulation in fluid dynamics: a practical introduction. SIAM.

Digital Library

[20]

Mohit Gupta and Srinivasa G. Narasimhan. 2007. Legendre Fluids: A Unified Framework for Analytic Reduced Space Modeling and Rendering of Participating Media. In Symposium on Computer Animation. 17--25.

Digital Library

[21]

Jonah Hall. 2004. The Day After Tomorrow Twister Sequence Toolkit: A Volumetric Tornado Pipeline. In ACM SIGGRAPH Sketches (Los Angeles, California). 143.

Digital Library

[22]

Milos Hasan, Edgar Velazquez-Armendariz, Fabio Pellacini, and Kavita Bala. 2008. Tensor Clustering for Rendering Many-Light Animations. Computer Graphics Forum 27, 4 (2008), 1105--1114.

Digital Library

[23]

Ronald D. Henderson. 2012. Scalable Fluid Simulation in Linear Time on Shared Memory Multiprocessors. In Proceedings of the Digital Production Symposium. 43--52.

Digital Library

[24]

David J Hill and Ronald D Henderson. 2016. Efficient fluid simulation on the surface of a sphere. ACM Transactions on Graphics (TOG) 35, 2 (2016), 1--9.

Digital Library

[25]

EL Hill. 1954. The theory of vector spherical harmonics. American Journal of Physics 22, 4 (1954), 211--214.

[26]

Weizhen Huang, Julian Iseringhausen, Tom Kneiphof, Ziyin Qu, Chenfanfu Jiang, and Matthias B Hullin. 2020. Chemomechanical simulation of soap film flow on spherical bubbles. ACM Transactions on Graphics (TOG) 39, 4 (2020), 41--1.

Digital Library

[27]

Aaron Demby Jones, Pradeep Sen, and Theodore Kim. 2016. Compressing Fluid Sub-spaces. In Symposium on Computer Animation. 77--84.

Digital Library

[28]

Florian Karpfinger, Boris Gurevich, and Andrey Bakulin. 2008. Modeling of wave dispersion along cylindrical structures using the spectral method. The Journal of the Acoustical Society of America 124, 2 (2008), 859--865.

[29]

Doyub Kim. 2017. Fluid engine development. CRC Press.

[30]

Theodore Kim and John Delaney. 2013. Subspace Fluid Re-simulation. ACM Trans. Graph. 32, 4, Article 62 (2013), 9 pages.

Digital Library

[31]

Granino Arthur Korn and Theresa M Korn. 2000. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. Courier Corporation.

[32]

Jakob Kühnen, Baofang Song, Davide Scarselli, Nazmi Burak Budanur, Michael Riedl, Ashley P Willis, Marc Avila, and Björn Hof. 2018. Destabilizing turbulence in pipe flow. Nature Physics 14, 4 (2018), 386--390.

[33]

Daniel Lecoanet, Geoffrey M Vasil, Keaton J Burns, Benjamin P Brown, and Jeffrey S Oishi. 2019. Tensor calculus in spherical coordinates using Jacobi polynomials. Part-II: Implementation and examples. J. Comput. Phys: X 3 (2019), 100012.

[34]

Christian Lessig. 2019. Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows. J. Math. Fluid Mech. 21, 18 (Feb 2019).

[35]

Beibei Liu, Gemma Mason, Julian Hodgson, Yiying Tong, and Mathieu Desbrun. 2015. Model-reduced Variational Fluid Simulation. ACM Trans. Graph. 34, 6, Article 244 (2015), 12 pages.

Digital Library

[36]

Benjamin Long and Erik Reinhard. 2009. Real-time Fluid Simulation Using Discrete Sine/Cosine Transforms. In Symp. on Interactive 3D Graphics and Games. 99--106.

Digital Library

[37]

Olivier Mercier and Derek Nowrouzezahrai. 2020. Local Bases for Model-reduced Smoke Simulations. In Comput. Graph. Forum, Vol. 39. Wiley Online Library, 9--22.

[38]

Martin J Mohlenkamp. 1999. A fast transform for spherical harmonics. Journal of Fourier analysis and applications 5, 2-3 (1999), 159--184.

[39]

Youhei Morinishi, Oleg V Vasilyev, and Takeshi Ogi. 2004. Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations. J. Comput. Phys. 197, 2 (2004), 686--710.

Digital Library

[40]

Matthias Müller, David Charypar, and Markus Gross. 2003. Particle-Based Fluid Simulation for Interactive Applications. In Proc. of Symposium on Computer Animation, D. Breen and M. Lin (Eds.).

Digital Library

[41]

Rahul Narain, Jason Sewall, Mark Carlson, and Ming C. Lin. 2008. Fast Animation of Turbulence Using Energy Transport and Procedural Synthesis. ACM Trans. Graph. 27, 5, Article 166 (Dec. 2008), 8 pages.

Digital Library

[42]

Steven A Orszag. 1974. Fourier Series on Spheres. Mon. Weather Rev. 102, 1 (1974), 56 -- 75.

[43]

Marcel Padilla, Albert Chern, Felix Knöppel, Ulrich Pinkall, and Peter Schröder. 2019. On Bubble Rings and Ink Chandeliers. ACM Trans. Graph. 38, 4, Article 129 (July 2019), 14 pages.

Digital Library

[44]

Sang Il Park and Myoung Jun Kim. 2005. Vortex fluid for gaseous phenomena. In Proc. of the ACM SIGGRAPH/Eurographics Symp. on Comput. Anim. 261--270.

Digital Library

[45]

Tobias Pfaff, Nils Thuerey, Jonathan Cohen, Sarah Tariq, and Markus Gross. 2010. Scalable Fluid Simulation Using Anisotropic Turbulence Particles. ACM Trans. Graph. 29, 6, Article 174 (Dec. 2010), 8 pages.

Digital Library

[46]

Tobias Pfaff, Nils Thuerey, Andrew Selle, and Markus Gross. 2009. Synthetic Turbulence Using Artificial Boundary Layers. ACM Trans. Graph. 28, 5 (Dec. 2009), 1--10.

Digital Library

[47]

Nick Rasmussen, Duc Quang Nguyen, Willi Geiger, and Ronald Fedkiw. 2003. Smoke Simulation for Large Scale Phenomena. ACM Trans. Graph. 22, 3 (July 2003), 703--707.

Digital Library

[48]

Osborne Reynolds. 1883. XXIX. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Philosophical Transactions of the Royal society of London 174 (1883), 935--982.

[49]

Yousef Saad. 2003. Iterative methods for sparse linear systems. SIAM.

Digital Library

[50]

Andrew Selle, Nick Rasmussen, and Ronald Fedkiw. 2005. A Vortex Particle Method for Smoke, Water and Explosions. ACM Trans. Graph. 24, 3 (July 2005), 910--914.

Digital Library

[51]

Isadore Silberman. 1954. Planetary Waves in the Atmosphere. Journal of Meteorology 1 (Feb 1954), 27--34.

[52]

Richard Mikael Slevinsky, Hadrien Montanelli, and Qiang Du. 2018. A spectral method for nonlocal diffusion operators on the sphere. J. Comput. Phys. 372 (2018), 893--911.

Digital Library

[53]

J. Stam. 1999. Stable Fluids. In Proceedings of SIGGRAPH. 121--128.

Digital Library

[54]

J. Stam. 2002. A Simple Fluid Solver Based on the FFT. J. Graph. Tools 6, 2 (Sept. 2002), 43--52.

Digital Library

[55]

J. Stam. 2003. Flows on surfaces of arbitrary topology. ACM Trans. Graph. 22, 3 (2003), 724--731.

Digital Library

[56]

Matt Stanton, Yu Sheng, Martin Wicke, Federico Perazzi, Amos Yuen, Srinivasa Narasimhan, and Adrien Treuille. 2013. Non-polynomial Galerkin Projection on Deforming Meshes. ACM Trans. Graph. 32, 4, Article 86 (July 2013), 14 pages.

Digital Library

[57]

Alexey Stomakhin, Craig Schroeder, Lawrence Chai, Joseph Teran, and Andrew Selle. 2013. A Material Point Method for Snow Simulation. ACM Trans. Graph. 32, 4, Article 102 (July 2013), 10 pages.

Digital Library

[58]

P. Tamain, H. Bufferand, G. Ciraolo, C. Colin, D. Galassi, Ph. Ghendrih, F. Schwander, and E. Serre. 2016. The TOKAM3X code for edge turbulence fluid simulations of tokamak plasmas in versatile magnetic geometries. J. Comput. Phys. 321 (2016), 606 -- 623.

Digital Library

[59]

Demetri Terzopoulos and Andrew Witkin. 1988. Physically based models with rigid and deformable components. IEEE CG&A 8, 6 (1988), 41--51.

Digital Library

[60]

Nils Thuerey and Tobias Pfaff. 2018. MantaFlow. http://mantaflow.com.

[61]

Adrien Treuille, Andrew Lewis, and Zoran Popović. 2006. Model Reduction for Real-time Fluids. ACM Trans. Graph. 25, 3 (July 2006), 826--834.

Digital Library

[62]

Geoffrey M Vasil, Daniel Lecoanet, Keaton J Burns, Jeffrey S Oishi, and Benjamin P Brown. 2019. Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: Mathematical analysis and derivations. J. Comput. Phys: X 3 (2019), 100013.

Digital Library

[63]

N. P. Wedi, M. Hamrud, and G. Mozdzynski. 2013. A Fast Spherical Harmonics Transform for Global NWP and Climate Models. Mon. Weather Rev. 141, 10 (oct 2013), 3450--3461.

[64]

Steffen Weissmann, Ulrich Pinkall, and Peter Schröder. 2014. Smoke Rings from Smoke. 4, Article 140 (July 2014), 8 pages.

Digital Library

[65]

Larry Yaeger, Craig Upson, and Robert Myers. 1986. Combining Physical and Visual Simulation---Creation of the Planet Jupiter for the Film "2010". In Proceedings of SIGGRAPH. 85--93.

Digital Library

[66]

Bowen Yang, William Corse, Jiecong Lu, Joshuah Wolper, and Chen-Fanfu Jiang. 2019. Real-Time Fluid Simulation on the Surface of a Sphere. Proceedings of the ACM on Computer Graphics and Interactive Techniques 2, 1 (2019), 1--17.

Digital Library

[67]

Can Yuksel, Domin Lee, and Ronald Henderson. 2012. Vortex of Awesomeness. In ACM SIGGRAPH Talks (Los Angeles, California).

[68]

Ben Zhu, Manaure Francisquez, and Barrett N. Rogers. 2018. GDB: A global 3D two-fluid model of plasma turbulence and transport in the tokamak edge. Computer Physics Communications 232 (2018), 46 -- 58.

[69]

Yongning Zhu and Robert Bridson. 2005. Animating Sand As a Fluid. ACM Trans. Graph. 24, 3 (July 2005), 965--972.

Digital Library

Cited By

Cen RRen B(2025)Layer-Based Simulation for Three-Dimensional Fluid Flow in Spherical CoordinatesIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2024.336052131:2(1435-1447)Online publication date: Feb-2025
https://doi.org/10.1109/TVCG.2024.3360521
Luo TLiu YPan S(2024)Collaborative Sequential Recommendations via Multi-view GNN-transformersACM Transactions on Information Systems10.1145/364943642:6(1-27)Online publication date: 25-Jun-2024
https://dl.acm.org/doi/10.1145/3649436
Wang XXu YLiu SRen BKosinka JTelea AWang JSong CChang JLi CZhang JBan X(2024)Physics-based fluid simulation in computer graphics: Survey, research trends, and challengesComputational Visual Media10.1007/s41095-023-0368-y10:5(803-858)Online publication date: 27-Apr-2024
https://doi.org/10.1007/s41095-023-0368-y
Show More Cited By

Index Terms

Spiral-spectral fluid simulation
1. Computing methodologies
  1. Computer graphics
    1. Animation
      1. Physical simulation

Recommendations

Multiple-Fluid SPH Simulation Using a Mixture Model

This article presents a versatile and robust SPH simulation approach for multiple-fluid flows. The spatial distribution of different phases or components is modeled using the volume fraction representation, the dynamics of multiple-fluid flows is ...
Realistic and stable simulation of turbulent details behind objects in smoothed-particle hydrodynamics fluids

This paper presents a novel realistic and stable turbulence synthesis method to simulate the turbulent details generated behind objects in smoothed particle hydrodynamics SPH fluids. Firstly, by approximating the boundary layer theory on the fly in SPH ...
Synthetic turbulence using artificial boundary layers

Turbulent vortices in fluid flows are crucial for a visually interesting appearance. Although there has been a significant amount of work on turbulence in graphics recently, these algorithms rely on the underlying simulation to resolve the flow around ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 40, Issue 6

December 2021

1351 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3478513

Issue’s Table of Contents

Copyright © 2021 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 10 December 2021

Published in TOG Volume 40, Issue 6

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Adobe Research

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

12
Total Citations
View Citations
749
Total Downloads

Downloads (Last 12 months)213
Downloads (Last 6 weeks)20

Reflects downloads up to 13 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Cen RRen B(2025)Layer-Based Simulation for Three-Dimensional Fluid Flow in Spherical CoordinatesIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2024.336052131:2(1435-1447)Online publication date: Feb-2025
https://doi.org/10.1109/TVCG.2024.3360521
Luo TLiu YPan S(2024)Collaborative Sequential Recommendations via Multi-view GNN-transformersACM Transactions on Information Systems10.1145/364943642:6(1-27)Online publication date: 25-Jun-2024
https://dl.acm.org/doi/10.1145/3649436
Wang XXu YLiu SRen BKosinka JTelea AWang JSong CChang JLi CZhang JBan X(2024)Physics-based fluid simulation in computer graphics: Survey, research trends, and challengesComputational Visual Media10.1007/s41095-023-0368-y10:5(803-858)Online publication date: 27-Apr-2024
https://doi.org/10.1007/s41095-023-0368-y
Zhang YLiu Y(2024)Point-of-interest recommendation based on LBSN with multi-aspect fusion of social and individual featuresNeural Computing and Applications10.1007/s00521-024-09711-036:20(12163-12184)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s00521-024-09711-0
Xu RHuang WZhao JChen MNie L(2023)A Spatial and Adversarial Representation Learning Approach for Land Use Classification with POIsACM Transactions on Intelligent Systems and Technology10.1145/362782414:6(1-25)Online publication date: 14-Nov-2023
https://dl.acm.org/doi/10.1145/3627824
Wang CZhu YLiu HZang TWang KYu J(2023)Multifaceted Relation-aware Meta-learning with Dual Customization for User Cold-start RecommendationACM Transactions on Knowledge Discovery from Data10.1145/359745817:9(1-27)Online publication date: 18-Jul-2023
https://dl.acm.org/doi/10.1145/3597458
Yin HNabizadeh MWu BWang SChern A(2023)Fluid CohomologyACM Transactions on Graphics10.1145/359240242:4(1-25)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592402
Bing JChen MYang MHuang WGong YNie L(2023)Pre-Trained Semantic Embeddings for POI Categories Based on Multiple ContextsIEEE Transactions on Knowledge and Data Engineering10.1109/TKDE.2022.321885135:9(8893-8904)Online publication date: 1-Sep-2023
https://dl.acm.org/doi/10.1109/TKDE.2022.3218851
Sheibani SShakeri HSheibani R(2023)Four-dimensional trust propagation model for improving the accuracy of recommender systemsThe Journal of Supercomputing10.1007/s11227-023-05278-079:15(16793-16820)Online publication date: 2-May-2023
https://dl.acm.org/doi/10.1007/s11227-023-05278-0
Bai ZZhang SLi PChang Y(2022)Personalized Point-of-Interest Recommendation with Relation-Enhanced Graph Convolutional NetworkProceedings of the 2022 11th International Conference on Networks, Communication and Computing10.1145/3579895.3579934(254-260)Online publication date: 9-Dec-2022
https://dl.acm.org/doi/10.1145/3579895.3579934
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Media

Figures

Other

Tables

View Issue’s Table of Contents