A Cut-Cell Geometric Multigrid Poisson Solver for Fluid Simulation
We present a novel multigrid scheme based on a cut‐cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved r...
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| Vydáno v: | Computer graphics forum Ročník 34; číslo 2; s. 481 - 491 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Oxford
Blackwell Publishing Ltd
01.05.2015
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| Témata: | |
| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | We present a novel multigrid scheme based on a cut‐cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved rate of convergence compared to previous methods. Existing fluid solvers with voxelized domains can directly benefit from this approach by only modifying the representation of the non‐fluid domain. The necessary building blocks are fully parallelizable and can therefore benefit from multi‐ and many‐core architectures. |
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| AbstractList | We present a novel multigrid scheme based on a cut‐cell formulation on regular staggered grids which generates compatible systems of linear equations on all levels of the multigrid hierarchy. This geometrically motivated formulation is derived from a finite volume approach and exhibits an improved rate of convergence compared to previous methods. Existing fluid solvers with voxelized domains can directly benefit from this approach by only modifying the representation of the non‐fluid domain. The necessary building blocks are fully parallelizable and can therefore benefit from multi‐ and many‐core architectures. |
| Author | Weber, Daniel Mueller-Roemer, Johannes Fellner, Dieter Stork, André |
| Author_xml | – sequence: 1 givenname: Daniel surname: Weber fullname: Weber, Daniel organization: TU Darmstadt & Fraunhofer IGD, Germany – sequence: 2 givenname: Johannes surname: Mueller-Roemer fullname: Mueller-Roemer, Johannes organization: TU Darmstadt & Fraunhofer IGD, Germany – sequence: 3 givenname: André surname: Stork fullname: Stork, André organization: TU Darmstadt & Fraunhofer IGD, Germany – sequence: 4 givenname: Dieter surname: Fellner fullname: Fellner, Dieter organization: TU Darmstadt & Fraunhofer IGD, Germany |
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| Copyright | 2015 The Author(s) Computer Graphics Forum © 2015 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2015 The Eurographics Association and John Wiley & Sons Ltd. |
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| References_xml | – reference: Ng Y.T., Min C., Gibou F.: An efficient fluid-solid coupling algorithm for single-phase flows. J. Comput. Phys. 228, 23 (Dec. 2009), 8807-8829. 2, 3, 4, 9 – reference: Batty C., Bertails F., Bridson R.: A fast variational framework for accurate solid-fluid coupling. ACM Trans. Graph. 26, 3 (July 2007). 2, 3 – reference: Selle A., Rasmussen N., Fedkiw R.: A vortex particle method for smoke, water and explosions. ACM Trans. Graph. 24, 3 (July 2005), 910-914. 3 – reference: Da F., Batty C., Grinspun E.: Multimaterial mesh-based surface tracking. ACM Trans. on Graphics (SIGGRAPH 2014) (2014). 3 – reference: Gibou F., Fedkiw R.P., Cheng L.-T., Kang M.: A second-order-accurate symmetric discretization of the poisson equation on irregular domains. J. Comput. Phys. 176, 1 (Feb. 2002), 205-227. 3 – reference: Wan J. W. L., dong Liu X.: A boundary condition-capturing multigrid approach to irregular boundary problems. SIAMJ. Sci. Comput. 25, 6 (2004), 1982-2003. 2 – reference: Hellrung Jr J.L., Wang L., Sifakis E., Teran J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. Journal of Computational Physics 231, 4 (2012), 2015-2048. 2 – reference: Pfaff T., Thuerey N., Gross M.: Lagrangian vortex sheets for animating fluids. ACM Trans. Graph. 31, 4 (July 2012), 112:1-112:8. 3 – reference: Trottenberg U., Schuller A.: Multigrid. Academic Press, Inc., Orlando, FL, USA, 2001. 2, 5, 9 – reference: Edwards E., Bridson R.: Detailed water with coarse grids: Combining surface meshes and adaptive discontinuous galerkin. ACM Trans. Graph. 33, 4 (July 2014), 136:1-136:9. 3 – reference: Harlow F.H., Welch J.E.: Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface. The Physics of Fluids 8, 12 (1965), 2182-2189. 3 – reference: Ferstl F., Westermann R., Dick C.: Large-scale liquid simulation on adaptive hexahedral grids. Visualization and Computer Graphics, IEEE Transactions on 20, 10 (Oct 2014), 1405-1417. 2, 9 – reference: Bridson R.: Fluid Simulation For Computer Graphics. Ak Peters Series. A K Peters, Limited, 2008. 1, 3, 6 – reference: Feldman B.E., O'Brien J.F., Klingner B.M.: Animating gases with hybrid meshes. ACM Trans. Graph. 24, 3 (July 2005), 904-909. 3 – reference: Lentine M., Zheng W., Fedkiw R.: A novel algorithm for incompressible flow using only a coarse grid projection. ACM Trans. Graph. 29 (July 2010), 114:1-114:9. 3, 9 – reference: Setaluri R., Aanjaneya M., Bauer S., Sifakis E.: Spgrid: A sparse paged grid structure applied to adaptive smoke simulation. ACM Trans. Graph. 33, 6 (Nov. 2014), 205:1-205:12. 2 – reference: Weber D., Bender J., Schnoes M., Stork A., Fellner D.: Efficient GPU data structures and methods to solve sparse linear systems in dynamics applications. Computer Graphics Forum 32, 1 (2013), 16-26. 6 – reference: Foster N., Metaxas D.: Realistic animation of liquids. Graphical models and image processing 58, 5 (1996), 471-483. 1, 3 – reference: Johansen H., Colella P.: A cartesian grid embedded boundary method for poisson's equation on irregular domains. Journal of Computational Physics 147, 1 (1998), 60-85. 2 – reference: Schwartz P., Barad M., Colella P., Ligocki T.: A cartesian grid embedded boundary method for the heat equation and poisson's equation in three dimensions. Journal of Computational Physics 211, 2 (2006), 531-550. 2 – reference: Chentanez N., Müller M.: Real-time Eulerian water simulation using a restricted tall cell grid. ACM Trans. Graph. 30, 4 (July 2011), 82:1-82:10. 1, 2, 4, 7 – reference: Wojtan C., Thürey N., Gross M., Turk G.: Deforming meshes that split and merge. ACM Trans. Graph. 28, 3 (July 2009), 76:1-76:10. 3 – reference: Zhu B., Lu W., Cong M., Kim B., Fedkiw R.: A new grid structure for domain extension. ACM Trans. Graph. 32, 4 (July 2013), 63:1-63:12. 3 – reference: Li Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 1 (Feb. 1998), 230-254. 9 – start-page: 219 end-page: 228 – volume: 25 start-page: 1982 issue: 6 year: 2004 end-page: 2003 article-title: A boundary condition‐capturing multigrid approach to irregular boundary problems publication-title: SIAMJ. Sci. Comput – start-page: 83 year: 2011 end-page: 90 – year: 2001 – year: 2003 – volume: 32 start-page: 16 issue: 1 year: 2013 end-page: 26 article-title: Efficient GPU data structures and methods to solve sparse linear systems in dynamics applications publication-title: Computer Graphics Forum – volume: 35 start-page: 230 issue: 1 year: 1998 end-page: 254 article-title: A fast iterative algorithm for elliptic interface problems publication-title: SIAM J. Numer. Anal – year: 2000 – start-page: 457 year: 2004 end-page: 462 – volume: 31 start-page: 112:1 issue: 4 year: 2012 end-page: 112:8 article-title: Lagrangian vortex sheets for animating fluids publication-title: ACM Trans. Graph – volume: 20 start-page: 1405 issue: 10 year: 2014 end-page: 1417 article-title: Large‐scale liquid simulation on adaptive hexahedral grids publication-title: Visualization and Computer Graphics, IEEE Transactions on – start-page: 121 year: 1999 end-page: 128 – volume: 228 start-page: 8807 issue: 23 year: 2009 end-page: 8829 article-title: An efficient fluid‐solid coupling algorithm for single‐phase flows publication-title: J. Comput. Phys – start-page: 121:1 year: 2009 end-page: 121:10 – volume: 231 start-page: 2015 issue: 4 year: 2012 end-page: 2048 article-title: A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions publication-title: Journal of Computational Physics – volume: 32 start-page: 63:1 issue: 4 year: 2013 end-page: 63:12 article-title: A new grid structure for domain extension publication-title: ACM Trans. Graph – volume: 8 start-page: 2182 issue: 12 year: 1965 end-page: 2189 article-title: Numerical calculation of time‐dependent viscous incompressible flow of fluid with a free surface publication-title: The Physics of Fluids – volume: 30 start-page: 82:1 issue: 4 year: 2011 end-page: 82:10 article-title: Real‐time Eulerian water simulation using a restricted tall cell grid publication-title: ACM Trans. Graph – start-page: 1 year: 2003 end-page: 1 – volume: 176 start-page: 205 issue: 1 year: 2002 end-page: 227 article-title: A second‐order‐accurate symmetric discretization of the poisson equation on irregular domains publication-title: J. Comput. Phys – volume: 29 start-page: 695 year: 2010 end-page: 704 – volume: 24 start-page: 904 issue: 3 year: 2005 end-page: 909 article-title: Animating gases with hybrid meshes publication-title: ACM Trans. Graph – volume: 33 start-page: 136:1 issue: 4 year: 2014 end-page: 136:9 article-title: Detailed water with coarse grids: Combining surface meshes and adaptive discontinuous galerkin publication-title: ACM Trans. Graph – volume: 211 start-page: 531 issue: 2 year: 2006 end-page: 550 article-title: A cartesian grid embedded boundary method for the heat equation and poisson's equation in three dimensions publication-title: Journal of Computational Physics – year: 2008 – volume: 26 issue: 3 year: 2007 article-title: A fast variational framework for accurate solid‐fluid coupling publication-title: ACM Trans. Graph – volume: 33 start-page: 205:1 issue: 6 year: 2014 end-page: 205:12 article-title: Spgrid: A sparse paged grid structure applied to adaptive smoke simulation publication-title: ACM Trans. Graph – volume: 3 year: 2014 article-title: Multimaterial mesh‐based surface tracking publication-title: ACM Trans. on Graphics (SIGGRAPH 2014) – volume: 58 start-page: 471 issue: 5 year: 1996 end-page: 483 article-title: Realistic animation of liquids publication-title: Graphical models and image processing – volume: 29 start-page: 114:1 year: 2010 end-page: 114:9 article-title: A novel algorithm for incompressible flow using only a coarse grid projection publication-title: ACM Trans. Graph – volume: 147 start-page: 60 issue: 1 year: 1998 end-page: 85 article-title: A cartesian grid embedded boundary method for poisson's equation on irregular domains publication-title: Journal of Computational Physics – start-page: 65 year: 2010 end-page: 74 – start-page: 15 year: 2001 end-page: 22 – volume: 24 start-page: 910 issue: 3 year: 2005 end-page: 914 article-title: A vortex particle method for smoke, water and explosions publication-title: ACM Trans. Graph – volume: 28 start-page: 76:1 issue: 3 year: 2009 end-page: 76:10 article-title: Deforming meshes that split and merge publication-title: ACM Trans. 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| SubjectTerms | Analysis Categories and Subject Descriptors (according to ACM CCS) Computational fluid dynamics Computer graphics Convergence Fluid flow Fluids Formulations Hierarchies I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Physically based modeling I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism -Animation Image processing systems Mathematical analysis Simulation Solvers Studies |
| Title | A Cut-Cell Geometric Multigrid Poisson Solver for Fluid Simulation |
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