A Schur Complement Preconditioner for Scalable Parallel Fluid Simulation

We present an algorithmically efficient and parallelized domain decomposition based approach to solving Poisson’s equation on irregular domains. Our technique employs the Schur complement method, which permits a high degree of parallel efficiency on multicore systems. We create a novel Schur complement preconditioner which achieves faster convergence, and requires less computation time and memory. This domain decomposition method allows us to apply different linear solvers for different regions of the flow. Subdomains with regular boundaries can be solved with an FFT-based Fast Poisson Solver. We can solve systems with 1,0243 degrees of freedom, and demonstrate its use for the pressure projection step of incompressible liquid and gas simulations. The results demonstrate considerable speedup over preconditioned conjugate gradient methods commonly employed to solve such problems, including a multigrid preconditioned conjugate gradient method.

[1]  Eftychios Sifakis,et al.  Physical simulation for animation and visual effects: parallelization and characterization for chip multiprocessors , 2007, ISCA '07.

[2]  Pradeep Dubey,et al.  Large-scale fluid simulation using velocity-vorticity domain decomposition , 2012, ACM Trans. Graph..

[3]  R.D. Falgout,et al.  An Introduction to Algebraic Multigrid Computing , 2006, Computing in Science & Engineering.

[4]  Robert Bridson,et al.  Animating sand as a fluid , 2005, ACM Trans. Graph..

[5]  Ronald Fedkiw,et al.  Chimera grids for water simulation , 2013, SCA '13.

[6]  Christopher Wojtan,et al.  Liquid surface tracking with error compensation , 2013, ACM Trans. Graph..

[7]  Jos Stam,et al.  Stable fluids , 1999, SIGGRAPH.

[8]  M. Berger,et al.  Adaptive mesh refinement for hyperbolic partial differential equations , 1982 .

[9]  Tarek P. Mathew,et al.  Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations , 2008, Lecture Notes in Computational Science and Engineering.

[10]  Will Tribbey,et al.  Numerical Recipes: The Art of Scientific Computing (3rd Edition) is written by William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, and published by Cambridge University Press, © 2007, hardback, ISBN 978-0-521-88068-8, 1235 pp. , 1987, SOEN.

[11]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[12]  Robert D. Falgout,et al.  Multigrid preconditioned conjugate gradients for the numerical simulation of groundwater flow on the Cray T3D , 1994 .

[13]  Ronald Fedkiw,et al.  Simulating water and smoke with an octree data structure , 2004, ACM Trans. Graph..

[14]  Christopher Wojtan,et al.  Highly adaptive liquid simulations on tetrahedral meshes , 2013, ACM Trans. Graph..

[15]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[16]  James F. O'Brien,et al.  Eurographics/acm Siggraph Symposium on Computer Animation (2007) Liquid Simulation on Lattice-based Tetrahedral Meshes , 2022 .

[17]  Jonathan M. Cohen,et al.  Low viscosity flow simulations for animation , 2008, SCA '08.

[18]  William H. Press,et al.  Numerical Recipes 3rd Edition: The Art of Scientific Computing , 2007 .

[19]  Osamu Tatebe,et al.  The multigrid preconditioned conjugate gradient method , 1993 .

[20]  C.-C. Jay Kuo,et al.  A Domain Decomposition Preconditioner Based on a Change to a Multilevel Nodal Basis , 1991, SIAM J. Sci. Comput..

[21]  Dimitris N. Metaxas,et al.  Realistic Animation of Liquids , 1996, Graphics Interface.

[22]  James F. O'Brien,et al.  Fluid animation with dynamic meshes , 2006, ACM Trans. Graph..

[23]  Rüdiger Westermann,et al.  Solving the Fluid Pressure Poisson Equation Using Multigrid—Evaluation and Improvements , 2016, IEEE Transactions on Visualization and Computer Graphics.

[24]  Robert Bridson,et al.  A fast variational framework for accurate solid-fluid coupling , 2007, ACM Trans. Graph..

[25]  Ronald D. Henderson.  Scalable fluid simulation in linear time on shared memory multiprocessors , 2012, DigiPro '12.

[26]  Christopher Wojtan,et al.  A Practical Method for High‐Resolution Embedded Liquid Surfaces , 2016, Comput. Graph. Forum.

[27]  J. Mandel.  Balancing domain decomposition , 1993 .

[28]  William F. Mitchell,et al.  A Collection of 2D Elliptic Problems for Testing Adaptive Algorithms , 2010 .

[29]  Rüdiger Westermann,et al.  Narrow Band FLIP for Liquid Simulations , 2016, Comput. Graph. Forum.

[30]  Robert Bridson,et al.  Fluid Simulation for Computer Graphics , 2008 .

[31]  O. Widlund,et al.  Iterative Methods for the Solution of Elliptic Problems on Regions, Partitioned Into Substructures , 2015 .

[32]  Hyeong-Seok Ko,et al.  Stretching and wiggling liquids , 2009, ACM Trans. Graph..

[33]  Eftychios Sifakis,et al.  A parallel multigrid Poisson solver for fluids simulation on large grids , 2010, SCA '10.

[34]  R. Glowinski,et al.  Variational formulation and algorithm for trace operation in domain decomposition calculations , 1988 .

[35]  Matthias Müller,et al.  A Multigrid Fluid Pressure Solver Handling Separating Solid Boundary Conditions , 2011, IEEE Transactions on Visualization and Computer Graphics.

[36]  Nicole Spillane,et al.  An Adaptive MultiPreconditioned Conjugate Gradient Algorithm , 2016, SIAM J. Sci. Comput..

[37]  Eftychios Sifakis,et al.  A scalable schur-complement fluids solver for heterogeneous compute platforms , 2016, ACM Trans. Graph..

[38]  Rüdiger Westermann,et al.  Large-Scale Liquid Simulation on Adaptive Hexahedral Grids , 2014, IEEE Transactions on Visualization and Computer Graphics.

[39]  Robert Bridson,et al.  A Multipreconditioned Conjugate Gradient Algorithm , 2005, SIAM J. Matrix Anal. Appl..

[40]  R. Henderson.  Nonlinear dynamics and pattern formation in turbulent wake transition , 1997, Journal of Fluid Mechanics.

[41]  Gene H. Golub,et al.  A Fast Poisson Solver for the Finite Difference Solution of the Incompressible Navier-Stokes Equations , 1998, SIAM J. Sci. Comput..

[42]  Yousef Saad,et al.  ILUT: A dual threshold incomplete LU factorization , 1994, Numer. Linear Algebra Appl..

[43]  X. Cai.  Overlapping Domain Decomposition Methods , 2003 .

[44]  Yang Yang,et al.  A Fast Iterated Orthogonal Projection Framework for Smoke Simulation , 2016, IEEE Transactions on Visualization and Computer Graphics.

[45]  Christopher Batty,et al.  Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids , 2010, Comput. Graph. Forum.

[46]  Tony F. Chan,et al.  The Interface Probing Technique in Domain Decomposition , 1992, SIAM J. Matrix Anal. Appl..

[47]  Tae-Yong Kim,et al.  Coupling 3D Eulerian, Heightfield and Particle Methods for Interactive Simulation of Large Scale Liquid Phenomena , 2014, Symposium on Computer Animation.

[48]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[49]  Ken Museth,et al.  VDB: High-resolution sparse volumes with dynamic topology , 2013, TOGS.