Generalized non-reflecting boundaries for fluid re-simulation

Abstract

When aiming to seamlessly integrate a fluid simulation into a larger scenario (like an open ocean), careful attention must be paid to boundary conditions. In particular, one must implement special "non-reflecting" boundary conditions, which dissipate out-going waves as they exit the simulation. Unfortunately, the state of the art in non-reflecting boundary conditions (perfectly-matched layers, or PMLs) only permits trivially simple inflow/outflow conditions, so there is no reliable way to integrate a fluid simulation into a more complicated environment like a stormy ocean or a turbulent river.

This paper introduces the first method for combining non-reflecting boundary conditions based on PMLs with inflow/outflow boundary conditions that vary arbitrarily throughout space and time. Our algorithm is a generalization of state-of-the-art mean-flow boundary conditions in the computational fluid dynamics literature, and it allows for seamless integration of a fluid simulation into much more complicated environments. Our method also opens the door for previously-unseen post-process effects like retroactively changing the location of solid obstacles, and locally increasing the visual detail of a pre-existing simulation.

References

1. Bécache, E., Fauqueux, S., and Joly, P. 2003. Stability of Perfectly Matched Layers, Group Velocities and Anisotropic Waves. In J. Comput. Phys. 188.2, pp. 399--433. issn: 0021-9991.
2. Berenger, J.-P. 1994. A Perfectly Matched Layer for the Absorption of Electromagnetic Waves. In J. Comput. Phys. 114.2, pp. 185--200. issn: 0021-9991.
3. Bhat, K. S., Seitz, S. M., Hodgins, J. K., and Khosla, P. K. 2004. Flow-based Video Synthesis and Editing. In ACM Transactions on Graphics (SIGGRAPH) 23.3, pp. 360--363.
4. Bridson, R. 2008. Fluid simulation for computer graphics. CRC Press.
5. Fedkiw, R., Stam, J., and Jensen, H. W. 2001. Visual simulation of smoke. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques. ACM, pp. 15--22.
6. Hagstrom, T., Goodrich, J., Nazarov, I., and Dodson, C. 2005. High-order methods and boundary conditions for simulating subsonic flows. In Proceedings of the 11th AIAA/CEAS Aeroacoustics Conference.
7. Hu, F. Q. 2001. A Stable, Perfectly Matched Layer for Linearized Euler Equations in Unsplit Physical Variables. In J. Comput. Phys. 173.2, pp. 455--480. issn: 0021-9991.
8. Hu, F. Q. 2006. On the construction of PML absorbing boundary condition for the non-linear Euler equations. In AIAA paper 798, p. 2006.
9. Hu, F. Q., Li, X., and Lin, D. 2008. Absorbing boundary conditions for nonlinear Euler and Navier--Stokes equations based on the perfectly matched layer technique. In Journal of Computational Physics 227.9, pp. 4398--4424.
10. Hu, F., Hussaini, M., and Manthey, J. 1996. Low-Dissipation and Low-Dispersion Runge-Kutta Schemes for Computational Acoustics. In J. Comput. Phys. 124.1, pp. 177--191. issn: 0021-9991.
11. Kim, T. and Delaney, J. 2013. Subspace fluid re-simulation. In ACM Transactions on Graphics (SIGGRAPH) 32.4, 62:1--62:9.
12. McNamara, A., Treuille, A., Popović, Z., and Stam, J. 2004. Fluid Control Using the Adjoint Method. In ACM Transactions on Graphics (SIGGRAPH) 23.3, pp. 449--456.
13. Museth, K. 2013. VDB: High-Resolution Sparse Volumes With Dynamic Topology. In ACM Transactions on Graphics (to appear) 32.3.
14. Nielsen, M. B. and Bridson, R. 2011. Guide Shapes for High Resolution Naturalistic Liquid Simulation. In ACM Transactions on Graphics (SIGGRAPH) 30.4, 83:1--83:8.
15. Osher, S. and Fedkiw, R. 2006. Level set methods and dynamic implicit surfaces. Vol. 153. Springer Science & Business Media.
16. Pan, Z., Huang, J., Tong, Y., Zheng, C., and Bao, H. 2013. Interactive Localized Liquid Motion Editing. In ACM Transactions on Graphics (SIGGRAPH Asia) 32.6.
17. Pighin, F., Cohen, J. M., and Shah, M. 2004. Modeling and editing flows using advected radial basis functions. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), pp. 223--232.
18. Raveendran, K., Wojtan, C., Thürey, N., and Turk, G. 2014. Blending Liquids. In ACM Transactions on Graphics (SIGGRAPH) 33.4, 137:1--137:10.
19. Shi, L. and Yu, Y. 2005. Taming Liquids for Rapidly Changing Targets. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), pp. 229--236.
20. Side Effects Software. 2016. Houdini. http://sidefx.com.
21. Söderström, A., Karlsson, M., and Museth, K. 2010. A PML-based Nonreflective Boundary for Free Surface Fluid Animation. In ACM Transactions on Graphics (TOG) 29.5, 136:1--136:17.
22. Söderström, A. and Museth, K. 2009. Non-reflective Boundary Conditions for Incompressible Free Surface Fluids. In SIGGRAPH 2009: Talks. SIGGRAPH '09. New Orleans, Louisiana: ACM, 4:1--4:1. isbn: 978-1-60558-834-6.
23. Srinivasan, R. and Malkawi, A. 2007. Adaptive Localization Method: An Approach to Real Time Airflow Simulation and Immersive Visualization. In Proceedings of the International Conference on Computer Graphics and Vision (GraphiCon).
24. Stam, J. 1999. Stable fluids. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques. ACM Press/Addison-Wesley Publishing Co., pp. 121--128.
25. Thürey, N., Keiser, R., Ruede, U., and Pauly, M. 2006. Detail-Preserving Fluid Control. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA), pp. 7--12.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation

    • Shape modeling

      • Volumetric models

  • Modeling and simulation

• Mathematics of computing

  • Mathematical analysis

    • Differential equations

      • Partial differential equations

    • Numerical analysis

      • Discretization