Abstract
This article presents a novel nonreflective boundary condition for the free surface incompressible Euler and Navier-Stokes equations. Boundaries of this type are very useful when, for example, simulating water flow around a ship moving over a wide ocean. Normally waves generated by the ship will reflect off of the boundaries of the simulation domain and as these reflected waves return towards the ship they will cause undesired interference patterns. By employing a Perfectly Matched Layer (PML) approach we have derived a boundary condition that absorbs incoming waves and thus efficiently prevents these undesired wave reflections. To solve the resulting boundary equations we present a fast and stable algorithm based on the stable fluids approach. Through numerical experiments we then show that our boundaries are significantly more effective than simpler reflection preventing techniques. We also provide a thorough analysis of the parameters involved in our boundary formulation and show how they effect wave absorption efficiency.
Supplemental Material
- Abarbanel, S. and Gottlieb, D. 1997. A mathematical analysis of the pml method. J. Comput. Phys. 134, 2, 357--363. Google ScholarDigital Library
- Adalsteinsson, D. and Sethian, J. A. 1995. A fast level set method for propagating interfaces. J. Comput. Phys. 118, 2, 269--277. Google ScholarDigital Library
- Baldauf, M. 2008. Stability analysis for linear discretisations of the advection equation with runge-kutta time integration. J. Comput. Phys. 227, 13, 6638--6659. Google ScholarDigital Library
- Bcache, E., Fauqueux, S., and Joly, P. 2003. Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 2, 399--433. Google ScholarDigital Library
- Bcache, E., Dhia, A.-S. B.-B., and Legendre, G. 2004. Perfectly matched layers for the convected helmholtz equation. SIAM J. Numer. Anal. 42, 1, 409--433. Google ScholarDigital Library
- Berenger, J.-P. 1994. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 2, 185--200. Google ScholarDigital Library
- Bcache, E., Fauqueux, S., and Joly, P. 2003. Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 2, 399--433. Google ScholarDigital Library
- Chew, W. C. and Weedon, W. H. 1994. A 3-d perfectly matched medium from modified maxwell's equations with stretched coordinates. Microwave Opt. Tech. Lett. 7, 599--604.Google ScholarCross Ref
- Chorin, A. 1968. Numerical solution of navier-stokes equations. Math. Comp. 22, 745--762.Google ScholarCross Ref
- Enright, D., Fedkiw, R., Ferziger, J., and Mitchell, I. 2002. A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 1, 83--116. Google ScholarDigital Library
- Fedkiw, R., Stam, J., and Jensen, H. W. 2001. Visual simulation of smoke. In Proceedings of ACM SIGGRAPH Conference. 15--22. Google ScholarDigital Library
- Foster, N. and Fedkiw, R. 2001. Practical animation of liquids. In Proceedings of ACM SIGGRAPH Conference. ACM Press, 23--30. Google ScholarDigital Library
- Gottlieb, S. and Shu, C.-W. 1998. Total variation diminishing runge-kutta schemes. Math. Comput. 67, 221, 73--85. Google ScholarDigital Library
- 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. AIAA--2005--2869.Google Scholar
- Hagstrom, T. and Nazarov, I. 2002. Absorbing layers and radiation boundary conditions for jet flow simulations. In Proceedings of the 8th AIAA/CEAS Aeroacoustics Conference and Exhibit. 2002--2606.Google Scholar
- Hagstrom, T. and Nazarov, I. 2003. Perfectly matched layers and radiation boundary conditions for shear flow calculations. In Proceedings of the 9th AIAA/CEAS Aeroacoustics Conference and Exhibit. 2003--3298.Google Scholar
- He, X. and Luo, L.-S. 1997. Lattice boltzmann model for the incompressible navierstokes equation. J. Statis. Phys. 88, 3, 927--944.Google ScholarCross Ref
- Hein, S., Hohage, T., Koch, W., and Schberl, J. 2007. Acoustic resonances in a high-lift configuration. J. Fluid Mech. 582, -1, 179--202.Google ScholarCross Ref
- Hu, F. 2006. On the construction of pml absorbing boundary condition for the non-linear euler equations. In Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit. AIAA--2006--798.Google ScholarCross Ref
- Hu, F. Q. 1995. On absorbing boundary conditions for linearized euler equations by a perfectly matched layer. J. Comput. Phys. 129, 201--219. Google ScholarDigital Library
- Hu, F. Q. 1996. On absorbing boundary conditions for linearized euler equations by a perfectly matched layer. J. Comput. Phys. 129, 1, 201--219. Google ScholarDigital Library
- Hu, F. Q. 2001a. A stable, perfectly matched layer for linearized euler equations in unslit physical variables. J. Comput. Phys. 173, 2, 455--480. Google ScholarDigital Library
- Hu, F. Q. 2001b. A stable, perfectly matched layer for linearized euler equations in unsplit physical variables. J. Comput. Phys. 173, 2, 455--480. Google ScholarDigital Library
- Hu, F. Q. 2008. Development of pml absorbing boundary conditions for computational aeroacoustics: A progress review. Comput. Fluids 37, 4, 336--348.Google ScholarCross Ref
- Hu, F. Q., Hussaini, M. Y., and Manthey, J. L. 1996. Low-dissipation and low-dispersion runge-kutta schemes for computational acoustics. J. Comput. Phys. 124, 1, 177--191. Google ScholarDigital Library
- Hu, F. Q., Li, X. D., and Lin, D. K. 2008. Absorbing boundary conditions for nonlinear euler and navier-stokes equations based on the perfectly matched layer technique. J. Comput. Phys. 227, 9, 4398--4424. Google ScholarDigital Library
- Johnson, S. G. 2007. Notes on perfectly matched layers. http://www-math.mit.edu/~stevenj/18.369/pml.pdfGoogle Scholar
- Li, X. D. and Gao, J. H. 2005. Numerical simulation of the generation mechanism of axisymmetric supersonic jet screech tones. Phys. Fluids 17, 8, 085105.Google ScholarCross Ref
- Li, X. D. and Gao, J. H. 2008. Numerical simulation of the three-dimensional screech phenomenon from a circular jet. Phys. Fluids 20, 3, 035101.Google ScholarCross Ref
- Liu, X., Osher, S., and Chan, T. 1994. Weighted essentially nonoscillatory schemes. J. Comput. Phys. 115, 200--212. Google ScholarDigital Library
- Monaghan, J. 1988. An introduction to sph. Comput. Phys. Comm. 48, 89--96.Google ScholarCross Ref
- Museth, K. and Leforestier, C. 1996. On the direct complex scaling of matrix elements expressed in a discrete variable representation: Application to molecular resonances. J. Chem. Phys. 104, 18, 7008--7014.Google ScholarCross Ref
- Neuhasuer, D. and Baer, M. 1989. The time?dependent schrždinger equation: Application of absorbing boundary conditions. J. Chem. Phys. 90, 8, 4351--4355.Google ScholarCross Ref
- Nielsen, M. B. and Museth, K. 2006. Dynamic Tubular Grid: An efficient data structure and algorithms for high resolution level sets. J. Sci. Comput. 26, 3, 261--299. Google ScholarDigital Library
- Nielsen, M. B., Nilsson, O., Söderström, A., and Museth, K. 2007. Out-of-core and compressed level set methods. ACM Trans. Graph. 26, 4, 16. Google ScholarDigital Library
- Osher, S. and Fedkiw, R. 2002. Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin.Google Scholar
- Özyörük, Y. 2009. Numerical prediction of aft radiation of turbofan tones through exhaust jets. J. Sound Vibrat. 325, 1-2, 122--144.Google ScholarCross Ref
- Reinhardt, W. P. 1982. Complex coordinates in the theory of atomic and molecular structure and dynamics. Ann. Rev. Phys. Chem. 33, 223--255.Google ScholarCross Ref
- Richards, S. K., Zhang, X., Chen, X. X., and Nelson, P. A. 2004. The evaluation of non-reflecting boundary conditions for duct acoustic computation. J. Sound Vibrat. 270, 3, 539--557.Google ScholarCross Ref
- Shu, C. and Osher, S. 1988. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439--471. Google ScholarDigital Library
- Söderström, A. and Museth, K. 2009. Nonreflective boundary conditions for incompressible free surface fluids. In Proceedings of the SIGGRAPH'09 Talks. ACM, New York 1--1. Google ScholarDigital Library
- Stam, J. 1999. Stable fluids. In Proceedings of SIGGRAPH Conference. 121--128. Google ScholarDigital Library
- Tam, C. K. and Webb, J. C. 1993. Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 2, 262--281. Google ScholarDigital Library
- Turkel, E. and Yefet, A. 1998. Absorbing pml boundary layers for wave-like equations. Appl. Numer. Math. 27, 4, 533--557. Google ScholarDigital Library
- Zhao, L. and Cangellaris, A. 1996. Gt-pml: generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids. IEEE Trans. Microw. Theory Tech. 44, 12, 2555--2563.Google ScholarCross Ref
- Zhu, Y. and Bridson, R. 2005. Animating sand as a fluid. In Proceedings of the ACM SIGGRAPH Papers. ACM, New York. 965--972. Google ScholarDigital Library
Index Terms
- A PML-based nonreflective boundary for free surface fluid animation
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