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Numerical Boundary Conditions for Specular Reflection in a Level-Sets-Based Wavefront Propagation Method

Published online by Cambridge University Press:  03 June 2015

Sheri L. Martinelli
Sheri L. Martinelli*
Affiliation:
Torpedo Systems Department, Naval Undersea Warfare Center, 1176 Howell Street, Newport, Rhode Island 02841, USA Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA
*
*Corresponding author.Email:sherimartinelli@alumni.brown.edu
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Abstract

We study the simulation of specular reflection in a level set method implementation for wavefront propagation in high frequency acoustics using WENO spatial operators. To implement WENO efficiently and maintain convergence rate, a rectangular grid is used over the physical space. When the physical domain does not conform to the rectangular grid, appropriate boundary conditions to represent reflection must be derived to apply at grid locations that are not coincident with the reflecting boundary. A related problem is the extraction of the normal vectors to the boundary, which are required to formulate the reflection condition. A separate level set method is applied to pre-compute the boundary normals which are then stored for use in the wavefront method. Two approaches to handling the reflection boundary condition are proposed and studied: one uses an approximation to the boundary location, and the other uses a local reflection principle. The second method is shown to produce superior results.

Keywords

65M0665M2265M2543.30.Gv43.30.ZkBoundary conditionsreflectionlevel set methodwavefront methodshigh-frequency acoustic propagationWENO
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 2 , August 2013 , pp. 509 - 536
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Roux, P., Cornuelle, B., Kuperman, W., and Hodgkiss, W., The structure of raylike arrivals in a shallow-water waveguide, J. Acoust. Soc. Amer., vol. 124, pp. 3430–3439, 2008.Google Scholar
[2]Singer, A., Nelson, J., and Kozat, S., Signal processing for underwater acoustic communications, Communications Magazine, IEEE, vol. 47, pp. 90–96, january 2009.Google Scholar
[3]Godin, O., Restless rays, steady wave fronts, J. Acoust. Soc. Amer., vol. 122, pp. 3353–3363, 2007.Google Scholar
[4]Martinelli, S., An application of the level set method to underwater acoustic propagation, Commun. Comput. Phys., vol. 12, no. 5, pp. 1359–1391, 2012.Google Scholar
[5]Osher, S., Cheng, L.-T., Kang, M., Shim, H., and Tsai, Y.-H., Geometric optics in a phase-space-based level set and Eulerian framework, J. Comput. Phys., vol. 179, pp. 622–648, 2002.Google Scholar
[6]Liu, X.-D., Osher, S., and Chan, T., Weighted essentially nonoscillatory schemes, J. Comput. Phys., vol. 115, pp. 200–212, 1994.CrossRefGoogle Scholar
[7]Gottlieb, S. and Shu, C., Total variation diminishing Runge-Kutta schemes, Mathematics of Computation, vol. 67, no. 221, pp. 73–85, 1998.Google Scholar
[8]Cheng, L.-T., Osher, S., Kang, M., Shim, H., and Tsai, Y.-H., Reflection in a level set framework for geometric optics, Computer Modeling in Engineering and Sciences, vol. 5, pp. 347–360, 2004.Google Scholar
[9]Shu, C., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Report 97-65, NASA/CR-97-206253, November 1997.Google Scholar
[10]Cockburn, B., Qian, J., Reitich, F., and Wang, J., An accurate spectral/discontinuous finite-element formulation of aphase-space-based level set approachto geometrical optics, J. Comput. Phys., vol. 208, pp. 175–195, 2005.CrossRefGoogle Scholar
[11]Forrer, H. and Jeltsch, R., A high-order boundary treatment for Cartesian-grid methods, J. Comput. Phys., vol. 140, pp. 259–277, 1998.Google Scholar
[12]Forrer, H. and Berger, M., Flow simulations on Cartesian grids involving complex moving geometries, in Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998) (Jeltsch, R., ed.), vol. 129 of Internat. Ser. Numer. Math., pp. 315–324, Birkhaäuser Verlag, 1999.Google Scholar
[13]Peng, D., Merriman, B., Osher, S., Zhao, H., and Kang, M., A PDE-based fast local level set method, J. Comput. Phys., vol. 155, pp. 410–438, 1999.Google Scholar
[14]Osher, S. and Fedkiw, R., Level Set Methods and Dynamic Implicit Surfaces, vol. 153 of Applied Mathematical Sciences. Springer, 2003.CrossRefGoogle Scholar
[15]Sussman, M., Smereka, P., and Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., vol. 114, pp. 146–159, 1994.CrossRefGoogle Scholar
[16]Galassi, M.et al., GNU Scientific Library Reference Manual. http://www.gnu.org/ software/gsl, 2011.Google Scholar
[17]Jensen, F., Kuperman, W., Porter, M., and Schmidt, H., Computational Ocean Acoustics. Modern Acoustics and Signal Processing, Springer, 2nd ed., 2011.Google Scholar
[18]Gustafsson, B., Kreiss, H.-O., and Sundstroäm, A., Stability theory of difference approximations for mixed initial boundary value problems. ii, Mathematics of Computation, vol. 26, no. 119, pp. 649–686, 1972.Google Scholar
[19]Trefethen, L., Stability of finite-difference models containing two boundaries or interfaces, Mathematics of Computation, vol. 45, no. 172, pp. 279–300, 1985.CrossRefGoogle Scholar
[20]Ramachandran, P. and Varoquaux, G., Mayavi: 3D Visualization of Scientific Data, Computing in Science & Engineering, vol. 13, no. 2, pp. 40–51, 2011.Google Scholar