Hostname: page-component-54dcc4c588-xh45t Total loading time: 0 Render date: 2025-09-14T05:36:58.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Micro-Differential Boundary Conditions Modelling the Absorption of Acoustic Waves by 2D Arbitrarily-Shaped Convex Surfaces

Published online by Cambridge University Press:  20 August 2015

Hélène Barucq ,
Julien Diaz  and
Véronique Duprat
Hélène Barucq*
Affiliation:
INRIA Bordeaux-Sud Ouest, Team-project MAGIQUE-3D LMA, CNRS UMR 5142, University of Pau, France
Julien Diaz*
Affiliation:
INRIA Bordeaux-Sud Ouest, Team-project MAGIQUE-3D LMA, CNRS UMR 5142, University of Pau, France
Véronique Duprat*
Affiliation:
INRIA Bordeaux-Sud Ouest, Team-project MAGIQUE-3D LMA, CNRS UMR 5142, University of Pau, France
*
Email address:helene.barucq@inria.fr
Email address:julien.diaz@inria.fr
Corresponding author.Email:veronique.duprat@univ-pau.fr
Get access

Abstract

We propose a new Absorbing Boundary Condition (ABC) for the acoustic wave equation which is derived from a micro-local diagonalization process formerly defined by M.E. Taylor and which does not depend on the geometry of the surface bearing the ABC. By considering the principal symbol of the wave equation both in the hyperbolic and the elliptic regions, we show that a second-order ABC can be constructed as the combination of an existing first-order ABC and a Fourier-Robin condition. We compare the new ABC with other ABCs and we show that it performs well in simple configurations and that it improves the accuracy of the numerical solution without increasing the computational burden.

Keywords

35L05Wave equationmicro-local diagonalizationabsorbing boundary conditionfinite element formulation

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 674 - 690
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Ainsworth, M., Monk, P., Muniz, W., Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation, J. Sci. Comput., 27(1-3) (2006), 5–40.Google Scholar
[2]Antoine, X., Barucq, H., Microlocal diagonalization of strictly hyperbolic pseudodifferential systems and application to the design of radiation conditions in electromagnetism, SIAM J. Appl. Math., 61 (2001), 1877–1905.CrossRefGoogle Scholar
[3]Antoine, X., Barucq, H., Bendali, A., Bayliss-Turkel like radiation conditions on surfaces of arbitrary shape, J. Math. Anal. Appl., 229 (1999), 184–211.Google Scholar
[4]Barucq, H., A new family of first-order boundary conditions for the Maxwell system: Derivation, well-posedness and long-time behavior, J. Math. Pure Appl., 82 (2002), 67–88.CrossRefGoogle Scholar
[5]Bayliss, A., Gunzburger, M., Turkel, E., Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (2002), 430–451.Google Scholar
[6]Bouche, D., Molinet, F., Méthodes asymptotiques en électromagnétisme, Springer-Verlag, 1994.Google Scholar
[7]Cagniard, L., Reflection and refraction of progressive seismic waves, McGraw-Hill, 1962.Google Scholar
[8]Dolean, V., Gander, M.J., Gerardo-Giorda, L., Optimized Schwarz methods for Maxwells equations, SIAM J. Sci. Comput., 31(3) (2009), 2193–2213.CrossRefGoogle Scholar
[9]Ehrhardt, M., Absorbing boundary conditions for hyperbolic systems, Numer. Math. Theor. Meth. Appl., 3 (2010), 295–337.CrossRefGoogle Scholar
[10]Engquist, B., Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629–651.Google Scholar
[11]Grote, M.J., Schneebeli, A., Schötzau, D., Discontinuous Galerkin finite element method for the wave equation, SIAM J. Numer. Anal., 44 (2006), 2408–2431.Google Scholar
[12]Hagstrom, T., Givoli, D., Warburton, T., Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comput. Appl. Math., 234 (2010), pp. 1988–1995.CrossRefGoogle Scholar
[13]Hagstrom, T., Mar-Or, A., Givoli, D., High-order local absorbing conditions for the wave equation: Extensions and improvements, J. Comput. Phys., 227 (2008), 3322–3357.CrossRefGoogle Scholar
[14]Hagstrom, T., Warburton, T., Complete radiation boundary conditions: Minimizing the long time error growth of local methods, SIAM J. Numer. Anal., 47 (2009), pp. 3678–3704.Google Scholar
[15]Higdon, R., Numerical absorbing boundary conditions for the wave equation, Math. Comp., 49 (1987), 65–90.Google Scholar
[16]Hörmander, L., Pseudodifferential operators and hypoelliptic equations, in Proc. Sym. Pure Math. X (Singular Integrals), 138–183, AMS, Providence, 1967.Google Scholar
[17]Reiner, R.C., Djellouli, R., Harari, I., The performance of local absorbing boundary conditions for acoustic scattering from elliptical shapes, Comp. Meth. Appl. Mech. Engrg., 195 (2006), 3622–3665.Google Scholar
[18]Taylor, M.E., Pseudodifferential Operators, Princeton University Press, NJ, 1981.Google Scholar