Skip to main content
SpringerLink
Log in

Approximating Unbounded Domains

  • Chapter
  • First Online:
Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations

Part of the book series: Scientific Computation ((SCIENTCOMP))

  • 1961 Accesses

Abstract

This chapter provides the construction and approximation of two approaches for the treatment of unbounded domains: first by absorbing boundary conditions (ABC), then by perfectly matched layers (PML) for the three wave equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Cerjan, C., Kosloff, D., Kosloff, R., Reshef, M.: A nonreflecting boundary condition for discrete acoustic and elastic wave equations. Geophysics 50(4), 705–708 (1985)

    Article  Google Scholar 

  2. Bayliss, A., Turkel, E.: Radiation boundary conditions for wave-like equations. Commun. Pure Appl. Math. 33, 707–725 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31(139), 629–651 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Engquist, B., Majda, A.: Radiation boundary conditions for acoustic and elastic wave calculations. Commun. Pure Appl. Anal. 32(3), 313–357 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Collino, F.: High order absorbing boundary conditions for wave propagation models: straight line boundary and corner cases. Second International Conference on Mathematical and Numerical Aspects of Wave Propagation (Newark, DE, 1993), pp. 161–171, SIAM, Philadelphia, PA (1993)

    Google Scholar 

  6. Bendali, A., Halpern, L.: Conditions aux limites absorbantes pour le système de Maxwell dans le vide en dimension trois d’espace. C.R. Acad. Sci. Paris Ser. I Math. 307(20):1011-1013 (1988)

    Google Scholar 

  7. Hall, W.F., Kabakian, A.V.: A sequence of absorbing boundary conditions for Maxwell’s equations. J. Comput. Phys. 194(1), 140–155 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Higdon, R.L.: Absorbing boundary conditions for acoustic and elastic waves in stratified media. J. Comput. Phys 101(2), 386–418 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sochacki, J.: Absorbing boundary conditions for the elastic wave equations. Appl. Math. Comput. 28(1), 1–14 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Bamberger, A., Joly, P., Roberts, J.E.: Second-order absorbing boundary conditions for the wave equation: a solution for the corner problem. SIAM J. Numer. Anal. 27(2), 323–352 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Halpern, L., Rauch, J.: Error analysis for absorbing boundary conditions. Numer. Math. 51(4), 459–467 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Trefethen, L.N., Halpern, L.: Well-posedness of one-way wave equations and absorbing boundary conditions. Math. Comp. 47(176), 421–435 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39(4), 319–326 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yee, K.: Numerical solutions of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14(3), 302–307 (1966)

    Article  MATH  Google Scholar 

  15. Bérenger, J.-P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bérenger, J.-P.: Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127(2), 363–379 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rappaport, C.: Perfectly matched absorbing conditions based on anisotropic lossy mapping of space. IEEE Microw. Guided Wave Lett. 5(3), 90–92 (1995)

    Article  MathSciNet  Google Scholar 

  18. Zhao, L., Cangellaris, A.C.: GT-PML: Generalized theory of perfectly matched layers and its application to reflectionless truncation of finite-difference time-domain grids. IEEE Trans. Microw. Theory Techn. 44(12), 2555–2563 (1996)

    Article  Google Scholar 

  19. Cohen, G., Fauqueux, S.: Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM J. Sci. Comput. 26(3), 864–884 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cohen, G., Imperiale, S.: Perfectly matched layer with mixed spectral elements for the propagation of linearized water waves. Commun. Comput. Phys. 11(2), 285–302 (2012)

    Article  MathSciNet  Google Scholar 

  21. Collino, F., Tsogka, C.: Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66(1), 294–307 (2001)

    Article  Google Scholar 

  22. Hesthaven, J.S.: On the analysis and construction of perfectly matched layers for the linearized Euler equations. J. Comput. Phys. 142(1), 129–147 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nataf, F.: A new construction of perfectly matched layers for the linearized Euler equations. J. Comput. Phys. 214(2), 757–772 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tam, C.K.W., Auriault, L., Cambuli, F.: Perfectly matched layer as an absorbing boundary condition for the linearized Euler equations in open and ducted domains. J. Comput. Phys. 144(1), 213–234 (1998)

    Article  MathSciNet  Google Scholar 

  25. Abarbanel, S., Gottlieb, D.: A mathematical analysis of the PML method. J. Comput. Phys. 134(2), 357–363 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Abarbanel, S., Gottlieb, D., Hesthaven, J.S.: Well-posed perfectly matched layers for advective acoustics. J. Comput. Phys. 154(2), 266–283 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Collino, F., Monk, P.: The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19(6), 2061–2090 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Halpern, L., Petit-Bergez, S., Rauch, J.: The analysis of matched layers. Conflu. Math. 3(2), 159–236 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lassas, M., Somersalo, E.: On the existence and convergence of the solution of PML equations. Computing 60(3), 229–241 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mittra, R., Pekel, U., Veihl, J.: A theoretical and numerical study of Berenger’s perfectly matched layer (PML) concept for mesh truncation in time and frequency domains. Approximations and numerical methods for the solution of Maxwell’s equations, pp. 1–19. Oxford University Press, Oxford (1995)

    Google Scholar 

  31. Kuzuoglu, M., Mittra, R.: Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers. IEEE Microw. Guided Wave Lett. 6(12), 447–449 (1996)

    Article  Google Scholar 

  32. Roden, J.A., Gedney, S.D.: Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media. Microw. Opt. Techn. Let. 27(5), 334–339 (2000)

    Article  Google Scholar 

  33. Meza-Fajardo, K., Papageorgiou, A.: A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: stability analysis. Bull. seism. Soc. Am. 98(4), 1811–1836 (2008)

    Article  Google Scholar 

  34. Tago, J., Métivier, L., Virieux, J.: SMART layers: a simple and robust alternative to PML approaches for elastodynamics. Geophys. J. Int. 199(2), 700–706 (2014)

    Article  Google Scholar 

  35. Bécache, E., Fauqueux, S., Joly, P.: Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188(2), 399–433 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kreiss, H.-O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Anal. 23, 277–298 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  37. Cohen, G., Fauqueux, S.: Mixed finite elements with mass-lumping for the transient wave equation. J. Comput. Acoust. 8(1), 171–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Inria, Saclay, France

    Gary Cohen

  2. The French Aerospace Lab, ONERA, Toulouse, France

    Sébastien Pernet

Authors
  1. Gary Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sébastien Pernet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gary Cohen .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Cohen, G., Pernet, S. (2017). Approximating Unbounded Domains. In: Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7761-2_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-7761-2_6

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-7759-9

  • Online ISBN: 978-94-017-7761-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation