Perfectly Matched Layers for Time-Harmonic Second Order Elliptic Problems

  • A. Bermúdez
  • L. Hervella-Nieto
  • A. Prieto
  • R. Rodríguez
Article

Abstract

The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for time-harmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the two-dimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using non-integrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.

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References

  1. 1.
    Abarbanel S, Gottlieb D (1998) On the construction and analysis of absorbing layers in CEM. Appl Numer Math 27(4):331–340 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abarbanel S, Gottlieb D, Hesthaven JS (1999) Well-posed perfectly matched layers for advective acoustics. J Comput Phys 154(2):266–283 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aguilar J, Combes J (1971) A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun Math Phys 22(4):269–279 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Allard JF (1993) Propagation of sound in porous media: modelling sound absorbing materials. Elsevier, New York Google Scholar
  5. 5.
    Astley RJ (2000) Infinite elements for wave problems: a review of current formulations and an assessment of accuracy. Int J Numer Methods Eng 49:951–976 MATHCrossRefGoogle Scholar
  6. 6.
    Asvadurov S, Druskin V, Guddati MN, Knizhnerman L (2004) On optimal finite-difference approximation of PML. SIAM J Numer Anal 41(1):287–305 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Bao G, Wu H (2006) Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell’s equations. SIAM J Numer Anal 43(5):2121–2143 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Basu U, Chopra AK (2003) Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput Methods Appl Mech Eng 192(11):1337–1375 MATHCrossRefGoogle Scholar
  9. 9.
    Bayliss A, Turkel E (1980) Radiation boundary conditions for wave-like equations. Commun Pure Appl Math 33:707–725 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bécache E, Bonnet-Benn Dhia AS, Legendre G (2004) Perfectly matched layers for the convected Helmholtz equation. SIAM J Numer Anal 42(1):409–433 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bécache E, Joly P (2002) On the analysis of Berenger’s perfectly matched layers for Maxwell’s equations. M2AN Math Model Numer Anal 36(1):87–119 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Beranek LL, Vér IL (1992) Noise and vibration control engineering: principles and applications. Wiley-Interscience, New York Google Scholar
  13. 13.
    Bérenger JP (1994) A perfectly matched layer for the absortion of electromagnetics waves. J Comput Phys 114(1):185–200 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bérenger JP (1996) Perfectly matched layer for the FDTD solution of wave-structure interaction problems. IEEE Trans Antennas Propag 44(1):110–117 CrossRefGoogle Scholar
  15. 15.
    Bérenger JP (1996) Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 127(2):363–379 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R (2004) An exact bounded PML for the Helmholtz equation. C R Math Acad Sci Paris 339(11):803–808 MATHMathSciNetGoogle Scholar
  17. 17.
    Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R (2007) An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J Comput Phys 223:469–488 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R (2007) Validation of acoustic models for time harmonic dissipative scattering problems. J Comput Acoust 15(1):95–121 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Bermúdez A, Hervella-Nieto L, Prieto A, Rodríguez R (2007) An exact bounded perfectly matched layer for time-harmonic scattering problems. SIAM J Sci Comput 30(1):312–338 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Bonet Ben-Dhia A, Goursaud B, Hazard C, Prieto A (2008) Finite element computation of leaky modes in stratified waveguides. In: Leger A, Deschamps M (eds) Ultrasonic wave propagation in non homogeneous media. Springer proceedings in physics, vol 128. Springer, Berlin, pp 73–86 CrossRefGoogle Scholar
  21. 21.
    Born M, Wolf E (1980) Principles of optics. Pergamon, New York Google Scholar
  22. 22.
    Bramble JH, Pasciak JE (2007) Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math Comput 76:597–614 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Brekhovskikh LM, Godin O (1999) Acoustics of layered media I: plane and quasi-plane waves. Springer, Berlin Google Scholar
  24. 24.
    Chandler-Wilde SN, Monk P (2009) The PML for rough surface scattering. Appl Numer Math 59(9):2131–2154 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Chew WC, Weedon WH (1994) A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. IEEE Microw Opt Technol Lett 7(13):599–604 CrossRefGoogle Scholar
  26. 26.
    Cohen G, Fauqueux S (2005) Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM J Sci Comput 26:864–884 MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Collino F, Monk P (1998) Optimizing the perfectly matched layer. Comput Methods Appl Mech Eng 164(1):157–171 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Collino F, Monk P (1998) The perfectly matched layer in curvilinear coordinates. SIAM J Sci Comput 19(6):2061–2090 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Collino F, Tsogka C (2001) Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66(1):294–307 CrossRefGoogle Scholar
  30. 30.
    Colton D, Kress R (1992) Inverse acoustic and electromagnetic scattering theory. Springer, Berlin MATHGoogle Scholar
  31. 31.
    Cummer SA (2004) Perfectly matched layer behavior in negative refractive index materials. IEEE Trans Antennas Propag 3:172–175 Google Scholar
  32. 32.
    Demkowicz L, Ihlenburg F (2001) Analysis of a coupled finite-infinite element method for exterior Helmholtz problems. Numer Math 88(1):43–73 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Engquist B, Majda A (1977) Absorbing boundary conditions for numerical simulation of waves. Math Comput 31:629–651 MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ervedoza S, Zuazua E (2008) Perfectly matched layers in 1-d: energy decay for continuous and semi-discrete waves. Numer Math 109(4):597–634 MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Feng K Finite element method and natural boundary reduction. In: Proceedings of the international congress of mathematicians, pp. 1439–1453 (1983) Google Scholar
  36. 36.
    Gedney SD (1996) An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans Antennas Propag 44(12):1630–1639 CrossRefGoogle Scholar
  37. 37.
    Gedney SD (1996) An anisotropic PML absorbing media for the FDTD simulation of fields in lossy and dispersive media. Electromagnetics 16(4):399–415 CrossRefGoogle Scholar
  38. 38.
    Givoli D (1991) Non-reflecting boundary conditions. J Comput Phys 94(1):1–29 MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Givoli D (1992) Numerical methods for problems in infinite domains. Elsevier, Amsterdam MATHGoogle Scholar
  40. 40.
    Givoli D, Neta B (2004) High-order non-reflecting boundary scheme for time-dependent waves. J Comput Phys 186(1):24–46 CrossRefMathSciNetGoogle Scholar
  41. 41.
    Hagstrom T (2003) New results on absorbing layers and radiation boundary conditions. In: Ainsworth M, Davies P, Duncan D, Martin P, Rynne B (eds) Topics in computational wave propagation: direct and inverse problems. Lecture notes in computational science and engineering. Springer, Berlin, pp 1–39 Google Scholar
  42. 42.
    Hagstrom T, Lau S (2007) Radiation boundary conditions for Maxwells equations: a review of accurate time-domain formulations. J Comput Math 25(3):305–336 MathSciNetGoogle Scholar
  43. 43.
    Harari I, Albocher U (2006) Studies of FE/PML for exterior problems of time-harmonic elastic waves. Comput Methods Appl Mech Eng 195(29–32):3854–3879 MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Harari I, Slavutin M, Turkel E (2000) Analytical and numerical studies of a finite element PML for the Helmholtz equation. J Comput Acoust 8(1):121–137 MathSciNetGoogle Scholar
  45. 45.
    Hislop P, Sigal I (1996) Introduction to spectral theory with applications to Schrödinger operators. Series in applied mathematical sciences. Springer, Berlin Google Scholar
  46. 46.
    Hohage T, Schmidt F, Zschiedrich L (2004) Solving time-harmonic scattering problems based on the pole condition, II: convergence of the PML method. SIAM J Math Anal 35(3):547–560 CrossRefMathSciNetGoogle Scholar
  47. 47.
    Hu FQ (2001) A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables. J Comput Phys 173(2):455–480 MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Ihlenburg F (1998) Finite element analysis of acoustic scattering. Springer, Berlin MATHCrossRefGoogle Scholar
  49. 49.
    Johnson S, Bienstman P, Skorobogatiy M, Ibanescu M, Lidorikis E, Joannopoulos J (2002) Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals. Phys Rev E 66:066608 CrossRefGoogle Scholar
  50. 50.
    van Joolen VJ, Neta B, Givoli D (2005) High-order Higdon-like boundary conditions for exterior transient wave problems. Int J Numer Methods Eng 63(7):1041–1068 MATHCrossRefGoogle Scholar
  51. 51.
    Kato T (1995) Perturbation theory for linear operators. Springer, Berlin MATHGoogle Scholar
  52. 52.
    Kim S, Pasciak J. Analysis of the spectrum of a Cartesian perfectly matched layer (PML) approximation to acoustic scattering problems. Preprint of the Department of Mathematics, Texas A&M University Google Scholar
  53. 53.
    Kormann J, Cobo P, Prieto A (2008) Perfectly matched layers for modelling seismic oceanography experiments. J Sound Vib 317(1–2):354–365 CrossRefGoogle Scholar
  54. 54.
    Kreiss HO, Lorenz J (1989) Initial-boundary problems and the Navier-Stokes equation. Academic Press, New York Google Scholar
  55. 55.
    Kuzuoglu M, Mittra R (1996) Frequency dependence of the constitutive parameters of causalperfectly matched anisotropic absorbers. IEEE Microw Guided Wave Lett 6(12):447–449 CrossRefGoogle Scholar
  56. 56.
    Lassas M, Liukkonen J, Somersalo E (2001) Complex Riemannian metric and absorbing boundary conditions. J Math Pures Appl 80(7):739–768 MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Lassas M, Somersalo E (1998) On the existence and convergence of the solution of PML equations. Computing 60(3):229–242 MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Lassas M, Somersalo E (2001) Analysis of the PML equations in general convex geometry. Proc R Soc Edinb Sect A 131(5):1183–1207 MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Lions JL, Métral J, Vacus O (2002) Well-posed absorbing layer for hyperbolic problems. Numer Math 92(3):535–562 MATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    Marcuse D (1974) Theory of dielectric optical waveguides. Academic Press, New York Google Scholar
  61. 61.
    Masmoudi M (1987) Numerical solution for exterior problems. Numer Math 51(1):87–101 MATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Navon IM, Neta B, Hussaini MY (2004) A perfectly matched layer approach to the linearized shallow water equations models. Mon Weather Rev 132(6):1369–1378 CrossRefGoogle Scholar
  63. 63.
    Nédélec JC (2000) Acoustic and electromagnetic equations. Integral representations for harmonic problems. Springer, Berlin Google Scholar
  64. 64.
    Oskooi A, Zhang L, Avniel Y, Johnson S (2008) The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers. Opt Express 16(15):11376–11392 CrossRefGoogle Scholar
  65. 65.
    Petropoulos PG (2000) Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates. SIAM J Appl Math 60(3):1037–1058 MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Prieto A (2007) Some contributions in time-harmonic dissipative acoustic problems. PhD thesis, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela Google Scholar
  67. 67.
    Qi Q, Geers TL (1998) Evaluation of the perfectly matched layer for computational acoustics. J Comput Phys 139(1):166–183 MATHCrossRefGoogle Scholar
  68. 68.
    Rappaport CM (1996) Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space. IEEE Trans Magn 32(3):968–974 CrossRefGoogle Scholar
  69. 69.
    Roden JA, Gedney SD (2000) Convolutional PML (CPML): an efficient FDTD implementation of the CFS-PML for arbitrary media. Microw Opt Technol Lett 27(5):334–338 CrossRefGoogle Scholar
  70. 70.
    Rudin W (1974) Real and complex analysis. Series in higher mathematics. McGraw-Hill, New York Google Scholar
  71. 71.
    Sacks ZS, Kingsland DM, Lee R, Lee JF (1995) A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans Antennas Propag 43(12):1460–1463 CrossRefGoogle Scholar
  72. 72.
    Shirron JJ, Babuska I (1998) A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems. Comput Methods Appl Mech Eng 164(1):121–139 MATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Simon B (1978) Resonances and complex scaling: a rigorous overview. Int J Quantum Chem 14(4):529–542 CrossRefGoogle Scholar
  74. 74.
    Sjögreen B, Petersson NA (2005) Perfectly matched layers for Maxwells equations in second order formulation. J Comput Phys 209(1):19–46 MATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Skelton EA, Adams SDM, Craster RV (2007) Guided elastic waves and perfectly matched layers. Wave Motion 44(7–8):573–592 CrossRefMathSciNetGoogle Scholar
  76. 76.
    Teixeira FL, Chew WC (1997) Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microw Guided Wave Lett 7(11):371–373 CrossRefGoogle Scholar
  77. 77.
    Teixeira FL, Chew WC (1998) Extension of the PML absorbing boundary condition to 3D spherical coordinates: scalar case. IEEE Trans Magn 34(5):2680–2683 Part 1 CrossRefGoogle Scholar
  78. 78.
    Turkel E, Yefet A (1998) Absorbing PML boundary layers for wave-like equations. Appl Numer Math 27(4):533–557 MATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    Ying LA (2006) Numerical methods for exterior problems. In: Peking University series in mathematics, vol. 2. World Scientific, Singapore Google Scholar
  80. 80.
    Zampolli M, Tesei A, Jensen F, Malm N, Blottman J (2007) A computationally efficient finite element model with perfectly matched layers applied to scattering from axially symmetric objects. J Acoust Soc Am 122(3):1472–1485 CrossRefGoogle Scholar
  81. 81.
    Zeng YQ, He JQ, Liu QH (2006) The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media. Geophysics 66(4):1258–1266 CrossRefGoogle Scholar
  82. 82.
    Zhao L, Cangellaris A (1996) A general approach for the development of unsplit-field time-domain implementations of perfectly matched layers for FDTD gridtruncation. IEEE Microw Guided Wave Lett 6(5):209–211 CrossRefGoogle Scholar
  83. 83.
    Zschiedrich L, Klose R, Schädle A, Schmidt F (2006) A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions. J Comput Appl Math 188(1):12–32 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2010

Authors and Affiliations

  • A. Bermúdez
    • 1
  • L. Hervella-Nieto
    • 2
  • A. Prieto
    • 1
    • 3
  • R. Rodríguez
    • 4
  1. 1.Departamento de Matemática AplicadaUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Departamento de MatemáticasUniversidade da CoruñaA CoruñaSpain
  3. 3.Applied and Computational Mathematics217-50 California Institute of TechnologyPasadenaUSA
  4. 4.CI2MA Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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