Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Hybrid perfectly-matched-layers for transient simulation of scalar elastic waves

  • Pakravan, Alireza (Department of Civil Engineering, New Mexico State University) ;
  • Kang, Jun Won (Department of Civil Engineering, Hongik University) ;
  • Newtson, Craig M. (Department of Civil Engineering, New Mexico State University) ;
  • Kallivokas, Loukas F. (Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin)
  • Received : 2014.03.05
  • Accepted : 2014.07.20
  • Published : 2014.08.25
⟨ Previous Next ⟩

Abstract

This paper presents a new formulation for forward scalar wave simulations in semi-infinite media. Perfectly-Matched-Layers (PMLs) are used as a wave absorbing boundary layer to surround a finite computational domain truncated from the semi-infinite domain. In this work, a hybrid formulation was developed for the simulation of scalar wave motion in two-dimensional PML-truncated domains. In this formulation, displacements and stresses are considered as unknowns in the PML domain, while only displacements are considered to be unknowns in the interior domain. This formulation reduces computational cost compared to fully-mixed formulations. To obtain governing wave equations in the PML region, complex coordinate stretching transformation was introduced to equilibrium, constitutive, and compatibility equations in the frequency domain. Then, equations were converted back to the time-domain using the inverse Fourier transform. The resulting equations are mixed (contain both displacements and stresses), and are coupled with the displacement-only equation in the regular domain. The Newmark method was used for the time integration of the semi-discrete equations.

Keywords

Acknowledgement

Supported by : Hongik University

References

  1. Basu, U. (2009), "Explicit finite element perfectly matched layer for transient three-dimensional elastic waves" Int. J. Numer. Method. Eng., 77, 151-176. https://doi.org/10.1002/nme.2397
  2. Basu, U. and Chopra, A.K. (2004), "Perfectly matched layers for transient elastodynamics of unbounded domains" , Int. J. Numer. Method. Eng., 59, 1039-1074. https://doi.org/10.1002/nme.896
  3. Becache, E., Joly, P. and Tsogka, C. (2002) "A new family of mixed finite elements for the linear elastodynamic problem" SIAM J. Numer. Anal., 39(6), 2109-2132. https://doi.org/10.1137/S0036142999359189
  4. Berenger, J.P. (1994), "A perfectly matched layer for the absorption of electromagnetic waves" J. Comput. Phys., 114(2), 185-200. https://doi.org/10.1006/jcph.1994.1159
  5. Brezzi, F. and Bathe, K.J. (1990) "A discourse on the stability conditions for mixed finite element formulations" Comput. Method. Appl. Mech. Eng., 82, 27-57. https://doi.org/10.1016/0045-7825(90)90157-H
  6. Chew, W.C. and Liu, Q.H. (1996), "Perfectly matched layers for elastodynamics: a new absorbing boundary condition" J. Comput. Acoust., 4(4), 341-359. https://doi.org/10.1142/S0218396X96000118
  7. Chew, W.C. and Weedon, W.H. (1994), "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates" Microw. Opt. Tech. Lett., 7(13), 599-604. https://doi.org/10.1002/mop.4650071304
  8. Drossaert, F.H. and Giannopoulos, A. (2007), "Complex frequency shifted convolution PML for FDTD modelling of elastic waves" Wave Motion, 44(7-8), 593-604. https://doi.org/10.1016/j.wavemoti.2007.03.003
  9. Frasca, L.P., Hughes, T.J.R., Loula, A.F.D. and Miranda, I. (1988) "A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation" Numerische Mathematik, 53, 123-141. https://doi.org/10.1007/BF01395881
  10. Hastings, F.D., Schneider, J.B. and Broschat, S.L. (1996), "Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation" J. Acoust. Soc. Am., 100(5), 3061-3069. https://doi.org/10.1121/1.417118
  11. Kang, J.W. and Kallivokas, L.F. (2010), "Mixed unsplit-field perfectly-matched-layers for transient simulations of scalar waves in heterogeneous domains" Comput. Geosci., 14, 623-648. https://doi.org/10.1007/s10596-009-9176-4
  12. Komatitsch, D. and Tromp, J (2003), "A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation" Geophys. J. Int., 154, 146-153. https://doi.org/10.1046/j.1365-246X.2003.01950.x
  13. Madsen, S.S., Krenk, S. and Hededal, O. (2013), "Perfectly matched layer (PML) for transient wave propagation in a moving frame of reference" Proceedings of the 4th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2013), Kos Island, Greece, June.
  14. Mahmoud, A. and Luo, Y (2009), "Application of a Perfectly Matched Layer Boundary Condition to Finite Element Modeling of Elastic Wave Scattering in Cracked Plates" Adv. Theor. Appl. Mech., 2(2), 75-92.
  15. Mahmoud, A., Rattanawangcharoen, N., Luo, Y. and Wang, Q. (2010) "FE-PML modeling of 3D scattering of transient elastic waves in cracked plate with rectangular cross section" Int. J. Struct. Stab. Dyn., 10(5), 1123-1139. https://doi.org/10.1142/S0219455410003932
  16. Martin, R., Komatitsch, D. and Gedney, S.D. (2008) "A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation" Comput. Model. Eng. Sci., 37(3), 274-304.
  17. Martin, R., Komatitsch, D., Gedney, S.D. and Bruthiaux, E. (2010) "A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using auxiliary differential equations (ADE-PML)" Comput. Model. Eng. Sci., 56(1), 17-40.
  18. Matzen, R. (2011), "An efficient finite element time-domain formulation for the elastic second-order wave equation: A non-split complex frequency shifted convolutional PML" Int. J. Numer. Method. Eng., 88(10), 951-973. https://doi.org/10.1002/nme.3205
  19. Sagiyama, K., Govindjee, S. and Persson, P.O. (2013), "An Efficient Time-Domain Perfectly Matched Layers Formulation for Elastodynamics on Spherical Domains" Report No. UCB/SEMM-2013/09, University of California at Berkeley, Berkeley, California, USA.
  20. Teixeira, F.L. and Chew, W.C. (2000), "Complex space approach to perfectly matched layers: a review and some new developments" Int. J. Numer. Model., 13, 441-455. https://doi.org/10.1002/1099-1204(200009/10)13:5<441::AID-JNM376>3.0.CO;2-J
  21. Turkel, E. and Yefet, A. (1998) "Absorbing PML boundary layers for wave-like equations" Appl. Numer. Math., 27, 533-557. https://doi.org/10.1016/S0168-9274(98)00026-9
  22. Xu, B.Q., Tsang, H.H. and Lo, S.H. (2013), "3-D convolutional perfectly matched layer models for dynamic soil-structure interaction analysis in the finite element time-domain" Proceedings of the 4th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2013), Kos Island, Greece, June.

Cited by

  1. Modeling of SH-waves in a fiber-reinforced anisotropic layer vol.10, pp.1, 2016, https://doi.org/10.12989/eas.2016.10.1.091
  2. A Gauss–Newton full-waveform inversion in PML-truncated domains using scalar probing waves vol.350, 2017, https://doi.org/10.1016/j.jcp.2017.09.017
  3. A Time-Domain Formulation of Elastic Waves in Heterogeneous Unbounded Domains vol.1, pp.3, 2019, https://doi.org/10.1007/s42493-019-00019-z
  4. Material profile reconstruction using plane electromagnetic waves in PML-truncated heterogeneous domains vol.96, pp.None, 2021, https://doi.org/10.1016/j.apm.2021.03.026