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Research Article

An Implementation Method of the Complex Frequency-Shifted Uniaxial/Multi-Axial PML Technique for Viscoelastic Seismic Wave Propagation

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Pages 885-905 | Received 24 Jun 2022, Accepted 23 May 2023, Published online: 05 Jun 2023

References

  • Basu, U., and A. K. Chopra. 2004. Perfectly matched layers for transient elastodynamics of unbounded domains. International Journal for Numerical Methods in Engineering 59 (8):1039–74. doi:10.1002/nme.896.
  • Bécache, E., S. Fauqueux, and P. Joy. 2003. Stability of perfectly matched layers, group velocities and anisotropic waves. Journal of Computational Physics 188 (2):399–433. doi:10.1016/S0021-9991(03)00184-0.
  • Bécache, E., P. Joly, and C. Tsogka. 2001. Fictitious domains, mixed finite elements and perfectly matched layers for 2-D elastic wave propagation. Journal of Computational Acoustics 9 (3):1175–201. doi:10.1142/S0218396X01000966.
  • Berenger, J. P. 1994. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 114 (2):185–200. doi:10.1006/jcph.1994.1159.
  • Beskos, D. E. 1997. Boundary element methods in dynamic analysis: Part II (1986-1996). Applied Mechanics Reviews 50 (3):149–97. doi:10.1115/1.3101695.
  • Bouchon, M. 1978. The importance of the surface or interface P wave in near-earthquake studies. Bulletin of the Seismological Society of America 68 (5):1293–311. doi:10.1785/BSSA0680061555.
  • Bouchon, M., and K. Aki. 1977. Discrete wave-number representation of seismic-source wave fields. Bulletin of the Seismological Society of America 67 (2):259–77. doi:10.1785/BSSA0670020259.
  • Carcione, J. M. 2007. Wave fields in real media: Wave propagation in anisotropic, anelastic, porous, and electromagnetic media. Amsterdam: Elsevier.
  • Cerjan, C., D. Kosloff, R. Kosloff, and M. Reshef. 1985. A nonreflecting boundary condition for discrete acoustic and elastic wave equations. Geophysics 50 (4):705–08. doi:10.1190/1.1441945.
  • Chern, A. 2019. A reflectionless discrete perfectly matched layer. Journal of Computational Physics 381:91–109. doi:10.1016/j.jcp.2018.12.026.
  • Chew, W. C., and Q. H. Liu. 1996. Perfectly matched layers for elastodynamics: A new absorbing boundary condition. Journal of Computational Acoustics 4 (4):341–59. doi:10.1142/S0218396X96000118.
  • Chew, W. C., and W. H. Weedon. 1994. A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microwave and Optical Technology Letters 7 (13):599–604. doi:10.1002/mop.4650071304.
  • Clayton, R., and B. Engquist. 1977. Absorbing boundary conditions for acoustic and elastic wave equations. Bulletin of the Seismological Society of America 67 (6):1529–40. doi:10.1785/BSSA0670061529.
  • Cohen, G., and S. Fauqueux. 2005. Mixed spectral finite elements for the linear elasticity system in unbounded domains. SIAM Journal on Scientific Computing 26 (3):864–84. doi:10.1137/S1064827502407457.
  • Collino, F., and C. Tsogka. 2001. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66 (1):294–307. doi:10.1190/1.1444908.
  • Connolly, D. P., A. Giannopoulo, and M. C. Forde. 2015. A higher order perfectly matched layer formulation for finite-difference time-domain seismic wave modeling. Geophysics 80 (1):T1–T16. doi:10.1190/geo2014-0157.1.
  • Correia, D., and J. M. Jin. 2005. On the development of a higher-order PML. IEEE Transactions on Antennas & Propagation 53 (12):4157–63. doi:10.1109/TAP.2005.859901.
  • Dmitriev, M. N., and V. V. Lisitsa. 2011. Application of M-PML reflectionless boundary conditions to the numerical simulation of wave propagation in anisotropic media, part I: Reflectivity. Numerical Analysis & Applications 4 (4):271–80. doi:10.1134/S199542391104001X.
  • Dmitriev, M. N., and V. V. Lisitsa. 2012. Application of M-PML absorbing boundary conditions to the numerical simulation of wave propagation in anisotropic media, part II: Stability. Numerical Analysis & Applications 5 (1):36–44. doi:10.1134/S1995423912010041.
  • Dominguez, J. 1993. Boundary elements in dynamics. USA: Computational Mechanics Publications.
  • Drossaert, F. H., and A. Giannopoulos. 2007a. Complex frequency shifted convolution PML for FDTD modelling of elastic waves. Wave Motion 44 (7–8):593–604. doi:10.1016/j.wavemoti.2007.03.003.
  • Drossaert, F. H., and A. Giannopoulos. 2007b. A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves. Geophysics 72 (2):T9–T17. doi:10.1190/1.2424888.
  • Emmerich, H., and M. Korn. 1987. Incorporation of attenuation into time-domain computations of seismic wave fields. Geophysics 52 (9):1252–64. doi:10.1190/1.1442386.
  • Feng, H., W. Zhang, J. Zhang, and X. Chen. 2017. Importance of double-pole CFS-PML for broad-band seismic wave simulation and optimal parameters selection. Geophysical Journal International 209 (2):1148–67. doi:10.1093/gji/ggx070.
  • Festa, G., and J. P. Vilotte. 2005. The Newmark scheme as velocity–stress time-staggering: An efficient PML implementation for spectral element simulations of elastodynamics. Geophysical Journal International 161 (3):789–812. doi:10.1111/j.1365-246X.2005.02601.x.
  • Gedney, S. D. 1998. The perfectly matched layer absorbing medium. In Advances in computational electrodynamics: The finite-difference time-domain method, ed. A. Taflove, 263–343. Boston: Artech House.
  • Hastings, F. D., J. B. Schneider, and S. L. Broschat. 1996. Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation. The Journal of the Acoustical Society of America 100 (5):3061–69. doi:10.1121/1.417118.
  • Higdon, R. L. 1991. Absorbing boundary conditions for elastic waves. Geophysics 56 (2):231–41. doi:10.1190/1.1443035.
  • Huang, L., Z. Liu, C. Wu, J. Liang, and Q. Pei. 2022. A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space. Engineering Analysis with Boundary Elements 135:167–81. doi:10.1016/j.enganabound.2021.11.012.
  • Komatitsch, D., and R. Martin. 2007. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics 72 (5):SM155–67. doi:10.1190/1.2757586.
  • Komatitsch, D., and J. Tromp. 2003. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation. Geophysical Journal International 154 (1):146–53. doi:10.1046/j.1365-246X.2003.01950.x.
  • Kosloff, R., and D. Kosloff. 1986. Absorbing boundaries for wave propagation problems. Journal of Computational Physics 63 (2):363–76. doi:10.1016/0021-9991(86)90199-3.
  • Kucukcoban, S., and L. F. Kallivokas. 2011. Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media. Computer Methods in Applied Mechanics and Engineering 200 (1–4):57–76. doi:10.1016/j.cma.2010.07.013.
  • Kucukcoban, S., and L. F. Kallivokas. 2013. A symmetric hybrid formulation for transient wave simulations in PML-truncated heterogeneous media. Wave Motion 50 (1):57–79. doi:10.1016/j.wavemoti.2012.06.004.
  • Kuzuoglu, M., and R. Mittra. 1996. Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers. IEEE Microwave and Guided Wave Letters 6 (12):447–49. doi:10.1109/75.544545.
  • Liang, J., Z. Liu, L. Huang, and G. Yang. 2019. The indirect boundary integral equation method for the broadband scattering of plane P, SV and Rayleigh waves by a hill topography. Engineering Analysis with Boundary Elements 98:184–202. doi:10.1016/j.enganabound.2018.09.018.
  • Lindman, E. L. 1975. Free-space boundary-conditions for the time-dependent wave-equation. Journal of Computational Physics 18 (1):66–78. doi:10.1016/0021-9991(75)90102-3.
  • Liu, S., X. Li, W. Wang, and Y. Liu. 2014. A mixed-grid finite element method with PML absorbing boundary conditions for seismic wave modeling. Journal of Geophysics and Engineering 11 (5):055009-1–13. doi:10.1088/1742-2132/11/5/055009.
  • Liu, T., K. Liu, and J. Zhang. 2004. Unstructured grid method for stress wave propagation in elastic media. Computer Methods in Applied Mechanics and Engineering 193 (23–26):2427–52. doi:10.1016/j.cma.2004.01.030.
  • Liu, T., Y. Luan, and W. Zhong. 2012. Earthquake responses of clusters of building structures caused by a near-field thrust fault. Soil Dynamics and Earthquake Engineering 42:56–70. doi:10.1016/j.soildyn.2012.06.002.
  • Liu, T., and W. Zhong. 2014. Earthquake responses of near-fault frame structure clusters due to thrust fault by using flexural wave method and viscoelastic model of earth medium. Soil Dynamics and Earthquake Engineering 61:57–62. doi:10.1016/j.soildyn.2014.01.023.
  • Liu, T., and W. Zhong. 2017. Earthquake responses of near-fault building clusters in mountain city considering viscoelasticity of earth medium and process of fault rupture. Soil Dynamics and Earthquake Engineering 99:137–41. doi:10.1016/j.soildyn.2017.05.012.
  • Luco, J. E., and F. C. P. De Barros. 1994. Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half‐space. Earthquake Engineering & Structural Dynamics 23 (3):321–40. doi:10.1002/eqe.4290230307.
  • Martin, R., and D. Komatitsch. 2009. An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation. Geophysical Journal International 179 (1):333–44. doi:10.1111/j.1365-246X.2009.04278.x.
  • Martin, R., D. Komatitsch, and S. D. Gedney. 2008. A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation. CMES-Computer Modeling Engineering & Science 37 (3):274–304.
  • 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. International Journal for Numerical Methods in Engineering 88 (10):951–73. doi:10.1002/nme.3205.
  • Meng, K., C. Cui, Z. Liang, H. Li, and H. Pei. 2020. A new approach for longitudinal vibration of a large-diameter floating pipe pile in visco-elastic soil considering the three-dimensional wave effects. Computers and Geotechnics 128:103840. doi:10.1016/j.compgeo.2020.103840.
  • Meza-Fajardo, K. C., and A. S. Papageorgiou. 2008. A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: Stability analysis. Bulletin of the Seismological Society of America 98 (4):1811–36. doi:10.1785/0120070223.
  • Meza-Fajardo, K. C., and A. S. Papageorgiou. 2010. On the stability of a non-convolutional perfectly matched layer for isotropic elastic media. Soil Dynamics and Earthquake Engineering 30 (3):68–81. doi:10.1016/j.soildyn.2009.09.002.
  • Meza‐Fajardo, K. C., and A. S. Papageorgiou. 2012. Study of the accuracy of the multiaxial perfectly matched layer for the elastic‐wave equation. Bulletin of the Seismological Society of America 102 (6):2458–67. doi:10.1785/0120120061.
  • Peng, C., and M. N. Toksöz. 1994. An optimal absorbing boundary condition for finite difference modeling of acoustic and elastic wave propagation. The Journal of the Acoustical Society of America 95 (2):733–45. doi:10.1121/1.408384.
  • Ping, P., Y. Zhang, and Y. Xu. 2014. A multiaxial perfectly matched layer (M-PML) for the long-time simulation of elastic wave propagation in the second-order equations. Journal of Applied Geophysics 101:124–35. doi:10.1016/j.jappgeo.2013.12.006.
  • Qin, Z., M. Lu, X. Zheng, Y. Yao, C. Zhang, and J. Song. 2009. The implementation of an improved NPML absorbing boundary condition in elastic wave modeling. Applied Geophysics 6 (2):113–21. doi:10.1007/s11770-009-0012-3.
  • Shi, R., S. Wang, and J. Zhao. 2012. An unsplit complex-frequency-shifted PML based on matched Z-transform for FDTD modelling of seismic wave equations. Journal of Geophysics and Engineering 9 (2):218–29. doi:10.1088/1742-2132/9/2/218.
  • Sochacki, J., R. Kubichek, J. George, W. R. Fletcher, and S. Smithson. 1987. Absorbing boundary conditions and surface waves. Geophysics 52 (1):60–71. doi:10.1190/1.1442241.
  • Stacey, R. 1988. Improved transparent boundary formulations for the elastic-wave equation. Bulletin of the Seismological Society of America 78 (6):2089–97. doi:10.1785/BSSA0780062089.
  • Vai, R., J. M. Castillo-Covarrubias, F. J. Sánchez-Sesma, D. Komatitsch, and J. P. Vilotte. 1999. Elastic wave propagation in an irregularly layered medium. Soil Dynamics and Earthquake Engineering 18 (1):11–18. doi:10.1016/S0267-7261(98)00027-X.
  • Wang, T., and X. Tang. 2003. Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach. Geophysics 68 (5):1749–55. doi:10.1190/1.1620648.
  • Zembaty, Z., G. Mutke, D. Nawrocki, and P. Bobra. 2017. Rotational ground-motion records from induced seismic events. Seismological Research Letters 88 (1):13–22. doi:10.1785/0220160131.
  • Zeng, C., J. Xia, R. D. Miller, and G. P. Tsoflias. 2011. Application of the multiaxial perfectly matched layer (M-PML) to near-surface seismic modeling with Rayleigh waves. Geophysics 76 (3):T43–52. doi:10.1190/1.3560019.
  • Zhang, N., J. Pan, Y. Gao, X. Chen, D. Dai, and Y. Zhang. 2022. Surface motion of an arbitrary number of semi-circular viscoelastic alluvial valleys for incident plane SH waves. Geophysical Journal International 228 (3):1607–20. doi:10.1093/gji/ggab414.
  • Zhang, W., and Y. Shen. 2010. Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling. Geophysics 75 (4):T141–54. doi:10.1190/1.3463431.
  • Zhang, Z., W. Zhang, and X. Chen. 2014. Complex frequency-shifted multi-axial perfectly matched layer for elastic wave modelling on curvilinear grids. Geophysical Journal International 198 (1):140–53. doi:10.1093/gji/ggu124.
  • Zhong, W., and T. Liu. 2016. Application of an investigated lump method to the simulation of ground motion for Beichuan town during the Wenchuan earthquake. Journal of Earthquake and Tsunami 10 (1):1650002-1–14. doi:10.1142/S1793431116500020.
  • Zhong, W., and T. Liu. 2022. A mesh grading technique for near-fault seismic wave propagation in large velocity-contrast viscoelastic earth media. Journal of Earthquake Engineering 26 (3):1388–415. doi:10.1080/13632469.2020.1719240.
  • Zhong, W., T. Liu, Y. Jiang, and B. Qian. 2023. Seismic responses of near-fault building clusters including in-plane rotational component in investigated lump model of earth medium. Soil Dynamics and Earthquake Engineering 167:107796. doi:10.1016/j.soildyn.2023.107796.
  • Zhou, F. X., Q. Ma, and B. B. Gao. 2016. Efficient unsplit perfectly matched layers for finite-element time-domain modeling of elastodynamics. Journal of Engineering Mechanics 142 (11):04016081-1–12. doi:10.1061/(ASCE)EM.1943-7889.0001145.
  • Zienkiewicz, O. C., R. L. Taylor, and J. Z. Zhu. 2005. The finite element method: Its basis and fundamentals, 6th ed. Amsterdam: Elsevier Butterworth-Heinemann.

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