Seismic motions in a non-homogeneous soil deposit with tunnels by a hybrid computational technique

  • Manolis, G.D. (Department of Civil Engineering, Aristotle University) ;
  • Makra, Konstantia (Institute of Engineering Seismology and Earthquake Engineering) ;
  • Dineva, Petia S. (Institute of Mechanics, Bulgarian Academy of Sciences) ;
  • Rangelov, Tsviatko V. (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences)
  • Received : 2012.07.03
  • Accepted : 2013.02.05
  • Published : 2013.08.07


We study seismically induced, anti-plane strain wave motion in a non-homogeneous geological region containing tunnels. Two different scenarios are considered: (a) The first models two tunnels in a finite geological region embedded within a laterally inhomogeneous, layered geological profile containing a seismic source. For this case, labelled as the first boundary-value problem (BVP 1), an efficient hybrid technique comprising the finite difference method (FDM) and the boundary element method (BEM) is developed and applied. Since the later method is based on the frequency-dependent fundamental solution of elastodynamics, the hybrid technique is defined in the frequency domain. Then, an inverse fast Fourier transformation (FFT) is used to recover time histories; (b) The second models a finite region with two tunnels, is embedded in a homogeneous half-plane, and is subjected to incident, time-harmonic SH-waves. This case, labelled as the second boundary-value problem (BVP 2), considers complex soil properties such as anisotropy, continuous inhomogeneity and poroelasticity. The computational approach is now the BEM alone, since solution of the surrounding half plane by the FDM is unnecessary. In sum, the hybrid FDM-BEM technique is able to quantify dependence of the signals that develop at the free surface to the following key parameters: seismic source properties and heterogeneous structure of the wave path (the FDM component) and near-surface geological deposits containing discontinuities in the form of tunnels (the BEM component). Finally, the hybrid technique is used for evaluating the seismic wave field that develops within a key geological cross-section of the Metro construction project in Thessaloniki, Greece, which includes the important Roman-era historical monument of Rotunda dating from the 3rd century A.D.


  1. Ahmad, S., Leyte, F. and Rajapakse, R.K.N.D. (2001), "BEM analysis of two-dimensional elastodynamic problems of anisotropic solids", J. Eng. Mech. - ASCE, 127, 149-156.
  2. Alterman, Z.S. and Karal, F.C. (1968), "Propagation of elastic waves in layered media by finite difference methods", B. Seismol. Soc. Am., 58, 367-398.
  3. Alvarez-Rubio, S., Benito, J.J., Sanchez-Sesma, F.J. and Alarcon, E. (2005), "The use of direct boundary element method for gaining insight into complex seismic site response", Comput. Struct., 83, 821-835.
  4. ANSYS Release 10.0 (2009), Structural mechanics package, Canonsburg, Pennsylvania
  5. Aznarez, J.J., Maezo, O. and Dominguez, J. (2006), "BE analysis of bottom sediments in dynamic fluid-structure interaction problems", Eng. Anal. Bound. Elem., 30, 124-136.
  6. Bardet, J.P. (1992), "A viscoelastic model for the dynamic behaviour of saturated poroelastic soils", Transac. ASME, 59, 128-135.
  7. Bielak, J. and Christiano, P. (1984), "On the effective seismic input for nonlinear soil- structure-interaction systems", Earthq. Eng. Struct. Dyn., 12, 107-119.
  8. Bielak, J., Loukakis, K., Hisada, Y. and Yoshimura, C. (2003) "Domain reduction method for three-dimensional earthquake modelling in localized regions. Part I: Theory", B. Seismol. Soc. Am., 93(2), 817-824.
  9. Biot, M. (1956), "Theory of propagation of elastic waves in a fluid-saturated porous solid", J. Acoust. Soc. Am., 28(4), 168-191
  10. Bouchon, M. and Sanchez-Sesma, F.J. (2007), "Boundary integral equations and boundary elements methods in elastodynamics", Adv. Geophys., 48, 157-189.
  11. CEN (2004), Eurocode 8: Design provisions of structures for earthquake resistance. Part 1: General rules, seismic actions and rules for buildings, Final Draft pr EN1998-1, European Committee for Standardization, Brussels.
  12. Christensen, R.M. (1971), Theory of viscoelasticity: an introduction, Academic Press, New York.
  13. Denda, M., Wang, C.Y. and Yong, Y.K. (2003), "2D time-harmonic BEM for solids of general anisotropy with application to eigenvalue problems", J. Sound Vib., 261, 247-276.
  14. Dineva, P. and Manolis, G.D. (2001), "Scattering of seismic waves by cracks in multi-layered geological regions: I. Mechanical model & II. Numerical results", Soil Dyn. Earthq. Eng., 21, 615-625 & 627-641.
  15. Dineva, P., Manolis, G.D. and Rangelov, T.V. (2006), "Sub-surface crack in inhomogeneous half-plane: wave scattering phenomena by BEM", Eng. Anal. Bound. Elem., 30(5), 350-362.
  16. Dineva, P., Rangelov, T.V. and Gross, D. (2005), "BEM for 2D steady-state problems in cracked anisotropic materials", Eng. Anal. Bound. Elem., 29, 689-698.
  17. Dineva, P., Rangelov, T.V. and Manolis, G.D. (2007), "Elastic wave propagation in a class of cracked functionally graded materials by BEM", Computat. Mech., 39(3), 293-308.
  18. Dravinski, M. and Niu, Y. (2002), "Three-dimensional time-harmonic Green's functions for a triclinic full-space using a symbolic computation system", Int. J. Numer. Methods Eng., 53, 455-472.
  19. EERI (2003), "Preliminary observations on the August 14, 2003 Lefkada Island (Western Greece) Earthquake", Special Earthquake Report,
  20. Fah, D. (1992), "A hybrid technique for the estimation of strong ground motion in sedimentary basins", PhD Thesis, ETH Nr. 9767, Swiss Federal Institute of Technology, Zurich, Switzerland.
  21. Fah, D., Suhadolc, P., Muller, S. and Panza, G.F. (1994), "A hybrid method for the estimation of ground motion in sedimentary basins: quantitative modelling for Mexico City", B. Seismol. Soc. Am., 84(2), 383-399.
  22. Fah, D., Suhadolc, P. and Panza, G.F. (1990), "Estimation of strong ground motion in laterally heterogeneous media: modal summation-finite differences", Proceedings of the 9th European Conference on Earthquake Engineering, Moscow, USSR, Vol. 4A, pp. 100-109.
  23. Fah, D., Suhadolc, P. and Panza, G.F. (1993), "Variability of seismic ground motion in complex media: the case of a sedimentary basin in the Friuli Italy area", J. Appl. Geophys., 30, 131-148.
  24. Galis, M., Moczo, P. and Kristek, J. (2008), "A 3-D hybrid finite-difference-finite-element viscoelastic modelling of seismic wave motion", Geophys. J. Int., 175, 153-184.
  25. Gatmiri, B. and Jabbari, E. (2005), "Time-domain Green's functions for unsaturated soils. Part I: Two-dimensional solution & Part II: Three-dimensional solution", Int. J. Solids Struct., 42(23), 5971-5990 & 5991-6002.
  26. Gatmiri, B., Maghoul, P. and Arson, C. (2009), "Site-specific spectral response of seismic movement due to geometrical and geotechnical characteristics of sites", Soil Dyn. Earthq. Eng., 29, 51-70.
  27. Goto, H., Ramirez-Guzman, L. and Bielak, J. (2010), "Simulation of spontaneous rupture based on a combined boundary integral equation method and finite element method approach: SH and P-SV cases", Geophys. J. Int., 183(2), 975-1004.
  28. Kattis, S.E., Beskos, D.E. and Cheng, A.H.D. (2003), "2D dynamic response of unlined and lined tunnels in poroelastic soil to harmonic body waves", Earthq. Eng. Struct. Dyn., 32, 97-110.
  29. Kaynia, A.M. and Banerjee, P.K. (1992), "Fundamental solutions of Biot's equations of dynamic poroelasticity", Int. J. Eng. Sci., 31(5), 817-830.
  30. Knopoff, R., Fredricks, R.F., Gangi, A.F. and Porter, L.D. (1957), "Surface amplitudes of reflected body waves", Geophysics, 22(4), 842-847.
  31. Kobayashi, S., Nishimura, N. and Kishima, T. (1986), A BIE analysis of wave propagation in anisotropic media, Boundary Elements VIII, Springer-Verlag, Berlin, 425-434.
  32. Lee, V.W. (1977), "On deformation near circular underground cavity subjected to incident plane SH-waves", Proceedings of Conference in Application of Computer Methods in Engineering, University of South California, Los Angeles, pp. 951-962.
  33. Lekhnitskii, S.G. (1963), Theory of elasticity of an anisotropic elastic body, Holden-Day, San Francisco.
  34. Lin, C.H, Lee, V.W. and Trifunac, M.D. (2001), "On the Reflection of Elastic Waves in a Poroelastic Half-space Saturated with Non-viscous Fluid", Report No. CE 01-04, Department of Civil Engineering, University of Southern California, Los Angeles.
  35. Lin, C.H., Lee, V.W. and Trifunac, M.D. (2005), "The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid", Soil Dyn. Earthq. Eng., 25, 205-223.
  36. Liu, H. and Zhang, C. (2003), "Internal stress calculation in 2D time domain BEM for wave propagation in anisotropic media", Int. J. Numer. Methods Biomed. Eng., (original title: Communications in Numerical Methods in Engineering), 19(8), 637-643.
  37. Luzon, F., Palencia, V.J., Morales, J., Sanchez-Sesma, F.J. and Garcia, J.M. (2002), "Evaluation of site effects in sedimentary basins", Fisica de la Tierra, 14, 183-214.
  38. Manolis, G.D. and Beskos, D.E. (1989), "Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity", Acta Mecanica, 76, 89-104.
  39. Manolis, G.D., Dineva, P. and Rangelov, T. (2004), "Wave scattering by cracks in inhomogeneous continua using BEM", Int. J. Solids Struct., 41(14), 3905-3927.
  40. Manolis, G.D., Dineva, P. and Rangelov, T.V. (2012) "Dynamic fracture analysis of a smoothly inhomogeneous plane containing defects by BEM", Eng. Anal. Bound. Elem., 36, 727-737.
  41. Manolis, G.D., Rangelov, T.V. and Dineva, P. (2007), "Free-field wave solutions in a half-plane exhibiting a special-type of continuous inhomogeneity", Wave Motion, 44, 304-321.
  42. Manolis, G.D., Rangelov, T.V. and Dineva, P. (2009), "Free-field dynamic response of an inhomogeneous half-space", Arch. Appl. Mech., 79, 595-603.
  43. Manolis, G.D. and Shaw, R.P. (1996), "Green's function for the vector wave equation in a mildly heterogeneous continuum", Wave Motion, 24, 59-83.
  44. Moczo, P. (1989), "Finite-difference technique for SH waves in 2-D media using irregular grids: application to the seismic response problem", Geophys.J. Int., 99, 321-329.
  45. Moczo, P. and Bard, P.Y. (1993), "Wave diffraction, amplification and differential motion near strong lateral discontinuities", B. Seismol. Soc. Am., 83(1), 85-106.
  46. Moczo P., Bystricky E., Kristek J., Carcione M. and Bouchon M. (1997), "Hybrid modelling of P-SV seismic motion at inhomogeneous viscoelastic topographic structures", B. Seismol. Soc. Am., 87(5), 1305-1323.
  47. Moczo, P., Kristek, J., Galis, M., Pazak, P. and Balazovjech, M. (2007), "The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion", Acta Physica Slovaca, 51(2), 177-406.
  48. Moczo, P., Labak, P., Kristek, J. and Hron, F. (1996), "Amplification and differential motion due to an antiplane 2D resonance in the sediment valleys embedded in a layer over the half-space", B. Seismol. Soc. Am., 86, 1434-1446.
  49. Morochnik, V. and Bardet, J.P. (1996), "Viscoelastic approximation of poroelastic media for wave scattering problems", Soil Dyn. Earthq. Eng., 15, 337-346.
  50. Oprsal, I., Matyska, C. and Irikura, K. (2009), "The source-box wave propagation hybrid methods: General formulation and implementation", Geophys. J. Int., 176, 555-564.
  51. Oprsal, I., Pakzad, M., Plicka, V. and Zahradnik, J. (1998), "Ground motion simulation by hybrid methods", In: K. Irikura, K. Kudo, H. Okada, T. Sasatani (Ed.), The Effects of Surface Geology on Seismic Motion, Proceedings of ESG'98, Yokohama, Japan, Vol. 2, Balkema, Rotterdam, 955-960.
  52. Oprsal, I., Plicka, V. and Zahradnik, J. (1998), "Kobe simulation by hybrid methods", In: K. Irikura, K. Kudo, H. Okada, T. Sasatani (Ed.), The Effects of Surface Geology on Seismic Motion, Proceedings of ESG'98, Yokohama, Japan, Vol. 3, Balkema, Rotterdam, 1451-1456.
  53. Oprsal, I. and Zahradnik, J. (2002), "Three-dimensional finite difference method and hybrid modeling of earthquake ground motion", J. Geophys. Res., 107(B8), 16-29.
  54. Petrovski, D. and Naumovski, N. (1979), Part I-Analytical methods in processing of strong motion accelerograms, Publication No. 66, Institute of Earthquake Engineering and Engineering Seismology, Skopje, F.Y.R. of Macedonia, pp.1-69.
  55. Rangelov, T., Dineva, P. and Gross, D. (2003), "A hypersingular traction boundary integral equation method for stress intensity factor computation in a finite cracked body", Eng. Anal. Bound. Elem., 27(1), 9-21
  56. Rangelov, T.V., Manolis, G.D. and Dineva, P. (2005), "Elastodynamic fundamental solutions for certain families of 2D inhomogeneous anisotropic domains: Basic derivations", Eur. J. Mech. A-Solids, 24, 820-836.
  57. Raptakis, D., Makra, K., Anastasiadis, A. and Pitilakis, K. (2004a), "Complex site effects in Thessaloniki (Greece): I. Soil structure and confrontation of observations with 1D analysis", B. Earthq. Eng., 2(3), 271-300.
  58. Raptakis, D., Makra, K., Anastasiadis, A. and Pitilakis, K. (2004b), "Complex site effects in Thessaloniki (Greece): II. 2D SH modeling and engineering insights", B. Earthq. Eng., 2(3), 301-327
  59. Rizzo, F.J. and Shippy, D. (1970), "A method for stress determination in plane anisotropic elastic bodies", J. Compos. Mater., 4, 36-61
  60. Robertson, J.O.A. and Chapman, C.H. (2000), "An efficient method for calculating finite-difference seismograms after model alterations", Geophysics, 65, 907-918
  61. Saez, A. and Dominguez, J. (1999), "BEM analysis of wave scattering in transversely isotropic solids", Int. J. Numer. Methods Eng., 44, 1283-1300<1283::AID-NME544>3.0.CO;2-O
  62. Schanz, M. and Pryl, D. (2004), "Dynamic fundamental solutions for compressible and incompressible modelled poroelastic continua", Int. J. Solids Struct., 41, 4047-4073.
  63. Skarlatoudis, A.A., Papazachos, C.B. and Theodoulidis, N. (2012), "Site-response study of Thessaloniki (Northern Greece) for the 4 July1978 M 5.1 aftershock of the June 1978 M 6.5 sequence using a 3D finite-difference approach", B. Seismol. Soc. Am., 102, 722-737.
  64. Smerzini, C., Aviles, J., Paolucci, R. and Sánchez-Sesma, F.J. (2009), "Effect of underground cavities on surface earthquake ground motion under SH wave propagation", Earthq. Eng. Struct. Dyn., 38, 1441-1460.
  65. Snyder, M.D. and Cruse, T.A. (1975), "Boundary integral analysis of anisotropic cracked plates", Int. J. Fract. Mech., 11, 315-328.
  66. Vrettos, C. (1991), "In-plane vibrations of soil deposits with variable shear modulus. II: Line load", Int. J. Numer. Anal. Methods Geomech., 14, 649-662
  67. Wang, C.Y. and Achenbach, J.D. (1995), "Three-dimensional time-harmonic elastodynamic Green's functions for anisotropic solids", Proceedings of the Royal Society of London A, 449, 441-458
  68. Wang, C.Y., Achenbach, J.D. and Hirose, S. (1996), "Two-dimensional time domain BEM for scattering of elastic waves in anisotropic solids", Int. J. Solids Struct., 33, 3843-3864.
  69. Wuttke, F., Dineva, P.S. and Schanz, T. (2011), "Seismic wave propagation in laterally inhomogeneous geological region via a new hybrid approach", J. Sound Vib., 330, 664-684.
  70. Zahradnik, J. and Moczo, P. (1996), "Hybrid seismic modelling based on discrete wave number and finite difference methods", PAGEOPH, 148(1-2), 21-38.
  71. Zhang, Ch. and Gross, D. (1998), On wave propagation in elastic solids with cracks, Computational Mechanics Publication, Southampton

Cited by

  1. Seismic response of buried metro tunnels by a hybrid FDM-BEM approach vol.13, pp.7, 2015,
  2. Elastic waves in continuous and discontinuous geological media by boundary integral equation methods: A review vol.70, 2015,
  3. Seismic response of laterally inhomogeneous geological region by boundary integral equations vol.202, pp.1, 2015,
  4. Effect of near field earthquake on the monuments adjacent to underground tunnels using hybrid FEA-ANN technique vol.10, pp.4, 2016,
  5. Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates pp.1432-0959, 2018,