Skip to main content

Simulation and characteristics analysis of a wavefield in a thermoelastic medium adopting the rotated staggered-grid pseudo-spectral method and L-S theory

Abstract

The simulation of wave propagation in high-temperature media requires thermoelastic theory. In this paper, we apply the rotated-staggered-grid pseudo-spectral method (RSG-PSM) to solving thermoelastic governing equations based on L-S theory. A time splitting method is used to solve the stiffness problem of the equations, and we introduce the rotated staggered pseudo-spectral operator and centered pseudo-spectral operator to compute the first-order spatial derivatives and second-order spatial derivatives, respectively. In the case of the heterogeneous-medium model, the Crank-Nicolson explicit method is used instead of the pseudo-spectral method to compute the wavefield. The properties and propagation of the thermal coupled wavefield are discussed, and we compare the simulation results obtained using the pseudo-spectral method, staggered-grid pseudo-spectral method, and RSG-PSM. In the case of an isotropic homogeneous medium, we obtain stable and highly accurate results using the time splitting method combined with the RSG-PSM. However, the algorithm cannot be applied with a large time step when the thermal conductivity changes dramatically, and the algorithm is unstable when the reference temperature has a gradient distribution. The optimal combined application of the mesh generation mode and numerical algorithm is explored, laying a foundation for the extension of these methods to thermoporoelasticity, thermoviscoelasticity, and anisotropy.

This is a preview of subscription content, access via your institution.

References

  1. Achenbach J D. 1973. Wave Propagation in Elastic Solids. Amsterdam: Elsevier. 129

    Google Scholar 

  2. Afanasiev M, Boehm C, van Driel M, Krischer L, Rietmann M, May D A, Knepley M G, Fichtner A. 2019. Modular and flexible spectral-element waveform modelling in two and three dimensions. Geophys J Int, 216: 1675–1692

    Article  Google Scholar 

  3. Berezovski A, Berezovski M. 2013. Influence of microstructure on thermoelastic wave propagation. Acta Mech, 224: 2623–2633

    Article  Google Scholar 

  4. Berezovski A, Engelbrecht J, Maugin G A. 2000. Thermoelastic wave propagation in inhomogeneous media. Archive Appl Mech-Ingenieur Archiv, 70: 694–706

    Article  Google Scholar 

  5. Berezovski A, Engelbrecht J, Maugin G A. 2003. Numerical simulation of two-dimensional wave propagation in functionally graded materials. Eur J Mech-A/Solids, 22: 257–265

    Article  Google Scholar 

  6. Berezovski A, Maugin G A. 2001. Simulation of thermoelastic wave propagation by means of a composite wave-propagation algorithm. J Comput Phys, 168: 249–264

    Article  Google Scholar 

  7. Berezovski A, Maugin G A. 2002. Thermoelastic wave and front propagation. J Thermal Stresses, 25: 719–743

    Article  Google Scholar 

  8. Biot M A. 1956. Thermoelasticity and irreversible thermodynamics. J Appl Phys, 27: 240–253

    Article  Google Scholar 

  9. Boley B A, Weiner J H. 1960. Theory of Thermal Stresses. Mineola, NY: Dover Publishing. 586

    Google Scholar 

  10. Bouchon M, Campillo M, Gaffet S. 1989. A boundary integral equation-discrete wavenumber representation method to study wave propagation in multilayered media having irregular interfaces. Geophysics, 54: 1134–1140

    Article  Google Scholar 

  11. Bracewell R N. 1978. The Fourier Transform and Its Applications. 2nd ed. New York: McGraw-Hill. 444

    Google Scholar 

  12. Brebbia C A. 1978. The Boundary Element Method for Engineers. London: Pentech Press. 189

    Google Scholar 

  13. Carcione J M. 2014. Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. 3rd ed. Amsterdam: Elsevier. 661

    Google Scholar 

  14. Carcione J M, Cavallini F, Wang E, Ba J, Fu L. 2019a. Physics and simulation of wave propagation in linear thermoporoelastic media. J Geophys Res-Solid Earth, 124: 8147–8166

    Article  Google Scholar 

  15. Carcione J M, Quiroga-Goode G. 1995. Some aspects of the physics and numerical modeling of Biot compressional waves. J Comp Acous, 3: 261–280

    Article  Google Scholar 

  16. Carcione J M, Wang Z W, Ling W, Salusti E, Ba J, Fu L Y. 2019b. Simulation of wave propagation in linear thermoelastic media. Geophysics, 84: T1–T11

    Article  Google Scholar 

  17. Chakraborty A, Gopalakrishnan S. 2004. Thermoelastic wave propagation in anisotropic layered media—A spectral element formulation. Int J Comput Methods, 1: 535–567

    Article  Google Scholar 

  18. Chen J K, Beraun J E, Grimes L E, Tzou D Y. 2003. Short-time thermal effects on thermomechanical response caused by pulsed lasers. J Thermophys Heat Transfer, 17: 35–42

    Article  Google Scholar 

  19. Chester M. 1963. Second sound in solids. Phys Rev, 131: 2013–2015

    Article  Google Scholar 

  20. Corrêa G J P, Spiegelman M, Carbotte S, Mutter J C. 2002. Centered and staggered Fourier derivatives and Hilbert transforms. Geophysics, 67: 1558–1563

    Article  Google Scholar 

  21. Deresiewicz H. 1957. Plane waves in a thermoelastic solid. J Acoust Soc Am, 29: 204–209

    Article  Google Scholar 

  22. Dong L G, Ma Z T, Cao J Z, Wang H Z, Geng J H, Lei B, Xu S Y. 2000. A staggered-grid high-order difference method of one-order elastic wave equation (in Chinese). Chin J Geophys, 43: 411–419

    Google Scholar 

  23. Francis P H. 1972. Thermo-mechanical effects in elastic wave propagation: A survey. J Sound Vib, 21: 181–192

    Article  Google Scholar 

  24. Fu L Y, Mou Y G. 1994. Boundary element method for elastic wave forward modeling (in Chinese). Chin J Geophys, 37: 521–529

    Google Scholar 

  25. Green A E, Lindsay K A. 1972. Thermoelasticity. J Elasticity, 2: 1–7

    Article  Google Scholar 

  26. Green A E, Naghdi P M. 1991. A re-examination of the basic postulates of thermomechanics. Proc R Soc Lond A, 432: 171–194

    Article  Google Scholar 

  27. Hosseini S M. 2014. Application of a hybrid mesh-free method for shock-induced thermoelastic wave propagation analysis in a layered functionally graded thick hollow cylinder with nonlinear grading patterns. Eng Anal Bound Elem, 43: 56–66

    Article  Google Scholar 

  28. Hosseini S M, Shahabian F, Sladek J, Sladek V. 2011. Stochastic meshless local Petrov-Galerkin (MLPG) method for thermo-elastic wave propagation analysis in functionally graded thick hollow cylinders. Comp Model Eng Sci, 71: 39–66

    Google Scholar 

  29. Hosseini zad S K, Komeili A, Akbarzadeh A H, Eslami M R. 2010. Numerical simulation of elastic and thermoelastic wave propagation in two-dimensional classical and generalized coupled thermoelasticity. In: ASME 2010 10th Biennial Conference on Engineering Systems Design and Analysis. 135–145

  30. Jain M K. 1984. Numerical Solution of Differential Equations. 2nd ed. New York: Wiley. 698

    Google Scholar 

  31. Kawase H. 1988. Time-domain response of a semi-circular canyon for incident SV, P, and Rayleigh waves calculated by the discrete wave-number boundary element method. Bull Seismol Soc Amer, 78: 1415–1437

    Article  Google Scholar 

  32. Kosloff D, Reshef M, Loewenthal D. 1984. Elastic wave calculations by the Fourier method. Bull Seismol Soc Amer, 74: 875–891

    Article  Google Scholar 

  33. Liu W T, Hu S T, Li J, Xiong Z H, Zhao Y A. 2008. Attenuation of seismic wave in thermal elastic medium and calculation of the corresponding Q value (in Chinese). Earthquake, 28: 39–45

    Google Scholar 

  34. Lord H W, Shulman Y. 1967. A generalized dynamical theory of thermoelasticity. J Mech Phys Solids, 15: 299–309

    Article  Google Scholar 

  35. Lu Y, Peng S, Cui X, Li D, Wang K, Xing Z. 2020. 3D FDTD anisotropic and dispersive modeling for GPR using rotated staggered grid method. Comput Geosci, 136: 104397

    Article  Google Scholar 

  36. Lysmer J, Drake L A. 1972. A finite element method for seismology. In: Methods in Computational Physics: Advances in Research and Applications. New York: Elsevier. 181–216

    Google Scholar 

  37. Madariaga R. 1976. Dynamics of an expanding circular fault. Bull Seismol Soc Amer, 66: 639–666

    Google Scholar 

  38. Marfurt K J. 1984. Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics, 49: 533–549

    Article  Google Scholar 

  39. Maurer M J. 1969. Relaxation model for heat conduction in metals. J Appl Phys, 40: 5123–5130

    Article  Google Scholar 

  40. Nettleton R E. 1960. Relaxation theory of thermal conduction in liquids. Phys Fluids, 3: 216–225

    Article  Google Scholar 

  41. Özdenvar T, McMechan G A. 1996. Causes and reduction of numerical artefacts in pseudo-spectral wavefield extrapolation. Geophys J Int, 126: 819–828

    Article  Google Scholar 

  42. Padovani E, Priolo E, Seriani G. 1994. Low and high order finite element method: Experience in seismic modeling. J Comp Acous, 2: 371–422

    Article  Google Scholar 

  43. Robertsson J O A. 1996. A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography. Geophysics, 61: 1921–1934

    Article  Google Scholar 

  44. Rudgers A J. 1990. Analysis of thermoacoustic wave propagation in elastic media. J Acoust Soc Am, 88: 1078–1094

    Article  Google Scholar 

  45. Saenger E H, Gold N, Shapiro S A. 2000. Modeling the propagation of elastic waves using a modified finite-difference grid. Wave Motion, 31: 77–92

    Article  Google Scholar 

  46. Sánchez-Sesma F J. 1983. Diffraction of elastic waves by three-dimensional surface irregularities. Bull Seismol Soc Amer, 73: 1621–1636

    Google Scholar 

  47. Sarma G S, Mallick K, Gadhinglajkar V R. 1998. Nonreflecting boundary condition in finite-element formulation for an elastic wave equation. Geophysics, 63: 1006–1016

    Article  Google Scholar 

  48. Serón F J, Sanz F J, Kindelán M, Badal J I. 1990. Finite-element method for elastic wave propagation. Commun Appl Numer Methods, 6: 359–368

    Article  Google Scholar 

  49. Sumi N. 2001. Numerical solutions of thermoelastic wave problems by the method of characteristics. J Thermal Stresses, 24: 509–530

    Article  Google Scholar 

  50. Veres I A, Berer T, Burgholzer P. 2013. Numerical modeling of thermoelastic generation of ultrasound by laser irradiation in the coupled thermoelasticity. Ultrasonics, 53: 141–149

    Article  Google Scholar 

  51. Virieux J. 1984. SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method. Geophysics, 49: 1933–1942

    Article  Google Scholar 

  52. Wang K, Peng S, Lu Y, Cui X. 2020. The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium. Geophysics, 85: T89–T100

    Article  Google Scholar 

  53. Wang Z W, Fu L Y, Wei J, Hou W, Ba J, Carcione J M. 2020. On the Green function of the Lord-Shulman thermoelasticity equations. Geophys J Int, 220: 393–403

    Article  Google Scholar 

  54. Witte D, Richards P G. 1987. Contributions to the pseudospectral method for computing synthetic seismograms. SEG Technical Program Expanded Abstracts 1987, Society of Exploration Geophysicists. 517–519

  55. Wu B N, Wu S Q, Huang Y, Feng B. 2012. Application of staggered grids in elastic wavefield modeling by pseudo-spectral method (in Chinese). Geophys Prospect Petrol, 51: 440–445+421

    Google Scholar 

  56. Zener C. 1948. Elasticity and Anelasticity of Metals. Chicago: University of Chicago Press. 170

    Google Scholar 

  57. Zou P, Cheng J B. 2016. Modified pseudo-spectral method for wave propagation modelling in arbitrary anisotropic media. In: 78th EAGE Conference and Exhibition 2016. Vienna. 1–5

  58. Zou P, Cheng J. 2018. Pseudo-spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media. Geophys Prospecting, 66: 47–61

    Article  Google Scholar 

Download references

Acknowledgements

We thank the anonymous reviewers for their critical comments on the original manuscript and Glenn Pennycook, MSc, for editing the English text of a draft of this manuscript. This study was supported by the National Natural Science Foundation of China (Grant Nos. 41874125, and 41430322) and the National Key Research and Development Project (Grant Nos. 2018YFC0603701, and 2017YFC06061301).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Cai Liu.

Electronic Supplementary Material

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Liu, C., Guo, Z. et al. Simulation and characteristics analysis of a wavefield in a thermoelastic medium adopting the rotated staggered-grid pseudo-spectral method and L-S theory. Sci. China Earth Sci. 64, 1390–1408 (2021). https://doi.org/10.1007/s11430-020-9742-x

Download citation

Keywords

  • Thermoelastic wave
  • L-S theory
  • Rotated-staggered-grid pseudo-spectral method
  • Seismic wavefield simulation
  • Time splitting method