شبیه‌سازی میرایی تشعشعی با استفاده از لایه کاملاً تطبیق یافته در مدل‌سازی عددی به روش اجزای محدود

نوع مقاله : Articles

نویسندگان

پژوهشکده مهندسی ژئوتکنیک، پژوهشگاه بین المللی زلزله شناسی و مهندسی زلزله، تهران، ایران

چکیده

در مسائل الاستودینامیک که پدیده انتشار امواج بخش عمده­ای از آن را تشکیل می­دهد، مدل‌سازی و محاسبات مربوط به محیط نامحدود، مستلزم استفاده از مرز مصنوعی برای لحاظ نمودن میرایی تشعشعی است. در پژوهش حاضر، برای تحلیل پاسخ دینامیکی یا لرزه­ای ساختگاه با استفاده از روش اجزای محدود در حوزه زمان، برنامه­ای به کمک نرم‌افزار MATLAB تهیه شد. در مدل‌سازی پدیده انتشار موج، لایه کاملاً تطبیق یافته1 (PML) که توانایی جذب و میرا نمودن امواج تحت هر زاویه برخورد و فرکانس را به لحاظ تئوریک دارد در روش اجزای محدود پیاده­سازی و برای در نظر گرفتن میرایی تشعشعی استفاده شده است. آنچه در مقاله حاضر ارائه می­شود، اعتبارسنجی و بررسی عملکرد لایه تطبیق یافته می­باشد که این امر از طریق حل سه مسئله مورد ارزیابی قرار گرفته است. در مسئله اول، ارتعاش شالوده صلب بدون جرم واقع بر سه حالت نیم­فضای بینهایت، لایه بر روی بستر صلب و لایه بر روی نیم­فضای بینهایت با استفاده از PML تحلیل و نتایج با مدل گسترده2 مقایسه شده است. در مسئله دوم، کارایی PML در مدل‌سازی انتشار امواج سطحی مورد ارزیابی قرار گرفته است. در مسئله سوم نیز، به بررسی بزرگنمایی امواج در دره نیم‌دایره­ای به کمک PML پرداخته شده است. نتایج حاصل، نشان­دهنده‌ی قابلیت مناسب PML در شبیه­سازی میرایی تشعشعی برای محیط نامحدود می­باشد که در واقع بیان­کننده‌ی جذب امواج برگشتی غیرواقعی و سازگار بودن PML با محیط اصلی است.

کلیدواژه‌ها

عنوان مقاله [English]

Radiation Damping Simulation in Finite Eelements Method Analysis Using Perfectly Matched Layer (PML)

نویسندگان [English]

  • Mohammad Davoodi
  • Abbas Pourdeilami
  • Mohammad Kazem Jafari
Geotechnical Engineering Department, International Institute of Earthquake Engineering and Seismology
چکیده [English]

In this research, perfectly matched layer has been implemented in the finite element method to simulate the radiation damping for soil-structure interaction analysis application. The perfectly matched layer (PML) has the ability to absorb and attenuate scattered waves under any angle of incidence and frequency, such that with the minimum dimensions of the modeling and the minimum amount of calculations, high-precision responses can be achieved. In order to time domain dynamic analysis by finite element method, a program is written utilizing MATLAB mathematical language, which is capable of analysis of different geometries, layering and dynamic/seismic loading in models with linear elastic behavior. The present program uses four-noded quadrilateral elements and uses the implicit Newmark method to solve the dynamic equation. The feature of theprogram is the implementation of PML, which can address the simulation of radiation damping in the finite element method correctly. This is done by rewriting the PML formulation, implementation in the finite element method, and step-by-step verifying the analysis of dynamic problems. First of all, to verify the dynamic analysis performance of the program, three simple examples have been solved, and the results show that theyare consistent with existing theories and the literature. Next, using PML, the problem of a rigid massless foundation vibration has been studied. Computing the impedance/compliance functions and comparing them with analytical or semi-analytical approaches existing in the technical texts, the efficiency and the precision of PML for surface loading conditions has been evaluated. In the frequency domain, the results are in good agreement with the previous studies. Besides, comparing the response from the reduced model (using PML) with the expected response from the extended mesh indicates that there is a complete match in the time domain. It is worth noting that this match is achieved while the model dimensions and the volume of data storage have been drastically reduced, but the accuracy of the answers has not varied. This reduction of dimension is such that if PML is located at a distance of up to a quarter of the foundation width, similar responses to larger models can be achieved.

کلیدواژه‌ها [English]

  • Perfectly Matched Layer
  • Radiation Damping
  • wave propagation
  • Finite elements method
  • Soil-Structure Interaction

مراجع

  1. Wilson, E.L. (2002) Three-dimensional static and dynamic analysis of structures a physical approach with emphasis on earthquake engineering. Computers and Structures, Inc. Third Edition.
  2. Szczesiak, T., Weber, B., and Bachmann, H. (1999) Non-uniform earthquake input for arch dam foundation interaction. Soil Dynamics & Earthquake Engineering, 18, 487–493.
  3. Wolf, J.P. (1985) Dynamic Soil Structure Interaction. NewJersey, Prentice Hall.
  4. Berenger, J.P. (1994) A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114, 185-200.
  5. Jiao, D., Jin, J. (2001) Time-domain finite-element modeling of dispersive media. IEEE Microwave and Wireless Components Letters, 11(5), 220–2.
  6. Kang, J.W., Kallivokas, L.F. (2010) Mixed unsplit-field perfectly matched layers for transient simulations of scalar waves in heterogeneous domains. Computational Geosciences, 14(4), 623–48.
  7. Nataf, F. (2006) A new approach to perfectly matched layers for the linearized Euler system. Journal of Computational Physics, 214(2), 757–72.
  8. Martin, R., Komatitsch, D., Ezziani, A. (2008) An unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave equation in poroelastic media. Geophysics, 73(4), T51-T61.
  9. Zheng, Y.Q., He, J.Q., and Liu, Q.H. (2001) The application of the perfectly matched layer in numerical modeling of wave propagation in poroelastic media. Geophysics, 66(4), 1258-1266.
  10. Chew, W.C. and Liu, Q. (1996) Perfectly matched layers for elastodynamics: a new absorbing boundary condition. Journal of Computational Acoustics, 4, 341-359.
  11. Chew, W.C., Jin, J.M. and Michielssen, E. (1997) Complex coordinate stretching as a generalized absorbing boundary condition. Microw. Opt. Tech. Let., 15(6), 363-369.
  12. Chew, W.C. and Weedon, W.H. (1994) A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates. Microw. Opt. Technol. Lett., 7, 599–604.
  13. Collino, F. and Tsogka, C. (2001) Application of the PML absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous
  14. media. Geophysics, 66(1), 294-307.
  15. Becache, E. Fauqueux, S., Joly, P. (2003) Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys., 188, 399-433.
  16. Festa, G. and Nielsen, S. (2003) PML absorbing boundaries. Bulletin of the Seismological Society of America, 93(2), 891-903.
  17. Basu, U. and Chopra, A.K. (2003) Perfectly matched layers for time-harmonic elastodynamics of unbounded domains theory and finite-element implementation. Comput. Method. Appl. Mech. Eng., 192, 1337-1375.
  18. Appelo, D. and Kreiss, G. (2006) A new absorbing layer for elastic waves. J. Comput. Phys., 215, 642-660.
  19. Harrari, I. and Albocher, U. (2006) Studies of FE-PML for exterior problems of time-harmonic elastic waves. Comput. Method. Appl. Mech. Eng., 195, 3854-3879.
  20. Ma, S. and Liu, P. (2006) Modeling of the perfectly matched layer absorbing boundaries and intrinsic attenuation in explicit finite-element method. B. Seismol. Soc. Am., 96(5), 1779-1794.
  21. Basu, U. (2009) Explicit finite element perfectly matched layer for transient three-dimensional elastic waves. Int. J. Numer. Method. Eng., 77, 151-176.
  22. Liu, J., Ma, J., and Yang, H. (2009) The study of perfectly matched layer absorbing boundaries for SH wave fields. App. Geophysics, 6(3), 267-274.
  23. Kim, S. and Pasciak, J.E. (2012) Analysis of Cartesian PML approximation to acoustic scattering problems in R2. Wave Motion, 49, 238-257.
  24. Lancioni, G. (2011) Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimensional medium. Comput. Method. Appl. Mech. Eng., 209, 209-212.
  25. Panji, M., Kamalian, M., Asgari Marnani, J., and Jafari, M.K. (2013) Transient analysis of wave propagation problems by half-plane BEM. Geophys. J. Int., 194(3), 1849-1865.
  26. Khazaee, A. and Lotfi, V. (2014) Application of perfectly matched layers in the transient analysis of dam-reservoir systems. Soil Dyn. Earthq. Eng., 60, 51-68.
  27. Farzanian, M., Arbabi, F., and Pak, R. (2016) PML solution of longitudinal wave propagation in heterogeneous media. Earthq. Eng. & Eng. Vib., 15(2), 357-368.
  28. Basu, U. and Chopra, A.K. (2004) Perfectly matched layers for transient elastodynamics of unbounded domains. Int. J. Numer. Method. Eng., 59, 1039-1074.
  29. Chouw, N., Le, R., Schmid, G. (1991) 'Impediment of surface waves in soil'. In: Mathematical and Numerical Aspects of Wave Propagation Phenomena.
  30. Wong, H.L. (1982) Effects of surface topography on the diffraction of P, SV and Rayleigh waves. Bull. Seismol. Soc. Am., 72, 1167-1183.
  31. Mossessian, T.K. and Dravinski, M. (1987) Application of a hybrid method for scattering of P, SV, and Reyleigh waves by near-surface irregularities. Bull. Seismol. Soc. Am., 77, 1784- 1803.

فایل ها

سابقه مقاله

  • تاریخ دریافت: 17 مرداد 1396
  • تاریخ بازنگری: 26 بهمن 1399
  • تاریخ پذیرش: 06 دی 1399

اشتراک گذاری

ارجاع به این مقاله

آمار