Document Type : Article

**Authors**

^{1}
Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

^{2}
Department of Mathematics, University of Colorado, Boulder, Colorado, USA

**Abstract**

In present study, the dynamic analysis of concrete arch dam-reservoir systems is formulated by FE-(FE-TE) approach. In this technique, dam and reservoir are discretized by solid and fluid finite elements. Moreover, the GN high-order condition imposed at the reservoir truncation boundary. This task is formulated by employing a truncation element at that boundary. It is emphasized that reservoir far-field is excluded from the discretized model. The formulation is initially explained in details. Subsequently, the response of idealized Morrow Point arch dam-reservoir system is obtained for two fully reflective and absorptive reservoir bottom/sidewalls conditions for all three types of excitations. Different Orders of GN condition are considered and convergence process is evaluated. Furthermore, the results are compared against exact solutions which are based on rigorous FE-(FE-HE) approach. It is shown that the technique converges prior to beginning of instability problems which is known to exist for high orders in GN condition. It must be emphasized that although time harmonic analysis is considered in the present study, the main part of formulation is explained in the context of time domain. Therefore, the approach can easily be extended for transient type of analysis.

**Keywords**

References:

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23. Lotfi, V. and Zenz, G. "Application of Wavenumber- TD approach for time harmonic analysis of concrete arch dam-reservoir systems", Coupled Systems Mechanics, 7(3), pp. 353-371 (2018).

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25. Lokke, A. and Chopra, A.K. "Direct finite element method for nonlinear earthquake analysis of concrete dams: Simplification, modeling, and practical application", Earthquake Eng. Struct. Dyn., 48(7), pp. 818- 842 (2019).

26. Lotfi, V. and Lotfi, A. "Application of Hagstrom- Warburton high-order truncation boundary condition on time harmonic analysis of concrete arch damreservoir systems", Engineering Computations, 38(7), pp. 2996-3020 (2021).

27. Mashayekhi, M. and Mostafaei, H. "Determining the critical intensity for crack initiation in concrete arch dams by endurance time method", International Journal of Numerical Methods in Civil Engineering, 5(2), pp. 21-32 (2020).

28. Mostafaei, H., Ghamami, M., and Aghabozorgi, P. "Modal identification of concrete arch dam by fully automated operational modal identification", Journal of Structures, 32, pp. 228-236 (2021).

29. Lotfi, V. and Sani, A.A. "Calculation of coupled modes of fluid-structure systems by pseudo symmetric subspace iteration method", Scientia Iranica, Transactions A-Civil Engineering, 26(4), pp. 2100-2107 (2019).

30. Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z., The Finite Element Method, Butterworth-Heinemann (2013).

31. Chopra, A.K. "Hydrodynamic pressure on dams during earthquake", J. Eng. Mech., -ASCE, 93, pp. 205- 223 (1967).

32. Chopra, A.K., Chakrabarti, P., and Gupta, S. "Earthquake response of concrete gravity dams including hydrodynamic and foundation interaction effects", Report No. EERC-80/01, University of California, Berkeley (1980).

33. Fenves, G. and Chopra, A.K. "Effects of reservoir bottom absorption and dam-water-foundation interaction on frequency response functions for concrete gravity dams", Earthq. Eng. Struct. D., 13, pp. 13-31 (1985).

2. Hall, J.F. and Chopra, A.K. "Dynamic analysis of arch dams including hydrodynamic effects", J. Eng. Mech. Div., ASCE, 109(1), pp. 149-163 (1983).

3. Fok, K.-L. and Chopra, A.K. "Frequency response functions for arch dams: hydrodynamic and foundation flexibility effects", Earthquake Eng. Struct. Dyn., 14, pp. 769-795 (1986).

4. Tan, H. and Chopra, A.K. "Earthquake analysis of arch dam including dam-water-foundation rock interaction", Earthquake Eng. Struct. Dyn., 24, pp. 1453- 1474 (1995).

5. Lotfi, V. "Direct frequency domain analysis of concrete arch dams based on FE-(FE-HE)-BE technique", Journal of Computers & Concrete, 1(3), pp. 285-302 (2004).

6. Sommerfeld, A., Partial Differential Equations in Physics, Academic press, NY (1949).

7. Sharan, S.K. "Time domain analysis of infinite fluid vibration", Int. J. Numer. Meth. Eng., 24(5), pp. 945- 958 (1987).

8. Berenger, J.P. "A perfectly matched layer for the absorption of electromagnetic waves", J. Comput. Phys., 114(2), pp. 185-200 (1994).

9. Chew, W.C. and Weedon, W.H. "A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates", Microw. Opt. Techn. Let., 7(13), pp. 599-604 (1994).

10. Basu, U. and Chopra, A.K. "Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation", Comput. Meth. Appl. Mech. Eng., 192(11-12), pp. 1337-1375 (2003).

11. Jiong, L., Jian-wei, M., and Hui-zhu, Y. "The study of perfectly matched layer absorbing boundaries for SH wave fields", Appl. Geophys., 6(3), pp. 267-274 (2009).

12. Zhen, Q., Minghui, L., Xiaodong, Z., et al. "The implementation of an improved NPML absorbing boundary condition in elastic wave modeling", Appl. Geophys., 6(2), pp. 113-121 (2009).

13. Kim, S. and Pasciak, J.E. "Analysis of Cartesian PML approximation to acoustic scattering problems in R2", Wave Motion, 49, pp. 238-257 (2012).

14. Khazaee, A. and Lotfi, V. "Application of perfectly matched layers in the transient analysis of damreservoir systems", Journal of Soil Dynamics and Earthquake Engineering, 60(1), pp. 51-68 (2014).

15. Khazaee, A. and Lotfi, V. "Time harmonic analysis of dam-foundation systems by perfectly matched layers", Journal of Structural Engineering and Mechanics, 50(3), pp. 349-364 (2014).

16. Higdon, R.L. "Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation", Math. Comput., 47(176), pp. 437-459 (1986).

17. Givoli, D. and Neta, B. "High order non-reflecting boundary scheme for time-dependent waves", J. Comput. Phys., 186(1), pp. 24-46 (2003).

18. Hagstrom, T. and Warburton, T. "A new auxiliary variable formulation of high order local radiation boundary condition: corner compatibility conditions and extensions to first-order systems", Wave Motion, 39(4), pp. 327-338 (2004).

19. Givoli, D., Hagstrom, T., and Patlashenko, I. "Finite element formulation with high-order absorbing conditions for time-dependent waves", Comput. Meth. Appl. M., 195(29-32), pp. 3666-3690 (2006).

20. Hagstrom, T., Mar-Or, A. and Givoli, D. "High-order local absorbing conditions for the wave equation: extensions and improvements", J. Comput. Phys., 227, pp. 3322-3357 (2008).

21. Rabinovich, D., Givoli, D., Bielak, J., et al. "A finite element scheme with a high order absorbing boundary condition for elastodynamics", Comput. Meth. Appl. Mech., 200, pp. 2048-2066 (2011).

22. Samii, A. and Lotfi, V. "High-order adjustable boundary condition for absorbing evanescent modes of waveguides and its application in coupled fluid-structure analysis", Wave Motion, 49(2), pp. 238-257 (2012).

23. Lotfi, V. and Zenz, G. "Application of Wavenumber- TD approach for time harmonic analysis of concrete arch dam-reservoir systems", Coupled Systems Mechanics, 7(3), pp. 353-371 (2018).

24. Lokke, A. and Chopra, A.K. "Direct finite element method for nonlinear earthquake analysis of 3-dimensional semi-unbounded dam-water-foundation rock systems", Earthquake Eng. Struct. Dyn., 47(5), pp. 1309-1328 (2018).

25. Lokke, A. and Chopra, A.K. "Direct finite element method for nonlinear earthquake analysis of concrete dams: Simplification, modeling, and practical application", Earthquake Eng. Struct. Dyn., 48(7), pp. 818- 842 (2019).

26. Lotfi, V. and Lotfi, A. "Application of Hagstrom- Warburton high-order truncation boundary condition on time harmonic analysis of concrete arch damreservoir systems", Engineering Computations, 38(7), pp. 2996-3020 (2021).

27. Mashayekhi, M. and Mostafaei, H. "Determining the critical intensity for crack initiation in concrete arch dams by endurance time method", International Journal of Numerical Methods in Civil Engineering, 5(2), pp. 21-32 (2020).

28. Mostafaei, H., Ghamami, M., and Aghabozorgi, P. "Modal identification of concrete arch dam by fully automated operational modal identification", Journal of Structures, 32, pp. 228-236 (2021).

29. Lotfi, V. and Sani, A.A. "Calculation of coupled modes of fluid-structure systems by pseudo symmetric subspace iteration method", Scientia Iranica, Transactions A-Civil Engineering, 26(4), pp. 2100-2107 (2019).

30. Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z., The Finite Element Method, Butterworth-Heinemann (2013).

31. Chopra, A.K. "Hydrodynamic pressure on dams during earthquake", J. Eng. Mech., -ASCE, 93, pp. 205- 223 (1967).

32. Chopra, A.K., Chakrabarti, P., and Gupta, S. "Earthquake response of concrete gravity dams including hydrodynamic and foundation interaction effects", Report No. EERC-80/01, University of California, Berkeley (1980).

33. Fenves, G. and Chopra, A.K. "Effects of reservoir bottom absorption and dam-water-foundation interaction on frequency response functions for concrete gravity dams", Earthq. Eng. Struct. D., 13, pp. 13-31 (1985).

Transactions on Civil Engineering (A)

November and December 2022Pages 2825-2836