Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Survey of cubic B-spline implicit time integration method in computational wave propagation

  • Rostami, S. (Department of Civil Engineering, Technical and Vocational University (TVU)) ;
  • Hooshmand, B. (Department of Civil Engineering, Shahid Bahonar University of Kerman) ;
  • Shojaee, S. (Department of Civil Engineering, Shahid Bahonar University of Kerman) ;
  • Hamzehei-Javaran, S. (Department of Civil Engineering, Shahid Bahonar University of Kerman)
  • Received : 2020.04.12
  • Accepted : 2021.06.29
  • Published : 2021.08.25
⟨ Previous Next ⟩

Abstract

Recently an implicit time integration method based-on Cubic B-spline has been presented for solving the problems in structural dynamics. This method benefits from high accuracy and desire stability. In this paper, the presented method is developed for analyzing the wave propagation problems and the results are compared to the methods in the literature including a series of Bathe family methods (Noh-Bathe, Standard Bathe β1/β2-Bathe and ρ∞-Bathe methods) and Newmark trapezoidal rule method. A numerical dispersion analysis is presented to evaluate the method in one and two wave propagation problems. Results indicate that the method under study has adequate performance in wave propagation as good as structural dynamic problems.

Keywords

References

  1. Alamatian, J. (2013), "New implicit higher order time integration for dynamic analysis", Struct. Eng. Mech., 48(5), 711-736. https://doi.org/10.12989/sem.2013.48.5.711.
  2. Bathe, K.J. (2007), "Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme", Comput. Struct., 85(7-8), 437-445. https://doi.org/10.1016/j.compstruc.2006.09.004.
  3. Bathe, K. (1996), Finite Element Procedures, Prentice-Hall Englewood Cliffs, NJ.
  4. Carrer, J.A.M. and Mansur, W.J. (2006), "Solution of the two-dimensional scalar wave equation by the time-domain boundary element method: lagrange truncation strategy in time integration", Struct. Eng. Mech., 23(3), 263-278. https://doi.org/10.12989/sem.2006.23.3.263.
  5. Chang, S.Y. (2010), "A new family of explicit methods for linear structural dynamics", Comput. Struct., 88(11-12), 755-772. https://doi.org/10.1016/j.compstruc.2010.03.002.
  6. Chang, S.Y. (2014), "Family of structure-dependent explicit methods for structural dynamics", J. Eng. Mech., 140(6), 06014005. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000748.
  7. Chang, S.Y. (2014), "Numerical dissipation for explicit, unconditionally stable time integration methods", Earthq. Struct., 7(2), 159-178. https://doi.org/10.12989/eas.2014.7.2.159.
  8. Fung, T. (2003), "Numerical dissipation in time-step integration algorithms for structural dynamic analysis", Prog. Struct. Eng. Mater., 5(3), 167-180. https://doi.org/10.1002/pse.149.
  9. Gopalakrishnan, S., Ruzzene, M. and Hanagud, S. (2011), Spectral Finite Element Method, Springer
  10. Guddati, M.N. and Yue, B. (2004), "Modified integration rules for reducing dispersion error in finite element methods", Comput. Meth. Appl. Mech. Eng., 193(3-5), 275-287. https://doi.org/10.1016/j.cma.2003.09.010.
  11. Hilber, H.M. and Hughes, T.J. (1978), "Collocation, dissipation and [overshoot] for time integration schemes in structural dynamics", Earthq. Eng. Struct. Dyn., 6(1), 99-117. https://doi.org/10.1002/eqe.4290060111.
  12. Kuhl, D. and Crisfield, M. (1999), "Energy-conserving and decaying algorithms in non-linear structural dynamics", Int. J. Numer. Meth. Eng., 45(5), 569-599. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5%3C569::AID-NME595%3E3.0.CO;2-A.
  13. Malakiyeh, M.M., Shojaee, S. and Bathe, K.J. (2019), "The Bathe time integration method revisited for prescribing desired numerical dissipation", Comput. Struct., 212, 289-298. https://doi.org/10.1016/j.compstruc.2018.10.008.
  14. Mullen, R. and Belytschko, T. (1982), "Dispersion analysis of finite element semidiscretizations of the two-dimensional wave equation", Int. J. Numer. Meth. Eng., 18(1), 11-29. https://doi.org/10.1002/nme.1620180103.
  15. Newmark, N.M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., 85(3), 67-94. https://doi.org/10.1061/JMCEA3.0000098
  16. Noh, G. and Bathe, K.J. (2019), "The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method", Comput. Struct., 212, 299-310. https://doi.org/10.1016/j.compstruc.2018.11.001.
  17. Noh, G., Ham, S. and Bathe, K.J. (2013), "Performance of an implicit time integration scheme in the analysis of wave propagations", Comput. Struct., 123, 93-105. https://doi.org/10.1016/j.compstruc.2013.02.006.
  18. Pezeshk, S. and Camp, C. (1995), "An explicit time-integration method for damped structural systems", Struct. Eng. Mech., 3(2), 145-162. https://doi.org/10.12989/sem.1995.3.2.145.
  19. Rezaiee-Pajand, M., Esfehani, S. and Karimi-Rad, M. (2018), "Highly accurate family of time integration method", Struct. Eng. Mech., 67(6), 603-616. https://doi.org/10.12989/sem.2018.67.6.603.
  20. Rogers, D. (2001), An Introduction to NURBS: With Historical Perspective. Morgan Kaufmann, San Francisco, CA.
  21. Rostami, S. and Shojaee, S. (2017), "Alpha-modification of cubic B-spline direct time integration method", Int. J. Struct. Stab. Dyn., 17(10), 1750118. https://doi.org/10.1142/S0219455417501188.
  22. Rostami, S. and Shojaee, S. (2018), "A family of cubic B-spline direct integration algorithms with controllable numerical dissipation and dispersion for structural dynamics", Iran. J. Sci. Technol. Tran. Civil Eng., 42(1), 17-32. https://doi.org/10.1007/s40996-017-0083-y.
  23. Rostami, S., Shojaee, S. and Moeinadini, A. (2012), "A parabolic acceleration time integration method for structural dynamics using quartic B-spline functions", Appl. Math. Model., 36(11), 5162-5182. https://doi.org/10.1016/j.apm.2011.11.047.
  24. Rostami, S., Shojaee, S. and Saffari, H. (2013), "An explicit time integration method for structural dynamics using cubic B-spline polynomial functions", Scientia Iranica, 20(1), 23-33. https://doi.org/10.1016/j.scient.2012.12.003.
  25. Shojaee, S., Rostami, S. and Abbasi, A. (2015), "An unconditionally stable implicit time integration algorithm: Modified quartic B-spline method", Comput. Struct., 153, 98-111. https://doi.org/10.1016/j.compstruc.2015.02.030.
  26. Shojaee, S., Rostami, S. and Moeinadini, A. (2011), "The numerical solution of dynamic response of SDOF systems using cubic B-spline polynomial functions", Struct. Eng. Mech., 38(2), 211-229. https://doi.org/10.12989/sem.2011.38.2.211.
  27. Wang, Y.C., Murti, V. and Valliappan, S. (1992), "Assessment of the accuracy of the Newmark method in transient analysis of wave propagation problems", Earthq. Eng. Struct. Dyn., 21(11), 987-1004. https://doi.org/10.1002/eqe.4290211104.
  28. Yasamani, K. and Mohammadzadeh, S. (2017), "A novel two sub-stepping implicit time integration algorithm for structural dynamics", Earthq. Struct., 13(3), 279-288. https://doi.org/10.12989/eas.2017.13.3.279.