DOI QR코드

DOI QR Code

Extended implicit integration process by utilizing nonlinear dynamics in finite element

  • 투고 : 2016.12.31
  • 심사 : 2017.08.04
  • 발행 : 2017.11.25

초록

This paper proposes a new direct numerical integration algorithm for solving equation of motion in structural dynamics problems with nonlinear stiffness. The new implicit method's degree of accuracy is higher than that of existing methods due to the higher order of the acceleration. Two parameters are defined, leading to a new family of unconditionally stable methods, which helps to take greater time steps in integration and eliminate concerns about the duration of solving. The method developed can be utilized for a number of solid plane finite elements, examples of which are given to compare the proposed method with existing ones. The results indicate the superiority of the proposed method.

키워드

참고문헌

  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), 437-445. https://doi.org/10.1016/j.compstruc.2006.09.004
  3. Bathe, K.J. and Baig, M.M.I. (2005), "On a composite implicit time integration procedure for nonlinear dynamics", Comput. Struct., 83(31), 2513-2524. https://doi.org/10.1016/j.compstruc.2005.08.001
  4. Bathe, K.J. and Noh, G. (2012), "Insight into an implicit time integration scheme for structural dynamics", Comput. Struct., 98, 1-6.
  5. Bathe, K. (1996), Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ.
  6. Bathe, K.J. and Wilson, E.L. (1973), "Stability and accuracy analysis of direct time integration methods", Earthq. Eng. Struct. Dyn., 1, 283-291.
  7. Bayat, M. and Pakar, I. (2017), "Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems", Struct. Eng. Mech., 61(5), 657-661. https://doi.org/10.12989/sem.2017.61.5.657
  8. Belytschko, T. and Lu, Y. (1993), "Explicit multi-time step integration for first and second order finite element semidiscretizations", Comput. Meth. Appl. Mech. Eng., 3-4(108), 353-383.
  9. Belytschko, T., Liu, W.K. and Moran, B. (2000), Nonlinear Finite Elements for Continua and Structures, 3rd Edition, Wiley, Chichester, U.K.
  10. Chang, S.Y. (2002), "Integrated equations of motion for direct integration methods", Struct. Eng. Mech., 13(5), 569-589. https://doi.org/10.12989/sem.2002.13.5.569
  11. Chang, S.Y. (2007), "Improved explicit method for structural dynamics", J. Eng. Mech., 133(7), 748-760. https://doi.org/10.1061/(ASCE)0733-9399(2007)133:7(748)
  12. Chang, S.Y. (2010), "A new family of explicit methods for linear structural dynamics", Comput. Struct., 88(11), 755-772. https://doi.org/10.1016/j.compstruc.2010.03.002
  13. Chang, S.Y., Wu, T.H. and Tran, N.C. (2015), "A family of dissipative structure-dependent integration methods", Struct. Eng. Mech., 55(4), 815-837. https://doi.org/10.12989/sem.2015.55.4.815
  14. Chen, S., Hansen, J.M. and Tortorelli, D.A. (2000), "Unconditionally energy stable implicit time integration: application to multibody system analysis and design", Int. J. Numer. Meth. Eng., 48(6), 791-822. https://doi.org/10.1002/(SICI)1097-0207(20000630)48:6<791::AID-NME859>3.0.CO;2-Z
  15. Chen, W.F. and Han, D.J. (2007), Plasticity for Structural Engineers, J. Ross Publishing.
  16. Chopra, A. (2007), Dynamics of Structures: Theory and Applications to Earthquake Engineering, 3rd Ed. Edition, Prentice-Hall, Upper Saddle River, NJ.
  17. Clough, R.W. and Penzien, J. (1983), Dynamics of Structures, McGraw Hill, New York.
  18. Crisfield, M. (1979), "A faster modified Newton-Raphson iteration", Comput. Meth. Appl. Mech. Eng., 20(3), 267-278. https://doi.org/10.1016/0045-7825(79)90002-1
  19. Dokainish, M. and Subbaraj, K. (1989), "A survey of direct time-integration methods in computational structural dynamics-I. Explicit methods", Comput. Struct., 32(6), 1371-1386. https://doi.org/10.1016/0045-7949(89)90314-3
  20. Felippa, C.A. and Park, K.C. (1979), "Direct time integration methods in nonlinear structural dynamics", Comput. Meth. Appl. Mech. Eng., 17-18(2), 277-313. https://doi.org/10.1016/0045-7825(79)90023-9
  21. Gao, Q., Wu, F., Zhang, H., Zhong, W., Howson, W. and Williams, F. (2012), "A fast precise integration method for structural dynamics problems", Struct. Eng. Mech., 43(1), 1-13. https://doi.org/10.12989/sem.2012.43.1.001
  22. Gholampour, A.A. and Ghassemieh, M. (2013), "Nonlinear structural dynamics analysis using weighted residual integration", Mech. Adv. Mater. Struct., 20, 199-216. https://doi.org/10.1080/15376494.2011.584146
  23. Gholampour, A.A., Ghassemieh, M. and Razavi, H. (2011), "A time stepping method in analysis of nonlinear structural dynamics", Appl. Comput. Mech., 5, 143-150.
  24. Goudreau, G.L. and Taylor, R.L. (1972), "Evaluation of numerical integration methods in elastodynamics", Comput. Meth. Appl. Mech. Eng., 2, 69-97.
  25. Hejranfar, K. and Parseh, K. (2016), "Numerical simulation of structural dynamics using a high-order compact finite-difference scheme", Appl. Math. Model., 40(3), 2431-2453. https://doi.org/10.1016/j.apm.2015.09.067
  26. Hilber, H.M. (1977), Analysis and Design of Numerical Integration Methods in Structural Dynamics, Earthquake Engineering Research Center, University of California, Berkeley.
  27. Houbolt, J.C. (1950), "A recurrence matric solution for the dynamic response of aircraft in gusts", NACA TN, 2060.
  28. Howe, R. (1991), "A new family of real-time redictor-corrector integration algorithms", Simul., 57(3), 177-186. https://doi.org/10.1177/003754979105700308
  29. Hughes, T. (1987), The Finite Element Methods, Eaglewood Cliffs, New Jersy, NJ.
  30. Hughes, T. and Belytschko, T. (1983), "A precis of developments in computational methods for transient analysis", J. Appl. Mech., 50(4b), 1033-1041. https://doi.org/10.1115/1.3167186
  31. Humar, J.L. (1990), Dynamics of Structures, Prentice-Hall, Englewood Cliffs, NJ.
  32. Keierleber, C. and Rosson, B. (2005), "Higher-order implicit dynamic time integration method", J. Struct. Eng., 131(8), 1267-1276. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:8(1267)
  33. Kim, J. and Kim, D. (2015), "A quadratic temporal finite element method for linear elastic structural dynamics", Math. Comput. Simul., 117, 68-88. https://doi.org/10.1016/j.matcom.2015.05.009
  34. Krieg, R. (1973), "Unconditional stability in numerical time integration methods", J. Appl. Mech., 40(2), 417-421. https://doi.org/10.1115/1.3422999
  35. Leontiev, V. (2007), "Extension of LMS formulations for L stable optimal integration methods with U0-V0 overshoot properties in structural dynamics: the level symmetric (LS) integration methods", Int. J. Numer. Meth. Eng., 71(13), 1598-1632. https://doi.org/10.1002/nme.2008
  36. Leontyev, V. (2010), "Direct time integration algorithm with controllable numerical dissipation for structural dynamics: two-step Lambda method", Appl. Numer. Math., 60(3), 277-292. https://doi.org/10.1016/j.apnum.2009.12.005
  37. Lindsay, P., Parks, M. and Prakash, A. (2016), "Enabling fast, stable and accurate peridynamic computations using multi-time-step integration", Comput. Meth. Appl. Mech. Eng., 306, 382-405. https://doi.org/10.1016/j.cma.2016.03.049
  38. Liu, Q., Ma, X., Bai, Z. and Zhuansun, X. (2013), "A Split-Step-Scheme-Based Precise Integration Time Domain Method for Solving Wave Equation", COMPEL: Int. J. Comput. Math. Elec. Electron. Eng., 33(1/2), 9-9.
  39. Lourderaj, U., Song, K., Windus, T. L., Zhuang, Y. and Hase, W.L. (2007), "Direct dynamics simulations using Hessian-based predictor-corrector integration algorithms", J. Chem. Phys., 126(4), 044105. https://doi.org/10.1063/1.2437214
  40. Newmark, N.M. (1959), "A method of computation for structural dynamics", J. Eng. Mech. Div., 85(3), 67-94.
  41. Park, K.C. (1977), "Practical aspects of numerical time integration", Comput. Struct., 7(3), 343-353. https://doi.org/10.1016/0045-7949(77)90072-4
  42. Paz, M. and Leigh, W. (2003), Structural Dynamics: Theory and Computation, Springer, Netherlands.
  43. 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
  44. Pezeshk, S. and Camp, C.V. (1995), "An explicit time integration technique for dynamic analysis", Int. J. Numer. Meth. Eng., 38(13), 2265-2281. https://doi.org/10.1002/nme.1620381308
  45. Razavi, S., Abolmaali, A. and Ghassemieh, M. (2007), "A weighted residual parabolic acceleration time integration method for problems in structural dynamics", Comput. Meth. Appl. Math. Comput. Meth. Appl. Math., 7(3), 227-238.
  46. Rezaiee-Pajand, M. and Alamatian, J. (2008), "Nonlinear dynamic analysis by dynamic relaxation method", Struct. Eng. Mech., 28(5), 549-570. https://doi.org/10.12989/sem.2008.28.5.549
  47. Rezaiee-Pajand, M. and Alamatian, J. (2008), "Numerical time integration for dynamic analysis using a new higher order predictor-corrector method", Eng. Comput., 25(6), 541-568. https://doi.org/10.1108/02644400810891544
  48. Sha, D., Zhou, X. and Tamma, K. (2003), "Time discretized operators. Part 2: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics", Comput. Meth. Appl. Mech. Eng., 192(3), 291-329. https://doi.org/10.1016/S0045-7825(02)00516-9
  49. Shing, P.S.B. and Mahin, S.A. (1985), "Computational aspects of a seismic performance test method using online computer control", Earthq. Eng. Struct. Dyn., 13(4), 507-526. https://doi.org/10.1002/eqe.4290130406
  50. Soares, D. (2016), "An implicit family of time marching procedures with adaptive dissipation control", Appl. Math. Model., 40(4), 3325-3341. https://doi.org/10.1016/j.apm.2015.10.027
  51. Subbaraj, K. and Dokainish, M. (1989), "A survey of direct timeintegration methods in computational structural dynamics-II. Implicit methods", Comput. Struct., 32(6), 1387-1401. https://doi.org/10.1016/0045-7949(89)90315-5
  52. Tamma, K.K. and Namburu, R.R. (1988), "A new finite element based Lax-Wendroff/Taylor-Galerkin methodology for computational dynamics", Comput. Meth. Appl. Mech. Eng., 71(2), 137-150. https://doi.org/10.1016/0045-7825(88)90082-5
  53. Wilson, E.L. (1962), Dynamic Response by Step-By-Step Matrix Analysis, Labortorio Nacional de Engenharia Civil, Lisbon, Portugal, Lisbon, Portugal.
  54. Wilson, E.L., Farhoomand, I. and Bathe, K.J. (1972), "Nonlinear dynamic analysis of complex structures", Int. J. Earthq. Eng. Struct. Dyn., 3(1), 241-252.
  55. Wood, W., Bossak, M. and Zienkiewicz, O. (1980), "An alpha modification of Newmark's method", Int. J. Numer. Meth. Eng., 15(10), 1562-1566. https://doi.org/10.1002/nme.1620151011
  56. Zhai, W.M. (1996), "Two simple fast integration methods for large scale dynamic problems in engineering", International Int. J. Numer. Meth. Eng., 39(24), 4199-4214. https://doi.org/10.1002/(SICI)1097-0207(19961230)39:24<4199::AID-NME39>3.0.CO;2-Y
  57. Zhong, W. and Zhu, J. (1996), "On a new fourth order self-adaptive time integration algorithm", Struct. Eng. Mech., 4(6), 589-600. https://doi.org/10.12989/sem.1996.4.6.589
  58. Zhou, X. and Tamma, K. (2004), "A new unified theory underlying time dependent linear first order systems: a prelude to algorithms by design", Int. J. Numer. Meth. Eng., 60(10), 1699-1740. https://doi.org/10.1002/nme.1019