A new explicit time integration method for structural dynamics. (English) Zbl 1359.65120
Summary: A family of new explicit time-integration method is proposed herein, which inherits the numerical characteristics of any existing implicit Runge-Kutta algorithms for a linear conservative system. Based on an exact derivation of the increment of mechanical energy, the method proposed is demonstrated to be unconditionally stable. Also, the stability condition of the proposed method is derived when applied to solving a nonlinear system. The characteristics of the proposed method are investigated by observing the mechanical-energy time history of a nonlinear conservative system. The numerical results can be explained by the stability condition derived in the nonlinear regime. Finally, the computational accuracy and efficiency between the Newmark time integration method and the proposed explicit method are compared in solving the dynamic response of a couple of linear oscillators.
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
Keywords:
explicit method; Runge-Kutta method; stability condition; time-integration method; unconditional stabilitySoftware:
RODASReferences:
| [1] | N. M. Newmark, J. Eng. Mech. Div. (ASCE) 85, 67 (1959). |
| [2] | E. L. Wilson, I. Farhoomand and K. J. Bathe, Earthq. Eng. Struct. Dyn. 1, 41 (1973). |
| [3] | H. M. Hilber, T. J. R. Hughes and R. L. Taylor, Earthq. Eng. Struct. Dyn. 5, 283 (1997), DOI: 10.1002/eqe.4290050306. genRefLink(16, ’rf3’, ’10.1002 |
| [4] | W. L. Wood, M. Bossak and O. C. Zienkiewicz, Int. J. Numer. Meth. Eng. 15, 1562 (1981), DOI: 10.1002/nme.1620151011. genRefLink(16, ’rf4’, ’10.1002 |
| [5] | C. Hoff and P. J. Pahl, Comput. Method Appl. M. 67, 367 (1988), DOI: 10.1016/0045-7825(88)90053-9. genRefLink(16, ’rf5’, ’10.1016 |
| [6] | C. Hoff and P. J. Pahl, Comput. Method. Appl. M. 67, 87 (1988), DOI: 10.1016/0045-7825(88)90070-9. genRefLink(16, ’rf6’, ’10.1016 |
| [7] | J. Chung and G. M. Hulbert, J. Appl. Mech. 60(6), 371 (1993), DOI: 10.1115/1.2900803. genRefLink(16, ’rf7’, ’10.1115 |
| [8] | T. Belytschko and D. F. Schoeberle, J. Appl. Mech. 19, 865 (1975). |
| [9] | K. K. Tamma, X. Zhou and D. Sha, Int. J. Numer. Meth. Eng. 50, 1619 (2001), DOI: 10.1002/nme.89. genRefLink(16, ’rf9’, ’10.1002 |
| [10] | X. Zhou and K. K. Tamma, Int. J. Numer. Meth. Eng. 59, 597 (2004), DOI: 10.1002/nme.873. genRefLink(16, ’rf10’, ’10.1002 |
| [11] | O. C. Zienkiewicz , The Finite Element Method ( McGraw-Hill , New York , 1977 ) . · Zbl 0435.73072 |
| [12] | T. Belytschko and T. J. R. Hughes , Computational Methods for Transient Analysis ( Elsevier , 1983 ) . · Zbl 0521.00025 |
| [13] | T. J. R. Hughes , The Finite Element Method ( Prentice-Hall , Englewood Cliffs, NJ, U.S.A. , 1987 ) . · Zbl 0634.73056 |
| [14] | S. Y. Chang, J. Eng. Mech. (ASCE) 128(9), 935 (2002), DOI: 10.1061/(ASCE)0733-9399(2002)128:9(935). genRefLink(16, ’rf14’, ’10.1061 |
| [15] | S. Y. Chang, J. Eng. Mech. (ASCE) 133(5), 541 (2007), DOI: 10.1061/(ASCE)0733-9399(2007)133:5(541). genRefLink(16, ’rf15’, ’10.1061 |
| [16] | T. C. Fung, Int. J. Numer. Meth. Eng. 39, 3475 (1996). genRefLink(16, ’rf16’, ’10.1002 |
| [17] | S. S. Kuo and J. D. Yau, Int. J. Struct. Stab. Dyn. 11(3), 473 (2011), DOI: 10.1142/S0219455411004178. [Abstract] genRefLink(128, ’rf17’, ’000290736600005’); |
| [18] | J. C. Simo and N. Tarnow, J. Appl. Math. Phy. 43, 757 (1992), DOI: 10.1007/BF00913408. genRefLink(16, ’rf18’, ’10.1007 |
| [19] | D. Kuhl and M. A. Crisfield, Int. J. Numer. Meth. Eng. 45, 569 (1999). genRefLink(16, ’rf19’, ’10.1002 |
| [20] | O. Bauchau and T. Joo, Int. J. Numer. Meth. Eng. 45, 693 (1999). genRefLink(16, ’rf20’, ’10.1002 |
| [21] | P. Betsch and P. Steinmann, Int. J. Numer. Meth. Eng. 50, 1931 (2001), DOI: 10.1002/nme.103. genRefLink(16, ’rf21’, ’10.1002 |
| [22] | A. Ibrahimbegovic and S. Mamouri, Comput. Method. Appl. M. 191, 4241 (2002), DOI: 10.1016/S0045-7825(02)00377-8. genRefLink(16, ’rf22’, ’10.1016 |
| [23] | X. Zhou, D. Sha and K. K. Tamma, Int. J. Numer. Meth. Eng. 59, 795 (2004), DOI: 10.1002/nme.878. genRefLink(16, ’rf23’, ’10.1002 |
| [24] | S. Krenk, Comput. Method. Appl. M. 195, 6110 (2006), DOI: 10.1016/j.cma.2005.12.001. genRefLink(16, ’rf24’, ’10.1016 |
| [25] | S. Lopez and K. Russo, Int. J. Struct. Stab. Dyn. 8(2), 257 (2008), DOI: 10.1142/S0219455408002648. [Abstract] genRefLink(128, ’rf25’, ’000257540400003’); |
| [26] | E. Hairer , S. P. Nørsett and G. Wanner , Solving Ordinary Differential Equations I: Nonstiff Problems ( Springer-Verlag , Berlin, New York , 1993 ) . · Zbl 0789.65048 |
| [27] | E. Hairer and G. Wanner , Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems ( Springer-Verlag , Berlin, New York , 1996 ) . genRefLink(16, ’rf27’, ’10.1007 · Zbl 0859.65067 |
| [28] | S. Y. Chang, Int. J. Struct. Stab. Dyn. 8(2), 321 (2008), DOI: 10.1142/S0219455408002673. [Abstract] genRefLink(128, ’rf28’, ’000257540400006’); |
| [29] | S. Y. Chang, Int. J. Numer. Meth. Eng. 77(8), 1100 (2009), DOI: 10.1002/nme.2452. genRefLink(16, ’rf29’, ’10.1002 |
| [30] | S. Y. Chang, J. Eng. Mech. (ASCE) 136(5), 599 (2010), DOI: 10.1061/(ASCE)EM.1943-7889.99. genRefLink(16, ’rf30’, ’10.1061 |
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