An improved time integration algorithm: a collocation time finite element approach. (English) Zbl 1535.74643
Summary: A time collocation finite element approach is employed to develop one- and two-step time integration schemes with algorithmic dissipation control capability. The newly developed time integration schemes are combined to obtain a new family of time integration algorithms using the concept employed by Baig and Bathe. The newly developed algorithm can effectively control the algorithmic dissipation by relating the collocation parameters with the spectral radius in the high frequency limit. The new algorithm provides better accuracy compared with the generalized-\(\alpha\) method for highly dissipative cases and includes the Baig and Bathe method within its family as a special case.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S22 | Isogeometric methods applied to problems in solid mechanics |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
Keywords:
implicit time integration scheme; composite time integration scheme; dissipation control; generalized-\(\alpha\) method; Baig and Bathe method; linear and nonlinear structural dynamicsSoftware:
FEAPpvReferences:
| [1] | Newmark, N. M., A method of computation for structural dynamics, J. Eng. Mech. Div. ASCE85 (1959) 67-94. |
| [2] | Goudreau, G. L. and Taylor, R. L., Evaluation of numerical integration method in eladtodynamics, Comput. Method Appl. Mech. Eng.2 (1972) 69-97. · Zbl 0255.73085 |
| [3] | Wilson, E. L., Farhoomand, I. and Bathe, K. J., Nonlinear dynamic analysis of complex structures, Earthq. Eng. Struct. Dyn.1 (1973) 241-252. |
| [4] | Hilber, H. M., Hughes, T. J. R. and Taylor, R. L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn.5 (1977) 283-292. |
| [5] | Hughes, T. J. R., Analysis of transient algorithms with particular reference to stability behavior. Computational Methods for Transient Analysis (1983), 67-155. · Zbl 0547.73070 |
| [6] | Wood, W. L., Bossak, M. and Zienkiewicz, O. C., An alpha modification of newmark’s methods, Int. J. Numer. Meth. Eng.15 (1981) 1562-1566. · Zbl 0441.73106 |
| [7] | Chung, J. and Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-alpha method, J. Appl. Mech.60 (1993) 271-275. · Zbl 0775.73337 |
| [8] | Baig, M. M. I. and Bathe, K. J., On direct time integration in large deformation dynamic analysis. Proceeding of the Third MIT Conference on Computational Fluid and Solid Mechanics, 2005, pp. 1044-1047. |
| [9] | Bathe, K. J. and Baig, M. M. I., On a composite implicit time integration procedure for nonlinear dynamics, Comput. Struct.83 (31) (2005) 2513-2524. |
| [10] | Bathe, K. J. and Noh, G., Insight into an implicit time integration scheme for structural dynamics, Comput. Struct.98-99 (2012) 1-6. |
| [11] | Idesman, A. V., A new high-order accurate continuous galerkin method for linear elastodynamics problems, Comput. Mech.40 (2) (2007) 261-279. · Zbl 1178.74167 |
| [12] | Fung, T. C., Solving initial value problems by differential quadrature method-part 2: Second-and higher-order equations, Int. J. Numer. Meth. Eng.50 (6) (2001) 1429-1454. · Zbl 1050.74056 |
| [13] | Reddy, J. N., An Introduction to the Finite Element Method (McGraw-Hill, NewYork, 2006). |
| [14] | Kuhl, D. and Crisfield, M. A., Energy-conserving and decaying algorithms in non-linear structural dynamics, Int. J. Numer. Meth. Eng.45 (1990) 569-599. · Zbl 0946.74078 |
| [15] | Kuhl, D. and Ramm, E., Constraint energy momentum algorithm and its application to non-linear dynamics of shells, Comput. Meth. Appl. Mech. Eng.136 (1996) 293-315. · Zbl 0918.73327 |
| [16] | Hilber, H. M. and Hughes, T. J. R., Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics, Earthq. Eng. Struct. Dyn.6 (1978) 99-118. |
| [17] | Hughes, T. J. R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Prentice-Hall, New Jersey, 1987). · Zbl 0634.73056 |
| [18] | Zienkiewicz, O. C., Taylor, R. L. and Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals (Elsevier, Amsterdam, 2005). · Zbl 1307.74005 |
| [19] | Oden, J. T., A general theory of finite elements ii. applications, Int. J. Numer. Meth. Eng.1 (3) (1969) 247-259. · Zbl 0263.73048 |
| [20] | Argyris, J. and Mlejnek, H. P., Dynamics of structures. Texts on computational mechanics, Vol. 5. (North Holland, Netherlands, 1991). · Zbl 0792.73001 |
| [21] | Hulbert, G. M., A unified set of single-step asymptotic annihilation algorithms for structural dynamics, Comput. Meth. Appl. Mech. Eng.113 (1) (1994) 1-9. · Zbl 0849.73067 |
| [22] | Fung, T. C., Unconditionally stable higher-order accurate hermitian time finite elements, Int. J. Numer. Meth. Eng.39 (20) (1996) 3475-3495. · Zbl 0884.73065 |
| [23] | Singh, K. M. and Kalra, M. S., Least-squares finite element schemes in the time domain, Comput. Meth. Appl. Mech. Eng.190 (1) (2000) 111-131. · Zbl 0980.65079 |
| [24] | Bathe, K. J. and Wilson, E. L., Stabilty and accuracy analysis of direct integration method, Earthq. Eng. Struct. Dyn.1 (1973) 283-291. |
| [25] | Hughes, T. J. R. and Tezduyar, T. E., Stability and accuracy analysis of some fully-discrete algorithms for the one-dimensional second-order wave equation, Comput. Struct.19 (1984) 665-668. |
| [26] | Bathe, K. J., Frontiers in finite element procedures & applications, Computational Methods for Engineering Science (Saxe-Coburg Publications, Scotland, 2014). |
| [27] | Bathe, K. J., Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme, Comput. Struct.85 (2007) 437-445. |
| [28] | Farlow, S. J., Partial Differential Equations for Scientists and Engineers (Wiley, New York, 1993). · Zbl 0851.35001 |
| [29] | Reddy, J. N., An Introduction to Nonlinear Finite Element Analysis (Oxford, New York, 2015). |
| [30] | Kim, W., Park, S. and Reddy, J. N., A cross weighted-residual time integration scheme for structural dynamics, Int. J. Struct. Stab. Dyn.14 (2013). |
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