0
Research Papers

Effective Higher-Order Time Integration Algorithms for the Analysis of Linear Structural Dynamics

[+] Author and Article Information
Wooram Kim

Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123
e-mail: kim.wooram@yahoo.com

J. N. Reddy

Oscar S. Wyatt Jr. Chair
Distinguished Professor
Life Fellow ASME
Department of Mechanical Engineering,
Texas A&M University,
MS 3123,
College Station, TX 77843-3123
e-mail: jnreddy@tamu.edu

1Present address: Department of Mechanical Engineering, Korea Army Academy at Yeongcheon, Yeongcheon 38900, South Korea.

2Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 17, 2017; final manuscript received May 21, 2017; published online June 7, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(7), 071009 (Jun 07, 2017) (13 pages) Paper No: JAM-17-1260; doi: 10.1115/1.4036822 History: Received May 17, 2017; Revised May 21, 2017

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Hulbert, G. M. , 1994, “ A Unified Set of Single-Step Asymptotic Annihilation Algorithms for Structural Dynamics,” Comput. Methods Appl. Mech. Eng., 113(1), pp. 1–9. [CrossRef]
Fung, T. C. , 1999, “ Weighting Parameters for Unconditionally Stable Higher-Order Accurate Time Step Integration Algorithms—Part 2: Second-Order Equations,” Int. J. Numer. Methods Eng., 45(8), pp. 971–1006. [CrossRef]
Idesman, A. V. , 2007, “ A New High-Order Accurate Continuous Galerkin Method for Linear Elastodynamics Problems,” Comput. Mech., 40(2), pp. 261–279. [CrossRef]
Fung, T. C. , 2003, “ Numerical Dissipation in Time-Step Integration Algorithms for Structural Dynamic Analysis,” Prog. Struct. Eng. Mater., 5(3), pp. 167–180. [CrossRef]
Ma, J. , 2015, “ A New Space-Time Finite Element Method for the Dynamic Analysis of Truss-Type Structures,” Ph.D. thesis, Edinburgh Napier University, Edinburgh, Scotland.
Kim, W. , and Reddy, J. N. , 2017, “ A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics,” ASME J. Appl. Mech., epub.
Zienkiewicz, O. C. , and Taylor, R. L. , 2005, The Finite Element Method for Solid and Structural Mechanics, Butterworth-Heinemann, Oxford, UK.
Fung, T. C. , 1996, “ Unconditionally Stable Higher-Order Accurate Hermitian Time Finite Elements,” Int. J. Numer. Methods Eng., 39(20), pp. 3475–3495. [CrossRef]
Fung, T. C. , 2001, “ Solving Initial Value Problems by Differential Quadrature Method—Part 2: Second-and Higher-Order Equations,” Int. J. Numer. Methods Eng., 50(6), pp. 1429–1454. [CrossRef]
Putcha, N. S. , and Reddy, J. N. , 1986, “ A Refined Mixed Shear Flexible Finite Element for the Nonlinear Analysis of Laminated Plates,” Comput. Struct., 22(4), pp. 529–538. [CrossRef]
Kim, W. , 2008, “ Unconventional Finite Element Models for Nonlinear Analysis of Beams and Plates,” Master's thesis, Texas A&M University, College Station, TX.
Kim, W. , and Reddy, J. N. , 2010, “ Novel Mixed Finite Element Models for Nonlinear Analysis of Plates,” Latin Am. J. Solids Struct., 7(2), pp. 201–226. [CrossRef]
Idesman, A. V. , Schmidt, M. , and Sierakowski, R. L. , 2008, “ A New Explicit Predictor–Multicorrector High-Order Accurate Method for Linear Elastodynamics,” J. Sound Vib., 310(1), pp. 217–229. [CrossRef]
Zienkiewicz, O. C. , Taylor, R. L. , and Zhu, J. Z. , 2005, The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann, Burlington, VT.
Argyris, J. , and Mlejnek, H. P. , 1991, “ Dynamics of Structures,” Texts on Computational Mechanics, Vol. 5, North-Holland, Amsterdam, The Netherlands.
Howard, G. F. , and Penny, J. , 1978, “ The Accuracy and Stability of Time Domain Finite Element Solutions,” J. Sound Vib., 61(4), pp. 585–595. [CrossRef]
Reddy, J. N. , 2017, Energy Principles and Variational Methods in Applied Mechanics, 3rd ed., Wiley, New York.
Gellert, M. , 1978, “ A New Algorithm for Integration of Dynamic Systems,” Comput. Struct., 9(4), pp. 401–408. [CrossRef]
Bathe, K. J. , and Noh, G. , 2012, “ Insight Into an Implicit Time Integration Scheme for Structural Dynamics,” Comput. Struct., 98–99, pp. 1–6. [CrossRef]
Hilber, H. M. , Hughes, T. J. R. , and Taylor, R. L. , 1977, “ Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,” Earthquake Eng. Struct. Dyn., 5(3), pp. 283–292. [CrossRef]
Hughes, T. J. R. , 1983, “ Analysis of Transient Algorithms With Particular Reference to Stability Behavior,” Computational Methods for Transient Analysis, Vol. 1, North Holland Publishing, Amsterdam, The Netherlands, pp. 67–155.
Leon, S. J. , 1980, Linear Algebra With Applications, Macmillan, New York.
Fung, T. C. , and Chow, S. K. , 2002, “ Solving Non-Linear Problems by Complex Time Step Methods,” Commun. Numer. Methods Eng., 18(4), pp. 287–303. [CrossRef]
Hilber, H. M. , 1976, “ Analysis and Design of Numerical Integration Methods in Structural Dynamics,” Ph.D. thesis, University of California, Berkeley, CA.
Chung, J. , 1992, “ Numerically Dissipative Time Integration Algorithms for Structural Dynamics,” Ph.D. thesis, University of Michigan, Ann Arbor, MI.
Newmark, N. M. , 1959, “ A Method of Computation for Structural Dynamics,” J. Eng. Mech. Div., 85(3), pp. 67–94.
Baig, M. M. I. , and Bathe, K. J. , 2005, “ On Direct Time Integration in Large Deformation Dynamic Analysis,” Third MIT Conference on Computational Fluid and Solid Mechanics, Boston, MA, June 14–17, pp. 1044–1047.
Kim, W. , Park, S. , and Reddy, J. N. , 2014, “ A Cross Weighted-Residual Time Integration Scheme for Structural Dynamics,” Int. J. Struct. Stab. Dyn., 14(6), p.1450023. [CrossRef]
Reddy, J. N. , 2006, An Introduction to the Finite Element Method, 3rd ed., McGraw-Hill, New York.
Fried, I. , and Malkus, D. S. , 1975, “ Finite Element Mass Matrix Lumping by Numerical Integration With no Convergence Rate Loss,” Int. J. Solids Struct., 11(4), pp. 461–466. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic presentation of time element obtained from pth-order Hermite interpolation functions

Grahic Jump Location
Fig. 2

Spectral radii versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

Grahic Jump Location
Fig. 3

Relative period errors versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

Grahic Jump Location
Fig. 4

Damping ratios versus Δt/T for current third-, fifth-, and seventh-order algorithms with varying values of μ

Grahic Jump Location
Fig. 5

Description of three-degrees-of-freedom spring system used by Bathe and Nho

Grahic Jump Location
Fig. 6

Displacement of second node for various methods

Grahic Jump Location
Fig. 7

Velocity of second node for various methods

Grahic Jump Location
Fig. 8

Acceleration of second node for various methods

Grahic Jump Location
Fig. 9

Acceleration of second node for various methods (enlarged image of Fig. 8 without the trapezoidal rule)

Grahic Jump Location
Fig. 10

Computational domain and mesh used to analyze 2D wave equation

Grahic Jump Location
Fig. 11

Comparison of center displacements at 0≤t≤5 for various methods

Grahic Jump Location
Fig. 12

Comparison of center displacements at 160≤t≤165 for various methods

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In