Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics

International Journal for Numerical Methods in Engineering

Volumn 59, Issue 5, 2004, Pages 597-668

  Zhou, X ^a   Tamma, K K ^a  

^a University of Minnesota   (United States)

Author keywords

Controllable numerical dissipative algorithms; Dynamic analysis; Linear multi step methods; Time integration algorithms

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; INTEGRATION; MATHEMATICAL MODELS; THEOREM PROVING;

LINEAR MULTI-STEP METHODS (LMS); STRUCTURAL DYNAMICS;

STRUCTURAL ANALYSIS;

STRUCTURAL ANALYSIS;

EID: 0347058540     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.873     Document Type: Article

References (34)

1. Tamma KK, Zhou X, Sha D. The time dimension: a theory of development/evolution, classification, characterization and design of computational algorithms for transient/dynamic applications. Archives in Computational Mechanics 2000; 7(2) 67-290.
2. Tamma KK, Zhou X, Sha D. A theory of development and design of generalized integration operators for computational structural dynamics. International Journal of Numerical and Mechanical Engineering 2000; 50:1619-1664.
3. Dahlquist G. A special stability problem for linear multistep methods. BIT 1963; 3:27.
4. Newmark NM. A method of computation for structural dynamics. Journal American Society of Civil Engineers 1959; 1:67.
5. Zienkiewicz OC. A new look at Newmark, Houbolt and other time stepping formulas: a weighted residual approach. Earthquake Engineering and Structural Dynamics 1977; 5:283.
6. Zienkiewicz OC, Wood WL, Hine NW, Taylor RL. A unified set of single-step algorithms, Part 1: general formulations and applications. International Journal for Numerical Methods in Engineering 1984; 20:1529.
7. Wood WL. A unified set of single-step algorithms, Part 2. International Journal for Numerical Methods in Engineering 1984; 20:2303.
8. Tamma KK, Namburu RR. Applicability and evaluation of an implicit self-starting unconditionally stable methodology for the dynamics of structures. Computers and Structures 1990; 34(6):835.
9. Sha D, Zhou X, Tamma KK. Time discretized operators: towards the theoretical design of a new generation of implicit and explicit representations of time operators for structural dynamics: Part 2. Computer Methods in Applied Mechanics and Engineering 2003; 192:291-329.
10. Wilson EL. A Computer Program for Dynamic Stress Analysis of Underground Structures. SESM: University California, Berkeley, 1968.
11. Goudreau GL, Taylor RL. Evaluation of numerical methods in elastodynamics. Computer Methods in Applied Mechanics and Engineering 1973; 2:69.
12. Argyris JH, Dunne PC, Angelopoulos T. Nonlinear oscillations using the finite element technique. Computer Methods in Applied Mechanics and Engineering 1973; 2:203.
13. Krieg RD, Key SW. Transient shell respones by numerical time integration. International Journal for Numerical Methods in Engineering 1973; 17:273-286.
14. Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 1977; 5:283.
15. Wood WL, Bossak M, Zienkiewicz OC. An alpha modification of Newmark's method. International Journal for Numerical Methods in Engineering 1980; 15:1562.
16. Hoff C, Pahl PJ. Development of an implicit method with numerical dissipation from a generalized single step algorithm for structural dynamics. Computer Methods in Applied Mechanics and Engineering 1988; 67:367.
17. Chung J, Hulbert G. A time integration method for structural dynamics with improved numerical dissipation: the generalized χ-method. Journal of Applied Mechanics 1993; 30:371.
18. Zhou X, Tamma KK, Kanapady R, Sha D. The time dimension: new and recent advances and a unified framework towards design of time discretized operators for structural dynamics. In European Congress on Computational Methods in Applied Science and Engineering (ECCOMAS) 2000, Barcelona, Spain. 11-14 September 2000.
19. Geradin M. A classification and discussion of integration operator for transient structural response. In AIAA 12th Aerospace Science Meeting, Washington, DC, January 30-February 1, 1974.
20. Krieg RD. Unconditional stability of numerical time integration methods. Journal of Applied Mechanics 1973; 40:417-421.
21. Henrici PK. Discrete Variable Methods in Ordinary Differential Equation. Wiley: New York, 1962.
22. Franklin JN. Matrix Theory. Prentice-Hall: Englewood Cliffs, NJ, 1968.
23. Lax PD, Richmyer RD. Servry of the stability of linear limit difference equations. Communications on Pure and Applied Mathematics 1956; 9:267-293.
24. Bazzi G, Anderheggen E. The ρ-family of algorithms for time-step integration with improved numerical dissipation. Earthquake Engineering and Structural Dynamics 1982; 10:537.
25. Hulbert GM, Chung J. On the unimportance of the spurious root of time integration algorithms for structural dynamics. Communications in Numerical Methods in Engineering 1994; 10:591-597.
26. Lambert JD. Computational Methods in Ordinary Differential Equations. Wiley: New York, 1977.
27. Routh EJ. A Treatise on the Stability of a Given State of Motion. Macmillan: London, 1877.
28. Routh EJ. The Advanced Part of a Treaties on the Dyanmics of a Rigid Body (6th edn.), Macmillan, 1905. Dover: New York (reprinted (1959)).
29. Hurwitz A. Über die Bedingungen, Unter Welchen Eine Gleichung Nur Wurzeln Mit Negativen Reellen Teilen Besitzt. Mathematische Annalen 1895; 46:273-284.
30. Hughes TJR. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. Prentice-Hall: Englewood Cliffs, NJ, 1987.
31. Wood WL. Practical Time Stepping Schemes. Clarendon Press: Oxford, 1990.
32. Belytschko T, Hughes TJR. Computational Methods in Transient Analysis. North Holland: Amsterdam, 1983.
33. Hoff C, Pahl PJ. Practical performance of the Θ-method and comparison with other dissipative algorithms in structural dynamics. Computer Methods in Applied Mechanics and Engineering 1988; 67:87.
34. Zhou X, Tamma KK. A new unified theory underlying time dependent linear first-order systems: a prelude to algorithms by design. International Journal for Numerical Methods in Engineering in press.