Abstract
Transient response analysis of structural and mechanical systems is now routinely carried out via a direct time integration procedure. Typically, the size of discrete equations that must be solved can range from 10 to 105, depending upon modeling accuracy desired and computational resources available. As a result of increasing demand for performing the transient response analysis of structural and mechanical systems in recent years, most of the existing structural analysis computer programs have a direct time integration capability. This increased demand on computational dynamics has been due to its programming simplicity, its capability for handling nonlinearities, and its adaptability to automatic error control at each time integration step, thus obviating the difficult task of including sufficient modes in modal-based dynamics analysis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Belytschko, J.R. Osias, and P.V. Marcal (eds.), Finite Element Analysis of Transient Nonlinear Structural Behavior, AMD Series, Vol. 14, ASME, New York, 1975.
J. Donea (ed.), Advanced Structural Dynamics, Applied Science, Essex, England, 1980.
T. Belytschko and T.J.R. Hughes (eds.), Computational Methods for Transient Analysis, North-Holland, Amsterdam, 1983.
W.K. Liu, T. Belytschko, and K.C. Park (eds.), Innovative Methods for Nonlinear Problems, Pineridge Press, Swansea, Great Britain, 1985.
K.C. Park, “Time integration of structural dynamics: A survey,” in Pressure Vessels and Piping Design Technology—A Decade of Progress, ASME, New York, 1982, bk. 4.2.
P.S. Jensen, “Transient analysis of structures by stiffly stable methods,” Comput. & Structures, 4(1974), 67 – 94.
N.M. Newmark, “A method of computation for structural dynamics,” Proc. ASCE, 85, EM3 (1959), 00. 67 – 94.
D.E. Johnson, “A proof of the stability of the Houbolt method,” AIAA J., 4(1966), 1450 – 1451.
R.E. Nickell, “On the stability of approximation operators in problems of structural dynamics,” Internat. J. Solids and Structures, 7(1971), 301 – 319.
K.J. Bathe, and E.L. Wilson, “Stability and accuracy analysis of direct integration methods,” Internat. J. Earthquake Engrg. Struct. Dynamics, 1(1973), 283 – 291.
R.D. Krieg and S.W. Key, “Transient shell response by numerical time integration,” Internat. J. Numer. Methods Engrg. 7(1973), 273 – 286.
G.L. Goudreau and R.L. Taylor, “Evaluation of numerical methods in elastodynamics,” J. Comput. Methods Appl. Mech. Engrg., 2(1973), 69 – 97.
J.H. Argyris, P.C. Dunne, and T. Angelopoulos, “Dynamic response by large step integration,” Internat. J. Earthquake Engrg. Struct. Dynamics, 2(1973), 185 – 203.
K.C. Park, “An improved stiffly stable method for direct integration of nonlinear structural dynamic equations,” J. Appl. Mech., 42(1975), 464 – 470.
H.M. Hilber and T.J.R. Hughes, “Collocation, dissipation and overshoot for time integration scheme in structural dynamics,” Internat. J. Earthquake Engrg. Struct. Dynamics, 6(1978), 99 – 117.
G. Hall and J.M. Watt (eds.), Modern Numerical Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1976, p. 128.
G. Dahlquist, “A special stability problem for linear multi-step methods,” BIT, 3(1963), 27 – 43.
P.S. Jensen, “Stiffly stable methods for undamped second-order equations of motion,” SIAM J. Numer. Anal., 13(1976), 549 – 563.
K.C. Park, “Practical aspects of numerical time integration,” Comput. & Structures, 7(1977), 343 – 353.
K.C. Park, “Evaluating time integration methods for nonlinear dynamics analysis,” in Finite Element Analysis of Transient Nonlinear Structural Behavior (T. Belytschko et al, ed.), ASME Applied Mechanics Symposia AMD, Vol. 14,1975, pp. 35–58.
W.H. Enright, T.E. Hull, and B. Lindberg, “Comparing numerical methods for stiff systems of O.D.E.’s,” BIT, 15 (1975), 10 – 48.
R.K. Brayton, F.G. Gustavson, and G.D. Hachtel, “A new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas,” Proc. IEEE, 60 (1972), 98–108.
C.W. Gear, “The automatic integration of stiff ordinary differential equations,” Proceedings of the IFIP Congress, 1969, pp. 187 – 193.
J.A. Stricklin, J.E. Martinez, J.R. Tillerson, J.R. Hong, and W.E. Haisler, “Nonlinear dynamic analysis of shells of revolution by the matrix displacement method,” AIAA J., 9 (1971), 629 – 636.
R.W.H. Wu and E.A. Witmer, “Nonlinear transient responses of structures by the spatial finite-element method,” AIAA J. 11(1973), 1087 – 1104.
J.F. McNamara, “Solution schemes for problems of nonlinear structural dynamics,” J. Pressure Vessels Tech. ASME, 96(1974), 96 – 102.
W.H. Enright and T.E. Hull, “Test results on initial value methods for nonstiff ordinary differential equations,” SIAM J. Numer. Anal., 13 (1976), 944 – 961.
K.C. Park and P.G. Underwood, “A variable-step central difference method for structural dynamics analysis: Theoretical aspects,” Comput. Methods Appl. Mech. Engrg., 22(1980), 241 – 258.
P.G. Underwood and K.C. Park, “A variable-step central difference method for structural dynamics analysis: Implementation and performance evaluation,” Comput. Methods Appl. Mech. Engrg., 23(1980), 259 – 279.
B. Lindberg, “Error estimates and stepsize strategy for the implicit midpoint rule with smoothing and extrapolation,” Report NA 7259, Department of Information Processing, Royal Institute of Technology, Stockholm, 1972.
H.D. Hibbit and B.I. Karlsson, “Analysis of pipe whip,” Paper 79-PVP-122, ASME, New York, 1979.
T. Belytschko and D.F. Schoeberle, “On the unconditional stability of an implicit algorithm for nonlinear structural dynamics,” J. Appl. Mech., 97(1975), 865 – 869.
V. Blaes, “Zur angenaherten Losung gewohnlicher Differentialgleichungen,” VDI-Z, 81(1937), 587 – 596.
B. Lindberg, “Characterization of optimal stepsize sequences for methods for stiff differential equations,” SIAM J. Numer. Anal., 14(1977), 859 – 887.
K.C. Park and P.G. Underwood, “An adaptive direct time integration package for structural dynamics analysis: Design considerations,” in preparation.
C.A. Felippa and K.C. Park, “Computational aspects of time integration procedures: Implementation,” J. Appl. Math. Mech., 45(1978), 595 – 602.
J.C. Houbolt, “A recurrence matrix solution for the dynamic response of an elastic aircraft,” J. Aeronaut. Sci., 17(1950), 540 – 550.
C.A. Felippa, “Procedures for computer analysis of large nonlinear structural systems,” in Large Engineering Systems(A. Wexler, ed.), Pergamon, Oxford, 1977, pp. 60 – 101.
C.A. Felippa, “Solution of nonlinear static equations,” in Large Deflection and Stability of Structures( K.J. Bathe, ed.), North-Holland, Amsterdam, 1986.
D. Bushnell, “A strategy for the solution of problems involving large deflections, plasticity and creeps,” Internat. J. Numer. Methods Engrg., 11(1977), 683 – 708.
H.L. Schreyer, R.F. Kulak, and J.M. Kramer, “Accurate numerical solutions for elastic-plastic models,” Preprint ASME Paper No. 79-PVP-107.
P.V. Marcal “A stiffness method foe elastic-plastic problem,” Interneal. J. Mech. Sci., (1965), 229 – 238.
J.C. Nagtegaal, D.M. Parks, and J.R. Rice, “On numerically accurate finite element solutions in the fully plastic range,” Comput. Methods Appl. Mech. Engrg., 4(1974), 153 – 177.
K.C. Park, “Partitioned transient analysis procedures for coupled-field problems: Stability analysis,” J. Appl. Math. Mech, 47, No. 2 (1980), 370 – 376.
T. Belytschko and R. Mullen, “Mesh partitions of explicit-implicit time integration,” in Formulations and Computational Algorithms in Finite Element Analysis, (J. Bathe et al. ed.), MIT Press, Cambridge, MA, 1977.
T. Belytschko and R. Mullen, “Stability of explicit-implicit mesh partitions in time integration,” Internat. J. Numer. Methods Engrg., 12(1978), 1575 – 1586.
T.J.R. Hughes and W.K. Liu, “Implicit-explicit finite elements in transient analysis: Stability theory,” J. Appl. Math. Mech., 45(1978), 375 – 378.
K.C. Park, C.A. Felippa, and J.A. DeRuntz, “Stabilization of staggered solution procedures for fluid-structure interaction analysis,” in Computational Methods for Fluid-Structure Interaction Problems(T. Belytschko and T.L. Geers, eds.), ASME Applied Mechanics Symposia, AMD Vol. 26, ASME, New York, 1977, pp. 94 – 124.
T. Belytschko, H.-J. Yen, and R. Mullen, “Mixed methods for time integration,” Comput. Methods Appl. Mech. Engrg., 17(1979), 259 – 275.
O.C. Zienkiewicz, E. Hinton, K.H. Leung, and R.E. Taylor, “Staggered time marching schemes in dynamic soil analysis and a selective explicit extrapolation algorithm,” in Innovative Numerical Analysis for the Applied Engineering Science(R. Shaw et al. eds.), The University of Virginia Press, Charlottesville, 1980, pp. 525 – 530.
K.C. Park and J.M. Housner, “Semi-explicit transient analysis procedures for structural dynamics problems,” Internat. J. Numer. Methods Engrg., 18(1982), 609 – 622.
D.M. Trujillo, “An unconditionally stable explicit algorithm for structural dynamics,” Internat. J. Numer. Methods Engrg., 11(1977), 1579– 1592.
J.E. Dennis Jr. and J.J. More, “Quasi-Newton methods, motivation and theory,” SIAM Rev., 19(1977), 46 – 89.
N.N. Yanenko, The Method of Fractional Steps, Springer-Verlag, New York, 1971.
H. Mathies and G. Strang, “The solution of nonlinear finite element equations,” Internat. J. Numer. Methods Engrg., 14(1979), 1613 – 1626.
M. Geradin and M. Hogge, “Quasi-Newton iteration in nonlinear structural dynamics,” Proc. SMiRT-5, Paper No. M7/1, Berlin, August 1979.
A.N. Konovalov, “Application of the method of splitting to the dynamic solution of problems of the theory of elasticity,” U.S.S.R. Comput. Math., 4(1964), 192 – 198.
S.P. Frankel, “Convergence rates of iterative treatments of partial differential equations,” Math. Tables, Aids to Comput., 4(1950), 65 – 70.
J.S. Brew and D.M. Brotton, “Nonlinear structural analysis by dynamic relaxation,” Internat. J. Numer. Methods Engrg., 3(1971), 463 – 483.
P.G. Underwood, “Dynamic relaxation,” in Computational Methods for Transient Dynamic Analysis (T. Belytschko and T.J.R. Hughes, eds.), North-Holland, Amsterdam, 1983, Chapter 5.
G.S. Patterson, “Large scale scientific computing: future directions,” Comput. Phys. Comm., 26(1982), 217 – 225.
H. Allik, J. Crawther, J. Goodhue, S. Moor, and R. Thomas, “Implementation of the finite element methods on the butterfly parallel processor,” BBN Lab. Inc., Cambrideg, MA, July 1985
G.A. Lyzenga, A. Raefsky, and B.H. Hager, “Finite elements and the method of conjugate gradient on a concurrent processor,” a CalTech Preprint, 1985.
B. Nour-Omid and K.C. Park, “Solving structural mechanics problems on the CalTech hypercube machine,” Comput. Methods Appl. Mech. Engrg., 61, (1981), 161 – 176.
P. Lotstedt and L.R. Petzold, “Numerical solution of nonlinear differential Equations with Algebraic Constraints,” Sandia National Laboratories, Report SAND 83-8877, Livermore, CA, 1983.
R.A. Wehage and E.J. Haug, “Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems,” ASME J. Mech. Design, 104(1982), 247 – 255.
P. Lötstedt, “On a penalty function method for the simulation of mechanical systems subject to constraints,” TRITA-NA-7919,1979, Royal Institute of Technology, Stockholm.
C. Fuehrer and O. Wallrapp, “A computer-oriented method for reducing linearized multibody system equations by incorporating constraints,” Comput. Methods Appl. Mech. Engrg., 46(1984), 169 – 175.
P.E. Nikravesh, x in Computer Aided Analysis and Optimization of Mechanical Systems Dynamics(E.J. Haug, ed.), NATO ASI series, F9, Springer-Verlag, Berlin, 1984, pp. 351 – 367.
K.C. Park, “Stabilization of computational procedures for constrained dynamical system: Formulation,” AIAA Paper No. AIAA-86-0926-CP, presented at the 27th SDM Conference, San Antonio, Texas, May 1986.
N. Kikuchi and J.T. Oden, “Contact problems in elastostatics,” in Finite Elements: Special Problems in Solid Mechanics( J.T. Oden and G. Carey, eds.), Prentice-Hall, Englewood Cliffs, NJ, 1984.
B. Nour-Omid and P. Wriggers, “Solution methods for contact problems,” Comput. Methods Appl. Mech. Engrg., 52(1983), 31 – 44.
J.T. Oden, “Exterior penalty methods for contact problems in elasticity,” in Nonlinear Finite Element Analysis in Structural Mechanics( W. Wunderlich, P. Stein, and K.J. Bathe, eds.), Springer-Verlag, Berlin, 1981.
T.J.R. Hughes, I. Levit, and J.M. Winget, “An element-by-element solution algorithm for problems of structural and solid mechanics,” Comput. Methods Appl. Mech. Engrg., 36 (1983), 223 – 239.
T. Belytschko, P. Smolinski, and W.-K. Liu, “Stability of multi-step partitioned integrators for first-order finite element systems,” Comput. Methods Appl. Mech. Engrg., 52(1985), 281 – 298.
M. Ortiz and B. Nour-Omid, “Unconditionally stable concurrent procedures for transient finite element analysis,” Comput. Methods Appl. Mech. Engrg. (1986), to appear.
N.F. Knight and W.J. Stroud, “Computational structural mechanics: A new activity at the NASA Langley Research Center,” NASA Tech. Mem. 87612, Hampton, VA, 1985.
C.G. Lotts (compiler), “Introduction to the CSM Testbed: NICE/SPAR,” NASA/ Langley Research Center, June 1986.
C.A. Felippa, “Architecture of a distributed analysis network for computational Mechanics,” Comput. & Structures, 13 (1981), 405–413.
C.A. Felippa, “A command language for applied mechanics processors,” 1–3, LMSC-D78511, Lockheed Palo Research Laboratory, Palo Alto, CA, 1983.
C.A. Felippa and G.M. Stanley, “NICE: A utility architecture for computational mechanics,” in Finite Elements for Nonlinear Problems(P.G. Bergan, K.J. Bathe, and W. Wunderlich, eds.), Springer-Verlag, New York, 1986, pp. 447 - V463.
T. Belytschko, “Transient analysis,” in Structural Mechanics Computer Programs(W. Pilkey et al. eds.), The University Press of Virginia, Charlottesville, 1974, pp. 255 – 276.
C.A. Felippa and K.C. Park, “Direct time integration methods in nonlinear structural dynamics,” Comput. Methods Appl. Mech. Engrg., 17/18(1979), 277 – 313.
T.J.R. Hughes and T. Belytschko, “A précis of developments in computational methods for transient analysis,” J. Appl. Math. Mech., 50(1985), 1033 – 1041.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this paper
Cite this paper
Park, K.C. (1988). Transient Analysis Methods in Computational Dynamics. In: Dwoyer, D.L., Hussaini, M.Y., Voigt, R.G. (eds) Finite Elements. ICASE/NASA LaRC Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3786-0_11
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3786-0_11
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8350-8
Online ISBN: 978-1-4612-3786-0
eBook Packages: Springer Book Archive