Practical Integration of Semidiscretized Nonlinear Equations of Motion: Proper Convergence for Systems with Piecewise Linear Behavior
Publication: Journal of Engineering Mechanics
Volume 139, Issue 2
Abstract
Time integration is the most versatile tool for analyzing semidiscretized equations of motion. The responses are approximations, with deviations from the exact responses mainly depending on the integration method and the integration step sizes. When repeating the analyses with smaller steps, the responses generally converge to the exact responses. However, the convergence trends are different in linear and nonlinear analyses. Whereas in linear analyses, by decreasing the sizes of integration steps, the errors decrease with a rate, depending on the orders of accuracy, in nonlinear analyses, the change in errors might be unpredictable. The main reason is the inconsistency between the integration steps sizes and the residuals of nonlinearity iterations. In this paper, based on careful selection of nonlinearity tolerances, a methodology and a method to overcome this inconsistency for semidiscretized systems with piecewise linear behavior are introduced. When the responses converge, except for systems with very complex behaviors, the proposed method leads to proper convergence, with tolerable computational costs. In addition, by implementing the proposed method, more reliable error estimations can be expected from convergence-based accuracy controlling methods.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The first author thanks Prof. A. Der Kiureghian and Prof. J. Retief, who illuminated the need for the study reported in this paper. Sincere appreciation of the authors are given to Mrs. L. Arzoumanian, and Mr. H. Doosti for their kind collaboration regarding the random simulations. The comments of the reviewers, causing considerable enhancements in the paper, the comments of Prof. F. Arbabi and Dr. M. Hosseini on the English of the paper, and the kind collaboration of the editor and editorial board and all in the ASCE Journal of Engineering Mechanics in different stages of this paper's revision, typesetting, and publication, are also sincerely acknowledged.
References
Allgower, E. L., and Georg, K. (1980). Numerical continuation methods, an introduction, Springer, New York.
Apostol, T. M. (1967). Calculus, Vol. I, Wiley, New York.
Argyris, J., and Mlejnek, J. P. (1991). Dynamics of structures, Elsevier, Amsterdam, Netherlands.
Aydinoglu, M. N., and Fahjan, Y. M. (2003). “A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P-delta effect.” Earthquake Eng. Struct. Dynam., 32(6), 871–890.
Babitsky, V. I., and Krupenin, V. L. (2001). Vibration of strongly nonlinear discontinuous systems, Springer, Berlin, Germany.
Basoz, N. I., Kiremidjian, A. S., King, S. A., and Law, K. H. (1999). “Statistical analysis of bridge damage data from the 1994 Northridge, CA California.” Earthq. Spectra, 15(1), 25–53.
Bathe, K. J. (1996). Finite element procedures, Prentice Hall, Upper Saddle River, NJ.
Bathe, K. J. (2007). “Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme.” Int. J. Comput. Struct., 85(7–8), 437–445.
Bathe, K. J., and Cimento, A. P. (1980). “Some practical procedures for the solution of nonlinear finite element equations.” Comput. Methods Appl. Mech. Eng., 22(1), 59–85.
Bauchau, O. A., Damilano, G., and Theron, N. J. (1995). “Numerical integration of nonlinear elastic multi-body systems.” Int. J. Numer. Methods Eng., 38(16), 2727–2751.
Belytschko, T., and Hughes, T. J. R. (1983). Computational methods for transient analysis, Elsevier, Amsterdam, Netherlands.
Belytschko, T., Liu, W. K., and Moran, B. (2000). Non-linear finite elements for continua and structures, Wiley, New York.
Belytschko, T., and Schoeberle, D. F. (1975). “On the unconditional stability of an implicit algorithm for structural dynamics.” J. Appl. Mech., 42(4), 865–869.
Bernal, D. (1991). “Locating events in step-by-step integration of equations of motion.” J. Struct. Eng., 117(2), 530–545.
Betsch, P., and Steinmann, P. (2001). “Conservation properties of a time FE method-part II: Time-stepping schemes for non-linear elastodynamics.” Int. J. Numer. Methods Eng., 50(8), 1931–1955.
Bornemann, P. B., Galvanetto, U., and Crisfield, M. A. (2002). “Some remarks on the numerical time integration of non-linear dynamical systems.” J. Sound Vibrat., 252(5), 935–944.
Brown, J. W., and Churchill, R. W. (1993). Fourier series and boundary value problems, McGraw Hill, Singapore.
Cardona, A., and Geradin, M. (1989). “Time integration of the equations of motion in mechanism analysis.” Int. J. Comput. Struct., 33(3), 801–820.
Chen, C. C., and Robinson, A. R. (1993). “Improved time-history analysis for structural dynamics. I: Treatment of rapid variation of excitation and material nonlinearity.” J. Eng. Mech., 119(12), 2496–2513.
Chen, X., Tamma, K. K., and Sha, D. (1995). “Virtual-pulse time integral methodology: A new approach for computational dynamics. Part 2. Theory for nonlinear structural dynamics.” Finite Elem. Anal. Des., 20(3), 195–204.
Cheney, W., and Kincaid, D. (1999). Numerical mathematics and computing, Brooks/Cole, Pacific Grove, CA.
Chopra, A. K. (1995). Dynamics of structures: theory and application to earthquake engineering, Prentice Hall, Upper Saddle River, NJ.
Chung, J., and Hulbert, G. M. (1993). “A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized- method.” J. Appl.Mech., 60(2), 371–375.
Chung, J., and Hulbert, G. M. (1994). “A family of single-step Houbolt time integration algorithms for structural dynamics.” Comput. Methods Appl. Mech. Eng., 118(1–2), 1–11.
Chung, J., and Lee, J. M. (1994). “A new family of explicit time integration methods for linear and non-linear structural dynamics.” Int. J. Numer. Methods Eng., 37(23), 3961–3976.
Clough, R. W. (1973). “Numerical integration of equations of motion.” Lectures on finite element methods in continuum mechanics, Univ. of Alabama, Tuscaloosa, AL, 525–533.
Clough, R. W., and Penzien, J. (1993). Dynamics of structures, McGraw Hill, Singapore.
Collatz, L. (1960). The numerical treatment of differential equations, Springer, Berlin, Germany.
Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J. (2002). Concepts and applications of finite element analysis, Wiley, New York.
Crisfield, M. A., Jelenic, G., Mi, Y., Zhong, H. J., and Fan, Z. (1997). “Some aspects of the non-linear finite element method.” Finite Elem. Anal. Des., 27(1), 19–40.
Deputy of Technical Affairs. (2007). Monograph 360, management and planning organization: Instruction for seismic rehabilitation of existing buildings, Deputy of Technical Affairs, Tehran, Iran (in Persian).
Farjoodi, J., Anvar, S. A., Mehrazin, H., and Soroushian, A. (2005). “‘On responses’ convergence in structural dynamic analysis. Part 1: The main reason of improper convergence in step-by-step analysis by some integration methods.” Proc., 2nd National Congress on Civil Engineering, Civil Engineering Congresses Council, Tehran, Iran (in Persian).
Farjoodi, J., and Soroushian, A. (2000). “An efficient method with more accuracy for nonlinear dynamic analysis.” Proc., 5th Int. Conf. on Civil Engineering, Vol. 2, Civil Engineering Congresses Council, Mashhad, Iran, 222–229 (in Persian).
Farjoodi, J., and Soroushian, A. (2003). “Shortcomings in numerical dynamic analysis of nonlinear systems.” Nashriyeh Fanni, 37(2), 269–281 (in Persian).
Farjoodi, J., and Soroushian, A. (2007). “A comparative study on the effects of impact and linearly-elastic/perfectly-plastic nonlinearities on structural dynamic behaviors.” Proc., 5th Int. Conf. on Seismology and Earthquake Engineering, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran (in Persian).
Gear, C. W. (1971). Numerical initial value problems in ordinary differential equations, Prentice Hall, Upper Saddle River, NJ.
Golub, G. H., and Ortega, J. M. (1992). Scientific computing and differential equations, Academic Press, New York.
Gratsch, T., and Bathe, K. J. (2005). “A posteriori error estimation techniques in practical finite element analysis.” Int. J. Comput. Struct., 83(4–5), 235–265.
Greenberg, M. D. (1978). Foundations of applied mathematics, Prentice Hall, Upper Saddle River, NJ.
Gupta, A. K. (1992). Response spectrum method: in seismic analysis and design of structures, CRC Press, Boca Raton, FL.
Hairer, E. (1996). Solving ordinary differential equations II stiff and differential-algebraic problems, Springer, Berlin, Germany.
Hauret, P., and Le Tallec, P. (2006). “Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact.” Comput. Methods Appl. Mech. Eng., 195(37–40), 4890–4916.
Henrici, P. (1962). Discrete variable methods in ordinary differential equations, Prentice Hall, Upper Saddle River, NJ.
Henrych, J. (1990). Finite models and methods of dynamics in structures, Elsevier, Amsterdam, Netherlands.
Hernandez, A., Pinto, C. H., Amezua, E., and Fernandez, H. (2003). “Analysis of the components of discretization error in nonlinear structural problems.” Finite Elem. Anal. Des., 39(9), 835–864.
Hinton, E. (1992). Introduction to non-linear finite element analysis, National Agency for Finite Element Methods and Standards, Glasgow, U.K.
Houbolt, J. C. (1950). “A recurrence matrix solution for the dynamic response of elastic aircraft.” J. Aeronaut. Sci., 17(9), 540–550.
Hughes, T. J. R. (1976). “Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics.” Int. J. Comput. Struct., 6(4–5), 313–324.
Hughes, T. J. R. (1987). The finite element method: Linear static and dynamic finite element analysis, Prentice Hall, Upper Saddle River, NJ.
Hughes, T. J. R., Pister, K. S., and Taylor, R. L. (1979). “Explicit-explicit finite elements in nonlinear transient analysis.” Comput. Methods Appl. Mech. Eng., 17/18(1), 159–182.
International Code Council (ICC). (2003). IBC - International Building Code, Club Hills, IL.
Jankowski, R., Wilde, K., and Fujino, Y. (2000). “Reduction of pounding effects in elevated bridges during earthquakes.” Earthquake Eng. Struct. Dynam., 29(2), 195–212.
Jeng, V., and Tzeng, W. L. (2000). “Assessment of seismic pounding hazard for Taipei city.” Eng. Structures, 22(5), 459–471.
Kardestuncer, H. (1987). Finite element handbook, McGraw Hill, New York.
Kuhl, D., and Crisfield, M. A. (1999). “Energy-conserving and decaying algorithms in nonlinear structural dynamics.” Int. J. Numer. Methods Eng., 45(5), 569–599.
Kuhl, D., and Ramm, E. (1996). “Conservation of energy and momentum for implicit single step time integration schemes.” Proc., 3rd European Conf. on Structural Dynamics, European Association for Structural Dynamics (EASD), Florence, Italy.
Lambert, J. D. (1973). Computational methods in ordinary differential equations, Wiley, New York.
Laursen, T. A., and Chawla, V. (1997). “Design of energy conserving algorithms for frictionless dynamic contact problems.” Int. J. Numer. Methods Eng., 40(5), 863–886.
Liu, J., and Wang, X. (2008). “An assessment of the differential quadrature time integration scheme for nonlinear dynamic equations.” J. Sound Vibrat., 314(1-2), 246–253.
Low, K. H. (1991). “Convergence of the numerical methods for problems of structural dynamics.” J. Sound Vibrat., 150(2), 342–349.
Mahin, S. A., and Lin, J. (1983). “Construction of inelastic response spectra for single degree-of-freedom systems.” Rep. UCB/EERC-83/17, Earthquake Engineering Research Center (EERC), Univ. of California, Berkeley, CA.
McNamara, J. F. (1974). “Solution schemes for problems of nonlinear structural dynamics.” J. Pressure Vessels, 96(2), 147–155.
Meirovitch, L. (2001). Fundamentals of vibrations, McGraw Hill, Singapore.
Monro, D. M. (1982). Fortran 77, Edward Arnold, London.
Naeim, F., and Zhogzhi, S. (2001). The seismic design handbook, Kluwer, New York, USA.
Nau, J. M. (1983). “Computation of inelastic spectra.” J. Eng. Mech., 109(1), 279–288.
Nau, J. M., and Hall, W. J. (1984). “Scaling methods for earthquake response spectra.” J. Struct. Eng., 110(7), 1533–1548.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Eng. Mech., 85(3), 67–94.
Noble, B., and Daniel, J. W. (1977). Applied linear algebra, Prentice Hall,Upper Saddle River, NJ.
Ortega, J. M., and Rheinboldt, W. C. (1970). Iterative solution of nonlinear equations in several variables, Academic Press, New York.
Park, K. C. (1975). “An improved stiffly stable method for direct integration of nonlinear structural dynamics equations.” J. Appl. Mech., 42(2), 464–470.
Penry, S. N., and Wood, W. L. (1985). “Comparison of some single-step methods for the numerical solution of the structural dynamic equation.” Int. J. Numer. Methods Eng., 21(11), 1941–1955.
Petzold, L. R., Jay, L. O., and Yen, J. (1997). “Numerical solution of highly oscillatory ordinary differential equations.” Acta Numerica, 6, 437–483.
Quateroni, A., Sacco, R., and Saleri, F. (2000). Numerical mathematics, Springer, New York.
Ralston, A., and Rabinowitz, P. (1978). First course in numerical analysis, McGraw Hill, New York.
Rashidi, S., and Saadeghvaziri, M. A. (1997). “Seismic modeling of multi-span simply supported bridges using Adina.” Int. J. Comput. Struct., 64(5–6), 1025–1039.
Reinhall, P. G., Caughey, T. K., Storti, T. K., and Storti, D. W. (1989). “Order and chaos in a discrete Duffing oscillator: Implications in numerical integration.” J. Appl. Mech., 56(1), 162–167.
Ruge, P. (1999). “A priori error estimation with adaptive time-stepping.” Commun. Numer. Methods Eng., 15(7), 479–491.
Schueller, G. I., and Pradlwarter, H. J. (1999). “On the stochastic response of non-linear FE models.” Arch. Appl. Mech., 69(9–10), 765–784.
Schweizerhof, K., Neumann, J., and Kizio, S. (2004). “On time integration error estimation and adaptive time stepping in structural dynamics.” Proc. Appl. Math. Mech., 4(1), 35–38.
Simo, J. C., and Tarnow, N. (1992). “The discrete energy-momentum methods: conserving algorithms for nonlinear elastodynamics.” Z. Angew. Math. Phys., 43(5), 757–792.
Soroushian, A. (2003). “New methods to maintain responses’ convergence and control responses’: Errors in the analysis of nonlinear dynamic models of structural systems.” Ph.D. thesis, Univ. of Tehran, Tehran, Iran (in Persian).
Soroushian, A. (2004). “Responses' convergence for time integration of highly non-linear models.” Proc., 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf.
Sorouhsian, A. (2005a). “A common reason of improper convergence in time integration of nonlinear semi-discretized equations of motion.” Proc., 13th Int. Conf. on Computational and Experimental Sciences, Chennai, India.
Sorouhsian, A. (2005b). “Two particular reasons for improper convergence in time integration of nonlinear semi-discretized equations of motion.” Proc., 13th Int. Conf. on Computational and Experimental Sciences, Chennai, India.
Soroushian, A. (2006). “Convergence from the standpoints of science and engineering.” 2nd Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy (LFME), Athens, Greece.
Soroushian, A. (2008). “A technique for time integration with steps larger than the excitation steps.” Commun. Numer. Methods Eng., 24(12), 2087–2111.
Soroushian, A. (2009). “Time integration with step sizes less depending on the steps of excitation: I. SDOF systems.” Rep. 7509-P89-1, Structural Engineering Research Center (SERC), International Institute of Earthquake Engineering and Seismology, Tehran, Iran (in Persian).
Soroushian, A. (2010a). “An enhanced fractional-time-stepping method for structural dynamics.” Proc., 17th Int. Congress on Sound and Vibration, International Institute of Acoustics and Vibration (IIAV), Cairo, Egypt.
Soroushian, A. (2010b). “Proper convergence, a concept new in science and important in engineering.” Proc., 4th Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.
Soroushian, A. (2010c). “Pseudo convergence and its implementation in engineering approximate computations.” Proc., 4th Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.
Soroushian, A., and Ahmadi, G. (2010). “A three parameter nonlinear dynamic system with exact closed form solution.” Proc., 17th Int. Congress on Sound and Vibration, International Institute of Acoustics and Vibration, Cairo, Egypt.
Soroushian, A., Arghavani, M., Rajabi, M., Saaed, A., and Sharifpour, M. M. (2011). “A proposition on the uniqueness of solutions for nonlinear structural dynamic models.” Proc., 10th Biennial Int. Conf. on Vibration Problems, Technical University of Liberec, Prague, Czech Republic.
Soroushian, A., and Eshghi, S. (2008). “The importance of appropriate selection of step size and nonlinearity tolerance in time history analyses.” Rep. 307-7711, Structural Engineering Research Center (SERC), International Institute of Earthquake Engineering and Seismology, Tehran, Iran (in Persian).
Soroushian, A., and Farjoodi, J. (2002). “An improvement in non-linear analysis.” Proc., 5th ASCE Engineering Mechanics Division Conf., ASCE, Reston, VA.
Soroushian, A., and Farjoodi, J. (2003a). “Convergence of the responses that time integration generates for contact problems.” Proc., 4th Int. Conf. on Seismology and Earthquake Engineering, International Institute of Earthquake Engineering and Seismology, Tehran, Iran (in Persian).
Soroushian, A., and Farjoodi, J. (2003b). “More reliable responses for time integration analyses.” Struct. Eng. Mech., 16(2), 219–240.
Soroushian, A., and Farjoodi, J. (2003c). “Responses convergence in time integration of nonlinear semi-discrete equations of motion.” Proc., 8th Int. Conf. on Recent Advances in Structural Dynamics, Institute of Sound and Vibration Research (ISVR), Southampton, U.K.
Soroushian, A., and Farjoodi, J. (2008). “A unified starting procedure for the Houbolt method.” Commun. Numer. Methods Eng., 24(1), 1–13.
Soroushian, A., Farjoodi, J., and Bargi, K. (2005a). “On responses’ convergence in structural dynamic analysis. Part 2: The particular reason of improper convergence in step-by-step analysis by some integration methods.” Proc., 2nd National Congress on Civil Engineering, Civil Engineering Congresses Council, Tehran, Iran (in Persian).
Soroushian, A., Farjoodi, J., and Mehrazin, H. (2004). “A nonlinearity measure for piecewisely linear structural dynamic models.” Proc., 11th Int. Congress on Sound and Vibration, International Institute of Acoustics and Vibration, St. Petersburg, Russia.
Soroushian, A., Farjoodi, J., and Mehrazin, H. (2006a). “A new measure for the nonlinear behavior of piecewisely linear structural dynamic models.” Proc., 13th Int. Congress on Sound and Vibration, International Institute of Acoustics and Vibration, Vienna, Austria.
Soroushian, A., Farjoodi, J., Mohammadi, S., and Rofooei, F. R. (2005b). “On responses’ convergence in structural dynamic analysis. Part 3: The role of computational facilities in time integration analysis.” Proc., 2nd National Congress on Civil Engineering, Civil Engineering Congresses Council, Tehran, Iran (in Persian).
Soroushian, A., Farjoodi, J., and Vahdani, S. (2005c). “On responses’ convergence in structural dynamic analysis. Part 4: The possibility of convergence to incorrect responses.” Proc., 2nd National Congress on Civil Engineering, Civil Engineering Congresses Council, Tehran, Iran (in Persian).
Soroushian, A., Farjoodi, J., and Wriggers, P. (2003). “Reliable convergence for dynamic linearly-elastic/perfectly-plastic systems analyzed with different time integration methods.” Proc., 10th Int. Congress on Sound and Vibration, International Institute of Acoustics and Vibration, Stockholm, Sweden.
Soroushian, A., Wriggers, P., and Farjoodi, J. (2005d). “On practical integration of semi-discretized nonlinear equations of motion. part 1: reasons for probable instability and improper convergence.” J. Sound Vibrat., 284(3–5), 705–731.
Soroushian, A., Wriggers, P., and Farjoodi, J. (2005e). “Time integration of nonlinear equations of motion - numerical instability or numerical inconsistency?” Proc., 5th EUROMECH Nonlinear Oscillations Conf., European Mechanics Society, Eindhoven, Netherlands.
Soroushian, A., Wriggers, P., and Farjoodi, J. (2006b). “A statement for the convergence of approximate responses and its application in structural dynamics.” Proc., 2nd Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.
Soroushian, A., Wriggers, P., and Farjoodi, J. (2009). “Asymptotic upper-bounds for the errors of Richardson Extrapolation with practical application in approximate computations.” Int. J. Numer. Methods Eng., 80(5), 565–595.
Soroushian, A., Wriggers, P., and Farjoodi, J. (2010). “Richardson extrapolation from the points of view of science and engineering with a practical consequence revisited.” Proc., 4th Int. Conf. From Scientific Computing to Computational Engineering, Laboratory of Fluid Mechanics and Energy, Athens, Greece.
Strikwerda, J. C. (1989). Finite difference schemes and partial differential equations, Wadsworth & Books/Cole, Pacific Grove, CA.
Warburton, G. B. (1989). “Formulae for errors for initial displacement and velocity problems using the Newmark method.” Earthquake Eng. Struct. Dynam., 18(4), 565–573.
Wood, W. L. (1990). Practical time stepping schemes, Oxford, New York.
Wood, W. L., and Oduor, M. E. (1988). “Stability properties of some algorithms for the solution of nonlinear dynamic vibration equation.” Commun. Appl. Numer. Methods, 4(2), 205–212.
Wriggers, P. (2002). Computational contact mechanics, Wiley, New York.
Xie, Y. M. (1996). “An assessment of time integration schemes for nonlinear dynamic equations.” J. Sound Vibrat., 192(1), 321–331.
Xie, Y. M., and Steven, G. P. (1994). “Instability, chaos, and growth and decay of energy of time-stepping schemes for nonlinear dynamic Equations.” Commun. Numer. Methods Eng., 10(5), 393–401.
Zhou, X., and Tamma, K. K. (2004). “A new unified theory underlying time dependent first-order systems: a prelude to algorithms by design.” Int. J. Numer. Methods Eng., 60(10), 1699–1740.
Zienkiewicz, O. C., and Xie, Y. M. (1991). “A simple error estimator and adaptive time stepping procedure for dynamic analysis.” Earthquake Eng. Struct. Dynam., 20(9), 871–887.
Zienkiewicz, O. C., Taylor, R. L., Zhu, J. Z., and Nithiarasu, P. (2005). The finite element method, Elsevier, Amsterdam, Netherlands.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 139 • Issue 2 • February 2013
Pages: 114 - 145
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Sep 5, 2010
Accepted: Mar 15, 2012
Published online: Mar 17, 2012
Published in print: Feb 1, 2013
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.