A General Rule for the Influence of Physical Damping on the Numerical Stability of Time Integration Analysis

Document Type : Research Paper


Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology, S. Lavasani (Farmaiyeh, North Dibajee, West Arghavan, No. 21,Tehran 19537, Iran


The influence of physical damping on the numerical stability of time integration analysis is an open question since decades ago. In this paper, it is shown that, under specific very general conditions, physical damping can be disregarded when studying the numerical stability. It is also shown that, provided the specific conditions are met, analysis of structural systems involved in extremely high linear-viscous damping is unconditionally stable. A secondary achievement is that, when the linear-viscous damping increases, the numerical damping may increase or decrease.


Main Subjects

[1] IBC. International Building Code, International Code Council, USA, 2003.
[2] Hart, G.C., Jain, A., Performance-based wind evaluation and strengthening of existing tall concrete buildings in the Los Angeles region: dampers, nonlinear time history analysis and structural reliability, Struct. Des. Tall Spec. 23(16), 2014 1256-1274.
[3] Sassi, M.A., Nonlinear Dynamic Analysis of Wind Turbine Towers Subject to Design Wind and Seismic Loads, PhD Thesis, Colorado School of Mines, USA, 2016.
[4] Betsch, P., Steinmann, P., Inherently energy conserving time finite elements for classical mechanics, J. Comput. Phys. 160(1), 2000, 88-116.
[5] Bruels, O., Golinval, J.C., The generalized-α method in mechatronic applications, Z. Angew. Math. Mech. 86(10), 2006, 748-758.
[6] Eggl, S., Dvorak, R., An introduction to common numerical integration codes used in dynamical astronomy. In The Dynamics of Small Solar System Bodies and Exoplanets, Springer, USA, 2010, 431-480.
[7] Faragó, I., Havasi, Á., Zlatev, Z. (Eds.) Advanced Numerical Methods for Complex Environmental Models: Needs and Availability, Bentham Science Publishers, ebook, 2013.
[8] Kontoe, S., Zdrakovic, L., Potts, D.M., An assessment of time integration schemes for dynamic geotechnical problems, Comput. Geotech. 35(2), 2008, 253–264.
[9] Kpodzo, K., Fourment, L., Lasne P., Montmotonnet, P., An accurate time integration scheme for arbitrary rotation motion: application to metal forming formulation, Int. J. Mater. Form. 9(1), 2016, 71-84.
[10] Lemieux, J.F, Knoll, D.A., Losch, M., Girard, C., A second-order accurate in time IMplicit–EXplicit (IMEX) integration scheme for sea ice dynamics, J. Comput. Phys. 263, 2014, 375-392.
[11] Meijaard, J.P., Efficient numerical integration of the equations of motion of non-smooth mechanical systems, Z. Angew. Math. Mech. 77(6), 1997, 419-427.
[12] Paultre, P., Dynamics of Structures, John Wiley & Sons, USA, 2010.
[13] Soroushian, A., Integration step size and its adequate selection in analysis of structural systems against earthquakes. In The Computational Methods in Earthquake Engineering, Vol. 3, Springer, Norway, 2017, 285-328.
[14] Tamma, K.K. and D'Costa, J.F., A new explicit variable time‐integration self‐starting methodology for computational structural dynamics, Int. J. Numer. Meth. Eng. 33(6), 1992, 1165-1180.
[15] Kadioglu, S.Y., and Knoll, D.A., A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems, J. Comput. Phys. 229(9), 2010, 3237-3249.
[16] Bonelli, A., Bursi O. S., Generalized-α methods for seismic structural testing, Earthq. Eng. Struct. D. 33(10) 1067-1102.
[17] Yousuf, M., High-order time stepping scheme for pricing American option under Bates model, Int. J. Comput. Math. 96(1), 2019, 18-32.
[18] Ghasemi, M., Sonner, S. Eberl, HY. J., Time adaptive numerical solution of a highly non-linear degenerate cross-diffusion system arising in multi-species biofilm modeling, Eur. J. Appl. Math. 29(6), 2018, 1035-1061.
[19] Clough, R. W. Penzien, J., Dynamics of Structures, McGraw Hill, Singapore, 1993.
[20] Geradin, M., Rixen, D.J., Mechanical Vibrations: Theory and Applications to Structural Dynamics, John Wiley & Sons, USA, 2015.
[21] Gavin, H., Structural Dynamics, Duke University, USA, Class Notes CE 283, 2001.
[22] Hughes, T.J. R., Pister, K.S., Taylor R.L., Implicit-explicit finite elements in nonlinear transient analysis, Comput. Methods Appl. Mech. Eng. 17/18(1), 1979, 159–182.
[23] Wriggers, P., Computational Contact Mechanics, John Wiley & Sons, New York, 2002.
[24] Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, NJ, 1971.
[25] Soroushian, A., Wriggers, P., Farjoodi, J., Asymptotic upperbounds for the errors of Richardson Extrapolation with practical application in approximate computations, Int. J. Numer. Meth. Eng. 80(5), 2009, 565–595.
[26] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, Prentice-Hall, NJ, 1962.
[27] Strikwerda, J.C., Finite Difference Schemes and Partial Differential Equations, Wadsworth & Books/Cole, Pacific Grove, CA, 1989.
[28] Richtmyer, R.D., Morton, K.W., Difference Methods for Initial-value Problems, Interscience Publishers, USA, 1967.
[29] Bathe, K.J., Wilson E.L., Stability and accuracy analysis of direct integration methods, Earthq. Eng. Struct. D. 1(3), 1972, 283-291.
[30] Lax, P.D., Richtmyer, R.D., Survey of the stability of linear finite difference equations, Commun. Pur. Appl. Math. 9(2), 1956, 267-293.
[31] Wood, W.L., Practical Time Stepping Schemes, Oxford, New York, 1990.
[32] Greenberg, M.G., Advanced Engineering Mathematics, Prentice-Hall, New Jersey, 1998.
[33] Belytschko, T., Hughes, T.J.R., (Eds.) Computational Methods for Transient Analysis, Elsevier, The Netherlands, 1983.
[34] Mengaldo, G., Wyszogrodzki, A., Diamantakis M., Lock, S. J.  Giraldo F. X. and Wedi, N. P. Current and emerging time-integration strategies in global numerical weather and climate prediction, Arch. Comput. Methods Eng. 1-22, 2018.
[35] NZS 1170, Structural Design Actions, Part 5: Earthquake Actions-New Zealand, New Zealand, 2004.
[36] Hughes, T.J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, NJ, 1987.
[37] Soroushian, A., Wriggers, P., Farjoodi, J., Time integration of nonlinear equations of motion - numerical instability or numerical inconsistency? In The Proc. 5th EUROMECH Nonlinear Oscillations Conference (ENOC 2005), European Mechanics Society, Eindhoven, The Netherlands, 2005.
[38] Han, B., Zdravkovic, L., Kontoe, S., Stability investigation of the Generalised-α time integration method for dynamic coupled consolidation analysis, Comput. Geotech. 64, 2015, 83-95.
[39] Rashidi, S., Saadeghvaziri M.A., Seismic modeling of multispan simply supported bridges using Adina, Int. J. Comput. Struct. 64(5–6), 1997, 1025–1039.
[40] Xie, Y.M., Steven, G. P., Instability, chaos, and growth and decay of energy of time-stepping schemes for nonlinear dynamic Equations, Commun. Numer. Methods Eng. 10(5), 1994, 393–401.
[41] Rose, I., Buffett, B., Heister, T., Stability and accuracy of free surface time integration in viscous flows, Phys. Earth Planet. Inter. 262, 2017, 90-100.
[42] Lee, K., A short note on time integration stability of dynamic frictional contact problems of elastic bodies, P. I. Mech. Eng. K-J. Mul. 230(2), 2016, 113-120.
[43] Cheng, M., Convergence and stability of step-by-step integration for model with negative-stiffness, Earthq. Eng. Struct. D. 16(2), 1988, 227-244.
[44] Elnashai, A.S., Sarno L.Di., Fundamentals of Earthquake Engineering, John Wiley & Sons, USA, 2008.
[45] Watkins, D. S., Fundamentals of Matrix Computation (2nd Ed.), John Wiley & Sons, USA, 2002.
[46] Houbolt, J. C., A recurrence matrix solution for the dynamic response of elastic aircraft, J. Aeronaut. Sci. 17(9), 1950, 540–550.
[47] Hart, G.C., Wong K., Structural Dynamics for Structural Engineers, John Wiley & Sons, USA, 1999.
[48] Soroushian, A., Development of an Algorithm and Computer Program to Evaluate the Numerical Stability and Consistency of New Time Integration Methods, International Institute of Earthquake Engineering and Seismology (IIEES), Iran, Report 7517, 2015. (in Persian)
[49] Piché, R., Nevalainen, P., Variable step Rosenbrock algorithm for transient response of damped structures, P. I. Mech. Eng. C-J. Mec. 213(2), 1998, 191-198.
[50] Zhu, M. Zhu, J.Q., Studies on stability of step-by-step methods under complex damping conditions, J. Earthq. Eng. Eng. Vib. 21(4), 2001, 59-62.
[51] Wang, J.T., Numerical stability of explicit finite element schemes for dynamic system with Rayleigh damping, Earthq. Eng. Eng. Vib. 22(6), 2002, 18-24.
[52] Wu, B., Bao, H., Ou., J., Tian, S., Stability and accuracy analysis of the central difference method for real-time substructure testing, Earthquake Eng. Struct. Dyn. 34(7), 2005, 705-718.
[53] Szabo, Z., Lukacs, A., Numerical stability analysis of a forced two-dof oscillator with bilinear damping, J. Comput. Nonlin. Dyn. 2(3), 2007, 211-217.
[54] Rezaiee-Pajand, M., Karimi-Rad, M., A new explicit time integration scheme for nonlinear dynamic analysis, Int. J. Struct. Stab. Dyn. 16(9), 2016, 1550054.
[55] Soroushian, A., A general rule for the effect of viscous damping on the numerical stability of time integration analyses. In The Proc. 12th World Congress on Computational Mechanics and 6th Asia-Pacific Congress on Computational Mechanics (WCCM XII & APCOM VI), IACM, Seoul, South Korea, 2016.
[56] Soroushian, A., A general rule for the effect of arbitrary damping on the numerical stability of time integration analyses. In The Proc., 7th International Conference on Computational Methods (ICCM2016), University of California at Berkeley, Berkeley, USA, 2016.
[57] Ding, Z., Li, L., Hu Y., A modified precise integration method for transient dynamic analysis in structural systems with multiple damping models, Mech. Syst. Sig. Process. 98, 2018, 613-633.
[58] Wood, W.L., On the effect of natural damping on the stability of a time-stepping scheme, Commun. Appl. Numer. M. 32(2), 1987, 141-144.
[59] Pourlatifi, S., A method to distinguish numerical and physical instability in analysis of structural systems, Msc Thesis, International Institute of Earthquake Engineering and Seismology, Iran, 2009.
[60] Zienkiewicz, O.C., Xie, Y.M., A simple error estimator and adaptive time stepping procedure for dynamic analysis, Earthq. Eng. Struct. D. 20(9) (1991) 871-887.
[61] Apostol, T.M., Calculus, Vol. I, John Wiley & Sons, New York, 1967.
[62] Kardestuncer, H., Finite Element Handbook, McGraw Hill, USA, 1987.
[63] Newmark, N.M., A method of computation for structural dynamics, J. Eng. Mech. 85(3), 1959, 67–94.
[64] Clough, R.W., Numerical integration of equations of motion. In The Lectures on Finite Element Methods in Continuum Mechanics, Univ. of Alabama, Tuscaloosa, AL, 1973, 525-533.
[65] Chung, J., Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-a method, J. Appl. Mech. 60(2), 1993, 371–375.
[66] Wilson, E.L. , A Computer Program for the Dynamic Stress Analysis of Underground Structures, University of California, Berkeley, Report 68-1,1968.
[67] Hilber, H.M., Hughes, T.J.R., Taylor R.L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. D. 5(3), 1977, 283–292.
[68] Hoff, C., Pahl, P.J., Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics, Comput. Methods Appl. Mech. Eng. 67(3), 1988, 367-385.
[69] Bathe, K.J., Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme, Int. J. Comput. Struct. 85(7-8), 2007, 437-445.
[70] Bathe, K.J., Noh, G., Insight into an implicit time integration scheme for structural dynamics, Int. J. Comput. Struct. 98-99(1), 2012, 1-6.
[71] Katsikadelis, J.T., A new direct time integration method for the equations of motion in structural dynamics, Z. Angew. Math. Mech. 94(9), 2014, 757-774.
[72] Rezaiee-Pajand, M., Karimi-Rad, M., A family of second-order fully explicit time integration schemes, Comp. Appl. Math. 37(3), 2017, 3431-3454.
[73] Lazan, B.J., Damping of Materials and Members in Structural Mechanics, Pergamon Press, UK, 1968.
[74] Allgower, E.L., Georg, K., Numerical Continuation Methods, An Introduction, Springer, New York, 1980.
[75] Kuhl D., Crisfield M.A., Energy-conserving and decaying algorithms in non-linear structural dynamics, Int. J. Numer. Meth. Eng. 45(5), 1999, 569–599.
[76] Soroushian, A., Wriggers, P., Farjoodi, J., Practical integration of semi-discretized nonlinear equations of motion: proper convergence for systems with piecewise linear behaviour, J. Eng. Mech., 139(2), 2013, 114-145.
[77] Soroushian, A., Farjoodi, J., Bargi, K., Rajabi, M., Saaed, A., Arghavani, M., Sharifpour, M.M., Two versions of the Wilson-θ time integration method. In The Proc. 10th International Conference on Vibration Problems (ICOVP 2011), Prague, Czech Republic, 2011.
[78] Rezaiee-Pajand, M. Karimi-Rad, M., An accurate predictor-corrector time integration method for structural dynamics, Int. J. Steel Struct., 17(3), 2017, 1033-1047.
[79] Rio G., Soive A. Grolleau V., Comparative study of numerical explicit time integration algorithms, Adv. Eng. Software 36(4), 2005, 252-265.