Time Integration Schemes in Dynamic Problems- Effect of Damping on Numerical Stability and Accuracy

A great deal of progress has been made in the past several decades towards the understanding and development of time integration methods in structural dynamics. These methods involves a step by step algorithm for transient analysis of linear and non-linear dynamic problems. It is essential to provide a comprehensive survey of various methodologies used to solve second order differential equations in a single article. Broadly, the methods include direct integration, mode superposition and response spectrum methods among others. The first two methods uses an integration scheme while response spectrum method is based on extreme response analysis. Both direct integration and mode superposition method use an integration scheme but the selection of a particular method depends on the problem and frequency content of the loading. A detailed survey of various integration schemes is presented in this paper. The stability and accuracy of these integration schemes has been studied by researches in the past. However, the effect of damping on the stability and accuracy of these schemes need to be investigated. A single degree of freedom system is used to check the stability and accuracy criteria of various integration schemes. Also, the effect of damping on these parameters is studied and results are presented.

[1]  M. Dokainish,et al.  A survey of direct time-integration methods in computational structural dynamics—I. Explicit methods , 1989 .

[2]  Ted Belytschko,et al.  An analysis of an unconditionally stable explicit method , 1983 .

[3]  Jan Ming Ko,et al.  Fatigue analysis and life prediction of bridges with structural health monitoring data — Part I: methodology and strategy , 2001 .

[4]  Massimiliano Lucchesi,et al.  The Numerical Method , 2008 .

[5]  K. Bathe,et al.  On a composite implicit time integration procedure for nonlinear dynamics , 2005 .

[6]  Mehdi Ghassemieh,et al.  New implicit method for analysis of problems in nonlinear structural dynamics , 2011 .

[7]  J. Ricles,et al.  Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation , 2014 .

[8]  Samuel W. Key,et al.  Transient shell response by numerical time integration , 1973 .

[9]  Mario Paz,et al.  Structural Dynamics: Theory and Computation , 1981 .

[10]  R. D. Krieg.  Unconditional Stability in Numerical Time Integration Methods , 1973 .

[11]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[12]  Anil K. Chopra,et al.  Dynamics of Structures: Theory and Applications to Earthquake Engineering , 1995 .

[13]  O. Zienkiewicz,et al.  A note on mass lumping and related processes in the finite element method , 1976 .

[14]  Edward L. Wilson,et al.  Three dimensional static and dynamic analysis of structures : A physical approach with emphasis on earthquake engineering , 2000 .

[15]  Erwin Fehlberg,et al.  Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme , 1970, Computing.

[16]  Ted Belytschko,et al.  A survey of numerical methods and computer programs for dynamic structural analysis , 1976 .

[17]  Carlos A. Felippa,et al.  DIRECT TIME INTEGRATION METHODS IN NONLINEAR STRUCTURAL DYNAMICS , 1979 .

[18]  T. Belytschko,et al.  A Précis of Developments in Computational Methods for Transient Analysis , 1983 .

[19]  Shuenn-Yih Chang,et al.  A new family of explicit methods for linear structural dynamics , 2010 .

[20]  L. Shampine.  Local error control in codes for ordinary differential equations , 1977 .

[21]  Tianyun Liu,et al.  An efficient time-integration method for nonlinear dynamic analysis of solids and structures , 2013 .

[22]  Åke Björck,et al.  Numerical Methods , 2021, Markov Renewal and Piecewise Deterministic Processes.

[23]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[24]  Shuenn-Yih Chang,et al.  A family of noniterative integration methods with desired numerical dissipation , 2014 .

[25]  C. P. Vendhan,et al.  Dynamic effects of wind loads on offshore deck structures — A critical evaluation of provisions and practices , 2000 .