Stochastic Newmark schemes for the discretization of hysteretic models

Abstract

The need to study and to obtain digital solutions of stochastic nonlineardifferential equations is a common situation in Seismic Engineering. This isthe case for the hysteretic models. These models do not have an exact solution andcan only be approximated by numerical methods. We discretize the solutions usingthe stochastic improved Euler scheme and the three parameter implicit stochasticNewmark schemes: a higher order and a lower order Newmark scheme. In the caseof hysteretic models subjected to gaussian white noises, we were able to reduce theproblem of approximating the solution to that of a linear system in each time stepavoiding the NewtonRaphson method in the same time steps. This allowed us tosave computational effort in the approximation of the response of the hystereticsystem and was achieved by giving explicitly the value of one of the parameters inthe equation of the Newmark scheme that corresponds to the hysteretic variablewhile keeping the equations of the displacement and velocity implicit. We comparethe performance of these two implicit Newmark schemes. In the simulationstudy for the Bouc-Wen model, we compare the solutions produced for the specificchoice of the parameters ( = 0.5, ß = 0.5) which are the values used by Roy andDash(2005) in the case of linear systems. We conclude that the standard deviationof the displacement obtained from the proposed higher order Newmark scheme islarger than that obtained from the proposed lower order Newmark scheme. Theproposed lower order Newmark scheme is computationally atractive to competewith the improved Euler scheme.