Stochastic Runge-Kutta Schemes for Discretization of Hysteretic Models

Abstract

The need to produce digital solutions of stochastic differential equations (SDE) is present in problems arising in many areas. This is the case in Seismic Engineering where hysteretic models are used (see Wan et al., Soil Dyn Earthquake Eng 21:75-81, 2001 for an example of a problem involving a bridge column). The simulation of the solutions of these nonlinear equations is based upon a discretization scheme. In the study of hysteretic models subject to Gaussian white noise, we aim to compare the response obtained by using two schemes in the discretization of the SDE, in terms of the second statistical moments of the displacement, with that obtained from solving numerically the ODE system satisfied by the moments that arises after the use of adapted Monte Carlo simulation and obtained by applying Ito's Formula. We analyze the single degree of freedom Noori--Baber--Wen model for different values of (a) the parameters of the non--linearity coefficient, (b) the parameters that characterize the type of hysteresis, (c) the parameters that take into account with the degradation effect of resistance, stiffness and the pinching effect. We conclude that when the discretization step is small, the estimates of the second moment are similar in both schemes meaning that the choice between the weakly convergence schemes is irrelevant. However the solutions obtained by using the Euler schemes are different from those obtained by approximately solving the equations of the moments. This difference is more relevant in situations where the allowed contribution of the dissipated energy is larger.