Digital simulation of multi-variate stochastic processes

Stochastic dynamic analysis of linear or nonlinear multi-degree-of-freedom systems excited by multi-variated processes is usually conducted by using digital Monte Carlo (MC) simulation. Since in structural systems few modal shapes contribute to the response in the nodal space, the computational burden of MC simulation is mainly related to the digital simulation of the input process. Usually, the generation of multi-variated samples of Gaussian input process is performed with the aid of the Shinozuka formula. However, since in this procedure the stochastic process is given as a summation of waves with random amplitude amplified by the square root of the power spectral density, the randomness is due to a random phase angle of each wave, therefore a very large number of waves is required to reach the Gaussianity, i.e. the process is only asymptotically stable. Moreover, the computational burden increases in case of multi-variated processes. The paper aims to drastically reduce the generation time of the input process through the use of a two-step procedure. In the first step, by using the Priestley formula, each wave is normally distributed. This first aspect allows to drastically reduce the computational effort for the mono-variate process since few waves are sufficient to reach the Gaussianity. In the second step, the multi-variate process is reduced as a summation of independent fully coherent vectors if the quadrature spectrum (q-spectrum) can be neglected. An application of digital simulation of the wind velocity field is discussed to prove the efficiency of the proposed approach.