MS5.4 - Computational methods for stochastic dynamics

Stochastic dynamic analysis of structures to turbulent wind loading has been widely recognised since its conception back in 1961, for its effectiveness and computational power with respect to classical deterministic time domain Monte Carlo simulations. It also allows to have a better understanding of how the structure interacts with the oncoming turbulent wind. Indeed, in its original formulation, its application assumes the linearised version of the loading, to be able to consider Gaussian random processes. In recent years, evidence along many studies have shown the importance of considering the actual non-Gaussian nature of the loading processes. In such context, higher-order stochastic – dynamic – analysis (HOSA) is required. If in the time domain it is based on the computation of Volterra Series of n increasing-order Volterra Kernels, in the frequency domain it requires the evaluation of some higher-order spectra, like the bispectrum at 3rd order, trispectrum at 4th, etc, each one of them providing an estimation of the statistical descriptor of the corresponding order. These are then used for a more accurate evaluation of the non-Gaussian Probability Density Function (PDF) of the random process, which coupled with the Extreme Value Theory, gives a tool for estimating the extreme values then used the for design/verification of civil structures based on Code Standards. A key step in the evaluation of the stochastic responses of structures to non Gaussian winds relies therefore in the bispectral analysis, which consists in multiplying the bispectrum of loads by the structural kernel and integrate the resulting bisepctrum in a 2-D frequency space. This extends what is usually done at second order by multiplying the power spectral density of the loads by the frequency response function and integrate the corresponding result to obtain the variance of the response. While it is clear that the third statistical moment of the response is obtained as the integration of a somewhat complex function (since the bispectrum is a function in a 2-D frequency space with many peaks), the efficient estimation of this integral still remains a challenge as soon as large multi degree-of-freedom structures are considered. In this context, this work aims at providing an optimised numerical algorithm for the computation of the skewness coefficients, linked to third and second order statistical descriptors of the structural response of large MDOFs structures. It is based on two simple ingredients : an efficient meshing of the 2-D frequency space and a proper orthogonal decomposition of the oncoming wind speed. The combination of these two specificities not only accelerate the structural analysis by several orders of magnitude, but also makes the analysis of large MDOF structures possible by reducing the required memory storage. The full paper will describe the proposed methodology and illustrate it with an example of a real case application.

An approximate analytical technique is developed for bounding the response and first-passage probability of lightly damped nonlinear oscillators endowed with fractional derivative elements and subjected to imprecise Gaussian loads. The direct identification of such bounds leads to a so-called double-loop approach. In this setting, reliability is repeatedly evaluated at different points of the epistemic parameter space during the solution of an optimization problem, which can lead to significant computational efforts. In this regard, using elements of the theory of fractional calculus, first, the nonlinear equation governing the dynamics of the oscillator is linearized by resorting to statistical linearization and averaging methodologies [1]. Next, the values of the epistemic uncertain parameters that produce bounds on the failure probability of the linearized system are derived by utilizing an operator norm-based framework [2]. The proposed technique can be construed as an extension of a recently developed approach for the reliability assessment of multi-DOF nonlinear systems subject to imprecise stochastic loads [3]. A hysteretic nonlinear oscillator with fractional derivative elements subject to imprecise stochastic loading is considered for demonstrating the efficiency of the technique. References [1] Roberts, J.B. and Spanos, P.D., 2003. Random vibration and statistical linearization. Courier Corporation. [2] Faes, M.G., Valdebenito, M.A., Moens, D. and Beer, M., 2021. Operator norm theory as an efficient tool to propagate hybrid uncertainties and calculate imprecise probabilities. Mechanical Systems and Signal Processing, 152, p.107482. [3] Ni, P., Jerez, D.J., Fragkoulis, V.C., Faes, M.G., Valdebenito, M.A. and Beer, M., 2022. Operator Norm-Based Statistical Linearization to Bound the First Excursion Probability of Nonlinear Structures Subjected to Imprecise Stochastic Loading. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 8(1), 04021086.