MS5.2 - Computational methods for stochastic dynamics

In the field of ground motion simulation, the stochastic site-based methodology relies on the existing database of ground shaking. Based on these methodologies, several properties of seismic signals are used to simulate seismic waves. These parameters could be evaluated either parametrically via linear or nonlinear regression techniques or non-parametrically via sophisticated machine-learning algorithms. Nonetheless, parametric models, which consist of a particular mathematical formulation, can be a source of large bias. In this study, we employ machine learning techniques to develop predictive models for two main input parameters of the stochastic site-based ground motion model: Arias intensity and significant duration, which control the time variation of the simulated ground shakings. The Arias intensity, defined by the integral of the square of the acceleration time series, and the significant duration, which is related to the strong shaking phase of an earthquake, are also of particular interest in structural and geotechnical engineering fields. For this purpose, the random forest approach is employed to develop prediction models for the European database. To guarantee the prediction accuracy of the models also for unseen future data, only 80 percent of the data is used for training, and the rest is reserved for testing the trained model. The model hyperparameters are tuned to control bias and variance tradeoffs by k-fold cross-validation. For each model, a set of hyperparameters is selected, and a possible range is given. Then, a Bayesian optimization technique is implemented to find the best set of these hyperparameters among the given range. All these models provided promising results compared to the prior models in the literature.

Incremental dynamic analysis (IDA) constitutes one of the most commonly employed methodologies for estimating the functional relationship between the Intensity Measures (IMs) and the selected Engineering Demand Parameters (EDPs). Subsequently, the information provided via this functional relationship is utilized in conjunction with appropriately defined limit states for quantifying system fragilities [1]. For an EDP-based criterion the twisting patterns of IDA curves can point multiple limit-state points, requiring the handling of this ambiguity on an ad hoc basis. Clearly, the mathematical entity of an IDA curve is interwoven with scaling as well as timing ambiguity. This peculiarity brings to the fore the need for studying the problem of the limit state exceedance (i.e. the onset of stiffness and strength degradation which signals the entrance of a structure into a limit/damage state) through the lens of the first-passage excursion time. A novel stochastic incremental dynamics (SIDA) methodology is developed for nonlinear structural systems exposed to a seismic excitation vector consistently aligned with contemporary aseismic codes provisions. Rendering to the concept of non-stationary stochastic processes, the vector of the imposed seismic excitations is characterized by evolutionary power spectra compatible in a stochastic sense with elastic response acceleration spectra of specified modal damping ratio and scaled ground acceleration [2]. The proposed technique can be construed as a two-stage approach. Firstly, relying on a statistical linearization treatment [3], the equivalent time-dependent natural frequencies (t) and modal damping ratios (t) are determined. Secondly, utilizing the time-dependent equivalent elements in conjunction with a combination of deterministic and stochastic averaging treatment [4,5], the system first-passage limit state exceedance probability density functions (PDFs) are determined for various limit states in an efficient and rigorous manner. The proposed SIDA technique has a number of noteworthy attributes such as (i) accounting for nonlinear/hysteretic system behavior; (ii) modeling the seismic excitation in the form of a vector of stochastic processes endowed with non-stationary characteristics; (iii) proposing a novel EDP that of the first-passage limit state exceedance which is naturally coupled with limit-state requirements; and (iv) providing reliably higher order statistics (PDF) of the selected EDP in a computationally efficient manner, by avoiding demanding nonlinear response time-history analyses in a Monte Carlo-based context. References: [1] Vamvatsikos D., Cornell C. A. Incremental dynamic analysis, Earthquake Engineering and Structural Dynamics 2002; 31:491-514. [2] Cacciola P., 2010. A stochastic approach for generating spectrum compatible fully nonstationary earthquakes, Computers & Structures 88, 889–901. [3] Roberts JB, Spanos PD. Random vibration and statistical linearization. New York: Dover Publications; 2003 [4] Mitseas, I.P., Kougioumtzoglou, I.A., Spanos P.D., Beer, M. Nonlinear MDOF system Survival Probability Determination Subject to Evolutionary Stochastic Excitation. Journal of Mechanical Engineering; 62 7-8, 440-451, 2016. [5] Kougioumtzoglou I. A., Ni P., Mitseas I. P., Fragkoulis V. C., Beer M., 2022. An approximate stochastic dynamics approach for design spectrum based response analysis of nonlinear structural systems with fractional derivative elements, International Journal of Non-Linear Mechanics 146, 104178, doi: 10.1016/j.ijnon linmec.2022.104178

The question of predicting the long-term evolution of a mechanical structure in operation (e.g. fatigue) faces the associated issue of computational cost, particularly when the problem is intended to be numerically solved in the time domain (which can be required to account for the history of loads applied on the structure). When fast phenomena have to be described for a very long time interval, a scale separation assumption allows the use of time homogenization methods such as in (Guennouni & Aubry. Comptes rendus de l’Académie des sciences, Série II. 1986), (Oskay & Fish. International Journal for Numerical Methods in Engineering. 2004) or (Puel & Aubry, International Journal for Multiscale Computational Engineering, 2014), which are based on a quasi-periodicity assumption. When fast loads can be described by stochastic processes, a so-called stochastic time homogenization method can be defined and be seen as a transposition to time of the classical stochastic space homogenization method (Sab, European Journal of Mechanics: A. Solids, 1992). The time-homogenized equations are determined by averaging the different quantities allowing to separate slow-evolving phenomena from fast-time components, as in (Puel & Sab, VII European Congress on Computational Methods in Applied Sciences and Engineering, 2016). The main difficulty is then to substitute to the ensemble expectations a time averaging over a suitable time interval in order to derive the time-homogenized equations. The aim of the talk is to give theoretical as well as practical insights of this specific method. Examples will focus on structures with typical nonlinear material behaviors (e.g. viscoplasticity, damage), with loadings that can be described as the superposition of a slow deterministic component and a fast stochastic component.