Pol D. Spanos

In this paper, the nonstationary stochastic response of fractional structural systems subjected to fractional excitations is examined. Fractional operators provide an effective mathematical construct to model energy dissipation and stiffness–damping coupling mechanisms that cannot be modeled by integer operators. In particular, the memory property of fractional operators makes them suitable for treating the non-Markovian features exhibited by systems with path-dependent responses, such as the damping and stiffness mechanisms in structural systems and soils under dynamic loading. In this paper, particular attention is devoted to fractional structural systems excited by a proposed stochastic ground motion model consisting of a nonstationary filtered fractional Kanai–Tajimi process, thus accounting for path-dependent energy dissipation in both the structure and the seismic waves’ propagating medium. The work attempts to elucidate the sensitivity and effects on probabilistic structural response predictions due to non-Markovian features in the excitation when both the system and the excitation are modeled using fractional operators. To analyze the fractional operators describing the structure and soil, the operators are discretized and approximated as the superposition of the solutions of a system of linear first-order ordinary differential equations, which, together with the system dynamics equations, are transformed into an equivalent first-order linear state-space model. The stochastic response evaluation for the system is conducted in closed form by solving the associated Lyapunov covariance equation. The analytical results from stochastic analyses are juxtaposed with pertinent Monte Carlo simulations to demonstrate the accuracy of the proposed method. Validation results using measured data from the 2023 Turkey–Syria earthquake demonstrated that the recorded ground motions exhibit fractional features with an energy rate of decay in the spectral density that is captured more accurately by the proposed fractional model than the traditional integer Kanai–Tajimi model.

Ocean wave action is a key environmental load on marine structures, and its numerical simulation is a critical issue in the field of ocean engineering. In stochastic dynamic response analysis and time-variant reliability analysis, say, by the dimension-reduced probability density evolution equation (DR-PDEE), accurate representation of wave excitation through filtering methods is essential. To advance the probability density level analysis in ocean engineering, it is necessary to develop filtering methods for wave fields that align with fundamental physical principles. In this paper, a phase-shifting method for calculating the phase difference between wave elevations at different points on the sea surface based on the Fourier transform and its inverse is first introduced. The method enables the generation of wave elevations in long-crested waves from a single-point signal. To simulate short-crested wave fields, the wave direction is discretized into multiple intervals, and a filtering simulation method based on the linear superposition principle along with the Mitsuyasu-type directional spreading function is proposed using a designed digital filter system. The validity of the proposed method is verified by illustrating three numerical examples: the long-crested wave case, the short-crested wave case, and the case of wave force calculation on two columns in a wave field. The necessity of adopting wave fields for calculating wave loads on large offshore structures is demonstrated. The proposed method can be applied in the dynamic response analysis of various ocean engineering structures.

The discretized paths-based sequential integration method (SIM) is a quite versatile approach for solving various problems, including barrier problems, first passage problems, reflecting barrier problems and so on. This method builds upon the Chapman–Kolmogorov equation and is not applicable to non-Markovian problems, as in the case of fractional Brownian motion (FBM). In this paper, it is shown that the loss of the Markovian property can be overcome by utilizing the self-similarity of the FBM. In order to apply the discretized paths-based SIM, we have to solve a specific stochastic boundary value problem, also called stochastic “bridge” problem, which involves selecting only the trajectories of the FBM that ends at an assigned value, say x ̄ at t k , at the beginning of the time interval t k − t k + 1 . It is shown that, due to self-similarity, the stochastic “bridge” problem may be solved only once, regardless of the value x ̄ at t k . It is also shown that the trajectories of the stochastic ”bridge” problem exhibit self-similarity, which circumvents the loss of Markovian property in FBM, thus allowing the discretized paths-based SIM to be employed without invoking the classical Chapman–Kolmogorov equation. Further, an application involving the classical first passage problem is presented.

The research titled Analysis of fractional dynamical systems using recursive Bayesian estimation methods and response data was presented at the Engineering Mechanics Institute Conference and Probabilistic Mechanics & Reliability Conference (EMI/PMC 2024), University of Illinois Urbana-Champaign, Chicago, IL, May 28-31 2024, as part of the Minisymposium Computational methods for stochastic engineering dynamics.

This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system's displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.

Abstract In this paper a novel procedure is developed for evolutionary cross power spectra (ECPS) estimation of bivariate nonstationary stochastic processes. Specifically, the ECPS is determined by estimating the statistical moments of energy-like response quantities of lightly damped linear filters excited by nonstationary stochastic processes. In this context, a smoothing procedure is incorporated by using the Savitzky-Golay (S-G) moving average filter to obtain reliable ECPS based even from a limited number of available records. Further, a refinement of the approach is proposed relying on polynomial based functions of the system output. Several numerical examples, including nonstationary processes with known spectra, and historic accelerograms are used to assess the reliability and accuracy of the proposed procedure.

Abstract A potential future challenge in the wave energy sector will involve the design and construction of massive wave power farms. That is, collections of several (> 100) wave energy converters (WEC) operating in identical environmental conditions at a distance comparable with typical water wave lengths. In this context, the WECs are likely to be influenced by each another by radiation force effects that are associated with the radiated wave field propagated by WECs operating in the surrounding wave field. These effects are commonly captured by the Cummins’ equation, where the radiation force is expressed as a convolution integral depending on the past values of the WEC response. Due to this mathematical representation, the time domain computation of the wave farm response can become computationally daunting. This article proposes one approach for computing efficiently the wave farm response in the time domain. Specifically, it demonstrates that the values of the radiation force components can be determined at each time step from their previous values by approximating the retardation function matrix elements via the Prony method. A notable advantage of this approach with respect to the ones available in the open literature is that it does not require either the storage of past response values or additional differential equations. Instead, it uses simple algebraic expressions for updating at each time instant the radiation force values. Obviously, this feature can induce significant computational efficiency in analyzing an actual wave farm facility. The reliability and efficiency of the proposed algorithm are assessed vis-à-vis direct time domain comparisons and Monte Carlo data concerning a wave farm composed by an array of U-Oscillating Water Columns. Notably, the proposed methodology can be applied to any linear or nonlinear dynamics problem governed by differential equations involving memory effects.

Extreme waves pose one of the major threats to marine structures. Furthermore, their non-stationary nature makes proper stochastic analysis of their responses a challenging problem. To address this issue, this paper proposes a method based on a novel linear filter wave model incorporated into the dimension-reduced probability density evolution equation (DR-PDEE). The linear filter system is capable of simulating random background waves conforming with the Joint North Sea Wave Project (JONSWAP) spectrum of any arbitrary sea state by adjusting the parameters of filters directly related to the parameters of the JONSWAP spectrum without reidentification. In particular, by conducting the digital filtering, wave kinematics at different depths below the sea surface can be reproduced conveniently, and therefore only one filter is adequate for the depthwise wave kinematics field. Extreme ocean waves are treated as the superposition of background waves and extreme crests according to the constrained quasi-determinism method, with randomness from both parts. Incorporating the filter into the equation of motion of the offshore structure of interest leads to an augmented high-dimensional stochastic system with multiple random variables. The DR-PDEE then is employed to reduce the dimensions of the equation governing the evolution of probability density of responses of the original complex system to two. Solving the DR-PDEE using the path integral method yields the probability function of the response at each time step. A numerical example involving the response of a National Renewable Energy Laboratory (NREL) 5-MW spar-type floating offshore wind turbine (FOWT) subjected to extreme waves was studied to assess the reliability of the proposed method. The method provides an effective tool for the determination of the stochastic extreme response of offshore structures, and provides a foundation for further dynamic analyses.

If disposing of this product, please recycle the paper.To my family, Katerina and Thanos, for their "functional" and "integral" roles in enhancing my life.

Abstract In this paper, a technique is developed for determining the nonstationary response statistics of linear oscillators endowed with fractional derivative elements. Notably, fractional operators are particularly effective in modeling solid mechanics problems as they offer the option of influencing both the elasticity and the energy dissipation capacity of the system. In this paper, particular attention is devoted to the case of fractional derivatives of rational order that approximates reasonably well any real order model. The oscillators are subjected to stationary stochastic excitations, and the pertinent nonstationary response statistical moments are determined by first introducing a finite number of oscillator response related states; this is afforded by the rational number order of the fractional operator. Next, the technique involves proceeding to treating the problem in the Laplace transform domain. This leads to multiple convolution integrals determined by representing the transfer function of the oscillator in a partial fraction form by a pole-residue formulation. In this manner, the response evolutionary power spectral density of the fractional oscillator is derived in a closed form, while nonstationary second-order statistics can be obtained by mundane numerical integration in the frequency domain. Applications to oscillators comprising one or two fractional derivative elements are presented, considering the case of a white noise excitation and of a random process possessing the classical Kanai–Tajimi spectrum. Reliability of the developed technique is assessed by juxtaposing its analytical results with pertinent Monte Carlo simulation data.

This paper presents the application of nonlinear Bayesian filtering–based system identification (SI) methods when employed to estimate the parameters of stochastic fractional dynamic systems. The objective is to demonstrate the capabilities and limitations of time-domain stochastic filtering–based SI for systems endowed with fractional derivative elements when the estimation is performed under different operating conditions. The conditions include measured forcing inputs (input-output identification), stochastic/unmeasured forcing inputs (output-only identification), and different types of measurements and levels of measurement noise, in the context of both linear and hysteretic fractional oscillators. The accuracy and estimation error of three methods was studied, namely, the unscented Kalman filter, the ensemble Kalman filter, and the particle filter. Baseline results that can be applied to the modeling, identification, and control of fractional structural and mechanical systems are provided. It is shown that nonlinear Bayesian filtering methods have the capability to accurately estimate the response and parameters of fractional oscillators, and that the coefficient and order of fractional elements are observable/identifiable from output response measurements.

An efficient technique to analyze non-stationary nonlinear random vibrations of dynamic systems endowed with a fractional derivative term is presented. The technique itself represents an extension of the concept of statistical linearization to this kind of systems, and it is applicable for both analytic and hysteretic system nonlinearities. The technique first resorts to harmonic balancing in deriving response-amplitude dependent equivalent damping and stiffness. This enables representation of the fractional derivative term as a linear combination of this system response displacement and velocity with amplitude dependent coefficients. Then, the expected values of these parameters are considered in proceeding to formulate a statistical linearization solution scheme. In this context, the solution procedure is completed by integrating in time the covariance Lyapunov equation associated with the derived equivalent linear system. The reliability of the proposed technique is tested by a series of germane Monte Carlo studies. This juxtaposition is also used to elucidate salient features of this technique, by varying the order of the fractional derivative term, and of the degree of the nonlinearity in the system. It also points out the versatility of the technique in determining the non-stationary values of auto-correlation and cross-correlations response parameters involving even the fractional derivative term.

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Abstract In this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.