Rollio delle navi in presenza di onde modellate come processi gaussiani e poissoniani agenti simultaneamente.

Obiettivo del presente lavoro è l’estensione del metodo della path integral solution (PIS) per lo studio della dinamica del rollio delle navi in presenza di onde modellate come processi gaussiani e poissionani agenti simultaneamente. Si è proceduto dapprima a mostrare come la PIS consenta di valutare l’evoluzione temporale della funzione densità di probabilità (PDF) del processo di risposta, applicando il metodo ad equazioni differenziali stocastiche soggette a forzanti esterne gaussiane e poissoniane. Successivamente si è trattato il caso di un sistema non lineare soggetto ad entrambi i rumori gaussiano e poissoniano agenti contestualmente. Si è infine affrontato sia analiticamente che num…

Statistical Methods for Parameter Identification of Temperature Dependent Viscoelastic Models

Probabilistic analysis of non-local random media

Computational stochastic methods have been devoted over the last years to analysis and quantification of the mechanical response of engineering systems involving random media. Specifically analysis of random, heterogeneous media is getting more and more important with the emergence of new complex materials requiring reliable methods to provide accurate probabilistic response. Advanced materials, often used at nano or meso-levels possess strong non-local characters showing that long-range forces between non-adjacent volume elements play an important role in mechanical response. Moreover long and short-range molecular interactions may have random nature due to unpredictable fabrication proces…

Nonlinear SDE Excited by External Lévy White Noise Processes

A numerical method for approximating the statistics of the solution of nonlinear stochastic systems excited by Gaussian and non-Gaussian external white noises is proposed. The differential equation governing the evolution in time of the characteristic function is resolved by the convolution quadrature method. This approach is especially suited for those problems in which the nonlinear drift term is not of polynomial form. In such cases the equation governing the evolution in time of the characteristic function is not a partial differential equation. Statistics are found by introducing an integral operator of Wiener-Hopf type, called the transformation operator, and applying the Lubich's con…

α-stable distributions for better performance of ACO in detecting damage on not well spaced frequency systems

Abstract In this paper, the Ant Colony Optimization (ACO) algorithm is modified through α -stable Levy variables and applied to the identification of incipient damage in structural components. The main feature of the proposed optimization is an improved ability, which derives from the heavy tails of the stable random variable, to escape from local minima. This aspect is relevant since the objective function used for damage detection may have many local minima which render very challenging the search of the global minimum corresponding to the damage parameter. As the optimization is performed on the structural response and does not require the extraction of modal components, the method is pa…

Incipient damage identification through characteristics of the analytical signal response

The analytical signal is a complex representation of a time domain signal: the real part is the time domain signal itself, while the imaginary part is its Hilbert transform. It has been observed that damage, even at a very low level, yields clearly detectable variations of analytical signal quantities such as phase and instantaneous frequency. This observation can represent a step toward a quick and effective tool to recognize the presence of incipient damage where other frequency-based techniques fail. In this paper a damage identification procedure based on an adimensional functional of the square of the difference between the characteristics of the analytical theoretical and measured sig…

Path Integral Solution Handled by Fractional Calculus

Fractional Spectral Moments for Digital Simulation of Multivariate Wind Velocity Fields

In this paper, a method for the digital simulation of wind velocity fields by Fractional Spectral Moment function is proposed. It is shown that by constructing a digital filter whose coefficients are the fractional spectral moments, it is possible to simulate samples of the target process as superposition of Riesz fractional derivatives of a Gaussian white noise processes. The key of this simulation technique is the generalized Taylor expansion proposed by the authors. The method is extended to multivariate processes and practical issues on the implementation of the method are reported.

Fractional mechanical model for the dynamics of non-local continuum

In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both…

Elastic waves propagation in 1D fractional non-local continuum

Aim of this paper is the study of waves propagation in a fractional, non-local 1D elastic continuum. The non-local effects are modeled introducing long-range central body interactions applied to the centroids of the infinitesimal volume elements of the continuum. These non-local interactions are proportional to a proper attenuation function and to the relative displacements between non-adjacent elements. It is shown that, assuming a power-law attenuation function, the governing equation of the elastic waves in the unbounded domain, is ruled by a Marchaud-type fractional differential equation. Wave propagation in bounded domain instead involves only the integral part of the Marchaud fraction…

Composite laminates buckling optimization through Levy based Ant Colony Optimization

In this paper, the authors propose the use of the Levy probability distribution as leading mechanism for solutions differentiation in an efficient and bio-inspired optimization algorithm, ant colony optimization in continuous domains, ACOR. In the classical ACOR, new solutions are constructed starting from one solution, selected from an archive, where Gaussian distribution is used for parameter diversification. In the proposed approach, the Levy probability distributions are properly introduced in the solution construction step, in order to couple the ACOR algorithm with the exploration properties of the Levy distribution. The proposed approach has been tested on mathematical test functions…

Stochastic dynamics of nonlinear systems with a fractional power-law nonlinear term: The fractional calculus approach

Fractional power-law nonlinear drift arises in many applications of engineering interest, as in structures with nonlinear fluid viscous–elastic dampers. The probabilistic characterization of such structures under external Gaussian white noise excitation is still an open problem. This paper addresses the solution of such a nonlinear system providing the equation governing the evolution of the characteristic function, which involves the Riesz fractional operator. An efficient numerical procedure to handle the problem is also proposed.

On the use of fractional calculus for the probabilistic characterization of random variables

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are o…

On the derivation of the Fokker-Plank equation by using of Fractional calculus

In this paper, fractional calculus has been used to find the spectral counterpart of the Fokker- Planck equations for non-linear systems driven by Lévy white noise processes. In particular it is shown that one can obtain the equation ruling the characteristic function of the response to a non-linear system, without using the Itô formula. Indeed, it is possible to reproduce the well-known results, already known in literature, by means of the characteristic function representation in terms of complex moments, recently proposed by the first two authors. The case of a-stable Lévy driven stochastic differential equation is also treated introducing an associated process constructed from the stabl…

Stochastic ship roll motion via path integral method

ABSTRACTThe response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple …

A modified Ant Colony damage identification algorithm for not well spaced frequency systems

Damage identification is of primary concern in many fields of civil engineering. Usually the damage is detected from the variation of structural response induced. When the damage level is very low, incipient damage, this variation is hardly seen. In the present work is studied the case of not well spaced frequency systems. Identification problem is formulated as a minimum problem of a functional expressed in term of damage parameters. The minimum problem is solved by heuristic algorithm, ACORL.

Sistemi non lineari eccitati da processi di Lévy risolti mediante momenti frazionari

Statistics of nonlinear stochastic dynamical systems under Lévy noises by a convolution quadrature approach

This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by L\'evy white noises. The proposed numerical procedure relies on the introduction of an integral transform of Wiener-Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: 1) Statistics of systems with several different drift terms can be handled in an efficie…

A microplane model for plane-stress masonry structures

Publisher Summary For a refined nonlinear finite element analysis of masonry structures, an accurate constitutive model that is able to reproduce the desired phenomenological material features is required. Constitutive models for quasi-brittle materials, as plain concrete, have been proposed in the chapter, which allow to reproduce the very complex response in the two- or three-dimensional state of stress. Usually, the constitutive relations proposed are based on some appropriate extensions of elastic-plastic continuum models and more recently on continuum damage models. It has been observed that for these tensorial-based constitutive relations to be effective often require a large number o…

FRACTIONAL MOMENTS AND PATH INTEGRAL SOLUTION FOR NON LINEAR SYSTEMS DRIVEN BY NORMAL WHITE NOISE

A new representation of power spectral density and correlation function by means of fractional spectral moments

In this paper, a new perspective for the representation of both the power spectral density and the correlation function by a unique class of function is introduced. We define the moments of order gamma (gamma being a complex number) of the one sided power spectral density and we call them Fractional Spectral Moments (FSM). These complex quantities remain finite also in the case in which the ordinary spectral moments diverge, and are able to represent the whole Power Spectral Density and the corresponding correlation function.

Path integral solution by fractional calculus

In this paper, the Path Integral solution is developed in terms of complex moments. The method is applied to nonlinear systems excited by normal white noise. Crucial point of the proposed procedure is the representation of the probability density of a random variable in terms of complex moments, recently proposed by the first two authors. Advantage of this procedure is that complex moments do not exhibit hierarchy. Extension of the proposed method to the study of multi degree of freedom systems is also discussed.

Probabilistic Analysis of Non-Local 1-D Continuum under Random Load

Ship Roll Motion under Stochastic Agencies Using Path Integral Method

The response of ship roll oscillation under random ice impulsive loads modeled by Poisson arrival process is very important in studying the safety of ships navigation in cold regions. Under both external and parametric random excitations the evolution of the probability density function of roll motion is evaluated using the path integral (PI) approach. The PI method relies on the Chapman-Kolmogorov equation, which governs the response transition probability density functions at two close intervals of time. Once the response probability density function at an early close time is specified, its value at later close time can be evaluated. The PI method is first demonstrated via simple dynamica…

A novel exact representation of stationary colored Gaussian processes (fractional differential approach)

A novel representation of functions, called generalized Taylor form, is applied to the filtering of white noise processes. It is shown that every Gaussian colored noise can be expressed as the output of a set of linear fractional stochastic differential equations whose solution is a weighted sum of fractional Brownian motions. The exact form of the weighting coefficients is given and it is shown that it is related to the fractional moments of the target spectral density of the colored noise.

Fractional calculus for the solution of non-linear stochastic oscillators with viscous dampers devices

Fluid viscoelastic dampers are of great interest in different fields of engineering. Examples of their applications can be found in seismic mitigation design of structures or in vibration absorption in airplane suspension. Such devices introduce a non-linear dissipative term in the equation of motion and therefore, the solution of even a single degree of freedom system excited by a white noise process, becomes prohibitive. The solution is usually obtained by approximated methods, like the stochastic linearization technique. In this paper it is shown that, by means of fractional operators, it is possible to find the solution of oscillators provided with fluid viscoelastic devices, approachin…

A representation of wind velocity by means of fractional spectral moments

This paper deals with the definition of a new function that is a link between Power Spectral Density (PSD) and correlation function, called the Fractional Spectral Moments function. This is defined as the moment of complex order g of the one-sided PSD. It is shown that by means of this complex function both the correlation function and PSD can be represented with great accuracy.

Damage identification by Lévy ant colony optimization

This paper deals with the identification of incipient damage in structural elements by non-destructive test based on experimentally measured structural dynamical response. By applycation of the Hilbert transform to the recorded signal the so-called phase of the analytical signal is recovered and a proper functional is constructed in such a way that its global minimum gives a measure of the damage level, meant as stiffness reduction. Minimization is achieved by applying a modified Ant Colony Optimization (ACO) for continuous variables, inspired by the ants’ forageing behavior. The modification consists in the application of a new perturbation operator, based on alpha stable Lévy distribution…

Fractional Derivatives in Interval Analysis

In this paper, interval fractional derivatives are presented. We consider uncertainty in both the order and the argument of the fractional operator. The approach proposed takes advantage of the property of Fourier and Laplace transforms with respect to the translation operator, in order to first define integral transform of interval functions. Subsequently, the main interval fractional integrals and derivatives, such as the Riemann–Liouville, Caputo, and Riesz, are defined based on their properties with respect to integral transforms. Moreover, uncertain-but-bounded linear fractional dynamical systems, relevant in modeling fractional viscoelasticity, excited by zero-mean stationary Gaussian…

Fractional moments of non-linear systems under Lévy white noise processes

Representation of Stationary Multivariate Gaussian Processes Fractional Differential Approach

In this paper, the fractional spectral moments method (H-FSM) is used to generate stationary Gaussian multivariate processes with assigned power spectral density matrix. To this aim, firstly the N-variate process is expressed as sum of N fully coherent normal random vectors, and then, the representation in terms of HFSM is used.

Non Linear Systems Driven by White Noise Processes and Handled by the Characteristic Function Equations

Lévy Flights for Ant Colony Optimization in Continuous Domains

In this paper, the authors propose the use of the Levy probability distribution as leading mechanism for solutions differentiation in an efficient and bio-inspired optimization algorithm, ant colony optimization in continuous domains, ACOR. In the classical ACOR, new solutions are constructed starting from one solution, selected from an archive, where Gaussian distribution is used for parameter diversification. In the proposed approach, the Levy probability distributions are properly introduced in the solution construction step, in order to couple the ACOR algorithm with the exploration properties of the Levy distribution.

Fractional calculus approach to the statistical characterization of random variables and vectors

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. A…

Fractional calculus approach for the representation of power spectral densities and correlation functions in wind engineering

The paper deals with the digital simulation of wind velocity samples by Fractional Spectral Moment function. It is shown that such a function represents a third useful way to characterize a stationary Gaussian stochastic process, alongside the power spectral density and the correlation function. The method is applied to wind velocity fields whose power spectra is given by the Kaimal’s, the Davenport’s and the Solari’s representation. It is shown that by constructing a digital filter whose coefficients are the fractional spectral moments, it is possible to simulate samples of the target process as superposition of Riesz fractional derivatives of Gaussian white noise processes.

Moment stability of parametrically perturbed systems via path integral solution