Alba Sofi

Approximate solution of the Fokker–Planck–Kolmogorov equation

Abstract The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.

A response surface approach for the static analysis of stochastic structures with geometrical nonlinearities

A response surface approach for the finite element analysis of uncertain structures undergoing large displacements is presented. This method is based on the use of ad hoc response surface functions built up by ratios of polynomials. As opposite to commonly used linear or quadratic polynomials, such functions are insensitive to the sampling point positions. Once the response surface form is defined, response statistics can be approximated by analytical relationships or statistical simulation taking full advantage of sensitivity analysis. Numerical investigations demonstrated that a remarkable accuracy is achieved in the evaluation of both statistical moments and probability density functions of the response.

Nonstationary response envelope probability densities of nonlinear oscillators

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigen-functions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

The mechanically based non-local elasticity: an overview of main results and future challenges

The mechanically based non-local elasticity has been used, recently, in wider and wider engineering applications involving small-size devices and/or materials with marked microstructures. The key feature of the model involves the presence of non-local effects as additional body forces acting on material masses and depending on their relative displacements. An overview of the main results of the theory is reported in this paper.

RESPONSE STATISTICS OF LINEAR STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS UNDER GAUSSIAN STOCHASTIC INPUT

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas. To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.

Dynamic Analysis of Prestressed Cables with Uncertain Pretension

This paper deals with finite element dynamic analysis of prestressed cables with uncertain pretension subjected to deterministic excitations. The theoretical model addressed for cable modeling is a two-dimensional finite-strain beam theory, which allows us to eliminate any restriction on the magnitude of displacements and rotations. The dynamic problem is formulated by referring the motion to the inertial frame, which leads to a simple uncoupled quadratic form for the kinetic energy. The effect of the externally applied stochastic pretension is approximately described by means of an uncertain ‘axial’ component of stress resultant, which is assumed constant along the cable in its dead load configuration. The so-called improved perturbation approach is employed to solve this stochastic problem, obtaining two coupled systems of nonlinear deterministic ordinary differential equations, governing the mean value and deviation of response. An efficient and accurate iterative procedure is proposed to obtain the solution of these equations. In order to investigate the influence of random pretension on structural response, few numerical applications are presented and results are discussed.

Stochastic Response of Structures With Uncertain-but-Bounded Parameters

The uncertainty plays an important role in structural engineering. It is largely recognized that the uncertainty may affect both external excitations and structural parameters. However, while the numerous available data permit to model with good accuracy the excitations as stochastic processes, unfortunately the data about the structural parameters are quite limited. It follows that the probabilistic approach cannot be realistically applied to represent structural uncertainties; indeed, it requires a wealth of data, often unavailable, to define the probability distribution density of the fluctuating structural parameters. Non probabilistic approaches can be alternatively used to treat these uncertainties. In this framework, the interval model seems today the most suitable analytical tool. The aim of this paper is to evaluate the range of the random response of linear structural systems, with slight variation of the uncertain-but-bounded parameters, subjected to stochastic Gaussian excitations by applying the so-called interval perturbation method.© 2009 ASME

Special Issue on Nonprobabilistic Treatments of Uncertainty: Recent Developments

This article is available in the ASME Digital Collection at http://dx.doi.org/10.1115/1.4031559.

Special Section on Nonprobabilistic Approaches for Handling Uncertainty in Engineering

This article is available in the ASME Digital Collection at http://dx.doi.org/10.1115/1.4030822.

Stochastic Sensitivity Analysis of Structural Systems With Interval Uncertainties

Interval sensitivity analysis of linear discretized structures with uncertain-but-bounded parameters subjected to stationary multi-correlated Gaussian stochastic processes is addressed. The proposed procedure relies on the use of the so-called Interval Rational Series Expansion (IRSE), recently proposed by the authors as an alternative explicit expression of the Neumann series expansion for the inverse of a matrix with a small rank-r modification and properly extended to handle also interval matrices. The IRSE allows to derive approximate explicit expressions of the interval sensitivities of the mean-value vector and Power Spectral Density (PSD) function matrix of the interval stationary stochastic response. The effectiveness of the proposed method is demonstrated through numerical results pertaining to a seismically excited three-storey frame structure with interval Young’s moduli of some columns.Copyright