Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: theory and computing

We study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preserved.

Stochastic Finite Element Method for Modelling Heat Transfer in the Building Envelope

Improving the energy efficiency of the buildings is one of the most effective and fastest ways of reduction of the carbon dioxide emission. However, in the assessment of the energy demand of the buildings, numerous factors are uncertain, i.e. layer thickness, material parameters, climatic conditions, etc. In the present study, mathematical model was developed for analyzing temperature distribution and heat flux in the wall with thermal conductivity of insulation as a random parameter. The results obtained employing a stochastic perturbation technique were compared against the results of the Monte Carlo simulation. Stochastic perturbation technique has been implemented using the tenth order Taylor series expansion. The direct differential method was used to determine the values of Taylor’s coefficient. The obtained results indicate good accordance of the stochastic perturbation technique with the Monte Carlo method. Afterwards, the expected value of the heat flux and its variance were studied for the reference year for a city in the Central Europe. Two cases of the external wall were investigated, in which the thermal insulation was localized either on the internal or the external side of the wall. Performed analyses serve as a good method for assessing the reliability of results obtained using standard, deterministic approach.

Stochastic Sensitivity Analysis of Cylindrical Shell

The paper deals with some chosen aspects of stochastic sensitivity structural analysis and its application in the engineering practice. The main aim of the study is to provide the generalized stochastic perturbation technique based on classical Taylor expansion with a single random variable. The study is illustrated by numerical results concerning an industrial thin shell structure modeled as a 3-D structure.

Symbolic Computing in Probabilistic and Stochastic Analysis

The main aim is to present recent developments in applications of symbolic computing in probabilistic and stochastic analysis, and this is done using the example of the well-known MAPLE system. The key theoretical methods discussed are (i) analytical derivations, (ii) the classical Monte-Carlo simulation approach, (iii) the stochastic perturbation technique, as well as (iv) some semi-analytical approaches. It is demonstrated in particular how to engage the basic symbolic tools implemented in any system to derive the basic equations for the stochastic perturbation technique and how to make an efficient implementation of the semi-analytical methods using an automatic differentiation and integration provided by the computer algebra program itself. The second important illustration is probabilistic extension of the finite element and finite difference methods coded in MAPLE, showing how to solve boundary value problems with random parameters in the environment of symbolic computing. The response function method belongs to the third group, where interference of classical deterministic software with the non-linear fitting numerical techniques available in various symbolic environments is displayed. We recover in this context the probabilistic structural response in engineering systems and show how to solve partial differential equations including Gaussian randomness in their coefficients.

Vibrations And Deformations Of Moderately Thick Plates In Stochastic Finite Element Method

The paper deals with some chosen aspects of stochastic dynamical analysis of moderately thick plates. The discretization of the governing equations is described by the finite element method. The main aim of the study is to provide the generalized stochastic perturbation technique based on classical Taylor expansion with a single random variable.

Reaction-Diffusion Problems with Stochastic Parameters Using the Generalized Stochastic Finite Difference Method

The main aim of this work is to demonstrate the new stochastic discrete computational methodology consisting of the generalized stochastic perturbation technique and of the classical Finite Difference Method for the regular grids to model reaction-diffusion problems with random time series. The generalized stochastic perturbation approach is based on the given order Taylor expansion of all random variables. A numerical algorithm is implemented here using the Direct Differentiation Method of the reaction-diffusion equation with respect to the height of a channel in 1D problem; further symbolic determination of the probabilistic moments and characteristics is completed by the computer algebra system MAPLE, v. 14. Computational illustration attached proves that it is possible to determine using this approach up to the fourth order probabilistic moments and coefficients as well as to consider time series with random coefficients for any dispersion of the input variables. Stochastic fluctuations of the input uncertainty source are defined here as the power time series with Gaussian random coefficients having given first two moments.