A translation model for non-stationary, non-Gaussian random processes

A model for simulation of non-stationary, non-Gaussian processes based on non-linear translation of Gaussian random vectors is presented. This method is a generalization of traditional translation processes that includes the capability of simulating samples with spatially or temporally varying marginal probability density functions. A formal development of the properties of the resulting process includes joint probability density function, correlation distortion and lower and upper bounds that depend on the target marginal distributions. Examples indicate the possibility of exactly matching a wide range of marginal pdfs and second order moments through a simple interpolating algorithm. Furthermore, the application of the method in simulating statistically inhomogeneous random media is investigated, using the specific case of binary translation with stationary and non-stationary target correlations.     

Translation models revisited

Translation models have been defined as memoryless mappings of Gaussian elements which match exactly/approximately target marginal distributions/correlations. We extend this class of translation models to include memoryless mappings of non-Gaussian elements. It is shown that quantities of interest inferred from equivalent translation models, i.e., models which share the same marginal distributions and have similar second moments, can differ significantly. It is suggested to construct families of equivalent translation models and select members of these families which are optimal for given quantities of interest.     

Evolution of crystallographic orientations in crystals subject to random and deterministic deformation

The response of polycrystals to deterministic and random deformations is investigated with respect to the evolution of the crystallographic orientation and its probability density function, the ODF. The crystals analyzed are highly simplified, being two-dimensional and with only two active slip systems. A differential equation is derived which governs the evolution of the orientation under arbitrary applied inelastic deformation. Evolution of the ODF is analyzed by either solution of a Fokker-Planck equation or Monte Carlo simulation. Two main results are presented. Applied deformation tends to reduce heterogeneity of polycrystals with an initially uniform ODF. The presence of randomness in the applied deformation leads the ODF to evolve to forms which are not obtained under any form of deterministic deformation.     

Simulation of random vectors with given moments

We extend in this article the simulation method of scalar random variables with given moments, which was developed initially by Devroye, to n-dimensional random vectors. Using the conditional distribution approach, we show that the vector simulation problem can be reduced to simulate scalar random variables with given moments, which are solutions of given linear systems.     

Modeling of atmospheric temperature fluctuations by translations of oscillatory random processes with application to spacecraft atmospheric re-entry

The presence of random fluctuations of air temperature within the Earth’s atmosphere is a well-documented phenomenon. During the past seventy years there have been numerous experimental efforts to accurately measure air temperature as a function of altitude and, through careful data analysis, provide statistics describing these fluctuations and the associated fluctuations in temperature gradients. In addition, several researchers suggest the presence of atmospheric layers or “sheets” where the statistics describing fluctuations in air temperature can vary significantly from layer to layer. Herein, we propose a model to represent fluctuations of air temperature within a layered atmosphere. The model is a special type of inhomogeneous non-Gaussian differentiable random process and can be calibrated to available data on the marginal statistics and spectral content of the fluctuating temperature field, as well as the associated first derivative of the process representing fluctuations in temperature gradients. Properties of the proposed model are presented, and statistical realizations of the fluctuating temperature field and its gradient are computed and presented for illustration. The random vibration response of a spacecraft falling to Earth through these fluctuating conditions is then considered to demonstrate the usefulness of the proposed model.     

Non-linear response statistics of composite laminates with random material properties under random loading

Laminated composite plates find extensive use in many engineering applications. Some of these incorporate large deflections that may not be in the linear range. The external loading may be random in nature. The laminate material properties show an inherent dispersion around a mean value. In this paper the static response of laminated composite flat plates to transverse random loading has been studied. The material properties have been taken as random variables for accurate prediction of the system behaviour. The basic formulation of the problem has been developed based on the classical laminate theory and the Von-Karman non-linear strain–displacement relationship. A first order perturbation technique has been used to obtain the second order response statistics. Typical results have been presented for a plate with all edges simply supported. A comparison has been drawn with Monte Carlo simulation results for validation of the proposed approach. The effects of side-to-thickness ratio, aspect ratio and change in standard deviation of input random variables have been investigated for cross-ply symmetric and anti-symmetric laminates.     

Solution of linear dynamic systems with uncertain properties by stochastic reduced order models

A novel method, referred to as the stochastic reduced order model (SROM) method, is proposed for finding statistics of the state of linear dynamic systems with random properties subjected to random noise. The method is conceptually simple, accurate, computationally efficient, and non-intrusive in the sense that it uses existing solvers for deterministic differential equations to find state properties.Bounds are developed on the discrepancy between the exact and the SROM solutions under some assumptions on system properties. The bounds show that the SROM solutions converge to the exact solutions as the SROM representation of the vector of random system parameters is refined. Numerical examples are presented to illustrate the implementation of the SROM method and demonstrate its accuracy and efficiency.     

Optimal representation of multi-dimensional random fields with a moderate number of samples: Application to stochastic mechanics

A significant amount of problems and applications in stochastic mechanics and engineering involve multi-dimensional random functions. The probabilistic analysis of these problems is usually computationally very expensive if a brute-force Monte Carlo method is used. Thus, a technique for the optimal selection of a moderate number of samples effectively representing the entire space of sample realizations is of paramount importance. Functional Quantization is a novel technique that has been proven to provide optimal approximations of random functions using a predetermined number of representative samples. The methodology is very easy to implement and it has been shown to work effectively for stationary and non-stationary one-dimensional random functions. This paper discusses the application of the Functional Quantization approach to the domain of multi-dimensional random functions and the applicability is demonstrated for the case of a 2D non-Gaussian field and a two-dimensional panel with uncertain Young modulus under plane stress.     

Free vibration of composite cylindrical panels with random material properties

Virtually in all structural systems, and in particular composites, there are uncertainties in the system parameters because of practical bounds on the quality control. In the present work the effect of variations in the mechanical properties of laminated composite cylindrical panels on its natural frequency has been obtained by modeling these as random variables. The transverse shear and rotatory inertia effects have been included in the governing equations. A perturbation approach is presented to obtain the mean and variance of the random natural frequencies. The effects of thickness ratios, edge support conditions and standard deviation of material properties on response of shallow square panels have been investigated. Results have been obtained by employing the finite element method. The approach has been validated by comparison of results with other approaches.