A Monte Carlo simulation model for stationary non-Gaussian processes

A class of stationary non-Gaussian processes, referred to as the class of mixtures of translation processes, is defined by their finite dimensional distributions consisting of mixtures of finite dimensional distributions of translation processes. The class of mixtures of translation processes includes translation processes and is useful for both Monte Carlo simulation and analytical studies. As for translation processes, the mixture of translation processes can have a wide range of marginal distributions and correlation functions. Moreover, these processes can match a broader range of second order correlation functions than translation processes. The paper also develops an algorithm for generating samples of any non-Gaussian process in the class of mixtures of translation processes. The algorithm is based on the sampling representation theorem for stochastic processes and properties of the conditional distributions. Examples are presented to illustrate the proposed Monte Carlo algorithm and compare features of translation processes and mixture of translation processes.     

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.     

An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics

A generalization for non-Gaussian random variables of the well-known Kazakov relationship is reported in this work. If applied to the stochastic linearization of non-linear systems under non-Gaussian excitations, this relationship allows us to define the significance of the linearized stiffness coefficient. It is the sum of that one known in the literature (the mean of the tangent stiffness) and of terms taking into account the non-Gaussianity of the response. Moreover, the relationship here given is used for finding alternative formulae between the moments and the quasi-moments. Lastly, it is used in the framework of the moment equation approach, coupled with a quasi-moment neglect closure, for solving non-linear systems under Gaussian or non-Gaussian forces. In this way an iterative procedure based on the solution of a linear differential equation system, in which the values of the response mean and variance are those of the precedent iteration, is originated. It reveals a good level of accuracy and a fast convergence.     

Numerical and experimental studies of linear systems subjected to non-Gaussian random excitations

This paper presents a numerical simulation scheme for generating symmetric non-Gaussian random processes governed by prescribed kurtosis and spectral density. The generated process is represented as a continuous stationary random signal with occasional spikes superimposed on a Gaussian random background. The generated time history data records are used to simulate random excitations acting on linear single-degree-of-freedom systems. The results of the numerical simulation are compared with those measured experimentally. For a wide-band random excitation with kurtosis close to 3, the response kurtosis is found to be very sensitive to small changes in the excitation kurtosis. This is manifested by the appearance of significant spikes in the time history records when the excitation records do not display any significant spikes. The influence of the system damping is also examined for narrow-band and wide-band random excitations, and some differences are reported in the results.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

Response analysis of nonlinear multi-degree-of-freedom systems with uncertain properties to non-Gaussian random excitations

The investigation reported in this paper is concerned with the development of an approach for response analysis of multi-degree-of-freedom (mdof) nonlinear systems with uncertain properties of large variations and under non-Gaussian nonstationary random excitations. The developed approach makes use of the stochastic central difference (SCD) method, time co-ordinate transformation (TCT), and adaptive time schemes (ATS). It is applicable to geometrically complicated systems idealized by the finite element method (FEM). For demonstration of its use and availability of results for direct comparison, a two-degree-of-freedom (tdof) nonlinear asymmetric system with uncertain natural frequencies and under Gaussian and non-Gaussian nonstationary random excitations is considered. Computed results obtained for the system with and without uncertain natural frequencies, and under Gaussian and non-Gaussian nonstationary random excitations are presented. It is concluded that the approach is relatively simple, accurate and efficient to apply.     

Efficient stationary multivariate non-Gaussian simulation based on a Hermite PDF model

An efficient stationary multivariate non-Gaussian simulation method is developed using spectral representation and third order Hermite polynomial translation. An approximate closed form relationship is employed to identify the Hermite translation parameters based on target skewness and kurtosis. This preserves a high degree of accuracy over the entire admissible range of the Hermite translation, and eliminates the need for iterative solution of the translation parameters. The Hermite PDF model is suitable for a wide range of strongly non-Gaussian stochastic process. In addition, an explicit bidirectional relationship between the target non-Gaussian and Gaussian correlation is developed to eliminate the need for iteration or numerical integration to identify the underlying Gaussian correlation. Examples apply the simulation method to both theoretical targets and experimental wind pressure data.     

New method for efficient Monte Carlo–Neumann solution of linear stochastic systems

The Neumann series is a well-known technique to aid the solution of uncertainty propagation problems. However, convergence of the Neumann series can be very slow, often turning its use highly inefficient. In this article, a λ convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann solutions of linear stochastic systems using first order Neumann expansions. The λ convergence parameter is found as solution to a distance minimization problem, for an approximation of the inverse of the system matrix using the Neumann series. The method presented herein is called Monte Carlo–Neumann with λ convergence, or simply MC–N λ method. The accuracy and efficiency of the MC–N λ method is demonstrated in application to stochastic beam bending problems.     

A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process

Some widely used methodologies for simulation of non-Gaussian processes rely on translation process theory which imposes certain compatibility conditions between the non-Gaussian power spectral density function (PSDF) and the non-Gaussian probability density function (PDF) of the process. In many practical applications, the non-Gaussian PSDF and PDF are assigned arbitrarily; therefore, in general they can be incompatible. Several techniques to approximate such incompatible non-Gaussian PSDF/PDF pairs with a compatible pair have been proposed that involve either some iterative scheme on simulated sample functions or some general optimization approach. Although some of these techniques produce satisfactory results, they can be time consuming because of their nature. In this paper, a new iterative methodology is developed that estimates a non-Gaussian PSDF that: (a) is compatible with the prescribed non-Gaussian PDF, and (b) closely approximates the prescribed incompatible non-Gaussian PSDF. The corresponding underlying Gaussian PSDF is also determined. The basic idea is to iteratively upgrade the underlying Gaussian PSDF using the directly computed (through translation process theory) non-Gaussian PSDF at each iteration, rather than through expensive ensemble averaging of PSDFs computed from generated non-Gaussian sample functions. The proposed iterative scheme possesses two major advantages: it is conceptually very simple and it converges extremely fast with minimal computational effort. Once the underlying Gaussian PSDF is determined, generation of non-Gaussian sample functions is straightforward without any need for iterations. Numerical examples are provided demonstrating the capabilities of the methodology.     

The Green’s functions of a vertically inhomogeneous soil with a random dynamic shear modulus

This paper deals with the study of the Green’s functions of a layered soil with random characteristics. The dynamic shear modulus of the soil is modelled as a non-Gaussian random process that varies in the vertical direction and is characterized by a marginal probability density function and a correlation function. The stochastic finite element method is applied to a hybrid thin layer — direct stiffness formulation in order to obtain the stochastic system equations, which are solved by means of a Monte Carlo simulation. The influence of the variations of the dynamic shear modulus on the Green’s functions is illustrated for different excitation frequencies and receiver positions.     

Simulation of non-Gaussian field applied to wind pressure fluctuations

A simulation algorithm to generate non-Gaussian wind pressure fields is proposed. This algorithm uses the correlation–distortion method based on translation vector processes. Conditions on the matrix of cross-covariance functions are given to assure the applicability of the model. The proposed method does not require iterative procedures and it is well suited when experimental data are available. In particular it requires cross-covariance functions and marginal distribution that can be directly estimated from data. To illustrate the procedure, the model is calibrated on experimental results obtained from wind tunnel tests on a tall building. The efficiency of the proposed methodology for reproducing the non-Gaussian nature of pressure fluctuations on separated flow regions is demonstrated.     

Response spectral density of linear systems with external and parametric non-Gaussian, delta-correlated excitation

An exact, closed form, analytical expressions for a response spectral density of a certain type of systems, subjected to non-Gaussian, stationary, delta-correlated noise are derived. A new, extended mean square stability conditions are derived for such systems.     

Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms

Mathematical justifications are given for a Monte Carlo simulation technique based on memoryless transformations of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.     

Elastic microcracked bodies with random properties

The stochastic properties of a ‘dense’ distribution of microcracks in an elastic body are analyzed by using a multifield continuum model describing the influence of the microcracks on the gross mechanical behavior of the body. Numerical examples are presented: the strain localization phenomena that have been already found in deterministic bodies are confirmed, and similar patterns are shown to exist also for the stochastic moments of the displacements. In particular, the patterns in the portraits of skewness and kurtosis become stronger when the correlation of the distance between neighboring microcracks increases. Such a distance is considered as a uni-variate non-Gaussian random field. Strain localization is an indicator toward the irreversible growth and coalescence of microcracks.