Stationary Gaussian Stochastic Process Research Articles

A nested non-intrusive stochastic isogeometric method for nonlinear thermal vibration of FGM plates under random loading

This paper introduces an adaptive nested non-invasive dynamic SIGA method based on RBFNN for the nonlinear thermal vibration analysis of FGM plates under random loading. This method is robust and accurate in high-dimensional input spaces regardless of the sample sequence, addresses the low convergence rate of QMCS, and solves the stochastic dynamic response probability density function under random loading. Firstly, the multidimensional input space for random loading is constructed based on several mutually independent random variables via the stationary Gaussian stochastic process simulation technique. The nonlinear FGM plate model subjected to random loading in the thermal environment is modeled using HSDT with von Kármán strain-displacement relation within the IGA framework. Subsequently, a nested non-invasive dynamic response surface analysis method, SIGA-RBFNN, is proposed based on the RBFNN technique. Analysis of the FGM plate response under random loading demonstrates that SIGA-RBFNN, compared to tensor-product-based SIGA methods, is less affected by sample sequence types. It effectively addresses high-dimensional stochasticity and offers significant advantages in robustness, accuracy, and efficiency. Finally, SIGA-RBFNN compensates for the low convergence speed of QMCS and successfully solves the FGM plate displacement response statistics and probability density function under random loading.

Probabilistic analysis of the steady state of weakly perturbed linear oscillators subject to a class of Gaussian inputs

This paper aims to probabilistically study a class of nonlinear oscillator subject to weak perturbations and driven by stationary zero-mean Gaussian stochastic processes. For the sake of generality in the analysis, we assume that the perturbed term is a polynomial of arbitrary degree in the spatial position, that contains, as a particular case, the important case of the Duffing equation. We then take advantage of the so-called stochastic equivalent linearization technique to construct an equivalent linear model so that its behavior consistently approximates, in the mean-square sense, that of the nonlinear oscillator. This approximation allows us to take extensive advantage of the probabilistic properties of the solution of the linear model and its first mean-square derivative to construct reliable approximations of the main statistical moments of the steady state. From this key information, we then apply the principle of maximum entropy to construct approximations of the probability density function of the steady state. We illustrate the superiority of the equivalent linearization technique over the perturbation method through some examples.

Statistics of the Stokes Parameters for the Signal Scattered by Layered Structures With an Arbitrary Number of Slightly Rough Interfaces

Horizontally stratified structures are commonly used to represent naturally occurring structures, such as soils. The electromagnetic signature of such a medium illuminated by a radar and the polarization state of the scattered wave are fully determined by the knowledge of the four Stokes parameters. In this paper, we determine the statistics of the four Stokes parameters for the signal scattered by layered structures with an arbitrary number of slightly rough interfaces. The rough interfaces are realizations of second-order stationary centered Gaussian stochastic processes and the layered structure is illuminated by an elliptically polarized monochromatic wave. The zenithal and azimuthal components of the far scattered electric field are derived from the first-order small perturbation method. The derivation leads to a multivariate Gaussian model for the underlying complex scattered amplitudes and we establish the closed-form expressions of the probability density function, the cumulative density function and the first- and second-order moments for the four Stokes parameters. For an observation direction outside the incidence plane, we establish the condition on the incidence wave parameters for which the zenithal and azimuthal components are uncorrelated. For an air / snow cover / frozen soil / unfrozen soil structure, we analyze the marginal probabilities and validate the theory by comparison with Monte-Carlo simulations. More generally, when the two complex components of the field scattered by the illuminated zone are Gaussian random variables, these statistics offer possibilities for in-depth investigating the polarization of scattering processes from random media.

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.

Simulation of ergodic multivariate stochastic processes: An enhanced spectral representation method

The spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian stochastic processes. In this study, an enhanced SRM for the simulation of ergodic multivariate stochastic processes is developed. Through shifting a frequency increment, the enhanced SRM offers a faster convergence rate of probabilistic characteristics in comparison with conventional SRM. The computational efficiency is enhanced as the Cholesky decomposition is only required at single-indexed frequencies. Additionally, a two-dimensional fast Fourier transform (FFT) algorithm is proposed to expedite simultaneously estimation of the double summation of cosine functions in both frequency and space dimensions. Accordingly, compared to the traditional FFT-based SRM, in which FFT is only used for the summation in the frequency dimension, the simulation efficiency is further enhanced significantly. The numerical example concerning the simulation of wind velocity field demonstrates that the proposed method offers a faster convergence rate and a higher level of efficiency.

Wind characteristics and flutter performance of a long-span suspension bridge located in a deep-cutting gorge

The wind characteristics as well as the flutter performance of a long-span suspension bridge located in a deep-cutting gorge terrain were investigated in the present study. To address the spatially variable or inhomogeneous wind fields (InWF) at the bridge site, a large terrain model surrounding the bridge site was installed in a large-scale wind tunnel. The mean wind speed, turbulence intensity, wind attack angle, wind power spectral density (PSD) and wind coherence all varying along the bridge girder were measured and analyzed in detail. Also, the stationarity and the cumulative distribution functions of the turbulent flow along the bridge girder were evaluated. Then, the flutter performance under four different InWF cases were comprehensively investigated. The results show that the turbulence components u as well as w along the bridge girder can be considered as stationary Gaussian stochastic processes. The wind PSDs significantly vary along the bridge girder, and the traditional wind PSDs obtained from the relatively flat terrains are not accurately applicable here. A two-parameter root coherence function model was developed to address the complex wind coherence over the deep-cutting gorge, and the wind coherence along the bridge girder was determined in three situations. When the wind fields along the bridge girder are assumed as homogeneous, the critical flutter velocity is overestimated by 10.1%. When inhomogeneous wind attack angles are considered, the contribution of the first symmetrical torsional mode to the critical flutter state become larger.

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

On test for checking hypothesis on expectation and covariance function of stochastic process

In this paper, we had constructed the goodness-of-fit tests incorporating several components, like expectation and covariance function, for identification of a non-centered stationary Gaussian stochastic process. For the construction of the corresponding estimators and investigation of their properties we had utilized the theory of Square Gaussian random variables.

The analysis of the wear process in building constructions with the use of the theory of stochastic functions.

One of the main causes of reduced reliability and durability of constructions, machinery, mechanisms, reinforced concrete structures, metal parts of the equipment is the wear. To develop the methods of the estimation of the reliability of the building constructions exposed to wear, the parametric failure models are used. The impact on the construction (the load coefficient) and the parameters of the load capacity are presented as the stationary or non-stationary Gaussian stochastic processes. By the proposed formulas for the reliability function, the example calculations are performed here below.

Construction of a criterion for testing hypothesis about covariance function of a stationary Gaussian stochastic process with unknown mean

ABSTRACTIn this paper, a new criterion is constructed for testing hypothesis about covariance function of Gaussian stationary stochastic process with an unknown mean. This criterion is based on the fact that we can estimate the deviation of covariance function from its estimator with a given accuracy and reliability in Lp metric.

A criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process

We consider a measurable stationary Gaussian stochastic process. A criterion for testing hypotheses about the covariance function of such a process using estimates for its norm in the space $L_p(\mathbb {T}),\,p\geq1$, is constructed.

Stochastic Control of Inertial Sea Wave Energy Converter

The ISWEC (inertial sea wave energy converter) is presented, its control problems are stated, and an optimal control strategy is introduced. As the aim of the device is energy conversion, the mean absorbed power by ISWEC is calculated for a plane 2D irregular sea state. The response of the WEC (wave energy converter) is driven by the sea-surface elevation, which is modeled by a stationary and homogeneous zero mean Gaussian stochastic process. System equations are linearized thus simplifying the numerical model of the device. The resulting response is obtained as the output of the coupled mechanic-hydrodynamic model of the device. A stochastic suboptimal controller, derived from optimal control theory, is defined and applied to ISWEC. Results of this approach have been compared with the ones obtained with a linear spring-damper controller, highlighting the capability to obtain a higher value of mean extracted power despite higher power peaks.

Numerical Approximation of Fractal Dimension of Gaussian Stochastic Processes

In this paper we propose a numerical method to estimate the fractal dimension of stationary Gaussian stochastic processes using the random Euler numerical scheme and based on an analytical formulation of the fractal dimension for filtered stochastic signals. The discretization of continuous time processes through this random scheme allows us to find, numerically, the expected value, variance and correlation functions at any point of time. This alternative method for estimating the fractal dimension is easy to implement and requires no sophisticated routines. We use simulated data sets for stationary processes of the type Random Ornstein Uhlenbeck to graphically illustrate the results and compare them with those obtained whit the box counting theorem.

Two Decades of the Atlantic Meridional Overturning Circulation: Anatomy, Variations, Extremes, Prediction, and Overcoming Its Limitations

Abstract The zonally integrated meridional volume transport in the North Atlantic [Atlantic meridional overturning circulation (AMOC)] is described in a 19-yr-long ocean-state estimate, one consistent with a diverse global dataset. Apart from a weak increasing trend at high northern latitudes, the AMOC appears statistically stable over the last 19 yr with fluctuations indistinguishable from those of a stationary Gaussian stochastic process. This characterization makes it possible to study (using highly developed tools) extreme values, predictability, and the statistical significance of apparent trends. Gaussian behavior is consistent with the central limit theorem for a process arising from numerous independent disturbances. In this case, generators include internal instabilities, changes in wind and buoyancy forcing fields, boundary waves, the Gulf Stream and deep western boundary current transports, the interior fraction in Sverdrup balance, and all similar phenomena arriving as summation effects from long distances and times. As a zonal integral through the sum of the large variety of physical processes in the three-dimensional ocean circulation, understanding of the AMOC, if it is of central climate importance, requires breaking it down into its unintegrated components over the entire basin.

Stochastic fracture mechanics using polynomial chaos

Crack propagation in metals has long been recognized as a stochastic process. As a consequence, crack propagation rates have been modeled as random variables or as random processes of the continuous. On the other hand, polynomial chaos is a known powerful tool to represent general second order random variables or processes. Hence, it is natural to use polynomial chaos to represent random crack propagation data: nevertheless, no such application has been found in the published literature. In the present article, the large replicate experimental results of Virkler et al. and Ghonem and Dore are used to illustrate how polynomial chaos can be used to obtain accurate representations of random crack propagation data. Hermite polynomials indexed in stationary Gaussian stochastic processes are used to represent the logarithm of crack propagation rates as a function of the logarithm of stress intensity factor ranges. As a result, crack propagation rates become log-normally distributed, as observed from experimental data. The Karhunen–Loève expansion is used to represent the Gaussian process in the polynomial chaos basis. The analytical polynomial chaos representations derived herein are shown to be very accurate, and can be employed in predicting the reliability of structural components subject to fatigue.

Adaptive inspection in coordinate metrology based on kriging models

Abstract The paper describes a model-based statistical methodology to design adaptive inspection plans for the geometric control of mechanical parts with Coordinate Measuring Machines (CMM). The inspection is adaptive because the design and measurement phases are not separate in time, as they usually are. Rather, they are carried out in a combined way: first designing the next measurement location, then measuring at that location and so on. This strategy is most informative as it allows for the exploitation of all of the currently available measurements. The next measurement point is selected by using predictions and prediction uncertainty of geometric deviations provided by non-parametric statistical models, known as kriging models. Based on stationary Gaussian stochastic processes, their merit is the ability to vary flexibly at each added point. The methodology is demonstrated in an illustrative case study, then its performance is compared to that of two statistical non adaptive plans, and two deterministic adaptive plans proposed in the literature. In each comparison kriging-based plans have proved to be superior in terms of the accuracy of the predicted geometric error and the inspection cost. The method is sufficiently general to enable technology transfer to different metrological sectors.

A Spectral Representation Model for Simulation of Multivariate Random Processes

A model is proposed to simulate multivariate weakly stationary Gaussian stochastic processes based on the spectral representation theorem. In this model, the amplitude, phase angle, and frequency involved in the harmonic function are random so that the generated samples are real stochastic processes. Three algorithms are then adopted to improve the simulation efficiency. A uniform cubic B-spline interpolation method is employed to fit the target factorized power spectral density function curves. A recursive algorithm for the Cholesky factorization is utilized to decompose the cross-power spectral density matrices. Some redundant cosine terms are cut off to decrease the computation quantity of superposition. Finally, an example involving simulation of turbulent wind velocity fluctuations is given to validate the capability and accuracy of the proposed model as well as the efficiency of the optimal algorithms.

Long-term correlation structure of wave loads using simulation

This paper proposes a new method for combining the lifetime wave-induced sectional forces and moments that are acting on the ship structure. The method is based on load simulation and can be used to determine the exceedance probabilities of any linear and nonlinear long-term load combination. It can also be used to determine the long-term correlation structure between these loads in the form of the long-term correlation coefficients. They are essential part of the load combination procedures in design and strength evaluations as well as in the fatigue and reliability analysis of ship structures. The simulation method treats the non-stationary wave elevations during the ship’s entire life (long-term) as a sequence of different stationary Gaussian stochastic processes. It uses the rejection sampling technique for the sea state generation, depending on the ship’s current position and the season. Ship’s operational profile is then determined conditional on the current sea state and the ship’s position along its route. The sampling technique significantly reduces the number of sea state–operational profile combinations required for achieving the convergence of the long-term statistical properties of the loads. This technique can even be used in combination with the existing long-term methods in order to reduce the number of required weightings of the short-term CDFs. The simulation method does, however, rely on the assumption that the ship is a linear system, but no assumptions are needed regarding the short-term CDF of the load peaks. The load time series are simulated from the load spectra in each sea state, taking into account the effects of loading condition, heading, speed, seasonality, voluntary as well as involuntary speed reduction in severe sea states and the short-crested nature of the ocean waves. During the simulation procedure, special care has been given to maintaining the correct phase relation between all the loads. Therefore, time series of various load combinations, including the nonlinear ones, can be obtained and their correlation structure examined. The simulation time can be significantly reduced (to the order of minutes rather than hours and days) by introducing the seasonal variations of the ocean waves into a single voyage simulation. The estimate of the long-term correlation coefficient, obtained by simulating only a single voyage with the correct representation of seasonality, approaches the true correlation coefficient in probability. This method can be applied to any ship and any route, or multiple routes as long as the percentage of the ship’s total lifetime spent in each of them is known. A study has been conducted to investigate the effects of ship type, route and the longitudinal position of the loads on the values of the correlation coefficients between six different sectional loads; vertical, horizontal and twisting moments, as well as shear, horizontal and axial forces. Three ocean-going ship types have been considered; bulk carrier, containership and tanker, all navigating on one of the three busy ship routes; North America–Europe, Asia–North America and Asia–Europe. Finally, the correlation coefficient estimates have been calculated for five different positions along the ship’s length to investigate the longitudinal variation of the correlation coefficient.

Noise modelling of improved detection scheme for displacement optic sensors

PurposeThe purpose of this paper is to propose modelling of the noise of an improved method for the measurement of small displacement and vibrations. It is based on a novel method for overcoming DC drift in RF subcarrier phase detection scheme.Design/methodology/approachThe method works in open loop and is characterized by low distortions, good signal‐to‐noise ratio and rather low cost.FindingsConsidering a stationary Gaussian stochastic process, the paper evaluated and modelled the power spectral density and the probability density against the phase error and the phase noise parameter.Practical implicationsThis offers an improvement of vibration, displacement and seismic sensors.Originality/valueBased on a novel method for overcoming DC drift in RF sub‐carrier phase detection scheme for fibre optic sensors, an improved method for displacement and vibration measurement is proposed. The presented method is characterized by rather low distortions in the modulation process and small distances measurement, low distortions and low‐cost electronic systems.

SIMULATION OF EXTERNAL PRESSURE FIELDS AND INTERNAL PRESSURES IN A LOW-RISE BUILDING MODEL IN TURBULENCE BOUNDARY LAYER

This paper describes a stochastic method for simulating external pressure fields and internal pressures in a low-rise building model in a turbulence boundary layer. The external wind pressure fields on the building model are simulated using a stochastic method that combines the advantages of both translation vector processes and a vector autoregressive (VAR) model for representing stationary Gaussian stochastic vector processes. The internal pressures in the building model are simulated using both the simulated external wind pressure fields and a numerical model for flow paths through the building envelope. Also, the internal pressures are measured in a wind tunnel experiment with a building model having leaky apertures. Results of this study demonstrate that the internal pressures in the low-rise building model can be simulated using both the external wind pressure fields, which are simulated by the stochastic method, and the numerical model for the flow paths through the envelope.