Equivalent linearization for systems driven by Lévy white noise

The equivalent linearization method is extended to the case of nonlinear systems driven by Lévy white noise. The classical objective function used for the determination of the equivalent linear system cannot be applied because the Lévy process generating the Lévy white noise has no variance and higher order moments. An alternative objective function based on the characteristic function of the state of the linear system is used for solution. Two simple examples are presented to illustrate the proposed extension of the equivalent linearization method. The examples show that the proposed equivalent linearization method provides approximations of quality similar to the classical equivalent linearization method.     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Simulation of a non-Gaussian stochastic process based on a combined distribution of the UHPM and the GBD

To simulate non-Gaussian stochastic processes based on the first four moments, various simulation methods are presented, in which the determination of the transformation model and the calculation of the correlation coefficients between non-Gaussian stochastic processes and Gaussian stochastic processes are the critical procedures in these simulation methods. However, some existing simulation methods are limited to specific ranges. Furthermore, their practical applications are affected negatively due to the expensive cost of determining the transformation model and the correlation coefficients between non-Gaussian and Gaussian stochastic processes. Therefore, an accurate and efficient simulation method of a non-Gaussian stochastic process with a broader range is proposed in this article. Since the simulation of non-Gaussian processes and the Nataf transformation of non-Gaussian variables have some similar characteristics, a new combined distribution is proposed based on the unified Hermite polynomial model (UHPM) and the generalized beta distribution (GBD). Then, the combined distribution is employed in the simulation of non-Gaussian stochastic processes, in which the transformation model is deduced by the combined distribution. The correlation coefficient transformation function (CCTF) between the Gaussian and non-Gaussian stochastic processes can be evaluated through the interpolation method. Furthermore, numerical examples are presented to show the accuracy and effectiveness of the proposed simulation method for non-Gaussian stochastic processes.     

Characteristic function equations for the state of dynamic systems with Gaussian, Poisson, and Lévy white noise

The Itô formula for semimartingales is applied to develop equations for the characteristic function of the state of linear and non-linear dynamic systems with Gaussian, Poisson, and Lévy white noise, viewed as the formal derivatives of Brownian, compound Poisson, and Lévy processes, respectively. These equations can be obtained if the drift and diffusion coefficient of a dynamic system are polynomials of the system state and the driving noise is Gaussian or Poisson. It was not possible to derive equations for the characteristic function for the state of systems driven by Lévy white noise. Numerical results are presented for dynamic systems with real-valued states driven by Gaussian, Poisson, and Lévy white noise processes.     

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide (i) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, (ii) simple solutions for linear random vibration problems with non-Gaussian input, and (iii) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.     

Stochastic response of linear and non-linear systems to α-stable Lévy white noises

The stochastic response of linear and non-linear systems to external α-stable Lévy white noises is investigated. In the literature, a differential equation in the characteristic function (CF) of the response has been recently derived for scalar systems only, within the theory of the so-called fractional Einstein–Smoluchowsky equations (FESEs). Herein, it is shown that the same equation may be built by rules of stochastic differential calculus, previously applied by one of the authors to systems driven by arbitrary delta-correlated processes. In this context, a straightforward formulation for multi-degree-of-freedom (MDOF) systems is also developed.Approximate CF solutions to the derived equation are sought for polynomial non-linearities, in stationary conditions. To this aim a wavelet representation is used, in conjunction with a weighted residual method. Numerical results prove in excellent agreement with exact solutions, when available, and digital simulation data.     

Signatures of active and passive optimized Lévy searching in jellyfish

Some of the strongest empirical support for Lévy search theory has come from telemetry data for the dive patterns of marine predators (sharks, bony fishes, sea turtles and penguins). The dive patterns of the unusually large jellyfish Rhizostoma octopus do, however, sit outside of current Lévy search theory which predicts that a single search strategy is optimal. When searching the water column, the movement patterns of these jellyfish change over time. Movement bouts can be approximated by a variety of Lévy and Brownian (exponential) walks. The adaptive value of this variation is not known. On some occasions movement pattern data are consistent with the jellyfish prospecting away from a preferred depth, not finding an improvement in conditions elsewhere and so returning to their original depth. This ‘bounce’ behaviour also sits outside of current Lévy walk search theory. Here, it is shown that the jellyfish movement patterns are consistent with their using optimized ‘fast simulated annealing’—a novel kind of Lévy walk search pattern—to locate the maximum prey concentration in the water column and/or to locate the strongest of many olfactory trails emanating from more distant prey. Fast simulated annealing is a powerful stochastic search algorithm for locating a global maximum that is hidden among many poorer local maxima in a large search space. This new finding shows that the notion of active optimized Lévy walk searching is not limited to the search for randomly and sparsely distributed resources, as previously thought, but can be extended to embrace other scenarios, including that of the jellyfish R. octopus. In the presence of convective currents, it could become energetically favourable to search the water column by riding the convective currents. Here, it is shown that these passive movements can be represented accurately by Lévy walks of the type occasionally seen in R. octopus. This result vividly illustrates that Lévy walks are not necessarily the result of selection pressures for advantageous searching behaviour but can instead arise freely and naturally from simple processes. It also shows that the family of Lévy walkers is vastly larger than previously thought and includes spores, pollens, seeds and minute wingless arthropods that on warm days disperse passively within the atmospheric boundary layer.     

Apparent power-law distributions in animal movements can arise from intraspecific interactions

Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.     

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Approximations are developed for the marginal and joint probability distributions for the extreme values, associated with a vector of non-Gaussian random processes. The component non-Gaussian processes are obtained as nonlinear transformations of a vector of stationary, mutually correlated, Gaussian random processes and are thus, mutually dependent. The multivariate counting process, associated with the number of level crossings by the component non-Gaussian processes, is modelled as a multivariate Poisson point process. An analytical formulation is developed for determining the parameters of the multivariate Poisson process. This, in turn, leads to the joint probability distribution of the extreme values of the non-Gaussian processes, over a given time duration. For problems not amenable for analytical solutions, an algorithm is developed to determine these parameters numerically. The proposed extreme value distributions have applications in time-variant reliability analysis of randomly vibrating structural systems. The method is illustrated through three numerical examples and their accuracy is examined with respect to estimates from full scale Monte Carlo simulations of vector non-Gaussian processes.     

Wave analysis by Slepian models

Ocean waves exhibit more or less a Gaussian distribution for the instantaneous water surface height, and there is a need to develop simple models for generation of the characteristic non-Gaussian statistics, namely the asymmetric distributions of water surface height and wave slope. We argue that a simple class of non-linear oscillators can reproduce some of the characteristic features of random water wave processes and linear or non-linear response to ocean waves. We describe the Slepian model for the Gaussian case, and explain the use of the regression approximation for level crossing distances and associated variables, such as wave period and amplitude. Finally we speculate about a generalization of the regression technique to the non-linear Markov process case.     

On the dynamic stochastic response of FE models

A numerical procedure to compute the mean and covariance matrix of the response of nonlinear structures modeled by large FE models is presented. Non-white, non-zero mean, non-stationary Gaussian distributed excitation is represented by the well known Karhunen–Loéve expansion, which allows to describe any type of non-white Gaussian excitation in contrast to filtered white noise which might not be easily adjusted to available statistical data. The solution procedure differs considerably from standard methodologies using a state space representation. In the proposed approach, step-by-step integration procedures developed for deterministic FE analysis are applied to compute the first two moments of the stochastic response.     

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.     

A generalized shaping filter method for higher order statistics

The shaping filter method has been a valuable tool for computing the stochastic response of linear dynamical systems to Gaussian stochastic inputs that are not delta-correlated. A straightforward extension of this method to the non-Gaussian case involves determining shaping filters that “match” higher order cumulant functions of the stochastic noise input. However it is easy to find simple examples of input signals for which this will not work because their cumulant functions cannot be realized as cumulant functions of the output of a linear system with delta-correlated input. This paper proposes an alternative shaping filter and it shows that it is applicable to an entire class of input signals for which the original shaping filter method fails.

A further extension of the shaping filter method is suggested, that involves the introduction of the concept of generalized moments that includes as particular cases both the moments and the cumulants.     

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.     

A simulation model for evaporation of defrosted water in household refrigeratorsSimulation de l''évaporation d''eau de dégivrage dans les réfrigérateurs domestiques

This paper presents a simple dynamic simulation model for predicting the evaporation rates of defrosted water from water trays in domestic refrigerators. The model was successfully tested for one domestic refrigerator. The model was used to investigate the effect of different variables (ambient temperature, relative humidity, air velocity, compressor heat, auxiliary condenser) by comparing eight different arrangements with each other on a consistent basis for effective evaporation of defrosted water. The model results indicate that the evaporation of water can be increased to the necessary rate by raising the water temperature and optimising the velocity of air over the water tray.

Résumé

Cette communication présente un modèle de simulation dynamique pour la prévision de la vitesse d'évaporation d'eau de dégivrage contenue dans les bacs de réfrigérateurs domestiques. On a utilisé ce modèle pour étudier les effets de diverses variables (température ambiante, humidité relative, vitesse d'air, chaleur dégagée par le compresseur et le condenseur auxiliaire) en comparant l'évaporation de l'eau de dégivrage dans huit dispositions différentes. Lés résultats indiquent qu'on peut accélerer la vitesse d'évaporation par l'augmentation de la température de l'eau et l'optimisation de la vitesse de l'air passant au dessus du bac à eau.     

Assessing Lévy walks as models of animal foraging

The hypothesis that the optimal search strategy is a Lévy walk (LW) or Lévy flight, originally suggested in 1995, has generated an explosion of interest and controversy. Long-standing empirical evidence supporting the LW hypothesis has been overturned, while new models and data are constantly being published. Statistical methods have been criticized and new methods put forward. In parallel with the empirical studies, theoretical search models have been developed. Some theories have been disproved while others remain. Here, we gather together the current state of the art on the role of LWs in optimal foraging theory. We examine the body of theory underpinning the subject. Then we present new results showing that deviations from the idealized one-dimensional search model greatly reduce or remove the advantage of LWs. The search strategy of an LW with exponent μ = 2 is therefore not as robust as is widely thought. We also review the available techniques, and their potential pitfalls, for analysing field data. It is becoming increasingly recognized that there is a wide range of mechanisms that can lead to the apparent observation of power-law patterns. The consequence of this is that the detection of such patterns in field data implies neither that the foragers in question are performing an LW, nor that they have evolved to do so. We conclude that LWs are neither a universal optimal search strategy, nor are they as widespread in nature as was once thought.