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.     

Uncertainty of the fatigue damage arising from a stochastic process with multiple frequency modes

Predicting the variance of the fatigue damage due to a stochastic load process is a difficult classical problem that dates back to the 1960s. For many years, the available analytical methods for tackling this problem have been limited to the linear oscillator response under Gaussian white noise excitation. In a recent prior work, the author developed an improved method for calculating the damage variance for a general narrowband Gaussian process. From a fatigue uncertainty perspective, a narrowband process is particularly crucial as the amplitude correlation magnifies the variance. This paper extends the analysis to a multimodal process comprising two or more narrowband components. The proposed method is tractable, involving a single summation for an arbitrary spectral density. Moreover, closed form expressions are available for two special cases, i.e. the components are all linear oscillator responses or the spectral density of each is rectangular. The equations also yield insight on the multilayered correlation mechanisms produced by different narrowband components. The accuracy of the method is verified by rainflow counting of simulated time history stresses.     

Crossing rate statistics of quadratic transformations of Gaussian processes

The main focus of this paper is the development of a numerical procedure for calculating the average crossing rates of a stochastic process that can be expressed as a sum of a linear and a nonlinear, quadratic transformation of a Gaussian process. Such a representation applies for instance to the motion response of a linear structure subjected to wind loading, when the loading model is proportional to the square of a Gaussian wind velocity process. It is also the standard model for expressing the total wave forces or horizontal excursion responses of a moored floating offshore platform in a random sea way. Knowledge of the crossing rate is a key to many important quantities related to response statistics and reliability applications. It is demonstrated how the proposed numerical procedure can be used for calculating the average crossing rate of the type of response processes considered.     

First excursion probabilities for linear systems by very efficient importance sampling

An analytical study of the failure region of the first excursion reliability problem for linear dynamical systems subjected to Gaussian white noise excitation is carried out with a view to constructing a suitable importance sampling density for computing the first excursion failure probability. Central to the study are ‘elementary failure regions’, which are defined as the failure region in the load space corresponding to the failure of a particular output response at a particular instant. Each elementary failure region is completely characterized by its design point, which can be computed readily using impulse response functions of the system. It is noted that the complexity of the first excursion problem stems from the structure of the union of the elementary failure regions. One important consequence of this union structure is that, in addition to the global design point, a large number of neighboring design points are important in accounting for the failure probability. Using information from the analytical study, an importance sampling density is proposed. Numerical examples are presented, which demonstrate that the efficiency of using the proposed importance sampling density to calculate system reliability is remarkable.     

An algorithm for accurate evaluation of the fatigue damage due to multimodal and broadband processes

In a recent work, the author developed an accurate method for calculating the fatigue damage in a bimodal Gaussian process. The novelty of this method lies in its ability to incorporate two critical effects, which have been unrecognized in prior studies. This paper extends the bimodal technique to multiple modes. Through case studies of multimodal processes, the algorithm is found to produce highly precise damage estimates (∼1% error) when benchmarked against simulations. This is true even when the component frequencies are very close. Further, the multimodal method is used to analyze general broadband processes, by partitioning a given spectrum. Since the accuracy is relatively insensitive to frequency spacing, the errors emanate primarily from the narrowband assumption, and this can be controlled by employing more segments. It is shown that with a sufficient level of discretization, the predicted damage for a variety of spectral shapes is negligibly conservative, generally within 2%.     

A method for accurate estimation of the fatigue damage induced by bimodal processes

This paper presents a method for calculating the fatigue damage from a stochastic bimodal process, in which the high frequency (HF) and low frequency (LF) components are narrowband Gaussian processes. Rainflow cycle counting identifies the following: small but numerous cycles, and large but fewer ones. In existing methods, the small-cycle amplitudes are assumed to be identical to that of the HF cycles, whereas the large-cycle amplitudes are approximated as the sum of the HF and LF amplitudes. The novelty of the present approach lies in the recognition and incorporation of two effects, which concern the reduction of the small-cycle amplitudes caused by the LF process, and the offset between the HF and LF peaks. Parametric studies are conducted, investigating a wide range of parameters. Using time domain simulation as a benchmark, the present method is found to provide a vast improvement over existing methods, with a root-mean-square error of 1%.     

Topological activity in Bragg elliptical twisted fibers

We have theoretically shown that Bragg twisted elliptical fibers manifest, in certain spectral regions, the property of topological activity--the ability to change in the reflected field the topological charge of incoming optical vortices and fundamental modes by two units. This property could be used for narrowband generation of optical vortices from Gaussian beams and for changing the topological charge of incoming optical vortices.     

A stochastic model for propagation through tissue

Attenuation of ultrasonic waves is often assumed linear with respect to frequency in biological applications whereas it is considered quadratic when the propagation occurs in the atmosphere or the water. In the latter case, other studies show that a Gaussian propagation duration can explain this attenuation behavior and provide a model for the energy loss in the stationary limit. The present paper defines an equivalent random propagation duration with Cauchy distribution, which is appropriate for the propagation of ultrasound through tissue. The model adds an unobserved noise that represents the signal deterioration. In addition, the model agrees with the mode downshift in the case of a narrowband signal.     

Counting level crossings by a stochastic process

The number of times, N, that a stochastic process has up-crossings of a fixed level within a fixed time interval, T, is investigated. Existing integral formulas for the moments of N for a stationary Gaussian process are shown also to apply to processes that are neither stationary nor Gaussian, and explicit formulas are given for approximating the probability distribution of N from the moment formulas. Particular attention is given to simplified results for the limiting situations of very small and very large values of T, and to the behavior of variance, skewness, and kurtosis of N. For small T, the number N approaches the well-known Poisson distribution, but the results for large T are significantly different. For many stationary processes it is shown that the variance of N tends to grow linearly with T when T is very large, but the large-T growth rate is sometimes much smaller than that of the small-T Poisson process. More detailed results and some numerical examples are presented for the special case of a stationary Gaussian process crossing its own mean value.     

Changes in the echoes from ultrasonic contrast agents with imaging parameters

Current harmonic imaging scanners transmit a narrowband signal that limits spatial resolution in order to differentiate the echoes from tissue from the echoes from microbubbles. Because spatial resolution is particularly important in applications, including mapping vessel density in tumors, we explore the use of wideband signals in contrast imaging. It is first demonstrated that microspheres can be destroyed using one or two pulses of ultrasound. Thus, temporal signal processing strategies that use the change in the echo over time can be used to differentiate echoes from bubbles and echoes from tissue. Echo parameters, including intensity and spectral shape for narrowband and wideband transmission, are then evaluated. Through these experiments, the echo intensity received from bubbles after wideband transmission is shown to be at least as large as that for narrowband transmission, and can be larger. In each case, the echo intensity increases in a nonlinear fashion in comparison with the transmitted signal intensity. Although the echo intensity at harmonic multiples of the transmitted wave center frequency can be larger for narrowband insonation, echoes received after wideband insonation demonstrate a broadband spectrum with significant amplitude over a very wide range of frequencies.     

Three-dimensional analysis of mode discrimination in vertical-cavity surface-emitting lasers

Optical mode discrimination in vertical-cavity surface-emitting lasers that contain distributed Bragg reflectors (DBRs) and a spatially limited gain medium is analyzed numerically. It is assumed that the output field is linearly polarized owing to gain selectivity. The analysis employs a three-dimensional model and an angular spectrum of plane-wave decomposition with the proper polarizations. Two types of round aperture are considered, namely, a Gaussian aperture and a ring-peak aperture that represents gain saturation. Coupled with the DBRs, the former aperture yields nearly Laguerre-Gaussian modes, whereas the latter aperture significantly distorts the mode shapes. In both cases, narrowband DBRs provide the best mode discrimination.     

Efficient direct position determination of orthogonal frequency division multiplexing signals

The authors present an efficient method for positioning of orthogonal frequency division multiplexing (OFDM) signals. The method, called direct position determination (DPD), enables one to determine the emitter position in a single search operation. In formerly published articles on DPD, the signal was assumed to be a narrowband Gaussian source, which was subjected to a rather simple flat-fading channel model. Here, the authors extend the DPD algorithm to handle the OFDM signals that propagate through a frequency-selective channel, and derive the least squares estimator (LSE) for the problem. In addition, the authors present a method for a fine 2D position estimation, which enables one to reduce the DPD computational complexity without compromising on its estimation accuracy.     

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.     

The Palaeocene-Eocene carbon isotope excursion: constraints from individual shell planktonic foraminifer records

The Palaeocene-Eocene thermal maximum (PETM) is characterized by a global negative carbon isotope excursion (CIE) and widespread dissolution of seafloor carbonate sediments. The latter feature supports the hypothesis that the PETM and CIE were caused by the rapid release of a large mass (greater than 2000Gt C) of 12C-enriched carbon. The source of this carbon, however, remains a mystery. Possible sources include volcanically driven thermal combustion of organic-rich sediment, dissociation of seafloor methane hydrates and desiccation and oxidation of soil/sediment organics. A key constraint on the source(s) is the rate at which the carbon was released. Fast rates would be consistent with a catastrophic event, e.g. massive methane hydrate dissociation, whereas slower rates might implicate other processes. The PETM carbon flux is currently constrained by high-resolution marine and terrestrial records of the CIE. In pelagic bulk carbonate records, the onset of the CIE is often expressed as a single- or multiple-step excursion extending over 10(4) years. Individual planktonic shell records, in contrast, always show a single-step CIE, with either pre-excursion or excursion isotope values, but no transition values. Benthic foraminifera records, which are less complete owing to extinction and diminutive assemblages, show a delayed excursion. Here, we compile and evaluate the individual planktonic shell isotope data from several localities. We find that the most expanded records consistently show a bimodal isotope distribution pattern regardless of location, water depth or depositional facies. This suggests one of several possibilities: (i) the isotopic composition of the surface ocean/atmosphere declined in a geologic instant (<500yr), (ii) that during the onset of the CIE, most shells of mixed-layer planktonic foraminifera were dissolved, or (iii) the abundances or shell production of these species temporarily declined, possibly due to initial pH changes.     

Inspection of commercial optical devices for data storage using a three Gaussian beam microscope interferometer

Recently, an interferometric profilometer based on the heterodyning of three Gaussian beams has been reported. This microscope interferometer, called a three Gaussian beam interferometer, has been used to profile high quality optical surfaces that exhibit constant reflectivity with high vertical resolution and lateral resolution near lambda. We report the use of this interferometer to measure the profiles of two commercially available optical surfaces for data storage, namely, the compact disk (CD-R) and the digital versatile disk (DVD-R). We include experimental results from a one-dimensional radial scan of these devices without data marks. The measurements are taken by placing the devices with the polycarbonate surface facing the probe beam of the interferometer. This microscope interferometer is unique when compared with other optical measuring instruments because it uses narrowband detection, filters out undesirable noisy signals, and because the amplitude of the output voltage signal is basically proportional to the local vertical height of the surface under test, thus detecting with high sensitivity. We show that the resulting profiles, measured with this interferometer across the polycarbonate layer, provide valuable information about the track profiles, making this interferometer a suitable tool for quality control of surface storage devices.     

地震作用的相关性对结构瞬态响应的影响

地震作用一般分解为水平运动分量和竖向运动分量,在这两个运动分量的作用下,结构发生大变形时,可能会经历由地震运动分量演变的外部激励和参数激励过程。由于运动分量间的相关性,推导出实际上这两个激励过程也是相关的,而且是完全相关的,但在过去的研究中,为了简化分析,常常假设这两个激励过程是完全独立的。该文以高斯白噪声和过滤高斯白噪声过程模拟地震动过程,以某一单层框架结构为研究对象,采用累积矩截断法,分析高斯白噪声和过滤高斯白噪声这两种地震动激励下单层框架结构的非平稳地震响应。同时考虑地震动分量间的相关性,得到更为精细化的结构随机地震响应,并分析这种相关性对结构响应的影响。结果表明:将地震动作用模拟为更接近实际的过滤高斯白噪声过程时,地震作用相关性对结构响应的影响更为明显,更为不可忽略。     

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.     

Dynamic analysis of bridge–vehicle system with uncertainties based on the finite element model

A new method of dynamic analysis on the bridge–vehicle interaction problem considering uncertainties is proposed in this paper. The bridge is modeled as a simply supported Euler–Bernoulli beam with Gaussian random elastic modulus and mass density of material with moving forces on top. These forces are time varying with a coefficient of variation at each time instance and they are considered as Gaussian random processes. The mathematical model of the bridge–vehicle system is established based on the finite element model in which the Gaussian random processes are represented by the Karhunen–Loéve expansion and the equations will be solved by the Newmark  -β$\beta$ method. The proposed method is compared with the Monte Carlo method in numerical simulations with good agreements for cases with different vehicle speed and level of uncertainties in the excitation and system parameters. The mean value and variance of the structural responses are found to be very accurate even with large uncertainties in the excitation forces. The proposed method is also found to have superior performance in the computational efficiency compared with the Monte Carlo method.     

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.     

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.