Exponential Autocorrelation Function Research Articles

A numerical method for generation of shot peened surfaces

Shot peening is a widely utilized surface modification technique, and the surface morphology of the treated materials plays a crucial role in determining their friction and fatigue properties. Surface morphology generation is a pivotal process in simulating shot-peened surface friction and other performance characteristics. However, there are limited available methods for shot-peened surfaces until now. In this paper, we present a method that directly specifies the height probability distribution and power spectrum to generate shot-peened surfaces. Compared to methods that specify height parameters, this method has more flexibility. This approach replaces the traditional Johnson transformation method with an optimization algorithm to generate height data that follows a high probability distribution, while the power spectrum is derived from the Fourier transform of an exponential autocorrelation function. The results show that this approach can generate shot-peened surfaces using any conventional height parameters, with a maximum difference in height parameters between the reconstructed surface and the original surface of no more than 18.63%. Compared to the Johnson transformation method, this method significantly reduces the errors in skewness and kurtosis of the generated surface. It offers rapid surface generation for shot-peened surfaces, leading to significant time and cost savings in experiments.

Enhancing the accuracy of the scale of fluctuation estimating for tunnel surrounding rock mass properties

The scale of fluctuation (SOF) is used to evaluate the correlation between two locations of geotechnical parameter, which is important for mineral exploration and engineering construction. In this study, the performances of conventional moment methods are firstly estimated, and then the exponential autocorrelation function method that can achieve the best performance in estimating SOF is selected. After that, the Torque Penetration Index (TPI) and the Field Penetration Index (FPI) obtained from the Yinsong Water Diversion Project are used to detect the scale effect of SOF and to quantify the influences of rock types and surrounding rock mass grades on SOF. As the spatial scale increases, the SOF values increase, and the uncertainty in the SOF of the surrounding rock mass properties increases. The SOF calculated using TPI is slightly higher than using FPI. The mean values of SOF for granite, tuff, diorite, and limestone are quite close at the 50 m–200 m scales, with coefficients of variation (COVs) of SOF range from about 0.2 to 0.4. There are differences in SOF between grade III and grade IV surrounding rock masses at the 50 m–500 m scales. The mean values of SOF in grade III surrounding rock mass are about 4%–15% higher than that in grade IV surrounding rock mass, and this difference in general increases as scale increases. The rock mass with better conditions has a stronger spatial correlation. The COVs of SOF for the two rock grades reached about 0.3–0.5, owning to the inherent spatial variability.

Probabilistic Analysis of Strip Footings Using the Subset Simulation Approach and the Influence of Spatial Variability of Two Clay Layers

Abstract This article studies the probabilistic analysis at the ultimate limit state (ULS) of a strip footing resting on a purely cohesive soil composed of two layers with spatial variability of the upper or lower layer and treats the simultaneous effect of the variability of the two layers on the probability of failure Pf value using the subset simulation approach. The results are examined by comparing the Pf values obtained with the Pf value corresponding to the reference case of two homogenous clay layers. In general, the subset simulation approach is used in cases of uncertain parameters modeled by random variables. But in this paper, it is employed with uncertain parameters modeled by random fields because the spatial variability of soil properties strongly affects the behavior of geotechnical structures and causes a significant change in the variability of their responses. The soil cohesion parameter is modelled as a non-Gaussian (log-normal) anisotropic random field using a square exponential autocorrelation function by the optimal linear expansion estimation (EOLE) and the deterministic model is based on numerical simulations using the finite difference software FLAC3D.

Elucidating appealing features of differentiable auto-correlation functions: A study on the modified exponential kernel

Research on stochastic processes in recent decades has pointed out that, in the context of modeling spatial or temporal uncertainties, auto-correlation functions that are differentiable at the origin have advantages over functions that are not differentiable. For instance, the non-differentiability of e.g., single exponential auto-correlation functions yields non-smooth sample paths. Such sample paths might not be physically possible or may yield numerical difficulties when used as random parameters in partial differential equations (such as encountered in e.g., mechanical equilibrium problems). Further, it is known that due to the non-differentiability of certain auto-correlation functions, more terms are required in the series expansion representations of the associated stochastic processes. This makes these representations less efficient from a computational standpoint.This paper elucidates some additional appealing features of auto-correlation functions which are differentiable at the origin. Further, it focuses on enhancing the arguments in favor of these functions already available in literature. Specifically, attention is placed on single exponential, modified exponential and squared exponential auto-correlation functions, which can be shown to be all part of the Whittle–Matérn family of functions. To start, it is shown that the power spectrum of differentiable kernels converges faster to zero with increasing frequency as compared to non-differentiable ones. This property allows capturing the same percentage of the total energy of the spectrum with a smaller cut-off frequency, and hence, less stochastic terms in the harmonic representation of stochastic processes. Further, this point is examined with regards to the Karhunen–Loève series expansion and first and second order Markov processes, generated by auto-regressive representations. The need for finite differentiability is stressed and illustrated.

Characterizing THz Scattering Loss in Nano-Scale SOI Waveguides Exhibiting Stochastic Surface Roughness with Exponential Autocorrelation

Electromagnetic (EM) scattering may be a significant source of degradation in signal and power integrity of high-contrast silicon-on-insulator (SOI) nano-scale interconnects, such as opto-electronic or optical interconnects operating at 100 s of THz where two-dimensional (2D) analytical models of dielectric slab waveguides are often used to approximate scattering loss. In this work, a formulation is presented to relate the scattering (propagation) loss to the scattering parameters (S-parameters) for the smooth waveguide; the results are correlated with results from the finite-difference time-domain (FDTD) method in 2D space. We propose a normalization factor to the previous 2D analytical formulation for the stochastic scattering loss based on physical parameters of waveguides exhibiting random surface roughness under the exponential autocorrelation function (ACF), and validate the results by comparing against numerical experiments via the 2D FDTD method, through simulation of hundreds of rough waveguides; additionally, results are compared to other 2D analytical and previous 3D experimental results. The FDTD environment is described and validated by comparing results of the smooth waveguide against analytical solutions for wave impedance, propagation constant, and S-parameters. Results show that the FDTD model is in agreement with the analytical solution for the smooth waveguide and is a reasonable approximation of the stochastic scattering loss for the rough waveguide.

Stochastic dynamic stiffness for damped taut membranes

An analytical stochastic dynamic stiffness formulation is developed for the dynamic analysis of damped membrane structures with parametric uncertainties. First, the exact general solution of a biaxially taut membrane in the frequency domain is derived, which is used as the frequency-dependent shape function. Both the material properties and the tension fields of the membrane are modelled as 2D random fields with an exponential autocorrelation function in both x and y directions. Then, the random fields are decomposed by Karhunen-Loève (KL) expansion. After a formulation procedure like the finite element method, the stochastic stiffness, and mass elemental matrices are derived based on the frequency-dependent shape function and the KL expansion, subsequently forming the stochastic dynamic stiffness matrix. The developed stochastic dynamic stiffness elements can be assembled to model membrane assemblies with general boundary conditions considering uncertainties. The proposed method can be utilized as a feasible technique for the efficient and accurate stochastic dynamic analysis in the whole frequency domain. The current research paves the way for stochastic dynamic stiffness formulation for other two-dimensional structures like plates and shells.

Complex s-Plane Modeling and 2D Characterization of the Stochastic Scattering Loss in Symmetric Dielectric Slab Waveguides Exhibiting Ergodic Surface-Roughness With an Exponential Autocorrelation Function

The symmetric dielectric slab waveguide serves as an important canonical structure for the high-speed high-bandwidth optical and opto-electronic silicon-on-insulator interconnects. A stochastic model of propagation loss, due to electromagnetic wave scattering with surface-roughness of the nano-scale waveguide excited at frequencies in the 100s of THz, provides useful insights for analysis and design optimization of optical interconnects. In this work, we propose an analytic solution for the stochastic scattering integral based on contour integration of the exponential autocorrelation function in the complex s-plane that improves performance at larger autocorrelation length values, compared to previous works. An expression is proposed for computing the upper-bound value of scattering loss. Results show that the proposed 2D formulation offers reasonable correlation against data from experimental measurements of loss by previous investigators, while several discrepancies are noted compared to previous 2D models. An expression is proposed for the transverse magnetic (TM) modal field amplitude, and several discrepancies in the transverse electric (TE) modal amplitudes are noted compared to previous works. A method is proposed for computation of the effective index (and propagation constant) for arbitrary cladding that improves convergence of the Newton search method, and it is compared to two other methods. Background discussions include the stochastic theory and assumptions that lead to the stationary and ergodic treatment of the surface-roughness, extension of the ensemble-average and time-average to the spatial domain, and effective thickness due to the Goos-Hänchen shift.

Simultaneous Retrieval of Surface Roughness Parameters for Bare Soils From Combined Active–Passive Microwave SMAP Observations

An active–passive microwave retrieval algorithm for simultaneous determination of soil surface roughness parameters [vertical root-mean-square (RMS) height ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${s}$ </tex-math></inline-formula> ) and horizontal correlation length ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${l}$ </tex-math></inline-formula> )] is presented for bare soils. The algorithm is based on active–passive microwave covariation, including the improved Integral Equation Method (I <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> EM), and is tested with global soil moisture active passive (SMAP) observations. The estimated retrieval results for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${s}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${l}$ </tex-math></inline-formula> are overall consistent with values in the literature, indicating the validity of the proposed algorithm. Sensitivity analyses showed that the developed roughness retrieval algorithm is independent of permittivity for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\varepsilon }_{s} &gt; 10$ </tex-math></inline-formula> [-]. Furthermore, the physical model basis of this approach (I <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> EM) allows the application of different autocorrelation functions (ACF), such as Gaussian and exponential ACFs. Global roughness retrieval results confirm bare areas in deserts such as Sahara or Gobi. However, the type of ACF used within roughness parameter estimation is important. Retrieval results for the Gaussian ACF describe a rougher surface than retrieval results for the exponential ACF. No correlations were found between roughness results and the amount of precipitation or the soil texture, which could be due to the coarse spatial resolution of the SMAP data. The extension of this approach to vegetated soils is planned as an add-on study.

Reliability Analysis of Layered Soil Slopes Considering Different Spatial Autocorrelation Structures

It is widely recognized that different geological formations often vary differently in space. Therefore, soil properties from different layers should be modeled by different autocorrelation functions (ACFs) to reflect such soil heterogeneity. However, the same ACFs are frequently used for different soil layers in slope reliability analysis for simplicity purpose in the literature. The present work is a study on the effects of ACFs on the reliability analysis of layered soil slopes, where the soil properties of different layers are considered by different ACFs. Five commonly used classical ACFs and the non-classical Whittle–Matérn model were investigated in this study. Cholesky decomposition and Monte Carlo simulation were used to simulate the spatial variability of the soil properties and estimate the probability of failure (Pf) of slopes, respectively. Illustrative examples with various parametric studies show that when the soil properties from different layers are characterized by the same ACFs, the Pf of the studied slopes is comparable with that estimated using different ACFs for different soil layers. This indicates that the type of ACF has only a small impact on the slope reliability assessment. However, the Pf may be underestimated by the single exponential ACF and overestimated by the cosine exponential ACF. The scale of fluctuation of the soil properties influences the slope reliability more than the ACFs. In addition, the smoothness parameter in the non-classical model has a significant influence on the reliability of the slope, where Pf increases with the increase of the smoothness parameter.

Testing the Validity of Greenwood and Tripp’s Sum Surface Assumption for Elastic-Plastic Contact

Abstract The complexity of modeling friction between rough surfaces has prompted many researchers to use Greenwood and Tripp’s sum surface assumption to simplify the analysis. This assumption approximates the contact between two rough surfaces as contact between their equivalent sum surface and a rigid plane. In this work, we develop detailed finite element models to test the sum surface assumption for surfaces with Gaussian and exponential autocorrelation functions. We consider surfaces with differing surface roughness and correlation length values. For each case, we conduct simulations of two rough surfaces interacting in compression followed by shear, and a corresponding equivalent surface model based on the sum surface assumption. Multiple realizations of each parameter combination are simulated to obtain a statistical picture of the responses. We find that (a) the sum surface assumption consistently under-predicts the static coefficient of friction and (b) the equivalent surface model is less accurate for surfaces with differing correlation length-to-surface roughness ratios.

Efficient reliability analysis of earth dam slope stability using extreme gradient boosting method

Reliability analysis approach provides a rational means to quantitatively evaluate the safety of geotechnical structures from a probabilistic perspective. However, it suffers from a known criticism of extensive computational requirements and poor efficiency, which hinders its application in the reliability analysis of earth dam slope stability. Until now, the effects of spatially variable soil properties on the earth dam slope reliability remain unclear. This calls for a novel method to perform reliability analysis of earth dam slope stability accounting for the spatial variability of soil properties. This paper develops an efficient extreme gradient boosting (XGBoost)-based reliability analysis approach for evaluating the earth dam slope failure probability. With the aid of the proposed approach, the failure probability of earth dam slope can be evaluated rationally and efficiently. The proposed approach is illustrated using a practical case adapted from Ashigong earth dam. Results show that the XGBoost-based reliability analysis approach is able to predict the earth dam slope failure probability with reasonable accuracy and efficiency. The coefficient of variations and scale of fluctuations of soil properties affect the earth dam slope failure probability significantly. Moreover, the earth dam slope failure probability is highly dependent on the selection of auto-correlation function (ACF), and the widely used single exponential ACF tends to provide an unconservative estimate in this study.

Numerical Simulation of Ultrasonic-Assisted Grinding Surfaces With Fast Fourier Transform

Abstract Numerical simulation of three-dimensional rough surfaces based on fast Fourier transform (FFT) is revisited. A more systematic approach, which is an extension of the current FFT-based simulation models, is proposed to approach surface reconstruction. Moreover, the simulation of the surfaces with machining signature by prescribing the parameters, take ultrasonic-assisted grinding as an example, has been taken as the research focus for the first time. The effectiveness is tested by three cases of simulation examples. Excepting the surface with exponential autocorrelation function, the simulation of surfaces with grinding machining marks is considered both by prescribing the theory autocorrelation function and by measuring a small area as a sample. The results show that the proposed method has great potential in engineering applications.

Simulation of radioelectronic means non-stationary bounce flow and its intensity estimation procedure

Unsteady bounce flow is a Poisson flow modulated in intensity by a stationery random process. Flows of this class, known as double stochastic flows, are found in application for describing in addition to the unsteady flow of failures of radio electronics means, a number of physical phenomena in the Earth’s magnetosphere, meteorology and medicine. The problem of registration is the choice of the optimal value of the linear filter parameter, which provides the minimum total root-mean-square error of estimating the flow rate. The obtained theoretical results, based on solving the Wiener-Hopf education, confirmed the presence of the optimal value of the filter parameter for a wide class of random process modulating the flow in intensity. However, when performing mathematical transformations, several approximations of transcendental functions were used, which influenced the accuracy of the results, but allowed to obtain solutions important for practical applications. As part of the study, an algorithm and a computer program for stimulating the implementation of a nonstationary flow and the procedure for forming the relative error of estimating its intensity have been developed. The simulation results combined with the theoretical results, obtained for the exponential autocorrelation function of the modulating random process are presented.

Probabilistic Risk Assessment of unsaturated Slope Failure Considering Spatial Variability of Hydraulic Parameters

Probabilistic risk assessment of slope failure evaluates the slope safety in a quantitative manner, which considers the failure probability and failure consequence simultaneously. However, risk assessment of unsaturated slope accounting for spatially variable soil-water characteristic curve (SWCC) model parameter and saturated hydraulic conductivity has been rarely reported. A probabilistic risk assessment approach is proposed in current study for rationally quantifying the unsaturated slope failure risk with the aid of Monte Carlo (MC) simulation. The SEEP/W and SLOPE/W modules contained in Geostudio software are applied to carry out deterministic analysis, where factor of safety (FS) of the unsaturated slope is calculated by Morgenstern–Price method. The spatially variable hydraulic parameters are characterized by their respective random fields that are transferred from the random void ratio field in this study, rather than generating them separately. The proposed approach is subsequently employed to an unsaturated slope example for exploring the influences of spatially variable void ratio. Results show that the unsaturated slope failure risk is considerably affected by the spatially variable void ratio, and the single exponential autocorrelation function (ACF) popularized in geotechnical engineering tends to underestimate the failure risk in the unsaturated slope risk assessment.

Reliability analysis of stability against piping and sliding in diversion dams, considering four cutoff wall configurations

The stability against piping and sliding, which is subject to numerous sources of uncertainty, is of great importance in the design of diversion dams. In this study, the performance of four cutoff wall configurations, including a single wall and two walls with half the length of the single wall, was evaluated stochastically using the random finite element method. The Cholesky decomposition technique in conjunction with three types of Auto-Correlation Function (ACF) was employed to generate numerous random fields. The results indicate that the probabilities of failure related to different cutoff wall configurations are similar, considering isotropic hydraulic conductivity. However, there are noticeable differences between the probabilities of failure of these configurations in anisotropic situations. Moreover, the use of a single cutoff wall on the upstream face of an impervious blanket provides the lowest probability of failure for piping. In addition, the exponential ACF ends up with greater exit hydraulic gradients than the second-order Markov and binary noise ACFs. In addition, the sliding stability of the ordinary and earthquake load combinations was examined stochastically using random field theory and Monte Carlo Simulation (MCS). The probability of failure appears to increase with an increase in the autocorrelation distance.

Numerical Generation of Compound Random Processes with an Arbitrary Autocorrelation Function

The generation of non-Gaussian random processes with a given autocorrelation function (ACF) is addressed. The generation is based on a compound process with two components. Both components are solutions of appropriate stochastic differential equations (SDEs). One of the components is a Gaussian process and the other one is non-Gaussian with an exponential ACF. The analytical study shows that a compound combination of these processes may be used for the generation of a non-Gaussian random process with a required ACF. The results are verified by two numerical examples.

Random field characterization of uniaxial compressive strength and elastic modulus for intact rocks

Rock properties exhibit spatial variabilities due to complex geological processes such as sedimentation, metamorphism, weathering, and tectogenesis. Although recognized as an important factor controlling the safety of geotechnical structures in rock engineering, the spatial variability of rock properties is rarely quantified. Hence, this study characterizes the autocorrelation structures and scales of fluctuation of two important parameters of intact rocks, i.e. uniaxial compressive strength (UCS) and elastic modulus (EM). UCS and EM data for sedimentary and igneous rocks are collected. The autocorrelation structures are selected using a Bayesian model class selection approach and the scales of fluctuation for these two parameters are estimated using a Bayesian updating method. The results show that the autocorrelation structures for UCS and EM could be best described by a single exponential autocorrelation function. The scales of fluctuation for UCS and EM respectively range from 0.3 m to 8.0 m and from 0.3 m to 8.4 m. These results serve as guidelines for selecting proper autocorrelation functions and autocorrelation distances for rock properties in reliability analyses and could also be used as prior information for quantifying the spatial variability of rock properties in a Bayesian framework.

On the numerical generation of positive-axis-defined distributions with an exponential autocorrelation function

Stochastic modeling commonly requires random process generation with an exponential autocorrelation function (ACF). These random processes may be represented as a solution of a stochastic differential equation (SDE) of the first order and usually have one-sided (positive-axis-defined) distributions. However, adoption of the SDE-based method faces serious limitations due to difficulties with the numerical solution. To overcome this issue we propose a tractable general numerical solution of the above-mentioned SDE that preserves solution positivity and accuracy, and validate it with numerical simulations.

Modeling of rough surfaces with given roughness parameters

Modeling of rough surfaces with given roughness parameters is studied, where surfaces with Gaussian height distribution and exponential auto-correlation function (ACF) are concerned. A large number of micro topography samples are randomly generated first using the rough surface simulation method with FFT. Then roughness parameters of the simulated roughness profiles are calculated according to parameter definition, and the relationship between roughness parameters and statistical distribution parameters is investigated. The effects of high-pass filtering with different cut-off lengths on the relationship are analyzed. Subsequently, computing formulae of roughness parameters based on standard deviation and correlation length are constructed with mathematical regression method. The constructed formulae are tested with measured data of actual topographies, and the influences of auto-correlation variations at different lag lengths on the change of roughness parameter are discussed. The constructed computing formulae provide an approach to active modeling of rough surfaces with given roughness parameters.

Comments on “Simple Generation of Gamma, Gamma–Gamma and K Distributions With Exponential Autocorrelation Function”

This paper is a comment on Simple generation of gamma, gamma-gamma and K distributions with exponential autocorrelation function”. The original article states that the simulation of a free-space optical (FSO) communication channel in the presence of strong turbulence typically requires the generation of channel states with a K or gamma-gamma distribution and a predefined autocorrelation function. In the comment it is concluded that the resulting autocorrelation function for gamma-gamma and K processes may be approximated as a simple exponent only under certain constraints on distribution parameters. Whenever these constraints are not met, the resulting autocorrelation function is still analytically tractable, but it is no longer a simple exponent.