Karhunen-Loeve Expansion Method Research Articles

Random Responses of Shield Tunnel to New Tunnel Undercrossing Considering Spatial Variability of Soil Elastic Modulus

This paper investigates the effect of spatial variability of soil elastic modulus on the longitudinal responses of the existing shield tunnel to the new tunnel undercrossing using a random two-stage analysis method (RTSAM). The Timoshenko–Winkler-based deterministic method considering longitudinal variation in the subgrade reaction coefficient and the random field of the soil elastic modulus discretized by the Karhunen–Loeve expansion method are combined to establish the RTSAM. Then, the proposed RTSAM is applied to carry out a random analysis based on an actual engineering case. Results show that the increases in the scale of fluctuation and the coefficient of variation of the soil elastic modulus lead to higher variabilities of tunnel responses. A decreasing pillar depth and mean value of the soil elastic modulus and an increasing skew angle strengthen the effect of the spatial variability of the soil elastic modulus on tunnel responses. The variabilities of tunnel responses under the random field of the soil elastic modulus are overestimated by the Euler–Bernoulli beam model. The results of this study provide references for the uncertainty analysis of the new tunneling-induced responses of the existing tunnel under the random field of soil properties.

Analysis of slope stochastic fields using a novel deep learning model with attention mechanism

This paper proposes a novel deep learning model incorporating attention mechanisms for the analysis of slope stochastic fields. Initially, a deep learning model is designed to digitally image the stochastic field features of soil strength variability. This is achieved by discretizing the slope soil stochastic field using the Karhunen-Loeve expansion method and transforming the discrete results into digital images. These images are then used to establish a Convolutional Neural Network (CNN) surrogate model that maps the implicit relationship between stochastic field images and slope functional function values, thus calculating the probability of slope failure. The precision of the CNN surrogate model is enhanced through Bayesian optimization and five-fold cross-validation. Moreover, to overcome the limitations of existing data-driven landslide stability prediction models, this study also introduces a Spatial-Temporal Attention (STA) mechanism. By combining the CNN with Long Short-Term Memory (LSTM) networks, the model can accurately approximate the actual results of slope stability calculations in scenarios of high-dimensional representation imaging of stochastic fields and low-probability slope instability. Consequently, this significantly improves the computational efficiency of slope reliability analysis considering stochastic field simulations.

Random analysis method for nonlinear interaction between shield tunnel and spatially variable soil

The spatial variability of soil properties is likely to have a remarkable influence on the longitudinal responses of shield tunnels. This paper proposes a one-dimensional random analysis method (RAM) for the interaction between the shield tunnel with circumferential joints and the spatially variable soil with nonlinear elastoplastic behaviors. A deterministic short beam-spring model with longitudinal variation in the initial subgrade reaction coefficient and the ultimate soil resistance is developed and solved by the finite difference method. The random field of the soil properties discretized by the Karhunen-Loeve expansion method is used to establish the RAM. A hypothetical example is further presented to demonstrate the applicability of the proposed method. The results imply that the soil nonlinearity may lead to a negative effect on the reliability control of the tunnel structure and should be considered to obtain more reasonable probabilistic evaluations of the tunnel deformations. The increasing joint rotational stiffness or shearing stiffness results in the lower variabilities of the maximum tunnel settlement, and reduces the probabilities of the maximum joint opening or joint dislocation exceeding the allowable value, respectively.

Uncertain Dynamic Characteristic Analysis for Structures with Spatially Dependent Random System Parameters.

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represent the random uncertainty of parameters, and the stochastic dynamic characteristics of different structural systems are then analyzed by incorporating the presented parameter setting model with finite element method. First, Gauss random field theory is used to describe the uncertainty of structural material parameters, the random parameters are then characterized as the standard deviation and correlation length of the random field, and the random field parameters are then discretized with the Karhunen-Loeve expansion method. Moreover, based on the discretized random parameters and finite element method, structural dynamic characteristics analysis is addressed, and the probability distribution density function of the random natural frequency is estimated based on multi-dimensional kernel density estimation method. Finally, the random field parameters of the structures are quantified by using the maximum likelihood estimation method to verify the effectiveness of the proposed method and the applicability of the constructed model. The results indicate that (1) for the perspective of maximum likelihood estimation, the parameter setting at the maximum value point is highly similar to the input parameters; (2) the random field considering more parameters reflects a more realistic structure.

Influence of Spatially Distributed Out-of-Plane CFRP Fiber Waviness on the Estimation of Knock-Down Factors Based on Stochastic Numerical Analysis

The presence of waviness defects in CFRP materials due to fiber undulation affects the structural performance of composite structures. Hence, without a reliable assessment of the resulting material properties, the full weight-saving potential cannot be exploited. Within the paper, a probabilistic numerical approach for improved estimation of material properties based on spatially distributed fiber waviness is presented. It makes use of a homogenization approach to derive viable knock-down factors for the different plies on the laminate level for reference material and is demonstrated for a representative tension loadcase. For the stochastic analysis, a random field is selected which describes the complex inner geometry of the plies in the laminate model and is numerically discretized by the Karhunen–Loeve expansion methods to fit into an FE model for the strength analysis. Conducted analysis studies reveal a substantial influence of randomly distributed waviness defects on the derived knock-down factors. Based on a topological analysis of the waviness fields, the reduction of the material properties was found to be weakly negatively correlated related to simple geometrical properties such as maximum amplitudes of the waviness field, which justifies the need for further subsequent sensitivity studies.

In-situ investigation on coal mine overburden dump slope and its seismic stability considering heterogeneity

Stability of the coal mine overburden (OB) dump slopes is many times compromised owing to the lesser availability of the in-situ characteristics of the OB dump materials and their spatial heterogeneity. In this study, multi-channel analysis of surface waves (MASW) investigation was carried out for the characterization of OB dump materials of Jambad opencast coal mine of India and the variation of heterogeneous in-situ characteristics has been reported in terms of statistical parameters. Next, the stability of a two-bench OB dump slope against seismic forces was investigated through finite element limit analyses coupled with random field theory utilizing Karhunen-Loeve expansion method for the spatial heterogeneity of the OB dump material. The effects of spatial heterogeneity of the OB dump material on the probability of failure and the critical pseudostatic seismic coefficient of the slope were investigated through Monte-Carlo simulations. The influence of spatial heterogeneity of OB dump materials were investigated with both isotropic and anisotropic correlation lengths. Finally, seismic slope stability analyses were also carried out using the discrete element method for the size heterogeneity of the OB dump materials. This study summarizes the effects of heterogeneity of material property and size on the seismic stability of OB dump slope.

Behavior of Shallow Circular Tunnels—Impact of the Soil Spatial Variability

Spatial variability is unavoidable for soils and it is important to consider such a feature in the design of geotechnical engineering as it may lead to some structure behaviors which cannot be predicted by a calculation assuming homogenous soils. This paper attempts to evaluate the performance of a shallow circular tunnel, in a context of the service limit state, considering the soil spatial variability. The Log-normal distributed random fields, generated by the Karhunen–Loeve expansion method, are used for the spatial modeling. A two-dimensional numerical model, based on the finite difference method, is constructed to deterministically estimate two quantities of interest (i.e., tunnel lining bending moment and surface settlement). The model is combined with the random fields and is implemented into the Monte Carlo simulation to investigate the effects of the soil spatial variability on the tunnel responses. The autocorrelation distance, an important parameter for random fields, is varied within multiple probabilistic analyses. For both of the two tunnel responses, their variabilities are increased with increasing the autocorrelation distance, while a minimum mean value can be observed with this parameter being approximately the tunnel radius. Such finding is very useful for practical designs. A sensitivity analysis is also conducted to show the importance of each random parameter.

Random Field Reliability Analysis for Time-Dependent Behaviour of Soft Soils Considering Spatial Variability of Elastic Visco-Plastic Parameters

• K-L expansion method was adopted to generate random field with different spatial correlation lengths • A critical spatial correlation length existed for predicting time-dependent settlement of soft soils • A single random variable analysis is not always the most conservative analysis in predicting settlement of soft soils • It is essential to perform random field analysis to reduce the risk of the design to be applied in construction. Low embankment strategy is one of the effective methods to control time-dependent settlement of soft soils in infrastructure construction projects. Spatial variability of soil characteristics is a crucial factor, affecting the reliability of predictions of the long-term settlement in soft soils. In this paper, the time-dependent behaviour of soft soils is analysed incorporating spatial variability of elastic visco-plastic model parameters. Standard Gaussian random fields for correlated elastic-plastic model parameter ( λ / V ) and the initial creep coefficient ( ψ 0 / V ) are generated adopting Karhunen-Loeve expansion method based on the spectral decomposition of correlation function into eigenvalues and eigenfunctions. Then the generated random fields are incorporated in the proposed non-linear elastic visco-plastic (EVP) creep model. The impacts of spatially variable elastic visco-plastic model parameters (i.e. ψ 0 / V and λ / V ) on long-term settlement predictions are evaluated through random field analysis (RF) with different spatial correlation lengths, and results are then compared to a single random variable (SRV) analysis. The probability of failure (PF) is calculated adopting RF and SRV analysis to determine the critical spatial correlation length, resulted in a maximum probability of failure. This study can be employed by design engineers to determine the critical spatial correlation length for safe design in the absence of adequate data to determine the exact spatial correlation length. The results also confirm that SRV analysis is not always the most conservative analysis in predicting time-dependent settlement of soft soils; and it is essential to perform RF analysis considering the spatial correlation length to reduce the risk and increase the reliability of the design to be applied in construction.

A three-dimensional stochastic progressive damage simulation model for polymer matrix-based laminate composites

This article proposes a three-dimensional (3D) stochastic progressive damage simulation model for the fiber-reinforced polymer matrix composite laminates. Constituents’ strength and fracture energy of each lamina were accounted as spatially varying random fields by 3D Karhunen–Loeve expansion method. Damage mechanisms modeled by continuum damage mechanics include internal damages of matrix and fiber. An anisotropic damage model was used for damage initiation and evolution of each lamina. By the proposed model, we investigated influences of variabilities in the lamina strength properties, and fracture energies along with delamination damage to the damage progression, strength, and post-failure behavior.

Robust project of resonant shunt circuit for passive vibration control of composite structures

Composite structures have been widely used in petroleum, aerospace and automotive industries for which structural components must be designed to support high levels of mechanical disturbances with typically high reliability levels. Moreover, the increasing high-speed and lightweight composite structures subjected to vibrations, and the interest in achieving vibration attenuation becomes capital importance as extensive vibrations can reduce structural life and contribute to mechanical failure. In this sense, smart materials can be used as an excellent alternative, being able to stabilize these structures. Thus, the use of shunted piezoceramics has received major attention in the last decades. The contribution intended herein is the proposition of a robust passive vibration control tool by using resonant shunt circuits. The stochastic finite element method is used, and the uncertain variables are modeled as Gaussian random fields and discretized in accordance with the Karhunen–Loeve expansion method. Numerical applications are presented, and the main features and capabilities of the proposed method are highlighted.

BER evaluation in a multi-channel graphene-silicon photonic crystal hybrid interconnect: a study of fast- and slow-light effects.

A comprehensive theoretical investigation on the bit-error ratio (BER) performance of multi-channel photonic interconnects operating in pulsed regimes is presented. Specifically, the optical link contains either a silicon photonic crystal (SiPhC) or a SiPhC-graphene (SiPhC-GRA) waveguide, possessing slow-light (SL) and fast-light (FL) regimes. A series of Gaussian pulses plus complex white noise are placed at input of each channel, with output signals demultiplexed and analyzed by a direct-detection receiver. Moreover, a rigorous theoretical model is proposed to measure signal propagation in SiPhC and SiPhC-GRA, which incorporates all crucial linear and nonlinear optical effects, as well as influences of free-carriers and SL effects. BER results of multi-channel systems are evaluated by utilizing the Fourier series Karhunen-Loeve expansion method. Our findings reveal that good BER performance is acquired at SiPhCs and SiPhC-GRAs in SL regimes but with their footprint about 2.5-fold smaller than FL waveguides. Moreover, the enhanced nonlinearity in SiPhC-GRAs induced by strong graphene-SiPhC coupling causes extra signal degradation than SiPhCs at the same length. This work provides additional insights into the coupling effect between SiPhCs operating in SL regimes and graphene, and their influence on WDM signal transmission, highlighting the potential applications of SiPhC-GRA interconnects in next-generation super-computing systems.

Soil spatial variability impact on the behavior of a reinforced earth wall

This article presents the soil spatial variability effect on the performance of a reinforced earth wall. The serviceability limit state is considered in the analysis. Both cases of isotropic and anisotropic non-normal random fields are implemented for the soil properties. The Karhunen-Loeve expansion method is used for the discretization of the random field. Numerical finite difference models are considered as deterministic models. The Monte Carlo simulation technique is used to obtain the deformation response variability of the reinforced soil retaining wall. The influences of the spatial variability response of the geotechnical system in terms of horizontal facing displacement is presented and discussed. The results obtained show that the spatial variability has an important influence on the facing horizontal displacement as well as on the failure probability.

Bit-error ratio calculation for single-channel silicon optical interconnects utilizing return-to-zero pulsed signals

We introduce an accurate and efficient approach to compute the transmission bit-error ratio (BER) of single-channel silicon photonic interconnects when the optical signal consists of arbitrarily shaped optical pulses. The silicon photonic systems investigated in this study contain an optical transmitter, a silicon optical interconnect, which can be either a strip single-mode silicon photonic waveguide (Si-PhW) or a silicon photonic crystal (PhC) waveguide (Si-PhCW), and a direct-detection receiver. The input signal consists of a superposition of a train of Gaussian pulses and white noise, its propagation in the silicon waveguide being described by a rigorous theoretical model based on a modified nonlinear Schrodinger equation, which captures all relevant linear and nonlinear optical effects. The statistical properties of the output signal and the corresponding BER were determined using the Fourier series Karhunen–Loeve expansion method. Our analysis reveals that in the case of Si-PhWs the pulse width is the parameter that most strongly influences the BER, whereas for Si-PhCWs the main factor affecting BER is the pulse group velocity. Finally, we show that similar values of BER are achieved in Si-PhCWs with length of about 2 orders of magnitude smaller than that of Si-PhWs, a further reduction of the silicon interconnect footprint being possible if the Si-PhCW is operated in the slow-light regime.

Variation of Time Domain Failure Probabilities of Jack-up with Wave Return Periods

This study evaluated failure probabilities of jack up units on the framework of time dependent reliability analysis using uncertainty from different sea states representing different return period of the design wave. Surface elevation for each sea state was represented by Karhunen-Loeve expansion method using the eigenfunctions of prolate spheroidal wave functions in order to obtain the wave load. The stochastic wave load was propagated on a simplified jack up model developed in commercial software to obtain the structural response due to the wave loading. Analysis of the stochastic response to determine the failure probability in excessive deck displacement in the framework of time dependent reliability analysis was performed by developing Matlab codes in a personal computer. Results from the study indicated that the failure probability increases with increase in the severity of the sea state representing a longer return period. Although the results obtained are in agreement with the results of a study of similar jack up model using time independent method at higher values of maximum allowable deck displacement, it is in contrast at lower values of the criteria where the study reported that failure probability decreases with increase in the severity of the sea state.

Dimension reduction of Karhunen-Loeve expansion for simulation of stochastic processes

Conventional Karhunen-Loeve expansions for simulation of stochastic processes often encounter the challenge of dealing with hundreds of random variables. For breaking through the barrier, a random function embedded Karhunen-Loeve expansion method is proposed in this paper. The updated scheme has a similar form to the conventional Karhunen-Loeve expansion, both involving a summation of a series of deterministic orthonormal basis and uncorrelated random variables. While the difference from the updated scheme lies in the dimension reduction of Karhunen-Loeve expansion through introducing random functions as a conditional constraint upon uncorrelated random variables. The random function is expressed as a single-elementary-random-variable orthogonal function in polynomial format (non-Gaussian variables) or trigonometric format (non-Gaussian and Gaussian variables). For illustrative purposes, the simulation of seismic ground motion is carried out using the updated scheme. Numerical investigations reveal that the Karhunen-Loeve expansion with random functions could gain desirable simulation results in case of a moderate sample number, except the Hermite polynomials and the Laguerre polynomials. It has the sound applicability and efficiency in simulation of stochastic processes. Besides, the updated scheme has the benefit of integrating with probability density evolution method, readily for the stochastic analysis of nonlinear structures.

Theoretical Comparative Analysis of BER in Multi-Channel Systems With Strip and Photonic Crystal Silicon Waveguides

We present a comparative analysis of the performance of multi-channel photonic systems containing strip or photonic crystal (PhC) silicon (Si) waveguides in the presence of white Gaussian noise. Specifically, we consider a multi-wavelength optical signal propagating either in a strip single-mode Si photonic waveguide (Si-PhW) or in a Si PhC waveguide (Si-PhCW), each channel consisting of a CW nonreturn-to-zero ON–OFF keying modulated signal. The output signal is demultiplexed and the signal in each channel is analyzed using direct-detection optical receivers. We describe the propagation of the optical signal in the Si-PhW and Si-PhCW using a linearized model and validate our main results by employing a rigorous theoretical model that incorporates all relevant linear and nonlinear optical effects and the mutual interaction between free-carriers and the optical field. The bit error rate (BER) of the transmitted signal is calculated by using both a time- and frequency-domain Karhunen–Loeve expansion method. Our analysis suggests that due to enhanced nonlinear effects, for comparable optical power, a similar BER is achieved in a PhC waveguide about $100\times$ shorter than a strip one, with a particularly strong signal degradation being observed in the slow-light regime of the Si-PhCW.

Analyzing the tidal frequency content using Karhunen-Loeve Expansion technique

The Karhunen-Loeve Expansion (KLE) technique has been recently applied by geophysicists for analyzing the temporal variations of dynamic systems such as the ocean-atmosphere interface, crustal deformation and fault systems. The application of this method to tidal data can provide a direct insight into the efficiency and reliability of this method in reconstructing a periodic signal. The comparison of the obtained results to those proposed by the least squares harmonic estimation (LSHE) as a newly developed method which is widely used in geodetic community for analyzing the GPS time series can be of signiocant interest for both geodesists and geophysicists. This paper applies the KLE method to the tidal time series of the Workington station in United Kingdom and compares the given results to those suggested by the LSHE method. The detection of the majority of the expected long period constituents and the larger number of the detected low frequency components by KLE as compared to the least squares harmonic estimation emphasizes on the efficiency and the predominance of this method to the LSHE technique.

Generalized Linear Random Vibration Analysis Using Autocovariance Orthogonal Decomposition

Application of a stationary Gaussian random process to describe a nondeterministic forcing function of a linear vibrating system is well studied and documented. Two algorithms, Karhunen-Loeve expansion method, and collocation technique for nonstationary and non-Gaussian forcing processes (narrow- or broadband, but not white noise) for linear dynamic systems are developed here. In the Karhunen-Loeve expansion method, the forcing random process autocovariance is decomposed using the well-known Karhunen-Loeve expansion. In the Karhunen-Loeve expansion, Galerkin projection (the weighted-residual method) and collocation technique (discretized covariance matrix) are used to get the eigenvalues and the eigenfunctions/eigenvectors of the autocovariance function, numerically. The steady-state and the transient response of a single-degree-of-freedom system for an exponential autocovariance (Gaussian random process) is obtained using three methods: 1) analytical, 2) semi-analytical and 3) numerical. In the semi-analytical method, the eigenvalues and the eigenfunctions of the exponential autocovariance function are obtained analytically by solving the Fredholm integral equation of the second kind and those definitions of the eigenfunctions and the eigenvalues are used to obtain the numerical response of the single-degree-of-freedom system. In the case of the steady-state response of the single-degree-of-freedom system, the convergence of the standard deviation of the response is shown to be a function of the number of Karhunen-Loeve expansion terms used in the expansion of the autocovariance of the forcing function. The transient analysis of the same single-degree-of-freedom system is carried out using an exponentially modulated nonstationary process. Comparison of the proposed methods with respect to the analytical solutions are presented for both the stationary and the nonstationary Gaussian excitations. The steady-state responses as well as the transient responses for non-Gaussian random processes (uniform, triangular, and beta) for the same single-degree-of-freedom system are also presented.

Micromagnetic modeling and experimental study of transition noise correlation in thin-film media

Micromagnetic modeling and time-domain experimental measurements are combined to study spatial noise correlations between two closely recorded transitions in oriented longitudinal thin-film media. The Karhunen–Loeve expansion method is utilized to characterize the noise correlations. The analysis shows that the noise correlation of the dipulses exhibits three principle modes: amplitude variation, shift in unison, and breathing. The supralinear increase of noise at small bit intervals in well-oriented longitudinal film media is dominated by the amplitude variation, which arises from large variation of the degree of percolation between adjacent transition boundaries. The results from the micromagnetic modeling shows excellent agreement with the results obtained from the experimental time-domain noise measurements.

Level-crossing approach to the compression of body surface potential maps

Body Surface Potential Mapping (BSPM) is a technique for the high-resolution study and analysis of the electrical events in the heart. The technical complications and the high cost of custom-designed systems have limited the use of BSPM to a small number of laboratories. Almost all efforts to simplify BSPM systems have concentrated so far on the reduction of the number of the electrodes employed with subsequent extrapolation of the measured information to the whole thoracic surface. In this report a new approach for simplifying the BSPM system is suggested. The system uses a different type of electrodes, which do not record the ECG signal, but only detect the time instants when the incoming ECG signal crosses some given threshold level (Level-Crossings (LC)). Different threshold levels may be attributed to different subsets of electrodes. An algorithm for the reconstruction of complete BSPM information from those LC is proposed, based on utilization of the Karhunen-Loeve expansion method. This approach is justified by treating the BSPM signals as samples of a three-dimensional random process. The complete recordings of 200 BSPMs (training set) with different cardiac pathologies is used to derive the set of basis functions for the reconstruction procedure. The reconstructions are then performed on the LC of 81 subjects (test set) with a distribution of pathologies similar to that of the training set. Best performance is found when four threshold levels (± 100, ± 500 μV) are selected, for four intermittent grids of 45 electrodes, thus comprising a whole net of 180 electrodes. The average RMS reconstruction error for a member of the rest set is about 136 μV, whereas the correlation between the original and estimated BSPMs is around 0.91. These results suggest that by implementing an efficient signal compression scheme, even with the limited information provided by the LC, a clinically useful reconstruction of the BSPM is obtained.