Stochastic Response Research Articles

Groundwater stress induced by shale resources development in the US: Evolution, response, and mitigation

Unconventional shale resources development consumes a huge amount of groundwater, yet it is short of systematic research on evolution, response, and mitigation of shale-induced groundwater stress (SGS). To fill this gap, we here propose a new SGS evaluation model based on an improved concept of groundwater footprint. We find that the SGS remains heavy within the shale regions of the US in 2017–2019. The projected SGS would keep growing in 2025–2050 within the regions except Permian, owing to comprehensive stochastic response to future intensified shale development and varied precipitation and temperature. Notwithstanding, comparison of the rate of risked returns indicates that shale development could be more economically and environmentally competitive than agricultural planting, demonstrating high suitability of shale development in part of regions such as Niobrara, Anadarko, Permian, and Eagle Ford. Joint policies regulating shale production, water intensity, and groundwater supply coefficient would be effective for mitigating future intensified SGS.

Stochastic analysis of excavation-induced wall deflection and box culvert settlement considering spatial variability of soil stiffness

In this study, a three-dimensional (3D) finite element modelling (FEM) analysis is carried out to investigate the effects of soil spatial variability on the response of retaining walls and an adjacent box culvert due to a braced excavation. The spatial variability of soil stiffness is modelled using a variogram and calibrated by high-quality experimental data. Multiple random field samples (RFSs) of soil stiffness are generated using geostatistical analysis and mapped onto a finite element mesh for stochastic analysis of excavation-induced structural responses by Monte Carlo simulation. It is found that the spatial variability of soil stiffness can be described by an exponential variogram, and the associated vertical correlation length is varied from 1.3 m to 1.6 m. It also reveals that the spatial variability of soil stiffness has a significant effect on the variations of retaining wall deflections and box culvert settlements. The ignorance of spatial variability in 3D FEM can result in an underestimation of lateral wall deflections and culvert settlements. Thus, the stochastic structural responses obtained from the 3D analysis could serve as an effective aid for probabilistic design and analysis of excavations.

A harmonic-wavelet-based representation for non-stationary stochastic excitations

Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.

Stochastic Digital-Twin Service Demand With Edge Response: An Incentive-Based Congestion Control Approach

The emergence of Digital Twin Edge Networks (DTENs) achieves the mapping of real physical entities to digital models of cyberspace. By offloading real-time mobile data to Mobile Edge Computing (MEC) servers for processing and modeling, communication-efficient Digital Twin (DT) services could be achieved. However, the spatio-temporal dynamic DT service demand stochastically generated by mobile users easily causes service congestion, which challenges the long-term DT service stability. Meanwhile, current DT services still lack long-term effective incentive designs for participants. To solve these issues, we design an incentive-based congestion control scheme for stochastic demand response in DTENs. First, we adopt the Lyapunov optimization theory to decompose the long-term congestion control decision into a sequence of online edge association decisions, with no need for future system information. We then present a contract-based incentive design to optimize the long-term profit of the DT service provider, comprehensively considering the delay sensitivity, incentive compatibility, and individual rationality. Finally, experimental simulations are carried out to verify the superiority of the proposed scheme with the base station dataset of Shanghai Telecom. Theoretical and simulation analysis demonstrates that compared with benchmarks, our scheme could effectively avoid long-term service congestion with an arbitrarily near-optimal profit.

Optimal design of SMA damper for vibration control of connected building under random seismic excitation

To alleviate the earthquake induced damages in closely spaced high-rise buildings, linear or non-linear dampers have been frequently employed. Out of different options, the frequently utilized non-linear damper is the yield damper. Yield damper shows good floor displacement control efficiency, but increases peak floor acceleration and leaves large residual deformation at the end of ground motion. Also, it shows degradation in the hysteresis loop; whereas to achieve a good acceleration and displacement control efficiency, it is imperative to have a stable hysteresis loop with negligible residual deformation. In this context, shape memory alloy (SMA) composed damper has been viewed as a potential solution to reduce floor displacement and acceleration, simultaneously. Also, it offers excellent recentering property and good seismic energy dissipation capacity. This research aims to use the above stated superior properties of SMA and investigate the seismic response control efficiency of optimally design SMA damper for connected buildings, and compare with the yield damper. Optimal stochastic responses of connected buildings are estimated under random earthquake, as modeled using the Kanai-Tajimi power spectra. To linearize the non-linear force–displacement behavior of SMA and yield damper, a statistical linearization approach is adopted. The RMS (root mean square) floor acceleration and RMS floor displacement ratios are determined as response outputs for the damper connected buildings. Studies results show that, optimally designed SMA damper is more effective in reducing acceleration of stiff building and displacement of the flexible building than the yield damper. Compared to the yield damper, SMA damper reduces the RMS acceleration ratios by 15 % for the flexible building and 35 % for the stiff building; and RMS displacement by 20 % for the flexible building and 0.5 % for the stiff building. Overall, SMA damper provide much better seismic response control efficiency for connected buildings than yield damper.

Stochastic Dynamics of Suspension System in Maglev Train: Governing Equations for Response Statistics and Reliability

The suspension system of the maglev train will inevitably be disturbed by random factors such as track irregularities, which will cause random vibration of the train and even affect the safety of the train. Therefore, the research on the response and reliability of suspension system under random disturbance is crucial to its safe operation. In this paper, the response and the reliability of a suspension system are investigated using the theory and methods of stochastic dynamics. First, the magnetic gap and vertical velocity of the suspension system are random due to the random disturbance. Thus, the stochastic response is investigated through the probability density function (PDF), which is governed by the Fokker–Planck–Kolmogorov (FPK) equation corresponding to suspension system. And the response statistics of the suspension system under different system parameters and disturbance intensities are analyzed by solving the corresponding FPK equation using the finite difference (FD) method. Second, random disturbance may lead to the vibration amplitude of the suspension system exceeding the safety domain and causing safety incident, which is a reliability problem in stochastic dynamical systems. The probability that response is still in the safety domain at a given time is the reliability function of the suspension system, which is governed by the backward Kolmogorov equation. The time that the response first passes through the safety domain is the first-passage time, and its n-order moment satisfies the generalized Pontryagin equation. Reliability of the suspension system is analyzed by solving these governing equations using the FD method. In addition, the results of the FD method in this paper are verified with those of Monte Carlo (MC) simulation, which shows the correctness of FD method.

Full characterization of building energy factor significance by novel integrated stochastic level-based sensitivity analysis with support vector network and multivariate clustering

Effective planning and targeted implementation of building energy efficiency programs explicitly require robust knowledge on significance pattern of involving factors for precision model-based estimates of potential energy savings. Though extensively adopted for factor significance characterization, prior sensitivity analysis (SA) frameworks fail to capture typical influences attributed to commonly used categorical factors. Moreover, they often assume the assessed factors are independent which can be invalid in many cases to generate misleading outcomes. To overcome these observable defects, this paper develops a novel integrated stochastic level-based sensitivity analysis approach for exhaustively patterning effectual significance of all available building factors of varied type while incorporating diverse factor interactions on validated reliability. While the integration of categorical factors is achieved through factor-type dependent variation levelling, the random factor interactions are apprehended by stochastic sampling and nonlinear support vector network computing to accommodate collinearity trap. Multivariate clustering identifies significance pattern via a composite multi-statistic sensitivity metric derived from stochastic energy responses. Case results of 289 U.S. residential houses validate its utility. This presented approach benefits to perform reliable full sensitivity investigations and analyses under collinearity for building energy systems with verifiable robustness.

New extreme statistics strategy for the extreme value predictions of ship parametric rolling motions through limited model test observations

This work proposes a new extreme statistics strategy to predict the extreme parametric rolling angle through minimal observations from model tests. The core of this strategy is a new synthetic moment method. Based on the Hermite transformation and the Markov chain, the synthetic moment method can provide extreme value predictions for stationary processes. Besides, this work introduces one specific simplification to solve the problem induced by the non-stationarity of the parametric roll. With the help of the new extreme statistics strategy, this work obtains the CDF and PDF of the extreme parametric rolling angle for the C11 container ship. These predictions are derived from only one observed model test time history. The data statistics of other model tests and a vast amount of numerical simulations verify the validities of these predictions. Compared with other methods, the new strategy shows advantages for small sampling predictions. It is possible to replace the widely used ACER method in case of insufficient samples. In addition, further research confirms that the new strategy's intrinsic probability model differs from the widely used GEV model, which is more reasonable for extreme value predictions in a finite time range. The newly presented strategy should effectively predict the extreme value for other complicated stochastic responses.

Nonstationary Stochastic Response of Hysteretic Systems Endowed With Fractional Derivative Elements

AbstractIn this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.

Stochastic Analysis of Impact Viscoelastic Energy Harvester with Unilateral Barrier Under Additive White Noise Excitation

Vibration impact is often used in the piezoelectric energy harvesting (PEH) system to increase the effective bandwidth of the harvester. Viscoelastic materials have been used successfully to mitigate vibration problems in various types of mechanical systems such as buildings, cars, aircraft and industrial equipment. However, less research has been done on the energy harvesting system with impact and viscoelastic force driven by random excitation. Stochastic response of an impact PEH system with viscoelastic force under Gaussian white noise excitation is investigated in this paper. Firstly, by transforming the variables, viscoelastic force can be substituted with the stiffness and damping terms to get an approximately equivalent system without viscoelastic term. Secondly, the approximate analytical solutions are acquired by the stochastic averaging method and nonsmooth coordinate transformation. The validity of this theoretical approach is confirmed by comparing the analytical solutions with the numerical solutions derived from the Monte Carlo method. Then, the effect of noise intensity and nonlinear damping coefficient on the stochastic response of the system is discussed. It is concluded that the restitution coefficient, viscoelastic component, relaxation time and linear damping coefficient can induce the occurrence of stochastic P-bifurcation. Finally, the roles of system parameters on the mean square voltage and average output power of the energy harvester are investigated respectively.

Reliability analysis of structures with inerter-based isolation layer under stochastic seismic excitations

Reliability analyses on various control devices are of vital importance in examining and quantifying their practical feasibility and effectiveness. In specific, an emerging base isolation technique using a tuned inerter damper (TID) draws researchers’ attention given some of its unique merits, such as the compact damper size, lightweight, and the fact that it will not compromise structural stiffness when the system is static and thus prevents the long-term creeping issue. To justify its feasibility and robustness in face of random earthquake excitations, this paper presents both (1) a simulation-based stochastic dynamic response analysis, and (2) a reliability analysis of a benchmark reinforced concrete (RC) structure with a TID isolation layer. It is found that although seismic randomness shall substantially affect the structural dynamic responses, the introduction of a TID isolation layer to the structure will significantly alleviate this effect and subsequently enhances the overall system robustness, which can be parallelly identified from the notable different features between their corresponding probability density evolutions. In addition, the reliability of the RC structure is also drastically improved owing to the introduction of the TID isolation layer, which highlights its effectiveness and sheds light on its great potential in future industrial applications.

Stochastic dynamic and reliability analysis of AP1000 nuclear power plants via DPIM subjected to mainshock-aftershock sequences

Earthquakes usually consist of a mainshock and a series of aftershocks, both of which are random. The mainshock can damage structures, and the subsequent aftershocks may cause further damage. However, no studies have considered the effects of stochastic seismic sequences on nuclear power plants (NPPs). We propose a new framework to analyze the stochastic dynamic response and dynamic reliability of AP1000 NPPs. First, stochastic mainshock-aftershock sequences are generated by combining a physical random function model, the narrowband harmonic group superposition method and the copula function. Then, the effects of aftershocks on NPPs are highlighted using acceleration and displacement as response indices and the tensile damage ratio (TDR) and plastic strain as damage indicators. Finally, probabilistic information and reliability of NPPs are obtained based on the direct probability integration method (DPIM) and the absorbing condition method (ACM). The results show that the dynamic response is larger for seismic sequences than a single mainshock, and seismic sequences cause more damage to the NPPs. When the peak ground acceleration (PGA) increases, the effect of the aftershock becomes more pronounced, and the response of the NPP becomes more random. Moreover, aftershocks reduce the reliability of NPPs, and the degree of reduction is related to the thresholds.

The two-level probabilistic description of the compressive constitutive parameters of concrete

The constitutive model of concrete is the critical basis for the nonlinear analyses of concrete structures. Due to the fact that the mechanical behavior of concrete exhibits remarkable randomness, the probabilistic modeling of the key parameters of the concrete constitutive model is of paramount significance. In the present study, a two-level probabilistic model is proposed to describe the dependent random constitutive parameters based on the compressive test results of several batches of concrete specimens of different strength grades. Both the local probabilistic dependence, i.e., the dependence of the parameters of intra-batch specimens, and the global probabilistic dependence, i.e., the dependence of the parameters of inter-batch specimens, are captured by the proposed method. To this end, the constitutive parameters of concrete are first standardized by their means and coefficients of variation (COVs). In this way, the local dependence can be represented by the same copula model for the concrete of different strength grades. The global dependence is expressed as the empirical formulae in terms of the means and COVs of the parameters. The full probabilistic model can then be obtained by synthesizing the local and global dependence models. The effectiveness of the proposed approach is demonstrated by comparing the generated samples with the test results. To illustrate the effect of the probabilistic dependence of the compressive constitutive parameters on the structural evaluation result, the nonlinear stochastic dynamic response analysis of a reinforced concrete frame is carried out. The results indicate that the probabilistic dependence of the constitutive parameters of concrete has a non-negligible effect on the structural response and reliability, and should be reasonably considered in practice.

Uncertainty quantification in inerter-based quasiperiodic lattices

Inerter-based periodic structures have attracted significant interest among the research community due to their wide range of applications. However, existing studies on quasiperiodic structures, especially with inerters, are limited. Therefore, this timely work investigates the dynamics of novel one-dimensional inerter-based quasiperiodic lattices with and without local resonators. The quasiperiodicity is introduced through the modulation of both spring and inerter properties resulting in the well-known Hofstadter-like butterfly spectrum having a fractal structure. Moreover, by considering the appropriate boundary conditions and many mass–spring-inerter subsystems, the bulk spectrum of the system is investigated demonstrating the existence of multiple edge states in lower and higher frequency ranges. The deterministic study showed that the effect of inertia amplification is to reduce the frequencies of the Hofstadter-like butterfly and to widen the low frequency band gaps. The second contribution of this work involves investigating the effect of parametric uncertainty for the acoustic-metamaterial-like chain based on Monte Carlo simulations. Moreover, to address the computational aspect of the problem, grey-box modeling using machine learning was performed. A Gaussian process model was trained on a limited dataset and was found to capture the stochastic responses of the lattices adequately. The statistical variation of parameters with different levels of uncertainty demonstrated significant effects on the Hofstadter-like butterfly, band gaps, corresponding edge states and frequency responses. The sensitivity of the dynamic behavior in quasiperiodic lattices to variabilities reveal the need to account for system uncertainties for their targeted performance as future vibration absorbers and energy harvesters. The study also paves the way to utilize the results as useful prior information for robust design optimization of real inerter-based quasiperiodic lattice devices.

A Reduced-Order Wiener Path Integral Formalism for Determining the Stochastic Response of Nonlinear Systems With Fractional Derivative Elements

Abstract A technique based on the Wiener path integral (WPI) is developed for determining the stochastic response of diverse nonlinear systems with fractional derivative elements. Specifically, a reduced-order WPI formulation is proposed, which can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the herein developed technique can determine, directly, any lower-dimensional joint response probability density function corresponding to a subset only of the response vector components. This is done by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in few only specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. An indicative numerical example is considered pertaining to a stochastically excited tuned mass-damper-inerter nonlinear system with a fractional derivative element. Comparisons with relevant Monte Carlo simulation data demonstrate the accuracy and computational efficiency of the technique.

Nonlinear stochastic dynamics of an array of coupled micromechanical oscillators

AbstractThe stochastic response of a multi‐degree‐of‐freedom nonlinear dynamical system is determined based on the recently developed Wiener path integral (WPI) technique. The system can be construed as a representative model of electrostatically coupled arrays of micromechanical oscillators, and relates to an experiment performed by Buks and Roukes. Compared to alternative modeling and solution treatments in the literature, the paper exhibits the following novelties. First, typically adopted linear, or higher‐order polynomial, approximations of the nonlinear electrostatic forces are circumvented. Second, for the first time, stochastic modeling is employed by considering a random excitation component representing the effect of diverse noise sources on the system dynamics. Third, the resulting high‐dimensional, nonlinear system of coupled stochastic differential equations governing the dynamics of the micromechanical array is solved based on the WPI technique for determining the response joint probability density function. Comparisons with pertinent Monte Carlo simulation data demonstrate a quite high degree of accuracy and computational efficiency exhibited by the WPI technique. Further, it is shown that the proposed model can capture, at least in a qualitative manner, the salient aspects of the frequency domain response of the associated experimental setup.

Profiles of Two JOMAE Associate Editors (The Fifth in a Continuing Series)

Profiles of Two JOMAE Associate Editors (The Fifth in a Continuing Series)

Stochastic Dynamic Response and Long-Term Settlement Performance of Superstructure–Underground Tunnel–Soil Systems Subjected to Subway-Traffic Excitation

The vibration impact force from the long-term operation of a subway affects the comfort of the residents living above the superstructure and the long-term settlement deformation of the tunnel foundation. A method for evaluating the dynamic responses of superstructure–tunnel systems is important, especially because of the randomness of vibration impact force. The coupling effect of the randomness of train-vibration excitation and the nonlinearity of the geotechnical properties that are subjected to dynamic action leads to challenges in the evaluation of the performance of superstructure–underground tunnel–soil systems under train vibration. In this study, a stochastic dynamic model of subway vibration-load excitation was established; then, the time histories of samples with rich probability characteristics in the same set system were generated. According to the nonlinear dynamic finite element analysis, several nonlinear dynamic responses of the deterministic superstructure–tunnel soil-foundation system were obtained. Finally, the probabilistic performance evolution of the superstructure–tunnel soil-foundation system was obtained by integrating the first-passage and probability density evolution theories, and the long-term deformation performance of the tunnel foundation was evaluated using time-varying reliability. This study presents a novel probabilistic method and a more objective performance index for the dynamic performance assessment of superstructure–underground tunnel–soil systems that are subjected to subway-traffic excitation.

Free vibration and random dynamic analyses for the composite cabin-like combined structure in aero-thermal environment

This work presents the numerical investigations on the modal behaviors and stochastic dynamics of a composite cabin-like combined structure considering the aero-thermal factor and various random loads based on the meshless approach. Following the thermo-elastic theory, the influence of the aero-thermal factor on the dynamic behaviors of the combined structure is included. The cabin-like combined structure is disassembled into the open elliptical, conical and cylindrical shells, and coupling conditions between adjacent subshells are imposed by the massless connection springs, in which the Hamilton's principle is adopted to establish free and stochastic vibration models of each subshell. The dynamic solutions for the combined structure system are calculated by employing the meshless technique combined with the pseudo excitation method (PEM). The numerical validity concerning meshless model is presented by performing several case studies related to the modal frequencies and stochastic responses. Additionally, some physical insights into the influences of aero-thermal effect, type of random loads, and geometric dimensions, etc., on the free as well as stochastic vibration properties of the composite cabin-like combined structure are provided.

A deep leaning method for dynamic vibration analysis of bridges subjected to uniform seismic excitation

The seismic response of a nonlinear, large-span cable-stayed bridge is a key problem in the assessment of structural safety. We propose an optimized deep learning network to study the random vibration of an uncertain bridge subjected to an earthquake. The dynamic formulas of the uncertain system were produced by transient analysis in the time domain to construct a numerical model comprising two functional modules: convolutional neural network (CNN) training with input rail irregularities and a long short-term memory (LSTM) layer for the bridge response time-history prediction. The LSTM cell was simulated by introducing the randomness of excitation and uncertain characteristics of parameters of the system into a portion of the cell, enabling the numerical model to convey the uncertainties of the bridge and obtained its stochastic response. The seismic response of railroad cable-stayed bridges under ground motion was calculated using the proposed method, taking into account the geometric nonlinearity of the bridges, and its accuracy and validity were verified. The effects of the distribution state of the seismic response samples, traveling wave effects, and intensity of the seismic spectrum were investigated, and the extremes of responses were analyzed based on Poisson's crossover assumption.