Stochastic Analysis Framework Research Articles

Multivariate copula-based framework for stochastic analysis of landslide runout distance

The increasing frequency of landslide disasters worldwide highlights the importance of accurate prediction of post-failure runout distances for effective risk management. However, the prediction of runout distances of gravitational mass flow remains challenging due to the inherent complex heterogeneities of geomaterials and their inter-correlated spatially varying mechanical properties. To address this challenge, this study proposes a novel multivariate stochastic method based on the combination of the Generalised Interpolation Material Point (GIMP) analysis and the multivariate copula-based approach. The method considers geotechnical uncertainties by simulating copula-based cross-correlated random fields from sparse field data and incorporating them into the GIMP analysis through Monte Carlo simulations (MCS). Two slope cases with similar geometries but different sources of probability information are presented to illustrate the effectiveness of the proposed method. The results show that considering the influence of multivariate random fields can significantly affect the post-failure analysis of landslides. Both slope cases show that over 40 % of all MCS samples exceed the deterministic case, which indicates that the current deterministic analysis notably underestimates the risk associated with large runout distances of landslides. This study highlights the necessity of considering the interdependency of spatial heterogeneity in post-failure modelling of landslides.

Machine learning aided uncertainty quantification for engineering structures involving material-geometric randomness and data imperfection

In real-world engineering, uncertainty is ubiquitous within material properties, structural geometry, load conditions, and the like. These uncertainties have substantial impacts on the estimation of structural performance. Furthermore, information or datasets in real life commonly contain imperfections, e.g., noise, outliers, or missing data. To quantify these impacts induced by uncertainties on structural behaviours and reduce the effects of data imperfections simultaneously, a machine learning-aided stochastic analysis framework is proposed. A novel supervised machine learning technique, namely the Capped Extended Support Vector Regression (CX-SVR) technique, is developed to effectively suppress the effects of outliers and noise in datasets. Its inherent convexity in optimization and capped strategy theoretically supports the accuracy of CX-SVR, especially in handling datasets with imperfections. Once the effective surrogate model is established, subsequent analyses, like sampling-based methods, can circumvent the cumbersome physical model, which is potentially the nest of computational burden and errors in engineering applications. The high robustness of the proposed approach can be summarized in four main aspects: unrestrictive selection of the system inputs and their statistical information, ‘perfect’ or ‘imperfect’ data, enough statistical information (including statistical moments, probability density functions, and cumulative distribution functions) of the system outputs, and physical problems from various engineering fields.

A Large-Scale Application of the Stochastic Approach for Estimating the Shear Strength of Natural Rock Discontinuities

Reliable shear strength determination of large in situ discontinuities is still a challenge faced by the rock mechanics field. This is principally due to the limited availability of surface roughness and morphology information of in situ discontinuities and the unresolved management of the ‘scale effect’ phenomenon. Recently, a stochastic approach for predicting the shear strength of large-scale discontinuities was established, encompassing random field theory, a semi-analytical shear strength model, and a stochastic analysis framework. A key aspect of the new approach is the application at field scale, thereby minimising or bypassing the scale effect. The approach has been validated at laboratory scale and an initial large-scale deterministic-based validation showed promising results. However, to date, no large-scale experimental-based validation has been undertaken. This paper presents the first rigorous application of the employed semi-analytical shear strength model and the stochastic approach on a 2 m-by-2 m discontinuity surface, with comparison of prediction to experimental shear strength data. The shear strength model was found to generally produce peak and residual predictions within a ± 10% relative error range, with good agreement between predicted and observed damage areas. It was observed that, applying the stochastic approach to seed traces with gradient statistics equivalent to that of the surface, produced predictions that closely resemble the experimental results. Whereas, predicting shear strength from different seed traces results in more variability of predictions, with many falling within ± 20% of the experimental data. The predictions of residual shear strength tended to be more accurate than peak shear strength.

Efficient stochastic analysis of unsaturated slopes subjected to various rainfall intensities and patterns

Rainfall infiltration poses a disastrous threat to the slope stability in many regions around the world. This paper proposes an extreme gradient boosting (XGBoost)-based stochastic analysis framework to estimate the rainfall-induced slope failure probability. An unsaturated slope under rainfall infiltration in spatially varying soils is selected in this study to investigate the influences of the spatial variability of soil properties (including effective cohesion c′, effective friction angle φ′ and saturated hydraulic conductivity ks), as well as rainfall intensity and rainfall pattern on the slope failure probability. Results show that the proposed framework in this study is capable of computing the failure probability with accuracy and high efficiency. The spatial variability of ks cannot be overlooked in the reliability analysis. Otherwise, the rainfall-induced slope failure probability will be underestimated. It is found that the rainfall intensity and rainfall pattern have significant effect on the probability of failure. Moreover, the failure probabilities under various rainfall intensities and patterns can be easily obtained with the aid of the proposed framework, which can provide timely guidance for the landslide emergency management departments.

PDEM-based stochastic analysis of a train-track-bridge system using dimension-reduced simulations of turbulent winds and track irregularities

Winds and track irregularities are inherently random in nature, but most of previous work did not fully consider this randomicity in computation of the dynamic response of train-track-bridge system. This study aims to develop an effective framework for stochastic analysis of the wind-train-track-bridge system based on the probability density evolution method. A dimension-reduced simulation scheme is applied since the involved system has high-dimensional random variables. Therefore, the representative samples of turbulent winds and track irregularities can be well expressed with only two random variables. In numerical examples, the accuracy of the framework is verified, and the influences of the mean wind velocity on the random vibration characteristics of the system are investigated. The results show that the dispersion of the lateral displacement of the bridge is larger than the vertical displacement, but both the vertical and lateral accelerations of the bridge have larger dispersions. The existence of lateral wind loads can reduce the dispersion of the lateral wheel–rail interaction forces of the wheel in the leeward direction. Considering the confidence levels of 75%, 85% and 95%, the train can run safely over the bridge in the wind velocity range of 0–25 m/s.

Intermittent fault detection for delayed stochastic systems over sensor networks

This paper is concerned with the intermittent fault (IF) detection problem for a class of linear discrete-time stochastic systems over sensor networks with constant time delay. By utilizing the lifting method, the distributed decoupled observers are proposed based on the output information of neighbor nodes and the node itself. In order to detect the appearing time and disappearing time of the IF, the truncated residuals are designed by introducing a sliding-time window. Furthermore, the IF detection and location thresholds are determined based on the hypothesis testing technique and the detectability of the IF is analyzed in the framework of stochastic analysis. Finally, a simulation example is presented to illustrate the effectiveness of the derived results.

Monte Carlo Simulation of Fractures Using Isogeometric Boundary Element Methods Based on POD-RBF

This paper presents a novel framework for stochastic analysis of linear elastic fracture problems. Monte Carlo simulation (MCs) is adopted to address the multi-dimensional uncertainties, whose computation cost is reduced by combination of Proper Orthogonal Decomposition (POD) and the Radial Basis Function (RBF). In order to avoid re-meshing and retain the geometric exactness, isogeometric boundary element method (IGABEM) is employed for simulation, in which the Non-Uniform Rational B-splines (NURBS) are employed for representing the crack surfaces and discretizing dual boundary integral equations. The stress intensity factors (SIFs) are extracted by M integral method. The numerical examples simulate several cracked structures with various uncertain parameters such as load effects, materials, geometric dimensions, and the results are verified by comparison with the analytical solutions.

Framework for fragility assessment of reinforced concrete portal frame subjected to elevated temperature

Despite realizing the importance of risk-based frameworks in fire engineering, current fire safety guidelines are largely prescriptive and over-conservative in nature, which do not consider the effect of uncertainties influencing the fire resistance of the structural systems. In this context, a probabilistic framework is presented to investigate vulnerability of a reinforced concrete (RC) portal frame subjected to fire. Here, a three-dimensional (3-D) nonlinear finite element (FE) model for the RC portal frame exposed to elevated temperature is developed, and transient thermo-mechanical analysis is carried out by duly incorporating temperature variation of thermal and mechanical properties of concrete and steel rebar. Probabilistic study is conducted by developing fragility curves based on limit state of deflection to quantify the vulnerability of the structural members exposed to fire loading. Parametric studies are conducted by varying the material properties of the RC portal frame to investigate the influence of uncertainties in the fire loading on its structural performance and fragility. The failure probability starts at a higher fire load density (450 MJ/m2) for the structure with higher concrete grade as compared to the relatively lower fire load density (200 MJ/m2) for lower grade. Therefore, the fire resistance of the RC structures has been recommended to be based on the strength of the member, rather than merely sizes of the members. The proposed stochastic analysis framework, thereby, provides a rational approach to improve the performance-based fire design of RC structures at member and structure levels by identifying the parameters affecting the structural fire resistance.

Bounds in Total Variation Distance for Discrete-time Processes on the Sequence Space

Let $\mathbb {P}$ and $\widetilde {\mathbb {P}}$ be the laws of two discrete-time stochastic processes defined on the sequence space $S^{\mathbb N}$, where S is a finite set of points. In this paper we derive a bound on the total variation distance $\mathrm {d}_{\text {TV}}(\mathbb {P},\widetilde {\mathbb {P}})$ in terms of the cylindrical projections of $\mathbb {P}$ and $\widetilde {\mathbb {P}}$. We apply the result to Markov chains with finite state space and random walks on $\mathbb {Z}$ with not necessarily independent increments, and we consider several examples. Our approach relies on the general framework of stochastic analysis for discrete-time obtuse random walks and the proof of our main result makes use of the predictable representation of multidimensional normal martingales. Along the way, we obtain a sufficient condition for the absolute continuity of $\widetilde {\mathbb {P}}$ with respect to $\mathbb {P}$ which is of interest in its own right.

Computational structural analysis of composites with spectral-based stochastic multi-scale method

Composite materials and structures may be characterized at different length scales, ranging from the micro-scale at the fiber–matrix level, meso-scale at the lamina–laminate level, to structural macro-scale. However, uncertainties in material properties and geometric parameters due to manufacturing, defects and assembly processes may occur at various length scales. This paper presents a computational framework for stochastic analysis of composites with consideration of stochastic parameters at micro- and meso-scales. The novelty of the proposed framework is the integration of the spectral stochastic finite element method and asymptotic homogenization method within a finite element technique, which was implemented through $$\hbox {ABAQUS}^{\circledR }$$ . This spectral stochastic homogenization method efficiently predicts the propagation of uncertainties from the constituent to ply levels. The derived probability distributions of effective properties were verified by Monte Carlo simulation. Another novelty is the study of the influence of stochastic parameters at both micro-scale and meso-scale on the failure prediction of composite structures, without assumptions of probabilistic characteristic of ply properties commonly used in a single-scale stochastic analysis. The up-scaled uncertainties combined with other randomness at meso-scale (strength properties and ply orientations) are provided as the input of meso-scale stochastic strength analysis of a quasi-isotropic laminate based on classical lamination theory (CLT) and ply discount. The probability distribution of first-ply failure and ultimate failure loads are obtained and their sensitivity factors with respect to input variations are presented.

Spectral stochastic isogeometric analysis of linear elasticity

The stochastic isogeometric analysis of linear elasticity problem is investigated in this study. The spectral stochastic analysis is introduced into isogeometric analysis (IGA), and a novel, yet robust, stochastic analysis framework, namely the spectral stochastic isogeometric analysis (SSIGA), is freshly proposed. Unlike traditional numerical solutions of the Karhunen–Loève (K-L) expansion, the non-uniform rational B-spline (NURBS) and T-spline basis functions are employed within the proposed framework of SSIGA, so the random fields acting on a continuous physical medium with complex geometry can be handled in an appropriate, physically feasible and efficient fashion. The polynomials chaos expansion (PCE) is implemented to represent the stochastic structural response (e.g., displacement, strain and stress), such that all corresponding statistical characteristics (e.g., mean and standard deviation) can be robustly acquired. Furthermore, by utilizing the nonparametric statistical analysis, both probability density functions (PDFs) and cumulative distribution functions (CDFs) of concerned structural displacements and stresses can be effectively established. Within the framework of IGA, by meticulously implementing the concept of the higher-order k-refinement, the proposed SSIGA provides a more legitimate and efficient stochastic computational approach for modern engineering structures which are complicated by both spatially dependent uncertainties and complex geometries.

Impacts of Voltage Control Methods on Distribution Circuit’s Photovoltaic (PV) Integration Limits

The widespread integration of photovoltaic (PV) units may result in a number of operational issues for the utility distribution system. The advances in smart-grid technologies with better communication and control capabilities may help to mitigate these challenges. The objective of this paper is to evaluate multiple voltage control methods and compare their effectiveness in mitigating the impacts of high levels of PV penetrations on distribution system voltages. A Monte Carlo based stochastic analysis framework is used to evaluate the impacts of PV integration, with and without voltage control. Both snapshot power flow and time-series analysis are conducted for the feeder with varying levels of PV penetrations. The methods are compared for their impacts on (1) the feeder’s PV hosting capacity; (2) the number of voltage violations and the magnitude of the largest bus voltage; (3) the net reactive power demand from the substation; and (4) the number of switching operations of feeder’s legacy voltage support devices i.e., capacitor banks and load tap changers (LTCs). The simulation results show that voltage control help in mitigating overvoltage concerns and increasing the feeder’s hosting capacity. Although, the legacy control solves the voltage concerns for primary feeders, a smart inverter control is required to mitigate both primary and secondary feeder voltage regulation issues. The smart inverter control, however, increases the feeder’s reactive power demand and the number of LTC and capacitor switching operations. For the 34.5-kV test circuit, it is observed that the reactive power demand increases from 0 to 6.8 MVAR on enabling Volt-VAR control for PV inverters. The total number of capacitor and LTC operations over a 1-year period also increases from 455 operations to 1991 operations with Volt-VAR control mode. It is also demonstrated that by simply changing the control mode of capacitor banks, a significant reduction in the unnecessary switching operations for the capacitor banks is observed.

Industrial applications of the PetriDotNet modelling and analysis tool

• PetriDotNet, a model checking and stochastic analysis framework is introduced. • Our formal analyses are applicable and beneficial, as demonstrated by examples. • Real-life case studies from railway, automotive and nuclear industries are presented. Since their invention, Petri nets have provided modelling and analysis methods to support the design of correct, reliable and robust systems. This motivated our work to develop PetriDotNet , a Petri net editor and analysis tool. In this paper we overview the supported modelling formalisms and the analysis methods included in PetriDotNet . Next, we present eight different industrial case studies, demonstrating the wide variety of scenarios where Petri nets and PetriDotNet can help the design, development and analysis of industrial systems. Our original goal with PetriDotNet was to provide an educational tool to our students, however our efforts led to a framework being able to serve both academic and industrial needs.

On Estimation and Sensitivity Analysis of Distribution Circuit's Photovoltaic Hosting Capacity

A high penetration of residential photovoltaic (PV) panels can potentially cause a number of operational issues in the distribution circuit, necessitating the requirement to determine the largest PV generation, a distribution circuit can accommodate. This paper presents a Monte Carlo-based hourly stochastic analysis framework to determine a circuit's PV accommodation limit or PV hosting capacity. First, a mathematical formulation of the hosting capacity problem for bus overvoltage concerns is presented, followed by the proposed analysis approach. In the proposed approach, the Monte Carlo method is used to simulate the scenarios of probable PV locations and sizes while the hourly analysis framework is used to include the daily variations in load and PV generation. Next, a method to evaluate the percentage accuracy of the hosting capacity results is presented. The sensitivity of PV hosting results to multiple parameters related to both feeder characteristics and simulation method are evaluated as well. The utility of the proposed framework is presented using an actual 12.47-kV distribution circuit. In sum, by presenting a PV hosting analysis method, accuracy assessment framework, and sensitivity analysis approach, this paper provides a thorough understanding of the PV hosting capacity problem.

Interval Fractile Levels for Stationary Stochastic Response of Linear Structures With Uncertainties

Abstract In the framework of stochastic analysis, the extreme response value of a structural system is completely described by its CDF. However, the CDF does not represent a direct design provision. A more meaningful parameter is the response level which has a specified probability, p, of not being exceeded during a specified time interval. This quantity, which is basically the inverse of the CDF, is referred to as a fractile of order p of the structural response. This study presents an analytical procedure for evaluating the lower bound and upper bound of the fractile of order p of the response of linear structures, with uncertain stiffness properties modeled as interval variables subjected to stationary stochastic excitations. The accuracy of the proposed approach is demonstrated by numerical results concerning a wind-excited truss structure with uncertain Young’s moduli.

Integrating aerial geophysical data in multiple-point statistics simulations to assist groundwater flow models

The process of accounting for heterogeneity has made significant advances in statistical research, primarily in the framework of stochastic analysis and the development of multiple-point statistics (MPS). Among MPS techniques, the direct sampling (DS) method is tested to determine its ability to delineate heterogeneity from aerial magnetics data in a regional sandstone aquifer intruded by low-permeability volcanic dykes in Northern Ireland, UK. The use of two two-dimensional bivariate training images aids in creating spatial probability distributions of heterogeneities of hydrogeological interest, despite relatively ‘noisy’ magnetics data (i.e. including hydrogeologically irrelevant urban noise and regional geologic effects). These distributions are incorporated into a hierarchy system where previously published density function and upscaling methods are applied to derive regional distributions of equivalent hydraulic conductivity tensor K. Several K models, as determined by several stochastic realisations of MPS dyke locations, are computed within groundwater flow models and evaluated by comparing modelled heads with field observations. Results show a significant improvement in model calibration when compared to a simplistic homogeneous and isotropic aquifer model that does not account for the dyke occurrence evidenced by airborne magnetic data. The best model is obtained when normal and reverse polarity dykes are computed separately within MPS simulations and when a probability threshold of 0.7 is applied. The presented stochastic approach also provides improvement when compared to a previously published deterministic anisotropic model based on the unprocessed (i.e. noisy) airborne magnetics. This demonstrates the potential of coupling MPS to airborne geophysical data for regional groundwater modelling.

A stochastic analysis framework for a steel frame structure using wireless sensor system measurements

This paper presents a stochastic analysis framework for estimating the system-level first-passage probability of the structural responses of multi-degree-of-freedom structural systems based on experimentally measured uncertainties. The uncertainties are quantified by comparing the measured structural responses using a wireless sensor system and the predicted responses from an analytical model. The wireless sensor network is designed based on a modular design method, and the experimental program details for the measurement of structural responses are provided using the developed wireless sensor network. This framework employs a Monte Carlo simulation (MCS)-based first-passage probability estimation technique in which a structural dynamic analysis is performed in each simulation realization. The framework is applied to a 16-story steel frame structure, and the first-passage probability of 16 locations and the series system passage probability of the entire system have been estimated. The effect of the dependency between the structural responses is considered, and the improvements that need to be made to the presented framework in the future works are discussed.

The influence of random element displacement on DOA estimates obtained with (Khatri-Rao-)root-MUSIC.

Although a wide range of direction of arrival (DOA) estimation algorithms has been described for a diverse range of array configurations, no specific stochastic analysis framework has been established to assess the probability density function of the error on DOA estimates due to random errors in the array geometry. Therefore, we propose a stochastic collocation method that relies on a generalized polynomial chaos expansion to connect the statistical distribution of random position errors to the resulting distribution of the DOA estimates. We apply this technique to the conventional root-MUSIC and the Khatri-Rao-root-MUSIC methods. According to Monte-Carlo simulations, this novel approach yields a speedup by a factor of more than 100 in terms of CPU-time for a one-dimensional case and by a factor of 56 for a two-dimensional case.

Convolutionless Nakajima-Zwanzig equations for stochastic analysis in nonlinear dynamical systems.

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection-reaction problems.

When can the environmental profile and emissions reduction be optimised independently of the pollutant level?

Consider a model for optimal timing of a policy measure which changes the emission rate, e.g. trading off the cost of reduction against the time-additive aggregate of environmental damage, the disutility from the pollutant stock the infrastructure contributes to. Intuitively, the optimal timing for an infinitesimal pollution source should reasonably not depend on its historical contribution to the stock, as this is negligible. Dropping the size assumption, we show how to reduce the minimisation problem to one not depending on the history of , under linear evolution and suitable linearity or additivity conditions on the damage functional. We employ a functional analysis framework which allows for delay equations, non-Markovian driving noise, a choice between discrete and continuous time, and a menu of integral concepts covering stochastic calculi less frequently used in resource and environmental economics. Examples are given under the common (Markovian Itô) stochastic analysis framework.