Crude Monte Carlo Simulation Research Articles

A dynamic importance function for accidental scenarios generation by RESTART in the computational risk assessment of cyber-physical infrastructures

The Computational Risk Assessment (CRA) of Cyber-Physical Systems (CPSs) calls for the analysis of accidental scenarios emerging from the complexities and interdependencies typical of CPSs. Generating these scenarios via crude Monte Carlo Simulation (MCS) is impractical due to the high computational demand of simulation codes of CPSs, considering the combinatorial number of possible scenarios. In this paper, we tailor the use of Repetitive Simulation Trials After Reaching Thresholds (RESTART), a rare-event simulation method of literature, to efficiently generate relevant accidental scenarios. The tailored RESTART is guided by a dynamic Importance Function (IF) originally introduced here to dynamically characterize the relevance of the scenarios with reference to the current topology of the CPS and the susceptibility of its components. Two case studies of increasing complexity are considered: a single power grid and a CPS consisting of an Integrated Power and Telecommunication (IP&TLC) infrastructure. Results show that RESTART mines out more relevant scenarios than crude MCS for a number of different IFs based on vulnerability metrics of literature, and thus particularly efficiently when the novel IF is adopted.

Seismic reliability analysis using Subset Simulation enhanced with an explorative adaptive conditional sampling algorithm

Reliability analysis of structures under earthquake loading represents a significant engineering challenge. This is due to the required and rather numerically involving non-linear dynamic analysis, the large computational burden when targeting small failure probabilities, and the synthetic earthquake model representation that may contain thousands of random variables. Subset Simulation is an efficient reliability analysis technique that can handle the challenge of a high-dimensional space with a reduced number of structural analysis calls compared to crude Monte Carlo Simulation. In this contribution, firstly, we investigate the conditions for which Subset Simulation performs efficiently. Thereafter we propose an enhancement to the existing Subset Simulation schemes that shows significant potentials for enhancing the strategy for the starting of the Markov Chain Monte Carlo simulations whenever a new level is reached in the Subset Simulation. Finally, the information gathered from the simulations is investigated to verify that Subset Simulation provides meaningful results from a physical point of view.

Deterministic and stochastic flexural behaviors of laminated composite thin-walled I-beams using a sinusoidal higher-order shear deformation theory

In this study, a novel sinusoidal higher-order shear deformation thin-walled beam theory is presented to examine the effects of material properties and external load uncertainty on static responses of laminated composite thin-walled beams with open sections. The solution for the deterministic flexural analysis is based on Hamilton’s principle and Ritz-type exponential shape function series. Several mechanical parameters of laminated composite materials are randomized and plugged into the beam solver to investigate the thin-walled beam’s stochastic flexural behaviors. The computational cost and accuracy of the polynomial chaos expansion (PCE) method with both projection and linear regression approaches are presented and evaluated by comparing its results with crude Monte Carlo simulation (MCS). This comparison allows for a thorough assessment of the PCE method’s performance. Additionally, a sensitivity analysis is conducted to compare the relative significance of the uncertainty in material properties and loads on the stochastic responses. The supervised training of the artificial neural network based on the MCS beam data is also conducted and compared to the PCE and MCS methods. The findings about the stochastic outputs are introduced in various statistical metrics and illustrations to demonstrate the influences of material properties’ randomness on different laminated composite thin-walled beam configurations.

Stochastic analysis of thin beams and plates incorporating von-Kármán nonlinear strains using meshless method

The current paper proposes methods for stochastic meshless analysis of thin beams and plates, by considering von-Kármán nonlinear strains. Here, Young’s modulus is assumed to have a spatial variation over the domain and is modeled as a homogeneous random field. Further, it is discretized utilizing moving least square based shape functions. Gaussian and lognormal properties are presumed to model the randomness related to Young’s modulus. Element-free Galerkin method is used as the meshless tool. The present study suggests employing perturbation method if the quantities of interest are limited to the first or second probabilistic moments of the responses. The nonlinear equations in the proposed perturbation formulations are solved using Newton-Raphson iterative scheme for finding out the deterministic part of the response, as well as its initial two derivatives with respect to random variables. Additionally, if there is an interest in higher probabilistic moments, probability density functions, cumulative distribution functions, and so forth, the study proposes a method using high-dimensional model representation (HDMR) in the meshless framework. In HDMR, the nonlinear connection between input variables and the output response is represented in terms of hierarchical correlated function expansions. These hierarchical functions are approximated over the response values evaluated using deterministic analysis on selected number and combinations of control points of random variables involved. Lagrange interpolation method is employed for the construction of response surfaces. Unlike Monte Carlo simulation (MCS) which needs numerous deterministic nonlinear analyses to be performed during each simulation, HDMR requires only a few deterministic nonlinear analyses to construct the component functions. Hence, simulations performed on the reduced order models of high-dimensional response can be used to estimate the probabilistic moments as well as density functions and cumulative distributions of response, with high computational efficiency compared to MCS. The probabilistic moments and distributions produced for a range of coefficient of variation of input random field up to 30%, for both the normal and lognormal input random fields are compared with those provided by crude MCS on the system of equations and are observed to be matching well.

Data-driven mid-air collision risk modelling using extreme-value theory

Mid-air collision risk estimation is crucial for maintaining aviation safety and improving the efficiency of air traffic procedures. This paper introduces a novel, data-driven methodology for estimating the probability of mid-air collisions between aircraft by combining Monte Carlo simulation and the Peaks Over Threshold approach from Extreme Value Theory. This innovative approach has substantial advantages over traditional methods. Firstly, it reduces the number of assumptions about the traffic flow compared to traditional analytical methods. In fact, data-driven techniques require fewer assumptions, as they inherently capture the structures of the traffic flow within the underlying data. Secondly, it converges faster than methods based on crude Monte Carlo simulation. Notably, by employing Extreme Value Theory, this approach enables efficient evaluation of low-probabilities, which are commonly found in collision risk modelling. The effectiveness of the proposed methodology is demonstrated through estimating the probability of a mid-air collision in a real-world practical example. The case study investigates the risk of collisions between departures and go-arounds in the terminal airspace at Zurich Airport, highlighting the potential for improved safety and efficiency in air traffic management.

Model order reduction in subset simulations using the proper orthogonal decomposition

AbstractThe crude Monte Carlo method is computationally expensive. Hence, incorporating model order reduction methods enabling reliability analysis for high‐dimensional problems is necessary. However, this strategy may result in an inaccurate estimation of the probability of failure for rare events for two reasons. First, the model order reduction, represented by the proper orthogonal decomposition (POD) here, requires response information in the form of snapshots a priori. To capture the essential nonlinear response behavior, we propose to update the proper orthogonal modes using extreme events. Second, the crude Monte Carlo simulation requires many samples to estimate low failure probabilities reliably. To this end, subset simulation found wide application in reliability analysis to reduce computational effort. Following this strategy, the proposed samples gradually move toward the failure region. Thus, incorporating updates of the modes is particularly promising in evaluating samples from the current subset region. This contribution shows the computational efficiency of POD within subset simulations. We then propose to leverage the estimation of the probability of failure by updating the modes within each subset.

Physical–statistical learning in resilience assessment for power generation systems

Upswing in extreme weather conditions and natural disasters in conjunction with the relentless penetration of the intermittent renewable energy have brought resilience of the power generation systems into sharp relief. In this study, we adopt a high-order physical model to characterize the full-detail sub-transient behaviors in synchronous generator dynamics, and consequently utilize basin stability (BS) to quantify system resilience against potentially large perturbations. This high-fidelity model has not been extensive probed in estimate of BS, largely owing to the tremendous computational overhead involved. We conduct sensitivity analysis to pick out the most critical system states, whose perturbation exerts huge impact and hence are sensitive on BS or system resilience. Following this, we develop a diversity-enhanced active learning framework to sequentially identify the informative perturbed states, which will be further evaluated by the high-fidelity sub-transient model. This approach only incurs a paltry of simulation effort compared to the crude Monte Carlo simulation but with comparable accuracy on BS estimation.

Stochastic vibration and buckling analysis of functionally graded sandwich thin-walled beams

Stochastic vibration and buckling analysis of functionally graded sandwich thin-walled beams with I-section based on the first-order shear deformation theory is for the first time proposed in this paper. The material properties of beams in both web and two flanges are assumed to be continuously varied in its thickness. Additionally, the constituent material properties are randomly changed according to the lognormal distributions. These stochastic variabilities are then propagated to the stochastic responses of the thin-walled beam through a beam solver with hybrid series-type approximation functions. To achieve efficient evaluations for stochastic responses including natural frequencies and critical buckling loads, polynomial chaos expansion (PCE) based surrogate model is developed. The efficiency and accuracy of PCE’s results are assessed by comparing with those of crude Monte Carlo simulation. Sensitivity analysis is carried out to compare the importance of the uncertainty in material properties to stochastic responses. New results reported in this paper can be interesting benchmarks for scientific and engineering community in the future.

An efficient Kriging model-based importance sampling method for estimating the failure probability-based parameter global sensitivity index with uncertain distribution parameters

The failure probability-based parameter global sensitivity index measures the importance of each uncertain distribution parameter on the uncertainty of failure probability. The crude Monte Carlo simulation for estimating the failure probability-based parameter global sensitivity index involves a complicated nested triple-loop process, and the huge amount of computational burden restricts the application of the failure probability-based parameter global sensitivity. For efficiently analyzing the failure probability-based parameter global sensitivity index, this paper proposes an efficient Kriging model-based importance sampling method. First, the standard normal auxiliary variables are inducted by using the equivalent probability transformation, in which the equivalent limit state function with respect to the uncertain distribution parameters and the standard normal auxiliary variables are constructed, so that the importance sampling samples of uncertain distribution parameters and the standard normal auxiliary variables can be generated based on the equivalent limit state function. Secondly, based on the importance sampling technique, the formula of single-loop method for estimating the failure probability-based parameter global sensitivity index is derived in which the importance sampling samples can be reused to obtain all uncertain distribution parameters' failure probability-based global sensitivity indices. Thirdly, the equivalent probability transformation is used again to transform the importance sampling samples in the high-dimensional space into the relatively low-dimensional input space, and the Kriging model of the original limit state function with relatively low-dimensional input space is adaptively constructed to recognize the states (failure or safety) of all high-dimensional importance sampling samples. Then, the failure probability-based parameter global sensitivity indices can be efficiently estimated. The results of three case studies verify the accuracy and efficiency of the proposed method.

Evaluation of sampling techniques for probabilistic structural integrity assessment of steam generator tubing

Steam generator (SG) tubes for nuclear rectors serve as both pressure and containment boundaries. Rupture of a SG tube is considered a significant event from the perspective of safety and potential for release of radiation. Therefore, SG tubes are designed and operated at high reliability to maintain the pressure boundary's structural integrity. Whereas probabilistic approaches become more widely used in the operational assessment to adequately account for uncertainties from various sources, the high reliability level exerts a high demand for computational efforts, which sometimes may be beyond present computational capabilities. Using the crude Monte Carlo simulation as the baseline approach, this paper investigates several variance reduction techniques that reduce the computational effort for the desired small failure probability of a SG tube rupture. It is shown that Latin hypercubic sampling, directional simulation and subset simulation can improve the accuracy of the failure probability estimate, and directional simulation can improve both accuracy and efficiency dramatically. The probabilistic operational assessment of a specific example shows the calculated failure frequency satisfies the probabilistic structural integrity performance criteria (SIPC) under the normal operating condition, which is consistent with the results when deterministic SIPC is applied.

EMCS-SVR: Hybrid efficient and accurate enhanced simulation approach coupled with adaptive SVR for structural reliability analysis

In structural reliability analysis, robust and efficient sampling methods that address low failure probabilities are vital challenges. In this paper, a novel dynamical adaptive enhanced simulation method coupled with support vector regression (SVR) is proposed. Firstly, a more general and efficient approximation formula is proposed as an improved scheme. Furthermore, a dynamical adaptive simulation strategy for Monte Carlo simulation and an active training methodology basis SVR are developed. The dynamical active region for improving the efficiency and robustness of structural reliability analysis is applied for training the SVR models which are utilized for accurate estimating the failure probability by simulation methods. Analytical methods and crude Monte Carlo simulation are used for comparison, validation and discussion with the proposed hybrid simulation method using four numerical examples and four engineering problems. Through coupling SVR with dynamical active region, an accurate failure probability prediction with robust and low-computational cost is achieved. The proposed adaptive strategy applied in hybrid enhanced simulation approaches provided the accurate results with low-computational burden and these hybrid methods are robust than the analytical approaches. The proposed methods have shown strong​ capability for application in engineering problems with complex nonlinear performance functions.

Quantifying rare events in spotting: How far do wildfires spread?

Spotting refers to the transport of burning pieces of firebrand by wind which, at the time of landing, may ignite new fires beyond the direct ignition zone of the main fire. Spot fires that occur far from the original burn unit are rare but have consequential ramifications since their prediction and control remains challenging. To facilitate their prediction, we examine three methods for quantifying the landing distribution of firebrands: crude Monte Carlo simulations, importance sampling, and large deviation theory (LDT). In particular, we propose an LDT method that accurately and parsimoniously quantifies the low probability events at the tail of the landing distribution. In contrast, Monte Carlo and importance sampling methods are most efficient in quantifying the high probability landing distances near the mode of the distribution. However, they become computationally intractable for quantifying the tail of the distribution due to the large sample size required. We also show that the most probable landing distance grows linearly with the mean characteristic velocity of the wind field. Furthermore, defining the relative landed mass as the proportion of mass landed at a given distance from the main fire, we derive an explicit formula which allows computing this quantity as a function of the landing distribution at a negligible computational cost. We numerically demonstrate our findings on two prescribed wind fields.

A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge

Barrier options based upon the extremum of more than one underlying prices do not allow for closed-form pricing formulas, and thus require numerical methods to evaluate. One example is the autocallable structured product with knock-in feature, which has gained a great deal of popularity in the recent decades. In order to increase numerical efficiency for pricing such products, this paper develops a semi-analytic valuation algorithm which is free from the computational burden and the monitoring bias of the crude Monte Carlo simulation. The basic idea is to combine the simulation of the underlying prices at certain time points and the exit (or non-exit) probability of the Brownian bridge. In the literature, the algorithm was developed to deal with a single-asset barrier option under the Black–Scholes model. Now we extend the framework to cover two-asset barrier options and autocallable product. For the purpose, we explore the non-exit probability of the two-dimensional Brownian bridge, which has not been researched before. Meanwhile, we employ the actuarial method of Esscher transform to simplify our calculation and improve our algorithm via importance sampling. We illustrate our algorithm with numerical examples.

Probabilistic analysis of long-term loss incorporating maximum entropy method and analytical higher-order moments

Quantifying economic losses of civil infrastructures subjected to various hazards under a life-cycle context is of vital importance for risk assessment and management. In previous studies, the expected long-term loss has been widely applied as a standard decision criterion during the life-cycle analysis. However, the expectation may not be informative enough to illustrate uncertainties associated with the long-term loss. Therefore, the higher-order moments and the probability distribution should be investigated. In this paper, a probabilistic analysis framework is proposed to construct the probability density function and cumulative distribution function of long-term loss by assessing the analytical statistical moments. The stochastic renewal process is utilized to assess the long-term loss by considering uncertainties associated with stochastic occurrence and frequency of the hazards. Based on the maximum entropy method, the proposed approach shows superior efficiency to assess the probability distribution of long-term loss than crude Monte Carlo simulation. The probability distribution can be essential information for decision-making process of risk management. An illustrative example is investigated to show the probability density function of long-term loss of civil infrastructure subjected to hurricane hazards. A good agreement of results obtained by the proposed approach and Monte Carlo simulation has verified the accuracy and effectiveness of the proposed method.

Contribution to efficient structural safety assessments: A comparative analysis of computational schemes

The design of structural elements requires the use of advanced analysis techniques that are capable to closely model reality throughout their expected lifetime. However, the use of stochastic principles might be necessary to address the impact of uncertainties in structural engineering. Probability-based safety methods, as Crude Monte-Carlo simulations, can evaluate structural safety with high precision in exchange of numerous simulations. However, for complex problems, Crude Monte-Carlo simulations can demand significant computational power and be very time-consuming. Sampling (or variance reducing) techniques – as Importance Sampling and Subset Sampling – can minimise such drawbacks and improve computational efficiency. This paper investigates the potential offered by a newly developed R-package built in the programming language R, to evaluate the computational performance of these sampling techniques. To this, these techniques are compared in terms of accuracy and computational efficiency. This investigation shows that both techniques are capable to generate a wide range of solutions with high precision and satisfactory computational efficiency. Finally, this study explores the potential of using the newly developed R-package to select probability-based safety methods based on the features of reliability problems. The R-package offers a robust platform to conduct and compare different safety assessments of complex structural problems.

Dynamic statistical responses of gear drive based on improved stochastic iteration method

Mechanical equipment is always influenced by environmental factors or random excitations, such as random collision, rock fall, sea wave and wind force. It is significant to analyze the statistical information of the system performance for the operation security and kinematic accuracy. Considering time-varying stiffness and nonlinearity, a stochastic iteration method is proposed for dynamic responses in the gear drive subjected to deterministic and stochastic excitations. Also, the convergence analysis is investigated. Considering the parametric correlation, the improved iterative schemes are analyzed for the mathematical expectation and standard deviation of any multi-stage gear drive with the backlash and mesh stiffness, respectively. Also, covariance and correlation coefficient are investigated to represent the dynamic relationship between each gear pair. The crude Monte Carlo simulation is performed to provide benchmark results, and the data of simulation have good coincidence with the iteration results. It is shown that auxiliary hole of gear drive can reduce the fluctuation of stochastic response in operation. The reasonable modification gear can decrease the dispersion of system response at the transient and stable stage.

Stochastic isogeometric analysis on arbitrary multipatch domains by spline dimensional decomposition

This paper presents a new stochastic method by integrating spline dimensional decomposition (SDD) of a high-dimensional random function and isogeometric analysis (IGA) on arbitrary multipatch geometries to solve stochastic boundary-value problems from linear elasticity. The method, referred to as SDD-mIGA, involves (1) analysis-suitable T-splines with significant approximating power for geometrical modeling, random field discretization, and stress analysis; (2) Bézier extraction operator for isogeometric mesh refinement; and (3) a novel Fourier-like expansion of a high-dimensional output function in terms of measure-consistent orthonormalized splines. The proposed method can handle arbitrary multipatch domains in IGA and uses standard least-squares regression to efficiently estimate the SDD expansion coefficients for uncertainty quantification applications. Analytical formulae have been derived to calculate the second-moment properties of an SDD-mIGA approximation for a general output random variable of interest. Numerical results, including those obtained for a 54-dimensional, industrial-scale problem, demonstrate that a low-order SDD-mIGA is capable of efficiently delivering accurate probabilistic solutions when compared with the benchmark results from crude Monte Carlo simulation.

Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method

In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions.

Stochastic Analysis of Lubrication in Misaligned Journal Bearings

Abstract Misalignment is common in hydrodynamic journal bearings and the causes of it can be diversified, making the lubrication performance exhibits stochasticity. Lubricant viscosity often heavily depends on temperature, which may vary during service and result in unexpected deviations. This article analyzes the stochastic lubrications of a cylindrical hydrodynamic journal bearing with misalignment under uncertainties. The stochastic Reynolds equation governing the misaligned journal bearing is discretized by the polynomial chaos expansion (PCE), an efficient uncertainty tracking tool, and then solved by the finite difference method to obtain sampled lubrication. The crude Monte Carlo simulation is used to verify the performance of the PCE frame. Various critical lubrication performance parameters are studied comprehensively by the ensemble mean, standard deviation, probability density function, and cumulative distribution function. Insightful inspections are provided on the stochastic results, and it is found that the misalignment and different stochastic parameters may cause significant effects on the lubrication performance. The new findings in the present study will guide the robust design and analysis of general hydrodynamic journal bearings.

Reliability Analysis of an Embankment Built and Reinforced in Soft Ground Using LE and FE

A coupling method is presented to integrate well-known geotechnical modeling commercial programs with a reliability one. Coupling enables us to use transformation methods, such as the first-order reliability method (FORM), to evaluate the reliability index (β) of a model, via the limit equilibrium (LEM) or the finite-element method (FEM). It was applied to a case study of a stage-constructed embankment on soft ground by considering two conditions, unreinforced and reinforced, to investigate its probabilistic stability. In addition, the value of β associated with a postconstruction settlement was acquired, and it showed one possibility that the coupling method is able to provide. The influence of the uncertainty level on reliability analysis was also analyzed. The outcomes were compared to the crude Monte Carlo simulation. As a result, the proposed approach was satisfactory, and the advantage of using it is presented; with a much lower computational cost than simulation methods, evaluation time may be up to 13.8 times faster than usual with great precision. Moreover, this approach can apply to other commercial programs widely used in practice.