Crude Monte Carlo Simulation Research Articles

Ring simulation: A novel simple and efficient simulation method for structural reliability analysis

In this paper, a novel simulation method, referred to as “ring simulation”, is proposed. The method estimates the failure probability only using the information from the importance interval of the radius, i.e., the hyper-spherical ring, where the contributions to the failure probability are dominant. A stepwise computing procedure is proposed to determine the importance interval and simultaneously obtain the failure probability, in which samples are uniformly generated in the importance interval. The variability of the proposed method for the evaluation of the failure probability is also investigated. The accuracy, efficiency, and robustness of the proposed method are demonstrated through four examples involving the multiple design points, existence of local minima, small failure probability, and highly nonlinear, implicit, and high-dimensional limit state functions. The results reveal that the proposed method is almost as simple and applicable as crude Monte Carlo simulation method, but more efficient. Compared with directional simulation, subset simulation, and weighted average simulation, the proposed method has shown its advantage.

A Bayesian surrogate constitutive model to estimate failure probability of elastomers

To calculate the uncertainty in the failure probability of elastomeric materials, a parametric and a non-parametric Bayesian-based stochastic constitutive model were evaluated. (i) A Bayesian linear regression calibration technique is created based on the Carroll model to construct a probabilistic hyper-elastic model in the parametric approach. The model was then calibrated using two methods: Maximum Likelihood Estimation (MLE) and Maximum a Posteriori (MAP) estimation, with the results compared. (ii) The Gaussian process (GP) is used in non-parametric hyper-parameters of the radial basis kernel computed using the limited-memory Broyden–Fletcher–Goldfarb–Shanno technique. Both models were trained and verified with regard to two sets of our experiments on silicon- and polyurethane-based elastomers to demonstrate their capabilities in modeling uncertainty propagation. Finally, failure probability analysis was carried out for these data sets using First Order Reliability Method (FORM) analysis and Crude Monte Carlo (CMC) simulation, with a limit state function based on the stochastic constitutive model at the failure point. Sensitivity analysis is also used to demonstrate the importance of Carroll model parameters in predicting failure likelihood. The results show that the parametric approach has great agreement with experimental data, not only for uncertainty quantification and model calibration, but also for calculating the failure probability of hyperelastic materials.

Reliability of an offshore wind turbine with an uncertain S-N curve

Probabilistic methods in wind turbine design are becoming more important in order to achieve a higher and more economic utilization of the structural resources. A design is thereby evaluated through its reliability by taking uncertainties into account. The influence of uncertainties emerging specifically within the fatigue limit state can be essential on a structure’s reliability. For that reason, this study investigates two different sources for uncertainties within probabilistic fatigue design. The influence of a reduced approach, where the random behavior of a structural failure is only represented through an uncertain critical Miner sum, is compared to an approach, where also the S-N curve is subjected to uncertainties. For this, an offshore wind turbine with monopile support structure is investigated. The reliability analysis is performed through crude Monte Carlo simulations which utilize a Gaussian Process regression model in order to determine the wind turbine’s structural response. The results show that both sources of uncertainties, the critical Miner sum and the S-N curve, contribute differently to the overall reliability. Uncertainties of the critical Miner sum have a rather small influence, whereas the implementation of an uncertain S-N curve leads to a noticeable increase of the reliability.

An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes

In recent years, methods were proposed so as to efficiently perform time-variant reliability analysis. However, importance sampling (IS) for time-variant reliability analysis is barely studied in the literature. In this paper, an IS framework is proposed. A multi-dimensional integral is first derived to define the time-variant cumulative probability of failure, which has the similar expression to the classical definition of time-invariant failure probability. An IS framework is then developed according to the fact that time-invariant random variables are commonly involved in time-variant reliability analysis. The basic idea of the proposed framework is to simultaneously apply time-invariant IS and crude Monte Carlo simulation on time-invariant random variables and stochastic processes, respectively. Thus, the probability of acquiring failure trajectories of time-variant performance function is increased. Two auxiliary probability density functions are proposed to implement the IS framework. However, auxiliary PDFs available for the framework are not limited to the proposed two. Three examples are studied in order to validate the effectiveness of the proposed IS framework.

Probabilistic Bearing Capacity Prediction of Square Footings on 3D Spatially Varying Cohesive Soils

The bearing capacity of square and/or rectangular footings in geotechnical foundation designs traditionally is determined based on experimental observations and/or deterministic analysis assuming uniform soil profiles. However, soils are spatially varying, and this spatial variability can significantly affect the bearing capacity of the foundation soils. Probability-based design methods can address this problem explicitly. However, a full three-dimensional (3D) probabilistic simulation, such as that involving the random finite-element method, generally is prohibitive, because it involves numerous Monte Carlo runs of a complicated nonlinear elastoplastic algorithm. This paper developed and validated an approximate analytical method based on local averaging theory and geometric averages of soil properties directly under the footing. It was found that the theoretical prediction of the first two moments of a square footing bearing capacity agrees very well with crude Monte Carlo simulation. The analytical prediction of the probability of a design failure was validated through simulation and can be used directly in reliability-based designs against bearing failure.

A Multilevel Simulation Method for Time-Variant Reliability Analysis

Crude Monte Carlo simulation (MCS) is the most robust and easily implemented method for performing time-variant reliability analysis (TRA). However, it is inefficient, especially for high reliability problems. This paper aims to present a random simulation method called the multilevel Monte Carlo (MLMC) method for TRA to enhance the computational efficiency of crude MCS while maintaining its accuracy and robustness. The proposed method first discretizes the time interval of interest using a geometric sequence of different timesteps. The cumulative probability of failure associated with the finest level can then be estimated by computing corrections using all levels. To assess the cumulative probability of failure in a way that minimizes the overall computational complexity, the number of random samples at each level is optimized. Moreover, the correction associated with each level is independently computed using crude MCS. Thereby, the proposed method can achieve the accuracy associated with the finest level at a much lower computational cost than that of crude MCS, and retains the robustness of crude MCS with respect to nonlinearity and dimensions. The effectiveness of the proposed method is validated by numerical examples.

The soft Monte Carlo method

This study introduces a novel probability concept, random probability density function (PDF), as an efficient alternative of the random sampling for probability/reliability analysis of multivariate problems. To this end, a solution is proposed for drawing 1-D random PDFs according to the joint PDF of random variables. It is shown that using this approach, few random PDFs could be used instead of millions of random samples for covering the sample space in probabilistic analysis. By drawing PDFs instead of samples, the expected value of multivariable functions can be represented as the expected value of several one-variable probability integrals and therefore, any capable 1-D integration technique can be henceforth used for accurate statistical moment estimation and reliability analysis of multivariable problems. The law of large numbers proves the correctness of the proposed approach and therefore, achieving correct solutions of complex probability and reliability problems will not be hereafter restricted to only using crude Monte Carlo simulation. Besides, a novel probability density simulator is proposed that can attain probability density and moments of multivariate functions in an efficient manner different from existing approaches. Accurate and efficient probability/reliability analyses of complex problems with small failure probabilities confirm that the proposed method can be a reliable and efficient alternative of the crude Monte Carlo method, especially for extremely rare-events and time-consuming problems. For such problems, the suggested approach can serve as a benchmark against which new methods can be compared.

Dynamic Reliability and Reliability-Based Sensitivity of Transmission System in Shearer

Transmission system in shearer, such as traction unit and cutting arm, is the essential mechanism in the mining process, which supports the whole shearer and shock vibration. Considering various uncertainties, the evolving process is established to describe the dynamic system response with uncertain factors. Considering the parametric randomness and material strength. the failure domain is regarded as the dynamic extreme value of transmission system response exceeding the threshold in operation. According to the saddle-point approximation method, the dynamic extreme distribution is investigated to measure the safe probability and parametric sensitivity of transmission system in shearer. The dynamic reliability and reliability-based sensitivity are analysed to evaluate the evolving process of transmission system in shearer. According to the transmission model of traction unit, the influence degree of each parameter’s fluctuation can be quantified based on the indices of dynamic mean-value reliability-based sensitivity and standard deviation reliability-based sensitivity. The crude Monte Carlo simulation is taken as the benchmark to validate the reasonability.

A Multilevel Monte Carlo Method for Performing Time-Variant Reliability Analysis

Even though a great number of methods have been developed for time-variant reliability analysis (TRA), crude Monte Carlo simulation (MCS) is still widely used alone or combined with other methods to enhance the efficiency and accuracy of TRA. Multilevel Monte Carlo (MLMC) is introduced to TRA herein in order to reduce the computational complexity of MCS. MLMC first discretizes the time interval of interest by a geometric sequence of different timesteps. The finer grid is associated with smaller discretization error but larger computational cost. Accounting for the computational complexity of each level and the contribution that each level makes to the time-variant failure probability, the allocation of computational resource is then optimized in order to achieve a satisfying estimation of failure probability at the minimal computational cost. MLMC retains the robustness of crude MCS with respect to nonlinearity and dimensions of the time-variant performance function. Two benchmark examples are adopted to validate the effectiveness of the proposed MLMC. The results show that achieving the same accuracy level of results, the proposed method saves considerably computational cost relative to crude MCS.

Importance Sampling for Time-Variant Reliability Analysis

Importance sampling methods are extensively used in time-independent reliability analysis. However, the kind of methods is barely studied in the field of time-variant reliability analysis. This article presents an importance sampling method for time-variant reliability analysis. It increases the probability of sampling failure trajectories of a time-variant performance function. To develop the method, the instantaneous performance function at a predefined time instant is regarded as a time-independent one. A time-independent importance sampling is first implemented on the instantaneous performance function in order to obtain instantaneous samples of stochastic processes and random variables. Then, conditional trajectories of stochastic processes are generated on the condition of instantaneous samples achieved above, which utilizes the correlationship among instantaneous uncertainties at different time instants associated with stochastic processes. Subsequently, trajectories of the time-variant performance function are obtained. Validation results show that comparing with crude Monte Carlo simulation, the proposed method remarkably increases the probability of sampling failure trajectories. The efficiency and accuracy of the proposed method are demonstrated.

A machine learning enhanced structural response prediction strategy due to seismic excitation

AbstractIn this contribution, we present a fast prediction approach to estimate response statistics of the crude Monte Carlo simulation using artificial neural networks. Hereby, the neural network is trained within an initial response subset, based on which a forecast can be evaluated in an early state of the Monte Carlo simulation.

An improved adaptive bivariate dimension-reduction method for efficient statistical moment and reliability evaluations

Statistical moment estimations of the performance function with the aim of balancing accuracy and efficiency still remains a challenge for moment-based structural reliability analysis. In this paper, an improved adaptive bivariate dimension-reduction method in terms of vectors (i-VBDRM) is proposed for efficient statistical moments and reliability evaluations. In the proposed method, the delineation of cross terms of two-dimensional functions involved in the bivariate dimension-reduction method (BDRM) is first implemented, where the random variables are classified into several sub-vectors after the non-normal to normal transformation. Then, the explicit expressions of the moments of multiple component sub-vector functions are formulated, where a novel cubature formula is proposed and the Gauss-Hermite quadrature is employed to evaluate the involved two- and one-dimensional Gaussian-weighted integrals. In that regard, the first-four central moments can be calculated with accuracy and efficiency. Then, the probability density function of the performance function is rebuilt by a flexible distribution model called the shifted generalized lognormal distribution based on the first-four central moments evaluated by the proposed method. To demonstrate the efficacy of the proposed method, four numerical examples are presented, where some other forms of BDRM, univariate dimension-reduction method and the crude Monte Carlo simulation are performed for comparisons. The results show that the proposed method can keep the trade-off of precision and efficiency for both the statistical moment assessments and structural reliability analysis.

Bilevel Data-Driven Modeling Framework for High-Dimensional Structural Optimization under Uncertainty Problems

Optimization under uncertainty (OUU) is a robust framework to obtain optimal designs for real engineering problems considering uncertainties. The numerical solution for large-scale problems involving millions of degrees-of-freedom is typically computation-intensive in nature. Also, OUU problems constitutes an uncertainty analysis, involving a computation-intensive numerical solver for large-scale systems. Hence, the solution of OUU problems are computationally demanding in nature. In this study, a bilevel data-driven modeling framework is proposed using proper orthogonal decomposition (POD) and polynomial chaos expansion (PCE) metamodels. A heuristic particle swarm optimization (PSO) technique is used for optimization. The effectiveness of the POD-PCE metamodel combined with PSO is demonstrated for two practical large-scale structural optimizations under uncertainty problems. From the case studies, it has been observed that the proposed method gives solutions that are almost hundreds and thousands of times faster as compared to the crude Monte Carlo simulation.

Asymptotic estimates for finite-time ruin probability in a discrete-time risk model with dependence structures and CMC simulations

Consider a discrete-time risk model with dependence structures, where claim sizes are assumed to follow a one-sided linear process whose innovations further obey a so-called bivariate upper tail independence. The stochastic discount factors follow a stationary causal process. Then, the insurer is said to be exposed to a stochastic economic environment that contains two kinds of risks, i.e. the insurance risk and financial risk. The two kinds of risks form a sequence of independent and identically distributed random pairs which are copies of a random pair with a common bivariate Sarmanov dependent distribution. When the distributions of the innovations belong to the intersection of the dominated-variation class and the long-tailed class, we derive some asymptotic formulas for the finite-time ruin probability. We also get conservative asymptotic bounds when the distributions of the innovations belong to the regular variation class. Finally, we verify our results through a Crude Monte Carlo simulation.

Dynamic Model-based Saddle-point Approximation for Reliability and Reliability-based Sensitivity Analysis

Dynamic reliability analysis must consider both dynamic performance and parametric randomness to evaluate system safety in operation. The saddle-point approximation method is utilized to establish the dynamic probability distribution for known or unknown types of random variables. The cumulative distribution function is used to establish the approximation dynamic reliability model based on a simplified formula with high accuracy. Dynamic reliability analysis is investigated to describe the system safety of the moving operation process. Additionally, dynamic reliability-based sensitivity is used to represent the influence of the parameters on the system's dynamic performance. The fixed-threshold model and load-strength interference model are considered to develop computational formulas of reliability analysis and the degree of effect of each parameter's fluctuation. Finally, the proposed method is evaluated and demonstrated by means of four numerical examples to analyse the system performance and the parameters’ effects. The crude Monte Carlo simulation is performed to provide benchmark results.

Model-free computation of risk contributions in credit portfolios

Abstract In this work, we propose a non-parametric density estimation technique for measuring the risk in a credit portfolio, aiming at efficiently computing the marginal risk contributions. The novel method is based on wavelets, and we derive closed-form expressions to calculate the Value-at-Risk (VaR), the Expected Shortfall (ES) as well as the individual risk contributions to VaR (VaRC) and ES (ESC). We consider the multi-factor Gaussian and t-copula models for driving the defaults. The results obtained along the numerical experiments show the impressive accuracy and speed of this method when compared with crude Monte Carlo simulation. The presented methodology applies in the same manner regardless of the used model, and the computational performance is invariant under a considerable change in the dimension of the selected model. The speed-up with respect to the classical Monte Carlo approach ranges from twenty-five to one-thousand depending on the used model.

Life safety in single-story steel frame buildings, Part II - probabilistic design

PurposeThis paper aims to investigate the probability of unacceptable consequences from structural fire damage in a typical Scandinavian single-story steel frame building and discusses it in relation to life safety. This paper is a complement to the paper “Life safety in single-story steel frame buildings, Part I – deterministic design” by Sandström (2019) which considers the same design philosophy but with a probabilistic design approach.Design/methodology/approachThe reliability of a single-story steel frame building is investigated by using crude Monte Carlo simulation by including consideration to the fire conditions.FindingsThe investigated building does not meet the safety levels as stipulated by EN 1990 for structural fire damage. However, by including consideration to the fire conditions in the compartment, it is shown that the life safety objective is not compromised by the structural fire damage, i.e. the structure remains intact as long as any individuals/firefighters can survive within the fire area compartment.Originality/valueThis paper presents practical application of a conceptual paper presenting a general approach to structural fire safety design and the life safety objective.

Probabilistic homogenization of hyper-elastic particulate composites with random interface

A principal aim of this study is determination of deterministic and basic probabilistic characteristics of the effective material properties and stresses in hyper-elastic particulate composites with spherical reinforcement and uncertain volume fraction of the interface defects. This is done on the basis of laboratory tests of Laripur LPR 5020 High Density Polyurethane (HDPU) using the FEM homogenization method applied on the cubic multi-particle Representative Volume Element (RVE). This 3D homogenization scheme is based upon numerical determination of strain energy density in the RVE under uniaxial stretch. Each particle is spherical, linear elastic and embedded in the hyper-elastic matrix. The interphase is a thin layer around particle and contains all the particle-matrix interface defects; these defects are of the semi-spherical shape lying with the diameters on the interface. A constitutive relation of the matrix is experimentally recovered through the uniaxial stretch test and computationally approximated with three best fitting available hyper-elastic material potentials. The coincidence of these results serves for verification of the numerical results. Uncertainty of volume fraction of interface defects is assumed to be Gaussian, whose coefficient of variation does not excess 0.25. A relation of the effective stress to this uncertain parameter is recovered as some polynomial basis via the Least Squares Method (LSM) with two different approaches. The first one introduces a functional based directly on the potentials with parameters related to the strain level and to the volume fraction of interface defects, while the alternative approach is based upon a bivariate polynomial. This polynomial order is optimized via simultaneous maximization of correlation and minimization of LSM error and variance and its order is optimized for minimum LSM error. The RVE meshing is completed with the use of 20-noded brick or 10-noded tetrahedral elements. Probabilistic computations have been carried out with three independent approaches, namely the Stochastic Finite Element Method (SFEM), the crude Monte-Carlo simulation and probabilistic semi-analytical method. Probabilistic characteristics of the effective constitutive tensor include expected values, coefficients of variation, skewness and kurtosis of the effective stress. It is investigated numerically (1) if the resulting homogenized characteristics are also Gaussian, (2) how fluctuation of the interface defects volume fraction affects an effective stress in context of the deterministic and probabilistic analysis and (3) how these characteristics vary together with an increase of the applied strain.

Reliability Assessment of Steel Lattice Tower Subjected to Random Wind Load by the Stochastic Finite-Element Method

This work concerns the stochastic finite-element method (SFEM) reliability analysis of a skeletal, very slender steel telecommunication tower calibrated with a full-scale pushover experiment carried out for the same tower. The basic SFEM implementation is completed with common application of the generalized stochastic perturbation technique (SPT) and of the response function method with polynomial basis of the statistically optimized order. Verification of the proposed stochastic technique is carried out for the first four probabilistic moments by a comparison with traditional crude Monte Carlo simulation and, alternatively, with semianalytical methodology using the same polynomial structural responses as SPT and the well-known probabilistic integral definitions. Characteristic mean wind speed is introduced here as the input random Gaussian variable, while the computational model is subjected to the wind profile linearly dependent upon this mean speed consistent with applicable European design codes. The SFEM analysis is carried out here including large deformations of the tower elements and dynamic excitations in each node induced by the given trial wind spectrum. Finally, second-order reliability method (SORM) analysis is provided on the basis of the resulting extreme normal forces in the tower legs and their counterparts taken directly from the full-scale pushover experiments. The obtained results show that reliability indices for these towers according to the engineering codes have larger magnitudes than these corresponding to the experimental statistics and SFEM.

An efficient Nataf transformation based probabilistic power flow for high-dimensional correlated uncertainty sources in operation

This paper proposes and introduces a novel approach for probabilistic power flow (PPF) analysis on large-scale power systems with such Pearson correlated high-dimensional uncertainty sources existing in normal operation. The proposed method is holistic since it covers the whole procedure of PPF analysis and improves its efficiency from each aspect. As one key step, the combination of Gauss-Hermite quadrature and simplified Newton-Raphson method based Nataf transformation is presented to accelerate the process of probabilistic modeling for wind speed, solar radiation, tidal current speed and load consumption, in particular the correlation relationship amongst these uncertainty sources. Besides, dimension adaptive sparse grid interpolation (DASGI) method and Monte Carlo simulation (MCS) method have been put to joint application to further boost the efficiency of PPF calculation, with the Pearson correlation issue handled in this procedure as the first time. In order to highlight the superiority of the proposed method, several existing advanced PPF analysis methods are used for comparison test, in which simple random samples based MCS method i.e. the crude MCS is assigned as the reference. The performance of the proposed approach is verified in IEEE 118-bus system involving with 114 random variables.