Dynamic pruning-based Bayesian support vector regression for reliability analysis

A computationally efficient procedure for design optimization of slot antennas is presented. We use space mapping as the main optimization engine, the underlying coarse model being coarse-discretization electromagnetic (EM) simulation data of the antenna structure of interest (low-fidelity model). In order to speed up the design process, the low-fidelity model is not used directly in the process; instead, the coarse-discretization simulation data - sampled only in the vicinity of their approximate optimum - are used to create an auxiliary response surface model through Bayesian support vector regression. The latter - after suitable space-mapping-based correction - serves as a prediction tool to find an accurate optimum design of the antenna. The proposed procedure is illustrated using two examples of slot antennas.

Methodical aspects of the reliability-based structural optimisation using stochastic quasigradient methods are considered. For an example of the simply supported reinforced concrete beam, the employment of the Lagrange multiplier method that belongs to the class of stochastic quasigradient methods is demonstrated. The classical optimum design goal to minimise structural cost or weight under the constraint on the structural failure probability is taken for consideration. Optimisation problems solved with the Lagrangemultiplier method are formulated in form of general stochastic programming problem. The mathematical expectation of the concrete volume reduced with respect to the in-place cost of the beam materials is taken as the objective function. Constraint function is the limitation placed on the beam failure probability. The beam is considered as a series structural system. Values of the prescribed allowable failure probability belongs to the interval in which the estimation of the failure probabilities by the simple Monte-Carlomethod is possible with an acceptable confidence. The time-independent case as well as the time-dependent one is considered in the optimisation problems. The generalisation on the time-dependent case is undertaken through the introduction into the constraint function of the quasi-linear distribution law of the random variables. In the time-dependent case, the objective function is associated with beginning and the constraint function with end of the service period. An expression of the stochastic gradient based on the differentiation under the integral sign is used for calculations with the Lagrange multiplier method. The stochastic gradient used is computationally more effective in comparison with stochastic finite-difference formulae usual in stochastic quasigradient methods because it requires only one computation of the structure in search iteration of the optimisation process. Three rules based on statistical argumentation are used for the stopping of the seat according to the procedure of the Lagrange multiplier method. The optimising of the beam shows that the Lagrange multiplier method is applicable for the optimal design of structures in that cases when the structural reliability can be estimated by means of the simple Monte-Carlo method. Additional research is needed for integration in the Lagrange multiplier method of statistical simulation techniques for the estimation of small structural failure probabilities.

Reliability analysis techniques combining with various surrogate models have attracted increasing attention because of their accuracy and great efficiency. However, they primarily focus on the structures with continuous response, while very rare researches on the reliability analysis for structures with discontinuous response are carried out. Furthermore, existing adaptive reliability analysis methods based on importance sampling (IS) still have some intractable defects when dealing with small failure probability, and there is no related research on reliability analysis for structures involving discontinuous response and small failure probability. Therefore, this paper proposes a novel reliability analysis method called AGPC-IS for such structures, which combines adaptive Gaussian process classification (GPC) and adaptive-kernel-density-estimation-based IS. In AGPC-IS, an efficient adaptive strategy for design of experiments (DoE), taking into consideration the classification uncertainty, the sampling uniformity and the regional classification accuracy improvement, is developed with the purpose of improving the accuracy of Gaussian process classifier. The adaptive kernel density estimation is introduced for constructing the quasi-optimal density function of IS. In addition, a novel and more precise stopping criterion is also developed from the perspective of the stability of failure probability estimation.The efficiency, superiority and practicability of AGPC-IS are verified by three examples

Surrogate SDOF models for probabilistic performance assessment of multistory buildings: Methodology and application for steel special moment frames

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

Although an accurate reliability assessment is essential to build a resilient infrastructure, it usually requires time-consuming computation. To reduce the computational burden, machine learning-based surrogate models have been used extensively to predict the probability of failure for structural designs. Nevertheless, the surrogate model still needs to compute and assess a certain number of training samples to achieve sufficient prediction accuracy. This paper proposes a new surrogate method for reliability analysis called Adaptive Hyperball Kriging Reliability Analysis (AHKRA). The AHKRA method revolves around using a hyperball-based sampling region. The radius of the hyperball represents the precision of reliability analysis. It is iteratively adjusted based on the number of samples required to evaluate the probability of failure with a target coefficient of variation. AHKRA adopts samples in a hyperball instead of an n-sigma rule-based sampling region to avoid the curse of dimensionality. The application of AHKRA in ten mathematical and two practical cases verifies its accuracy, efficiency, and robustness as it outperforms previous Kriging-based methods.

Decoupled reliability-based optimization using Markov chain Monte Carlo in augmented space

A logical framework is developed for load and resistance factor design (LRFD) of slopes based on reliability analysis. LRFD of slopes with resistance factors developed in this manner ensures that a target probability of slope failure is not exceeded. Three different target probabilities of failure (0.0001, 0.001 and 0.01) are considered in this report. The ultimate limit state for slope stability (formation of a slip surface and considerable movement along this slip surface) is defined using the Bishop simplified method with a factor of safety equal to unity. Gaussian random field theory is used to generate random realizations of a slope with values of strength and unit weight at any given point of the slope that differ from their mean by a random amount. A slope stability analysis is then performed for each slope realization to find the most critical slip surface and the corresponding driving and resisting moments. The probability of slope failure is calculated by counting the number of slope realizations for which the factor of safety did not exceed 1 and dividing that number by the total number of realizations. The mean of the soil parameters is adjusted and this process repeated until the calculated probability of failure reaches to the target probability of failure. Optimal resistance and load factors are obtained by dividing the resisting and driving moments corresponding to the most probable ultimate limit state by the nominal values of resisting and driving moments. The main goal of this study was to provide specific values of resistance and load factors to implement in limit states and load resistance design of slopes in the context of transportation infrastructure. This report discusses the concepts of load and resistance factors, target probability of failure and the ultimate limit state equation in the context of slope stability analysis. It then presents a detailed algorithm for resistance factor calculation by using reliability analysis. Six cases of real slopes designed and constructed by the Indiana Department of Transportation (INDOT) are examined by using undrained shear strengths in order to illustrate the LRFD procedure and validate the recommended resistance and load factors.

ABSTRACTIn the present study, reliability analysis of near surface disposal facility is performed, by assessing the probability of sequential failure of the multi barrier system using the contaminant transport model. The concentration and dose rate of the radionuclide evolve with time hence there is a need for time dependent reliability analysis. Due to the low values of expected probabilities of failure, an enhanced Monte Carlo (EMC) method and Subset simulation is employed. The Result of the analysis show that, the EMC method is useful to evaluate the probability of failure associated with the barrier system which has low probability of failure.

For engineering, there are two major challenges in reliability analysis. First, to ensure the accuracy of simulation results, mechanical products are usually defined implicitly by complex numerical models that require time-consuming. Second, the mechanical products are fortunately designed with a large safety margin, which leads to a low failure probability. This paper proposes an efficient and high-precision adaptive active learning algorithm based on the Kriging surrogate model to deal with the problems with low failure probability and time-consuming numerical models. In order to solve the problem with multiple failure regions, the adaptive kernel-density estimation is introduced and improved. Meanwhile, a new criterion for selecting points based on the current Kriging model is proposed to improve the computational efficiency. The criterion for choosing the best sampling points considers not only the probability of misjudging the sign of the response value at a point by the Kriging model but also the distribution information at that point. In order to prevent the distance between the selected training points from too close, the correlation between training points is limited to avoid information redundancy and improve the computation efficiency of the algorithm. Finally, the efficiency and accuracy of the proposed method are verified compared with other algorithms through two academic examples and one engineering application.

Integration of computation and testing for reliability estimation

This paper develops a new modified Bayesian Kriging (MBKG) surrogate modeling method for problems in which simulation analyses are inherently noisy and thus standard Kriging approaches fail to properly represent the responses. The purpose is to develop a method that can be used to carry out reliability analysis to predict probability of failure. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are presented. Using the full conditional distributions with a Gibbs sampling algorithm, Markov chain Monte Carlo is used to fit the MBKG surrogate model. A sequential sampling method that uses the posterior credible sets for inserting new design of experiment (DoE) sample points is proposed. The sequential sampling method is developed in such a way that the newly added DoE sample points will provide the maximum amount of information possible to the MBKG surrogate model, making it an efficient and effective way to reduce the number of DoE sample points needed. Therefore, the proposed method improves the posterior distribution of the probability of failure efficiently. To demonstrate the developed MBKG and sequential sampling methods, a 2-D mathematical example with added random noise is used. It is shown how, with the use of the sequential sample method, the posterior distribution of the probability of failure converges to capture the true probability of failure. A 3-D multibody dynamics (MBD) engineering block-car example illustrates an application of the new method to a simple engineering example for which standard Kriging methods fail.

Knowing the probability of failure in real-time i.e. the probability that a system violates some constraints is decisive for aerospace applications in the presence of state uncertainty. However, it often involves stochastic prediction of the constraints on the upcoming system’s trajectory, which is computationally demanding. In addition, the trajectory prediction depends on the accuracy of the current state estimate, e.g. the navigation system, whose errors are usually not accounted for. This paper studies the impact of the state estimate error on the failure probability calculation, obtained by Monte Carlo trajectory sampling. The failure probability error is first shown to be made of two terms depending on the current state estimation error and the number of Monte Carlo samples. Then, an iterative Least Square estimator is introduced to refine the failure probability estimation without significantly increasing the computational load. It is theoretically shown to converge and lowers the failure probability estimation error of 33% in simulation. The approach is illustrated on a constrained stochastic control application: the atmospheric reentry of a vehicle inspired from SpaceX’s Starship (SXS). The proposed method allows the failure probability estimate to be more accurate despite the unavoidable disturbances yielded by the state estimation algorithm and the Monte Carlo discrete sampling, in particular when the available computational load is limited onboard.

A novel fidelity selection strategy-guided multifidelity kriging algorithm for structural reliability analysis

Traffic congestion which has been a common problem faced by almost all of the metropolitan cities across the world is caused by non-recurring traffic incidents such as accidents and vehicle breakdowns in many cases. Therefore, accurate prediction of the incident duration has become an essential part of Intelligent Transport Systems (ITS). Although the data-driven deterministic regression models can serve this purpose, the performances of these models are subject to varying test conditions. In this study, we consider two different data-sets, one from Singapore and the other from the Netherlands, and explore stochastic models (Bayesian Support Vector Regression, i.e. BSVR and Gaussian Process, i.e. GP) that not only perform prediction but also provide error bars as confidence measures along with the predicted values. Moreover, we also verify whether these methods can anticipate when the predictions are unreliable. To this end, we analyze the specificity and sensitivity of the two methods for detecting prediction errors with different tolerance levels. We observe that for 70% specificity, the sensitivity of BSVR and GP is 67% and 78% respectively, for the incidents in Singapore. On the other hand, for the incidents in the Netherlands, the two methods have 55% and 60% sensitivity respectively, for the same level of specificity. Therefore, our proposed method of Bayesian models with error bars leads to more reliable predictions of the traffic incident duration, as well as forecasts to some extent when the predictions are unreliable.