Active learning methods have demonstrated their successful applications in structural reliability problems. Most active learning surrogate models are regression-based, while the classification-based surrogate model has been less discussed. This paper proposes an active learning Bayesian network classifier (ALBC) method to fill this gap. Unlike regression models that predict continuous responses, Bayesian network classifiers directly categorise inputs into discrete classes (safe/failed). This classification approach is particularly suitable for reliability analysis where the primary concern is determining whether a structure will fail under given conditions. A learning function based on misclassification probability is formulated to identify update points that refine the decision boundary with an error expectation function for the sample population work as a stopping criterion for the model iteration. Three mathematical examples and a box-beam buckling failure problem are used to extensively investigate the feasibility and performance of the proposed method. This approach is shown to effectively refine the Bayesian classifier to achieve efficiency and accuracy for structural reliability applications.

ABSTRACT Ground anchors are crucial components in various construction and engineering applications. They play a critical role in retaining structures and, therefore, design guidelines have established the necessity of comprehensive testing campaigns to derive the anchors characteristic resistance. The latter is a specified percentile within a presumed statistical distribution. In principle, a limited number of investigation tests cannot be used to estimate the characteristic values. To overcome this limitation, in a simplified way, the design codes suggest reducing the resistance found in experimental results by a factor to estimate the anchor characteristic resistance to be used in the design. In this paper, the authors propose a new approach for interpreting ground anchor test results and determining the statistical distribution of ground anchor resistance. The approach is based on the use of Bayesian updating, formulated as a structural reliability problem, and on the definition of a simplified phenomenological model relating the imposed load and the measured anchor (creep) displacements. This distribution can be used to determine a “proven” anchor characteristic resistance, which can then be used to update the anchor design.

Structural reliability analyses may become very computationally demanding, especially when numerical simulations are employed to represent the structural behavior, and/or these analyses are used within structural optimization procedures. To overcome this problem, surrogate models have been largely used in the last decades, helping to avoid evaluations of the demanding parts of the computational code and to reduce the overall computational demand. However, the efficiency of the surrogates is usually compromised when dealing with high dimensional problems. In fact, high dimensionality imposes some difficulties not only to surrogate models but also for some structural reliability methods available in the literature. For these reasons, the present paper proposes to investigate the application of Artificial Neural Networks to reduce the dimensionality of structural reliability problems. A proper dimensionality reduction may help visualizing and understanding the problem and may assist surrogate models and reliability methods which would otherwise lose accuracy, precision and/or efficiency when applied to high dimensional problems.

The paper proposes a parallel extension of the Direct Optimized Probability Computation Method. This method can be used as an alternative to Monte Carlo – based approaches for problems of structural reliability. It does not depend on any kind of randomly generated numbers, but it requires a much higher number of computations thus it can benefit from its parallelization. The proposed parallel Direct Optimized Probability Computation method is developed and studied on the problem of fatigue damage prediction. Description of the parallel algorithm is provided, and the functionality of the method is shown in an example case. The results are also compared to the results of the more common Monte Carlo – based approach.

The classic formulation of structural reliability problems, such as that used in the GRG (Generalized Reduced Gradient) and FORM (First Order Reliability Method) methods, is defined as an optimization problem, where the random variables are the design variables, the reliability index is the objective function and the equality constraint is given by the performance function, calculated as the safety margin.The reliability index can be defined as the smallest distance, in the space of reduced variables, between the performance function and the origin of the system.So the reliability problem is usually formulated as: determine the design variables (random variables) that minimize the objective function (reliability index) subject to the equality constraint (safety margin).As the objective function is the distance from the project to the origin, in the space of the reduced variables, it doesn't matter in the equation whether these variables have positive or negative values.This can cause problems for the solution, as the sign of these variables significantly interferes with the calculation of the probability of failure of the model being analyzed.Examples will be shown where this formulation is not valid.The paper concludes that the most appropriate formulations are those based on defining the reliability index as the ratio between the mean and standard deviation of the performance function.Formulations such as the Monte Carlo (MC) process use this definition and therefore do not affect the results obtained and are more reliable, especially in more complex problems with a significant number of random variables.Examples using the GRG method and the Monte Carlo process will be presented and the discrepancies between the GRG method and the coherent results given by the Monte Carlo process in some classical problems will be shown.Suggestions for future studies will also be presented.

Abstract Multifidelity surrogate modeling offers a cost-effective approach to reducing extensive evaluations of expensive physics-based simulations for reliability prediction. However, considering spatial uncertainties in multifidelity surrogate modeling remains extremely challenging due to the curse of dimensionality. To address this challenge, this paper introduces a deep learning-based multifidelity surrogate modeling approach that fuses multifidelity datasets for high-dimensional reliability analysis of complex structures. It first involves a heterogeneous dimension transformation approach to bridge the gap in terms of input format between the low-fidelity and high-fidelity domains. Then, an explainable deep convolutional dimension-reduction network (ConvDR) is proposed to effectively reduce the dimensionality of the structural reliability problems. To obtain a meaningful low-dimensional space, a new knowledge reasoning-based loss regularization mechanism is integrated with the covariance matrix adaptation evolution strategy (CMA-ES) to encourage an unbiased linear pattern in the latent space for reliability prediction. Then, the high-fidelity data can be utilized for bias modeling using Gaussian process (GP) regression. Finally, Monte Carlo simulation (MCS) is employed for the propagation of high-dimensional spatial uncertainties. Two structural examples are utilized to validate the effectiveness of the proposed method.

A clustering-based partially stratified sampling for high-dimensional structural reliability assessment

Abstract The first-order reliability method (FORM) is simple and efficient for solving structural reliability problems but may have large errors and converge slowly or even result in divergence when dealing with strongly nonlinear performance functions. For this case, the existing second-order reliability method (SORM) achieves higher computational accuracy but with a consequent decrease in efficiency. To achieve a better balance between accuracy and efficiency, this paper proposes an improved FORM and an improved SORM. First, an improved modified symmetric rank 1 (IMSR1) algorithm, in which the line search strategy for step length is unnecessary, is proposed for iterations of the FORM, and an adaptive Kriging model with a rational update criterion is presented to improve the efficiency of the FORM. Then, an improved FORM with high efficiency and good convergence is proposed. Second, due to the good precision of the adaptive Kriging model at the final design point, the Hessian matrix is available easily without additional computational effort, and an improved SORM with the same efficiency as the improved FORM is presented naturally. Finally, the accuracy, efficiency, and convergence of the proposed methods are verified by numerical and engineering examples.

Structural reliability analysis based on improved wolf pack algorithm AK-SS

Bi-fidelity Kriging model for reliability analysis of the ultimate strength of stiffened panels

Adaptive Kriging-based Bayesian updating of model and reliability

Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic reliability index method, the extreme value of the limit state function is obtained using the properties of Bernstein polynomials, thus avoiding the need for a lot of sampling to solve the reliability analysis problem. The method is applied to numerical examples and engineering applications such as experiments, and the results show that the method has higher computational efficiency and accuracy than the traditional linear approximation method, especially for some reliability problems with higher nonlinearity. Moreover, this method can effectively improve the reliability of results and reduce the cost of calculation in practical engineering problems.

The shaft of the slanted axial-flow pump device is inclined, so, the stress on the impeller is complex, which leads to the prominent problem of structural reliability. In order to study the mechanical characteristics and fatigue life of the slanted axial-flow pump, the stress-strain characteristics and variation law of impeller under different flow rate conditions are analyzed by fluid-structure interaction method, at the same time, the stress-strain characteristics of impellers with geometrically similar and five different structural sizes when nD value is equal are solved, finally, the fatigue analysis of blades is carried out based on Miner’s linear fatigue cumulative damage theory to predict the fatigue life of impellers. The results show that under different flow rate conditions, the maximum equivalent stress of the impeller blade is concentrated at the blade root and gradually attenuates to the blade flange. The maximum deformation of the blade is located at the blade rim and moves from the inlet side to the outlet side with the increase of flow rate. When the nD value is constant, the maximum equivalent stress and maximum deformation displacement of the blade increase with the increase of the impeller size. Under various flow rate conditions, the service life of the impeller reaches 108, and the safety factor of each part of the blade is greater than 15, when operating within the working condition range of 0.8 Qbep∼1.2 Qbep, the impeller can operate continuously for 14 years without fatigue damage.

Adaptive stratified sampling for structural reliability analysis

Sample regeneration algorithm for structural failure probability function estimation

Abstract The multi‐output structural system with implicit function widely exists in actual engineering, which refers that the multiple output responses of structural system can be obtained by one experiment or finite element simulation. Considering the correlation of multiple output responses and the small failure probability involved in multi‐output structural system, this article proposes a novel active learning Kriging (ALK) based reliability method for multi‐output structural system by combining multiple response Gaussian process (MRGP) and importance sampling (IS). First, due to the Kriging model can only construct the surrogate model under the single‐output variable, the MRGP model is introduced to substitute the Kriging model and thus the correlation in multiple output responses can be efficiently described by a correlation matrix in MRGP model. Second, for the case that the distance information of new iterated sample point is not considered by the commonly used learning functions (U‐function, EFF‐function and H‐function) in ALK, three improved learning functions are correspondingly proposed. Finally, aiming at the problem that the small failure probability leads to the increasing of candidate sample pool and further results in low computational efficiency, the IS method is combined with the MRGP model to efficiently accomplish the reliability analysis for multi‐output structural system. Several examples are also provided to demonstrate the effectiveness of the proposed method.

Aiming at the characteristics of high computational cost, implicit expression and high nonlinearity of performance functions corresponding to large and complex structures, this paper proposes a support-vector-machine- (SVM) based grasshopper optimization algorithm (GOA) for structural reliability analysis. With this method, the reliability problem is transformed into an optimization problem. On the basis of using the finite element method (FEM) to generate a small number of samples, the SVM model is used to construct a surrogate model of the performance function, and an explicit expression of the implicit nonlinear performance function under the condition of small samples is realized. Then, the GOA is used to search for the most probable point (MPP), and a reasonable iterative method is constructed. The MPP information of each iteration step is used to dynamically improve the reconstruction accuracy of the surrogate model in the region that contributes most to the failure probability. Finally, with the MPP after the iteration as the sampling center, the importance sampling method (ISM) is used to further infer the structural failure probability. The feasibility of the method is verified by four numerical cases. Then, the method is applied to a long-span bridge. The results show that the method has significant advantages in computational accuracy and computational efficiency and is suitable for solving structural reliability problems of complex engineering.

Vectorial surrogate modeling method for multi-objective reliability design

Model updating for slope stability assessment in spatially varying soil parameters using multi-type observations

Active learning for structural reliability: Survey, general framework and benchmark