Structural Reliability Problems Research Articles

Deep Learning-Based Multifidelity Surrogate Modeling for High-Dimensional Reliability Prediction

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

Assessing structural reliability problem with high-dimensional random inputs is still challenging due to the “curse of dimensionality”. In this paper, this challenge is addressed by extending the Generalized Distribution Reconstruction Method via Characteristic Function Inversion (GDRM-CFI). Specifically, a clustering-based partially stratified sampling method is proposed for selecting high-dimensional points to numerically evaluate the characteristic function (CF) curve of complex high-dimensional problems. An improved number-theoretical method (i-NTM) is used to establish a uniform, efficient point set, ensuring determinism and reducing variability. Subsequently, a partial stratification approach partitions the high-dimensional space into orthogonal two-dimensional subspaces. The fundamental point set is projected into each subspace, and the k-means clustering algorithm identifies centroids within each, acting as representative points. The complete set of representative points from all subspaces formulates the high-dimensional point set. Numerical examples are investigated, which demonstrate the proposed method is effective for high-dimensional structural reliability assessment.

Improved FORM and SORM Based on Improved Modified Symmetric Rank 1 Algorithm and Adaptive Kriging Model

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

To improve the accuracy and efficiency of solving structural reliability problems with highly nonlinear black box functions, this paper proposes an active learning Kriging approach for structural reliability analysis based on an improved learning strategy of the Improved Wolf Pack Algorithm (IWPA).Firstly, we optimize the kriging surrogate model by IWPA. Secondly, a parallel plus point strategy named IWPA-NLF which is based on IWPA and a new learning function (NLF) is used to update the kriging surrogate model. Finally, the subset simulation (SS) method is used to calculate the reliability. A numerical example and three engineering examples are used to verify the proposed method in this paper. The results show that the proposed method in this paper has a strong function fitting ability and can calculate the accurate failure probability with fewer performance function calls.

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

A method based on a Bi-fidelity Kriging model is proposed for structural reliability analysis. It is based on adding low-fidelity data samples to the model to predict high-fidelity values, thus saving computational effort. Distance Correlation develops the correlation between the low and high-fidelity functions, initially proposed to assess the correlation between two variables. The bi-fidelity Kriging response surface model's efficiency as a surrogate model will be assessed for structural reliability problems that demand high computational costs, such as nonlinear finite element analysis structural models. The efficiency assessment is performed by comparing the accuracy of the failure probability predictions based on the Subset Simulation and First-order reliability method using the Bi-fidelity Kriging model as a surrogate for the performance function. The idea is illustrated by considering a representative component of marine structures analyzed by finite element analysis to create bi-fidelity scenarios to assess structural reliability with many variables. The results show that the proposed multi-fidelity method can provide an accurate failure probability estimation with less computational cost.

Adaptive Kriging-based Bayesian updating of model and reliability

Bayesian updating is a powerful tool to reassess and calibrate models and their reliability as new observations emerge, and the Bayesian updating with structural reliability method (BUS) is an efficient approach that reformulates it as a structural reliability problem. However, the efficiency and accuracy of BUS depend on a constant c determined by the maximum of likelihood function. To efficiently complete Bayesian updating with new observations related to implicit performance function, a method that combines adaptive Kriging with Bayesian updating is proposed. The proposed method involves three stages. Firstly, an innovatively advanced expected improvement (AEI) learning function is proposed to train the Kriging model of the likelihood function for estimating c, in which the convergence criterion and the strategy of selecting new training point guarantee the accuracy and efficiency of estimating c. Secondly, a new learning function based on expectation and variance of contribution uncertainty function (EVCUF) is proposed to adaptively train the Kriging model of the performance function constructed in BUS to extract posterior samples and complete Bayesian updating of model. By simultaneously taking the expectation and variance of the contribution of the candidate sample to improving accuracy of the Kriging model into consideration, the EVCUF learning function ensures the robust and efficient convergence of the Kriging model. Finally, based on the training points of the previous two stages, the traditional U learning function is employed to subsequentially update Kriging model of the performance function for classifying posterior samples and completing Bayesian updating of reliability. Additionally, a reduction strategy of the candidate sample pool is proposed to improve the efficiency of the proposed method. After demonstrating the basic principle and advantage of the proposed method, three examples are introduced to verify the efficiency and accuracy of the proposed method.

Structural Interval Reliability Algorithm Based on Bernstein Polynomials and燛vidence Theory

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.

Dynamic stress characteristics and fatigue life analysis of a slanted axial-flow pump

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

An adaptive framework of stratified sampling is established for estimating the probability of failure in structural reliability analysis in combination with variance reduction employed by strata. Among a variety of variance reduction techniques, stratification of Latin hypercube type makes it possible to apply importance sampling and control variates by strata in an adaptive manner without the need to track differences in computing cost across strata. By batching the resulting paralleled tasks, the proposed algorithm adjusts the allocation of computing budget only occasionally yet effectively taking full account of decreasing stratum variances. A variety of structural reliability problems with sparse structures lie within the scope of the proposed framework in the sense that stratification with gradual adjustment of computing budget avoids wasting the resource on possibly many empty strata, while importance sampling and control variates improve Monte Carlo methods on non-empty strata. We provide numerical results to illustrate the wide applicability and effectiveness of the proposed framework.

Sample regeneration algorithm for structural failure probability function estimation

An efficient strategy to approximate the failure probability function in structural reliability problems is proposed. The failure probability function (FPF) is defined as the failure probability of the structure expressed as a function of the design parameters, which in this study are considered to be distribution parameters of random variables representing uncertain model quantities. The task of determining the FPF is commonly numerically demanding since repeated reliability analyses are required. The proposed strategy is based on the concept of augmented reliability analysis, which only requires a single run of a simulation-based reliability method. This paper introduces a new sample regeneration algorithm that allows to generate the required failure samples of design parameters without any additional evaluation of the structural response. In this way, efficiency is further improved while ensuring high accuracy in the estimation of the FPF. To illustrate the efficiency and effectiveness of the method, case studies involving a turbine disk and an aircraft inner flap are included in this study.

A novel approach for multi‐output structural system reliability problem with small failure probability

AbstractThe 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.

A Novel Support-Vector-Machine-Based Grasshopper Optimization Algorithm for Structural Reliability Analysis

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

To improve the computational cost and accuracy for the multi-objective reliability design of complex structures, a synchronous modeling concept is proposed by introducing matrix thought into the surrogate modeling methods, the purpose of which is to realize the transformation from single-objective reliability design to comprehensive (multiple objectives) reliability design. With respect to the synchronous modeling concept, vectorial surrogate modeling (VSM) method is developed to synchronously establish an overall model of complex structures with multiple objectives. Besides, the types of examples including a numerical example and four engineering cases are adopted to highlight the abilities of the developed VSM method in accuracy, computational efforts and capability. It is illustrated that (1) the developed VSM method is workable in the synchronous modeling and comprehensive reliability design of complex structures with many objectives; (2) the proposed VSM method has the superior performances in both computational efficiency and accuracy for high-dimensional nonlinear problems, by comparison of methods; (3) By comparing with the analytical approaches, the efficient results based on computational times and elevating the performance are provided using the proposed method for highly nonlinear problems; (4) the comprehensive reliability level of turbine blisk deformation, stress and strain is 0.9974 obtained by the VSM method, by considering the effect of fluid-thermal-structural interaction, when the allowable values are 4.3698 × 10−3 m, 1.0389 × 109 Pa and 6.4298 × 10−3 m/m (determined by 3-sigma level); (5) the modeling and simulation efficiencies of the VSM method are improved by 42.56% and 0.31% relative to the RSM. The VSM method are validated to hold high accuracy and high efficiency in modeling characteristics and simulation performance of high-dimensional structural reliability problems. The efforts of this study are to provide a promising synchronous modeling concept and methodology with the VSM method for the multi-objective reliability design of complex systems, which contributes to enrich mechanical reliability theory.

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

With site-specific information, model updating can calibrate the probability distributions of material parameters, and the updated distributions can be further utilized to facilitate a more realistic slope reliability assessment. Many model updating methods have been proposed, while the inherent spatial variability of material properties is scarcely discussed in the literature, mainly due to the curse of dimensionality when considering thousands of random variables. The BUS (Bayesian updating with structural reliability methods) algorithm can well tackle the high-dimensional problem by converting it into an equivalent structural reliability problem. The BUS algorithm can integrate monitoring data and field observations to back analyse stability parameters. However, when generating a conditional random field using a large number of in-situ test data, the BUS algorithm becomes less efficient because low acceptance probability often occurs, while the Kriging algorithm is applicable for this situation. This study proposes an effective approach for model updating that combines Kriging-based conditional random field with the BUS algorithm to integrate multi-type observations. To illustrate the effectiveness of the proposed approach, an undrained saturated clay slope with a spatially varying soil parameter is taken as an example. The results indicate that the proposed approach is able to update the probability distribution of spatially varying soil parameters and update the slope reliability using multi-type observations with reasonable calculation efficiency.

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

Active learning methods have recently surged in the literature due to their ability to solve complex structural reliability problems within an affordable computational cost. These methods are designed by adaptively building an inexpensive surrogate of the original limit-state function. Examples of such surrogates include Gaussian process models which have been adopted in many contributions, the most popular ones being the efficient global reliability analysis (EGRA) and the active Kriging Monte Carlo simulation (AK-MCS), two milestone contributions in the field. In this paper, we first conduct a survey of the recent literature, showing that most of the proposed methods actually span from modifying one or more aspects of the two aforementioned methods. We then propose a generalized modular framework to build on-the-fly efficient active learning strategies by combining the following four ingredients or modules: surrogate model, reliability estimation algorithm, learning function and stopping criterion. Using this framework, we devise 39 strategies for the solution of 20 reliability benchmark problems. The results of this extensive benchmark (more than 12,000 reliability problems solved) are analyzed under various criteria leading to a synthesized set of recommendations for practitioners. These may be refined with a priori knowledge about the feature of the problem to solve, i.e. dimensionality and magnitude of the failure probability. This benchmark has eventually highlighted the importance of using surrogates in conjunction with sophisticated reliability estimation algorithms as a way to enhance the efficiency of the latter.

A novel response surface method for structural reliability

The response surface method is a common method to solve structural reliability problems. The core principle of this method is the approximation of the actual limit state function using the response surface function for subsequent analytical calculations. The existing response surface method for structural reliability has poor approximation ability and waste of the actual limit state function of the experimental points. These problems lead to low computational efficiency and inability to satisfy high-precision engineering requirements. This paper proposes a novel response surface method for structural reliability. The efficiency and accuracy are improved by the new experimental point selection strategy of the proposed method. This method has the following obvious advantages: the experimental point is close to the actual limit state function, the actual number of calculations is reduced, and the effect of the constructed response surface function approximation is improved. In addition, the single response surface function form is not used in the proposed method to improve the iterative efficiency. Compared with other response surface methods for structural reliability, three calculation examples prove the accuracy and efficiency of the proposed method.

Structural dynamic reliability analysis of super large-scale lattice domes during earthquakes using the stochastic finite element method

For large-scale dome structures subjected to earthquake ground motions, the need for accurate and efficient approaches that account for uncertainties in design, material properties, loads, damping, and manufacturing processes has grown significantly. In uncertainty analyses, the theory of probability, uncertainty quantification, and reliability analysis approaches are commonly used. However, because of computing efficiency and capability, the space scale of dome structures is severely limited. As a result, the computational issues of the complex dynamic reliability that super-large-scale dome structures encounter during earthquakes are challenging. The uncertainties of the dynamic demands of a super-large-scale lattice dome with multiple variables are quantified using a stochastic finite element method (SFEM) with nonlinear time-history analysis developed in this paper, and this method can consider the randomness of variables in more detail for a structure. The dynamic reliability is investigated in depth using the efficient Latin hypercube sampling (LHS) technique to reduce the number of repetitive time-consuming calculations. To improve the efficiency and accuracy of the analysis, an optimization approach based on the genetic algorithm (GA) is used to obtain the probability of structural failure and the reliability index. The results show that these methods are highly efficient for super-large-scale structures. Finally, a sensitivity analysis based on variable randomness is conducted to further evaluate the effects of variables on structural performance using various evaluation methods. This paper provides an efficient solution for structural dynamic reliability problems of super-large-scale structures in the SFEM framework.

System reliability analyses of static and dynamic structures via direct probability integral method

Structural system reliability analysis is of important significance for evaluating the safety of structures with many components. Since a structural system can be acted by static or dynamic load, a unified and efficient method is required to assess the system reliability of static or dynamic load induced structures. In this study, the direct probability integral method (DPIM) is proposed to uniformly attack system reliability problems of static and dynamic structures. Firstly, the static and first-passage dynamic reliability formulas of the series, parallel and mixed systems are established by the joint probability density function (PDF) of multiple performance functions. Based on the probability density integral equation (PDIE) of performance functions, the DPIM is proposed along the two approaches, i.e., DPIM-S and DPIM-H. The former computes the system reliability using the PDF of extreme value mapping of performance functions, which is obtained by smoothing Dirac delta function. In the latter, the system reliability formulas with Heaviside function are analytically derived by the PDIE of multiple performance functions. Specially, the role of smoothing of Dirac delta function in DPIM for stochastic response and reliability analyses is revealed. Finally, five typical examples, including two mathematical examples and three static and dynamic structural systems, demonstrate high efficiency and accuracy of the DPIM for system reliability computation. Because of omitting the smoothing of Dirac delta function, the DPIM-H takes less CPU time for solving system reliabilities of static and dynamic structures than DPIM-S, while the DPIM-S has a significant advantage of obtaining the PDF of performance function.

Structural Reliability Analysis via the Multivariate Gegenbauer Polynomial-Based Sparse Surrogate Model

Structural reliability analysis is usually realized based on a multivariate performance function that depicts failure mechanisms of a structural system. The intensively computational cost of the brutal-force Monte-Carlo simulation motivates proposing a Gegenbauer polynomial-based surrogate model for effective structural reliability analysis in this paper. By utilizing the orthogonal matching pursuit algorithm to detect significant explanatory variables at first, a small number of samples are used to determine a reliable approximation result of the structural performance function. Several numerical examples in the literature are presented to demonstrate potential applications of the Gegenbauer polynomial-based sparse surrogate model. Accurate results have justified the effectiveness of the proposed approach in dealing with various structural reliability problems.

Accurate Structural Reliability Analysis Using an Improved Line-Sampling-Method-Based Slime Mold Algorithm

Line sampling (LS) is a robust and powerful simulation technique to reduce the computational burden provided by Monte Carlo simulation (MCS) for the reliability analysis of engineering structures. However, when dealing with highly nonlinear and implicit limit-state functions, LS yields instable results as nonconvergence or divergence. In this study, a novel framework that integrates the LS method with the slime mold algorithm (LS-SMA) is proposed to solve complex structural reliability problems. SMA is a new metaheuristic population-based algorithm inspired by the behavior and morphological changes in slime molds that can well solve multivariable optimization problems. In the proposed method, the determination of the important direction of LS is formulated as an unconstrained optimization problem according to the LS theory. Then SMA is employed to solve this optimization problem to decrease the computational cost. Thus, the LS-SMA is able to overcome the drawbacks of LS such as the local convergence and divergence. Seven numerical problems were utilized to investigate the LS-SMA applicability, where its performance was compared with MCS, subset simulation (SS), importance sampling (IS), LS, first-order reliability method (FORM), and first-order control variate method (FOCM). The results demonstrate that the proposed LS-SMA can be applied with high efficiency for solving the reliability problems that involve highly nonlinear or dimensional and complex implicit limit-state functions.