Complex Performance Functions Research Articles

Drivers’ Steering Behavior in Curve by Means of New Indicators

The design of road geometry is based on a rather elementary assumption that the user strictly follows the lane axis. Based on this hypothesis, the ideal trend of some factors related to the driver’s performance, such as steering angle and speed, can be derived to optimize the most appropriate design choices. In practice, driving behavior differs from the assumed one and produces trends in these variables, which are very different from the ideal functions. The purpose of this research is therefore to propose synthetic performance indicators useful for highlighting the real characteristics of users’ driving behavior during road travel. Toward this aim, some driving experiments along four different curves in a simulated environment were studied in order to evidence possible criticisms. The proposed indicators showed a remarkable ability to represent and synthesize even very complex performance function trends. The proposed performance indicators can have multiple uses, such as, for example, in statistical analyses—which are generally carried out at a later stage—or constitute sufficient information to guide the decisions of infrastructure managers. In the long term, in a “smart road” perspective, they can be used by road administrators for information exchange among users (with each other and with the infrastructure) to improve road operation and safety.

A novel adaptive approach based on over-distance interval searching and precise step size determining for first-order structural reliability analysis

The first-order second-moment reliability method (FORM) is a prevalent method in the reliability analyzing community owing to its accuracy and efficiency. However, it may encounter unstable solutions such as periodic oscillation and non-convergence when a high-dimensional problem with complex performance functions is involved. In this study, a novel adaptive approach based on over-distance interval searching and precise step size determining is presented to work out a solution for the above issues. First, the relationship between the state of the iterative rotation angle and the ideal step size was noticed. Therefore, a rotation control condition is proposed to dictate the selection of step size for the algorithm. After that, the determination of the step size of the proposed method here can be realized in two stages: In the first stage, an over-distance step size interval searching approach is established to define an initial exploration interval of step size with a length of 1. In the second stage, a determination approach of adaptive precise step size is then proposed to modify the fixed step sizes to make it adaptable to changes, therefore, get the final step sizes for the proposed method. Together, an over-distance adaptive precise searching approach of step size that can generate self-adaptive step sizes both greater and less than 1 is performed by interplaying the above two stages, thus preventing an excessive amount of redundant workload and speeding up the convergence while reducing the risk of oscillations. Besides, an iterative direction modification approach is given to orient the first iteration more accurately. In addition to that, the proposed method is tested on a offshore jacket platform and other seven examples to verify its performance. The results demonstrate that the proposed method provides excellent robustness and efficiency for both low and high-nonlinearity problems, showing significant merits.

A Kriging-based adaptive adding point strategy for structural reliability analysis

Aiming at the problem of structural reliability analysis with complex performance function in practical engineering, an adaptive adding point strategy for structural reliability analysis is proposed by combining Kriging surrogate model and learning function. In the process of structural reliability analysis, in order to reduce the computational cost, the surrogate model is usually used to fit the implicit performance function. Existing learning functions rarely take into account the problem of sample point aggregation due to too many sample points selected from the failure domain (or safety domain) during the fitting process. In order to overcome this defect, a new method can ensure that the added sample points are evenly distributed on both sides of the limit state function, preventing the aggregation of sample points and causing information redundancy, thereby improving the fitting accuracy of the model, accelerating the convergence speed of the sample points and saving the sample space. Through the analysis of four-branch series system, nonlinear oscillator and truss structure, the results show that the algorithm needs less samples than other methods, and the reliability calculation accuracy is higher, which verifies the correctness and efficiency of the proposed method.

Hybrid uncertainty propagation and reliability analysis using direct probability integral method and exponential convex model

Uncertainty propagation and reliability evaluation, being the crucial parts of engineering system analysis, play vital roles in safety assessment. How to reasonably consider the complex multisource uncertainty behavior in both static and dynamic systems is paramount to ensuring their safe operation. However, there is a significant lack of research on aleatory and epistemic uncertainties for both static and dynamic systems. To this end, a new hybrid exponential model is proposed by combining probabilistic and non-probabilistic exponential models, which aims to accurately measure the uncertainty propagation and reliability evaluation problem with aleatory and epistemic uncertainties for static and dynamic systems. The proposed hybrid exponential model consists of nested double optimization loops. The outer loop performs a probabilistic analysis based on the direct probability integral method, and the inner loop performs a non-probabilistic computation. Then, a new hybrid exponential probability integral method is developed to effectively perform uncertainty propagation and reliability analysis. Finally, four examples, including two static and two dynamic examples with complex performance functions, are tested. The results indicate that the proposed hybrid exponential model offers a universal tool for uncertainty quantification in static and dynamic systems. Moreover, the hybrid exponential probability integral method can accurately and efficiently obtain the upper and lower bounds of the probability density function and cumulative distribution function.

The adaptive weight learning function for reliability analysis and its application to multiple active learning methods

AbstractIn engineering, evaluating the failure probability of structures with complex performance functions is a challenging task. Some active learning methods based on the Kriging model gain considerable attention for structural reliability analysis. The existing learning functions offer new learning criterions to select sample points, but the probability density function (PDF) of sample points is ignored in some learning functions. To decrease selecting sample points in low PDF regions, this study presents a new adaptive weight learning function (WLF). The learning function WLF constituted by the adaptive orient function and the joint PDF can be applicable to multiple learning functions. Depending on the importance degree of candidate sample points, the sample points can be assigned different weight by learning function WLF. When the leaning function WLF is applicated to the existing active learning methods, the sample point with a higher PDF in neighborhood of the limit state function (LSF) can be given a larger weight to preferentially select into design of experiments (DoE). Therefore, the learning function WLF can help the multiple learning functions to select informative sample points with high PDF, which can further improve the efficiency of these learning functions. Four examples are used to illustrate the accuracy and efficiency as well as the suitability of the learning function WLF.

Reliability analysis based improved directional simulation using Harris Hawks optimization algorithm for engineering systems

In this paper, a new framework for accurate reliability analysis is proposed based on improving the directional simulation by using metaheuristic algorithms. Usually for highly nonlinear and complex performance functions, finding the unit vector direction requires very high calculations or impossible practically. Hence, the novel improved version incorporates the Harris Hawks Optimization algorithm, where the unit vector of direction is formulated as a constrained optimization problem and estimated using metaheuristic algorithms. Given that metaheuristic algorithms have been introduced to solve unconstrained problems, the penalty function method is used to convert the constrained problem into an unconstrained problem. The applicability of the proposed framework is firstly tested on five highly nonlinear benchmark functions and then applied to solve four high-dimensional engineering problems. The performance of six simulations-based reliability analysis methods and the first-order reliability method were compared with the proposed method. Besides the feasibility of other metaheuristic algorithms were investigated. The results show high-performance abilities of the improved version of the directional simulation for solving highly nonlinear engineering problems.

Novel efficient method for structural reliability analysis using hybrid nonlinear conjugate map-based support vector regression

The estimation of the failure probability for complex systems is a crucial issue for sustainability. Reliability analysis methods are needed to be developed to provide accurate estimations of the safety levels for the complex systems and structures of today. In this paper, a novel hybrid framework for the reliability analysis of engineering systems and structures is extended to reduce the computational burden. The proposed hybrid framework is named as SVR–CFORM and consists of coupling two parts: the first is an enhanced first-order reliability method (FORM) using nonlinear conjugate map (CFORM); the second is an artificial intelligence technique called support vector regression (SVR). The conjugate FORM (CFORM) is adaptively formulated to improve the robustness of the original iterative FORM algorithm, whereas the SVR technique is used to enhance the efficiency of the reliability analysis by reducing the computational burden. The performance of the proposed SVR–CFORM formulation is compared in terms of efficiency and robustness with several FORM formulas (i.e. HL–RF, directional stability transformation method, conjugate HL–RF and finite step length) through different numerical/structural reliability examples. Results indicate that the proposed SVR–CFORM formulation is more accurate and efficient than other reliability methods. Based on the comparative analysis results, the SVR technique can highly reduce the computational costs and accurately model the response of complex performance functions, while the iterative CFORM formulation found to provide stable and robust reliability index results compared to the others reliability methods.

Efficient sample reduction strategy based on adaptive Kriging for estimating failure credibility

Failure credibility is popular in measuring safety degree of structure under fuzzy uncertainty, but the heavy computational cost is still a challenge in estimating the failure credibility. To alleviate this issue, an iterative method combining adaptive Kriging and fuzzy simulation (AK-FS) has been developed by Ling et al. However, for the problem with complex performance function, a large candidate sampling pool is needed in the AK-FS, which makes the training process of the Kriging model fairly time consuming. In order to improve the estimation efficiency of failure credibility through reducing the size of candidate sampling pool in AK-FS, an efficient sample reduction strategy based on adaptive Kriging (SR-AK) is proposed in this paper. In the SR-AK, the estimation of failure credibility is transformed into searching two active points in candidate sampling pool. After updating the Kriging model in each circle, current active points can be easily identified. Then, according to the properties of the active points and the prediction characteristics of Kriging model, the samples in current candidate sampling pool can be divided into two sets, i.e., the samples affect the estimation of active points and the samples have no effect on it. Obviously, the samples in the latter set can be deleted from current candidate sampling pool to reduce its size. By using this sample reduction strategy, the process for training Kriging model is accelerated circle by circle, which is very helpful to save the analysis time and improve the computational efficiency in estimating failure credibility. Four examples are employed to demonstrate the performance of the proposed SR-AK in fuzzy safety degree analysis.

Backward Recursive Method for Structural Reliability Calculation Based on the Universal Generating Function

For structural performance functions with characteristics of multivariable, non-normality, small failure probability and nonlinearity, it is usually difficult to pursue satisfactory accuracy by means of traditional probabilistic methods such as first-order reliability method and second-order reliability method. According to the idea of adaptive subdivision, a backward recursive method based on the universal generating function is suggested for structural reliability calculation. In the sensitive region, the variable space is subdivided partially, the discrete interval length is reduced, and the random variables are discretized nonuniformly and adaptively by the backward recursion operation. Compared with the current method of forward recursion operator and the corresponding similar items merging operation, the backward recursion operator can not only avoid low accuracy owing to subdividing the state combinations in the sensitive region, but avert the high cost of calculation owing to merging state combinations in the nonsensitive region of the variable space as well. Three examples are presented to demonstrate the rationality of the backward recursive algorithm. The results indicate that the accuracy of the proposed method is significantly higher than that of traditional methods, and the computational efficiency of the proposed method meets the engineering requirement. The research work provides a new path for reliability calculation with consideration of complex performance functions.

Reliability analysis of pumping station for sewage network using hybrid neural networks - genetic algorithm and method of moment

The reliability of pumping stations is of great importance for the robust design of wastewater networks. In order to strike a balance between safe design and energy consumption of the pumping system, a complex performance function using multiple-failure mode under various uncertainties is required. In the present research to evaluate the failure probability of wastewater pumping station, a hybrid reliability analysis framework using artificial neural network (ANN) coupled by moment methods is proposed. Drawing upon genetic algorithm (GA), ANN model is trained to approximate the failure domain of the pump. The ANN is approximated input variables captured by the optimal condition of the rotational speed of the pump in turn it is obtained by GA. The main strength of the reliability method is to diminish the computational burden with accurate predictions of safety margin in the pumping systems of Zabol. The machine learning-based backpropagation (BP) for training feedforward neural networks using GA and gradient methods are compared for the regressed process of ANN models. This hybrid reliability framework involves three main levels including i) the pump rotational speed is minimized using GA as optimal hydraulic parameters such as inlet flow, static head, pumping head and outlet flow rate, ii) the limit sate function is approximated using ANN for optimal rotating speed pumps and iii) reliability analysis is disused using MCS and method of the moment for sewage pumping stations located at Zabol (Iran) station. Given the results, the machine learning-based GA for training ANN model provides accurate predictions compared to the ANN-based gradient method. Three moments of reliability method is an efficient and accurate approach to evaluate the reliable conditions of the pumping system.

New efficient and robust method for structural reliability analysis and its application in reliability-based design optimization

Due to its advantages of simplicity and high efficiency, Hasofer–Lind and Rackwitz–Fiessler (HL–RF) method is widely used in structural reliability analysis. However, it may encounter difficulties in convergence, such as periodic oscillation, bifurcation and chaos when complex performance functions are involved. In this study, a new efficient and robust method is proposed to deal with the convergence problem and is applied to reliability-based design optimization (RBDO). The strategy is to modify the search direction of HL–RF to make it adaptable to changes, thereby reducing the risk of periodic oscillations. A step size adjustment formula is then established to adaptively adjust the finite-step size to prevent bifurcation or chaos. In addition, the proposed method is integrated into double loop method (DLM) to strengthen the robustness and efficiency of DLM for complex RBDO problems. The efficiency and robustness of the proposed method are demonstrated by comparison with other existing first order reliability methods (FORMs) through five numerical examples and two structural examples. Two complex RBDO problems also show that DLM based on the proposed method has the ability to solve complex RBDO problems.

M5 model tree and Monte Carlo simulation for efficient structural reliability analysis

• CFORM is proposed to enhance the robustness of FORM-based MPP search. • M5Tree model with MCS is developed to improve the accuracy and efficiency of structural reliability analysis. • The performances of CFORM and M5Tree+MC are illustrated using several reliability examples. • The CFORM is more efficient than MCS and robust than the HL-RF method. • The M5Tree+MC method provides an accurate result compared to CFORM, more efficiently. Practically, the performance of many engineering problems can be defined using a complex implicit limit state function. Approximation of the accurate failure probability is very time-consuming and inefficient based on Monte Carlo simulation (MCS) for complex performance functions. M5 model tree (M5Tree) model is robust approach for simulation and prediction phenomena, which provides ability to dealing with complex implicit problems by dividing them into smaller problems. By improving the efficiency of reliability method using accurate approximated failure probability, an efficient reliability method using the MCS and M5Tree is proposed to calibrate the performance function and estimate the failure probability, respectively. The superiorities including simplicity and accuracy of M5Tree meta-model are investigated to evaluate the actual performance function through five nonlinear complex mathematical and structural reliability problems. The proposed reliability method-based MCS and M5Tree improved the computational efforts for evaluating the performance function in reliability analysis. The M5Tree significantly increased the efficiency of reliability analysis with accurate failure probability.

Assessing small failure probabilities by AK–SS: An active learning method combining Kriging and Subset Simulation

With complex performance functions and time-demanding computation of structural responses, the estimation of small failure probabilities is a challenging problem in engineering. Although Subset Simulation (SS) is a powerful tool for small probabilities, the computation amount is still large for time-consuming numerical procedures. Metamodelling is an important approach to increase the computational efficiency for engineering problems, however, a larger set of sample points is required for higher accuracy. This is a time-consuming task when the performance function needs to be numerically evaluated. To address this issue, AK–SS: an active learning method combining Kriging model and SS is proposed in this paper. The efficiency of this new method relies upon the advantages of SS in evaluating small failure probabilities and the Kriging model with active learning and updating characteristic for approximating the true performance function. The proposed method is applied to several benchmark functions in the literature, and to the reliability analysis of a shield tunnel, which requires finite element analysis. The results demonstrated that as compared to the other approaches in literature, AK–SS can provide accurate solutions more efficiently, making it a promising approach for structural reliability analyses involving small failure probabilities, high-dimensional performance functions, and time-consuming simulation codes in practical engineering.

New algorithm of mine slope reliability based on limiting state hyper-plane and its engineering application

Due to the influence of joint fissure, mining intensity, designed slope angle, underground water and rainfall, the failure process of mine slope project is extremely complicated. The current safety factor calculation method has certain limitations, and it would be difficult to obtain the reliability index when the performance function of reliability analysis is implicit or has high order terms. Therefore, with the help of the logistic equation of chaos theory, a new algorithm of mine slope reliability based on limiting state hyper-plane is proposed. It is shown that by using this new reliability algorithm the calculation of partial derivative of performance function is avoided, and it has the advantages of being simple and easy to program. The new algorithm is suitable for calculating the reliability index of complex performance function containing high order terms. Furthermore, the limiting state hyper-plane models of both simplified Bishop’s and Janbu’s method adaptive to slope project are obtained, and have achieved satisfactory effect in the study of mine slope stability in Dexing copper open pit.

Modeling multivariate distributions using Monte Carlo simulation for structural reliability analysis with complex performance function

The simulation of multivariate distributions has not been investigated extensively. This article aims to propose Monte Carlo simulation (MCS)-based procedures for modeling the joint probability distributions and estimating the probabilities of failure of complex performance functions. Two approximate methods, namely method P and method S, often used to construct multivariate distributions with given marginal distributions and covariance, are introduced. The MCS-based procedures are proposed to simulate the theoretical multivariate distributions or approximate multivariate distributions constructed by methods P and S, which are further used to compute the probabilities of failure of complex performance functions. Four illustrative examples with known theoretical joint probability distributions are investigated to examine the accuracy of the proposed procedures in modeling the multivariate distributions and estimating the probabilities of failure. The results indicate that the bivariate distributions can be effectively simulated by the proposed procedures, which can evaluate the reliability of complex performance functions efficiently. These provide a useful tool for solving the reliability problems with complex performance functions involving correlated random variables under incomplete probability information. The performance of the simulation procedures associated with the two approximate methods highly depends on the level of probability of failure, the form of performance function, and the degree of correlation between variables.

Sensitivity of Active Two-Ports

A possible measure for estimating the sensitivity of a complex performance function of a system with respect to variations in a complex variable, including the case where the function is not differentiable, is presented by defining the magnitude of the relative change in the complex function divided by the magnitude of the relative change in the complex variable. If the performance function is differentiable in the complex variable, the present definition equals the absolute value of the conventional defining equation of sensitivity. The sensitivity of a real performance function to changes in a complex variable is obtained as the special case of the present definition. As the application of the present definition to the class of function for which the Cauchy-Riemann equations are not satisfied, the sensitivities of the power wave transfer ratio and the transducer power gain to variations in the terminating immittances and in each component of circuit parameters are obtained. Then, in order to evaluate graphically the changes in these maximum sensitivities due to those in the terminating immittances, various sensitivity charts that are the distribution maps of maximum sensitivities on the Smith chart or the power wave reflection coefficient plane are developed.