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Time-variant reliability analysis plays a vital role in improving the validity and practicability of product reliability evaluation over a specific time interval. Sampling-based extreme value method is the most direct way to implement accurate reliability assessment. Its adoption for time-variant reliability analysis, however, is limited due to the computational burden caused by repeatedly evaluating performance function. This paper proposes a semi-analytical extreme value method to improve the computational efficiency of extreme value method. The time-variant performance function is transformed into dependent instantaneous performance functions in which the stochastic processes are discretized by the expansion optimal linear estimation method to simulate the dependence among different time instants. Each instantaneous function is separately approximated by Taylor series expansion at the most probable point through instantaneous reliability analysis. Based on the approximated performance functions, the computational cost of sampling-based extreme value method is significantly reduced. Results of three numerical examples demonstrate the efficacy of the proposed method.

Reliability-based design optimization (RBDO) offers a powerful tool to deal with the structural design with heterogeneous interval parameters concurrently. However, it is time-consuming in the practical engineering design. Therefore, a novel sequential moving asymptote method (SMAM) is proposed to improve the computational efficiency for convex model in this study, in which the nested double-loop optimization problem is decoupled to a sequence of deterministic suboptimization problems based on the method of moving asymptotes. In addition, the sensitivity of reliability index is derived, so the finite difference for the nested optimization loop can be avoided to tremendously improve the computational efficiency. Then, the accuracy of the SMAM is proved based on the error analysis. Furthermore, the Kreisselmeier-Steinhauser (KS) function is used to assemble the multiple constraints to deal with the parallel and series RBDO problems. One benchmark mathematical example, three numerical examples, and one complex civil engineering example, i.e., tower crane, are tested to demonstrate the efficiency of the proposed method by comparison with other existing methods, and the results indicate that SMAM offers a general and effective tool for non-probabilistic reliability analysis and optimization.

The ever-increasing demands for resource-saving, engineering technology progress, and environmental protection stimulate the progress of the progressive design method. As an excellent promising design method for dealing with the inevitable uncertainty factors, reliability-based design optimization (RBDO) is capable of offering reliable and robust results and minimizing the cost under the prescribed uncertainty level, which can provide a trade-off between economy and safety. However, the primary challenges, including global convergence capacity and complicated mixed design variable type, hinder the wider application of RBDO. This study presents a comprehensive work on the application of ten popular and recent metaheuristic algorithms of five engineering problems. Furthermore, we focus on the RBDO equip with metaheuristic algorithms about its global convergence, robustness, accuracy, and computational speed. This paper also presents the major difference of convergence property between metaheuristic algorithms and gradient algorithms. The detailed statement of this study presents the state-of-the-art in RBDO to demonstrate its crucial technologies and great challenges, as well as the beneficial future development direction.

There are many uncertain factors in aerospace structures, such as variations in manufacturing tolerance, material properties and environmental aspects. Although conventional robust design optimizat...

The convex model is commonly applied to quantify the uncertain-but-bounded parameters. However, the typical interval and ellipsoid models may lead to inaccurate approximation for existing experimental data, which may incur either too risk or conservative for safety assessment. To this end, this study aims to create a novel data-driven exponential convex model to achieve accurate approximation for experiment data, in which the dimension reduction minimum volume method plays the key role. Furthermore, a novel relaxed exponential nominal value method (RENVM) is developed to evaluate the corresponding non-probabilistic reliability index robustly and efficiently, and the sensitivities are also derived based on the straight forward perturbation method to guarantee its efficiency. Through numerical and experimental studies, the accuracy and validity of the proposed data-driven exponential convex model are validated compared to the interval and ellipsoid models, and the robustness and efficiency of the proposed RENVM are also demonstrated for solving both linear and nonlinear problems.

In this paper, a novel direct probability integral method (DPIM) is proposed to address stochastic response and dynamic reliability analyses of nonlinear multidegree-of-freedom (MDOF) syste...

In current years, the metamodel-based reliability analysis method has been developed to assess the failure probability for engineering problems involving time-consuming computational model. Despite the fact that some sequential metamodel-based reliability analysis methods have improved the computational efficiency, there still exists a certain possibility to further reduce the computational effort without loss of accuracy. In this study, an active weight learning method based upon the Kriging model is well proposed for reliability analysis. An active weight learning function based on the optimization theory is built to replace the traditional learning function, in which the important degrees of sampling points on the limit state function are assigned as different weight indices. The Kriging surrogate model is updated according to the proposed active weight learning function. In addition, the proposed strategy is extended to solve the system reliability problem, which can effectively avoid the nonlinearity of composite function in the traditional approach. A novel stopping criterion is also exploited to guarantee the convergence of the proposed method. Five numerical examples are provided to verify the effectiveness of the proposed method and convergence strategy. Results indicate that the proposed method can significantly improve the computational efficiency of reliability analysis without sacrificing computational accuracy.

Reducing the vibration of smart structures based on adaptive active vibration control (AAVC) has been extensively studied in recent years, because it is a light weight and effective method for reducing low-frequency structural vibrations. The multi-channel AAVC methodology based on the filtered-X least mean square (FXLMS) algorithm is widely implemented in active control applications owing to its self-adjustment ability to adapt to dynamically varying structures. In this paper, a new multi-channel FXLMS with online secondary path modelling (SPM) is designed based on the auxiliary random noise technique. Through a defined indirect error signal, the proposed variable step-size (VSS) strategies can ensure that every online SPM filter and every active control filter have their own exclusive step size to update the coefficients. Moreover, the proposed auxiliary noise power scheduling (ANPS) strategy can ensure that the variation rules of auxiliary noise power applied to different secondary paths are different. A complete multi-channel AAVC real-time experimental system based on NI compact RIO is set up to conduct the experimental investigation. A series of AAVC control experiments on a piezoelectric smart cantilever plate with PZT sensors and actuators are conducted to demonstrate the validity and efficiency of the proposed method. The experimental results show that the vibration of the smart cantilever plate could be effectively attenuated with a high convergence rate. The proposed methodology has an important advantage in applications where active vibration control of piezoelectric smart structures is required.

The single-loop approach (SLA) for reliability-based design optimization (RBDO) is one of the most efficient schemes for optimization problems with linear and weak nonlinear probabilistic constraints. However, it may produce unstable results or increase computational efforts when using the ordinary search direction to determine the optimum design in RBDO problems with highly nonlinear probabilistic constraints. The conjugate gradient (CG) is a promising sensitivity vector for locating the most probable point (MPP) of highly nonlinear concave performance functions. However, the MPP computation using the CG may require a high computational burden for convex constraints To overcome the drawbacks of the SLA the adaptive conjugate single-loop approach (AC-SLA) is proposed for RBDO problems with a large variety of nonlinear constraints. The sensitivity vector of the probabilistic constraints is adaptively computed using the CG vector with a dynamical conjugate scalar factor (DCF). The DCF is adjusted within the range from 0 to 2 using two adaptive coefficients, which are adapted based on the new and previous points. Moreover, the Lyapunov exponents are developed as a general tool for detecting the robustness of different MPP approximation algorithms. The method is also applied to solve a reliability-based topology optimization (RBTO) problem. The ability of the AC-SLA in six RBDO benchmark problems, one applicable to an RBDO aircraft engineering problem and one for RBTO problem, is compared in terms of both robustness and efficiency using the SLA, performance measure approach (PMA), reliability index approach (RIA), and sequential optimization and reliability assessment (SORA). The numerical experiments demonstrate that the convergence capability of the proposed AC-SLA is significantly superior to the PMA, RIA, and SORA. The computational effort of the AC-SLA is significantly reduced, with stable results.

The deep Q-network (DQN) is one of the most successful reinforcement learning algorithms, but it has some drawbacks such as slow convergence and instability. In contrast, the traditional reinforcement learning algorithms with linear function approximation usually have faster convergence and better stability, although they easily suffer from the curse of dimensionality. In recent years, many improvements to DQN have been made, but they seldom make use of the advantage of traditional algorithms to improve DQN. In this paper, we propose a novel Q-learning algorithm with linear function approximation, called the minibatch recursive least squares Q-learning (MRLS-Q). Different from the traditional Q-learning algorithm with linear function approximation, the learning mechanism and model structure of MRLS-Q are more similar to those of DQNs with only one input layer and one linear output layer. It uses the experience replay and the minibatch training mode and uses the agent’s states rather than the agent’s state-action pairs as the inputs. As a result, it can be used alone for low-dimensional problems and can be seamlessly integrated into DQN as the last layer for high-dimensional problems as well. In addition, MRLS-Q uses our proposed average RLS optimization technique, so that it can achieve better convergence performance whether it is used alone or integrated with DQN. At the end of this paper, we demonstrate the effectiveness of MRLS-Q on the CartPole problem and four Atari games and investigate the influences of its hyperparameters experimentally.

With the time-consuming computations incurred by nested double-loop strategy and multiple performance functions, the enhancement of computational efficiency for the non-probabilistic reliability estimation and optimization is a challenging problem in the assessment of structural safety. In this study, a novel importance learning method (ILM) is proposed on the basis of active learning technique using Kriging metamodel, which builds the Kriging model accurately and efficiently by considering the influence of the most concerned point. To further accelerate the convergence rate of non-probabilistic reliability analysis, a new stopping criterion is constructed to ensure accuracy of the Kriging model. For solving the non-probabilistic reliability-based design optimization (NRBDO) problems with multiple non-probabilistic constraints, a new active learning function is further developed based upon the ILM for dealing with this problem efficiently. The proposed ILM is verified by two non-probabilistic reliability estimation examples and three NRBDO examples. Comparing with the existing active learning methods, the optimal results calculated by the proposed ILM show high performance in terms of efficiency and accuracy.

Second-order reliability method (SORM) can provide sufficient accuracy for evaluating the probabilistic constraints in reliability-based design optimization (RBDO). However, the application of SORM in RBDO significantly increases the computational burden, as it is necessary to calculate the second-order sensitivities of the performance function. In order to achieve equal efficiency to that of the first-order reliability method-based RBDO approach, enhanced sequential approximate programming (ESAP) is proposed by implementing the SORM-based RBDO method. Based on the diagonal quadratic approximation method, the Hessian matrix is calculated without generating additional computational costs for providing the design sensitivity analysis of probabilistic constraints within the same iterations. Furthermore, ESAP is applied to the reliability-based topology optimization domain, and five numerical benchmark RBDO problems with two complex engineering examples are studied. The proposed ESAP is compared with other RBDO methods, including the reliability index approach, performance measure approach, sequential optimization and reliability assessment method, and SAP, and the results demonstrate the superiority of the proposed ESAP.

Non-probabilistic reliability-based design optimization (NRBDO) is an essential tool to deal with an optimization model with uncertain-but-bounded variables. A major disadvantage of NRBDO is that it conventionally suffers from excessively high computational costs. In the present study, a new target performance approach (TPA) is proposed to eliminate high computational cost incurred by complicated equality constraints in non-probabilistic reliability analysis in which the target performance constraint is applied to substitute for the non-probabilistic reliability index constraint. To further enhance the optimization efficiency, a new NRBDO algorithm is developed based on the proposed TPA in which the target performance iterative method (TPIM) is established to search for the target concern point while a convex approximate method is used to further improve the convergence. A mathematical example, three numerical examples, and two complex engineering examples, i.e., an axially compressed stiffened shell in a launch vehicle and a tower crane in civil engineering, are utilized to demonstrate the effectiveness of the proposed method in comparison with that of other existing methods. The results indicate that the proposed method is promising in terms of complicated engineering problems in the absence of prior knowledge of uncertain parameters.

In this study, we attempt to propose a new super parametric convex model by giving the mathematical definition, in which an effective minimum volume method is constructed to give a reasonable enveloping of limited experimental samples by selecting a proper super parameter. Two novel reliability calculation algorithms, including nominal value method and advanced nominal value method, are proposed to evaluate the non-probabilistic reliability index. To investigate the influence of non-probabilistic convex model type on non-probabilistic reliability-based design optimization, an effective approach based on advanced nominal value method is further developed. Four examples, including two numerical examples and two engineering applications, are tested to demonstrate the superiority of the proposed non-probabilistic reliability analysis and optimization technique.

Cuckoo Search (CS) algorithm has shortcomings of weak local search ability, slow convergence speed, and low accuracy. In order to overcome these disadvantages, an improved CS algorithm based on Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (CS-BFGS) is proposed for solving inverse geometry heat conduction problems, and the physical field is the steady-state heat conduction. Firstly, the unknown initial boundary is evolved by Levy flights and elimination mechanism. Then the BFGS algorithm is applied to minimize the objective function. Finally, the influences of random errors, measurement point number, and measurement point position on the inverse results are investigated. The results show that the CS-BFGS algorithm has higher accuracy and faster convergence speed than BFGS and CS algorithm. With the decrease of measurement errors, the increase of measurement point number, and measurement point position closer to the inverse boundary, the results become more accurate.

Due to having a large number of uncertain variables originating from various sources in time-variant reliability problem, the methods for taking account of these uncertainties are of high significance in engineering. In this study, a novel hybrid time-variant reliability model is established based on a probabilistic model and a super parametric convex model . The probabilistic and non-probabilistic parameters can be addressed simultaneously using this model. The stochastic processes are discretized using the expansion optimal linear estimation method to capture the correlation among different time nodes. In addition, the hybrid reliability iteration method is presented to obtain the upper and lower bounds of failure probability simultaneously. Subsequently, a new relaxed reliability iteration method is proposed to solve the hybrid time-variant reliability model. The relaxed method is adopted to ensure the robustness and efficiency of the proposed method. Three examples are discussed to demonstrate the validity and usage of the presented methodology.

In the reliability analysis of mechanical systems, sampling method is widely used due to the universality and practicability. However, the computation of high-dimensional problems encounters tremendous numerical difficulties, especially when the performance function is highly nonlinear. In this study, an augmented weighted simulation method (AWSM) is proposed in order to tackle this difficulty. The basic idea of AWSM is introducing a series of intermediate events into weighted simulation method (WSM), in which a new optimization method is constructed to reasonably determine each intermediate event. In this way, the failure event is divided to a sequence of conditional events, and the failure probability is accordingly converted to the product of conditional probabilities. Furthermore, a space reduction strategy is proposed to increase the probability of the samples generated in each conditional event, which greatly improves the sampling efficiency. Also, the coefficient of variation of AWSM is derived. Two mathematical examples and four engineering examples are tested, and the results demonstrate the efficiency and accuracy of the proposed method for high-dimensional problems.

Firefly algorithm (FA) is an effective meta-heuristic method, which has drawn extensive attentions and inspired many variants. However, it is difficult to balance the relation of the exploration and exploitation of FA. To this end, FA with self-adaptive strategy (SAFA) is proposed to improve the performance of the algorithm for constrained engineering design problems. Firstly, the characteristics of the standard FA are analyzed, and the essence of FA easily trapped into the local optima is revealed by theory analysis and case study. Secondly, a self-adaptive strategy for attraction model is proposed to improve the exploitation ability, and another self-adaptive strategy for stochastic model is developed to ensure a better balance between the exploration and exploitation. Thirdly, a self-adaptive penalty function is presented to treat constraint conditions. Then, the initial parameter setting is investigated, and the proper initial parameter value corresponding to the optimal performance of SAFA is obtained. And then, SAFA and other FA variants are used to optimize a set of classical and Congress on Evolutionary Computation (CEC) 2015 benchmark functions. Meanwhile, the contributions of self-adaptive strategies for attraction model and stochastic model are investigated with experimental analysis, respectively Finally, SAFA and other meta-heuristic methods are employed to solve constrained engineering design problems. The effects of SAFA with standard and self-adaptive penalty function are compared. Numerical experimental results show that self-adaptive strategies for attraction model and stochastic model improve the performance of FA, and SAFA obtains the best solutions on most of the benchmark functions. In addition, SAFA has the advantage in solving problems with different kind of dimensions, and maintains reasonable population diversity throughout the iteration compared with the other FA variants. Meanwhile, SAFA can reduce the computational complexity and ensure the solution accuracy in solving constrained optimization problems.

The efficiency and robustness are two key performance indexes for first-order reliability method (FORM). In this study, two different algorithms, including the adaptive stability transformation method (ASTM) and enhanced adaptive stability transformation method (EASTM), are proposed to improve the efficiency and robustness of FORM. In both ASTM and EASTM, the computation of most probable failure point (MPFP) is converted to find a series of most probable target points (MPTPs), in which the chaos control factor is properly selected by proposing two different algorithms. Four benchmark examples with normal and non-normal random variables and one practical engineering application example are tested to verify the effectiveness of the proposed algorithms. The results illustrate that the proposed algorithms not only are more efficient than other advanced FORM algorithms, but also are very robust.

For solution of reliability-based design optimization (RBDO) problems, single loop approach (SLA) shows high efficiency. Thus SLA is extensively used in RBDO. However, the iteration solution procedure by SLA is often oscillatory and non-convergent for RBDO with nonlinear performance function. This prevents the application of SLA to engineering design problems. In this paper, the chaotic single loop approach (CLSA) is proposed to achieve the convergence control of original iterative algorithm in SLA. The modification involves automated selection of the chaos control factor by solving a novel one-dimensional optimization model. Additionally, a new oscillation-checking method is constructed to detect the oscillation of iterative process of design variables. The computational capability of CLSA is demonstrated through five benchmark examples and one stiffened shell application. The comparison of numerical results indicates that CSLA is more efficient than the double loop approach and the decoupled approach. CSLA also solves the RBDO problems with highly nonlinear performance function and non-normal random variables stably.

The second-order reliability method (SORM) is an effective tool to evaluate the reliability, but its application for reliability-based design optimization faces unbearable computational cost. In this study, a new hybrid sequential approximate programming (HSAP) method is developed to calculate the optimum efficiently by developing a distance-checking criterion and a convex approximate method. Since the distance-checking criterion identifies the feasibility of the probabilistic constraint effectively, the proposed method combines the efficiency of the sequential approximate programming method and the accuracy of SORM. The convex approximate method is also constructed using the sensitivity and function value of the probabilistic constraint. So no additional computational cost is required in the optimization process. Five illustrative examples, including two mathematical examples and thee practical engineering examples, demonstrate the efficiency and accuracy of the proposed HSAP.

Currently, most works on layout optimization problem of continuum structure embedded with movable holes are all carried out to maximize the stiffness of the overall system. In this work, the embedding problem is solved for minimum stress design for the first time. To this end, we propose an effective hybrid methodology under SIMP-based computational framework. The material density characterizing the topology of the load transfer path and the geometric parameters defining the rigid body motion (translation and rotational) of the embedded holes are considered as design variables and are optimized simultaneously to minimize the aggregated maximum stress. To unite these two seemingly different representations into an optimization model, we mapped the embedded holes into a density field on a fixed grid using a Sigmoid activation function. Then, a new SIMP-like material interpolation scheme that considers embedding holes is introduced for stiffness penalization and stress penalization. The optimization model for solving the minimum stress problem embedded with movable holes and its sensitivity analysis are detailed. Finally, two typical examples are performed to illustrate the effectiveness of the proposed methodology.

This study presents a hybrid reliability-based topology optimization (RBTO) method for handling epistemic and aleatory uncertainties. First, we establish a new triple-nested RBTO model based on fuzzy and probabilistic theory for describing the multi-source uncertainties. Subsequently, an efficient single-loop optimization method is proposed to degrade the triple-nested optimization problem into a deterministic optimization problem using the Karush–Kuhn–Tucker optimality condition. Furthermore, the sensitivities of the hybrid reliability constraint with respect to the random probabilistic variables, fuzzy variables, and deterministic design variables are derived using the adjoint variable method. Finally, a cantilever beam example, an L-shape beam design and a 3D example are tested to verify the validity of the proposed single-loop method.

Due to the nested optimization loop structure and time-demanding computation of structural response, the computational accuracy and cost of reliability-based design optimization (RBDO) have become a challenging issue in engineering application. Kriging-model-based approach is an effective tool to improve the computational efficiency in the practical RBDO problems; however, a larger number of sample points are required for meeting high computational accuracy requirements in traditional methods. In this paper, an adaptive directional boundary sampling (ADBS) method is developed in order to greatly reduce the computational sample points with a reasonable accuracy, in which the sample points are added along the ideal descending direction of objective function. Furthermore, only sample points located near the constraint boundary are mainly selected in the vicinity of the optimum point according to the strategy of multi-objective optimization; thus, substantial number of sample points located in the failure region is neglected, resulting in the improved performance of computational efficiency. Four numerical examples and one engineering application are provided for demonstrating the efficiency and accuracy of the proposed sampling method.

To achieve an optimal design of complicated structures with stochastic parameters, the reliability-based design optimization (RBDO) usually needs to handle the nested double optimization loops, which results in unbearable computational cost. In this paper, a new active learning method for RBDO combining with Kriging metamodel and accelerated chaotic single loop approach (AK-ACSLA) is developed, in which the most probable learning function (MPLF) is proposed to search the most probable point instead of the limit state function in entire design space with an active learning behavior. To ensure the high efficiency, the system’s most probable learning function (SMPLF) is further constructed to solve the RBDO problem of series system with multiple probabilistic constraints, and then the ACSLA is proposed by taking full advantage of chaos feedback control methodology for guaranteeing the validity of AK-ACSLA. Nonlinear mathematical examples and complex RBDO engineering examples illustrate the high efficiency and accuracy of AK-ACSLA through comparison with both existing gradient-based methods and active learning methods.

In the practical engineering applications of stiffened shell, the initial imperfection is inevitable and it could cause significant reduction in the load-carrying capacity of stiffened shell. The light-weight optimization of stiffened shell is generally performed under the constraint of fixed maximum load-carrying capacity. However, the load-carrying capacity of stiffened shell has been improved continuously as the promotion of manufacturing technology, which causes the previous strategies of light-weight optimization become conservative and outdated. Therefore, an improved hybrid framework of multi-objective optimization of stiffened shell with imperfection is necessary and presented in this paper, which focus on developing a general posterior design method to determine the optimal weight according to the different collapse loads. A new adaptive update criterion based on the Kriging model is developed to improve the efficiency and accuracy of the hybrid framework. The present optimal results provide a set of the Pareto optimal points and form a Pareto front, from which new posterior design can be achieved.