Nonlinear Performance Functions Research Articles

Hybrid adaptive moment estimation based performance measure approach for complex reliability-based design optimization

The complexity of engineering problems often poses challenges, especially in reliability-based design optimization, where the nonlinearity of the performance function is not known in advance. The substantial costs and oscillating results significantly impede the engineering application for highly nonlinear problems. To address this issue, a hybrid adaptive moment estimation (HAME) method is proposed in this work. Given that the performance measure approach is heavily influenced by the gradient of performance function, this method employs the statistical moment information of the gradient, thereby accelerating the search for the minimum performance target point. By combining gradient mean and modified chaos control as new search directions, oscillation phenomena are resolved. To filter the aggregated gradient calculations, the gradient variance and state variable are introduced to approximate the gradients for weakly nonlinear performance functions or in later iterations by the dynamic identification of the nonlinearity of the performance function. The proposed method eliminates the limitation of taking the upper bound for the chaos control factor and addresses the balance issue between efficiency and robustness. The advantages of the proposed method are demonstrated through inverse reliability analysis problems, several highly nonlinear classic RBDO examples, and engineering applications of the crank rod mechanism of an engine.

Effects of limit state data on constructing accurate surrogate models for structural reliability analyses

Engineering problems are mainly defined in implicit processes; hence, the fully probabilistic analyses, e.g., Monte Carlo simulations (MCS), are expensive to implement. In practice, two approaches to overcome the issues are either reducing the size of simulations or developing surrogate models for actual problems. The latter does not sacrifice the size of MCS and requires less insight into probabilistic calculation; hence, it is preferable to most engineers. This study proposes an efficient framework to develop reliable and accurate surrogate models by considering data at the limit state margins (LS data). Effects of involving LS data in the training process and performances of the proposed metamodels are investigated for most issues relating to reliability analyses, including nonlinear performance functions, multiple failure modes, and implicitly defined problems. Two machine learning algorithms, including artificial neural networks and the Gaussian process, are employed to prove the ability of the proposed method. Investigations reveal that the limit state data plays a vital role in developing accurate surrogate models for reliability analyses, and accumulating them into the training dataset helps quickly construct accurate metamodels. This work contributes a practical framework for reliability analyses because the LS data can be detected easily without insight into probabilistic calculations.

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.

How Useful is Learning in Mitigating Mismatch Between Digital Twins and Physical Systems?

In the control of complex systems, we observe two diametrical trends: model-based control derived from digital twins, and model-free control through AI. There are also attempts to bridge the gap between the two by incorporating learning-based AI algorithms into digital twins to mitigate mismatches between the digital twin model and the physical system. One of the most straightforward approaches to this is direct input adaptation. In this paper, we ask whether it is useful to employ a generic learning algorithm in such a setting, and our conclusion is “not very”. We denote an algorithm to be more useful than another algorithm based on three aspects: 1) it requires fewer data samples to reach a desired minimal performance, 2) it achieves better performance for a reasonable number of data samples, and 3) it accumulates less regret. In our evaluation, we randomly sample problems from an industrially relevant geometry assurance context and measure the aforementioned performance indicators of 16 different algorithms. Our conclusion is that blackbox optimization algorithms, designed to leverage specific properties of the problem, generally perform better than generic learning algorithms, once again finding that “there is no free lunch”. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Digital twins have the potential to improve productivity and quality in complex systems such as manufacturing systems. Their impact on system performance hinges on the accuracy of their digital models around the system’s operating points. Difficult to measure phenomena, such as wear and tear of equipment, however, may cause a mismatch between the digital twin model and the physical system. In this paper, we formalize this problem and compare 16 potential solution strategies under practical aspects. We argue that readily available off-the-shelf blackbox optimization algorithms may prove more useful for this problem, than more recent learning-based approaches. Specifically, gradient-based algorithms will perform best in systems with high-dimensional, continuous, and non-linear performance functions – even in the presence of white measurement noise.

AK-SEUR: An adaptive Kriging-based learning function for structural reliability analysis through sample-based expected uncertainty reduction

Reliability Analysis (RA) is a critical aspect of structural design and performance evaluation aiming to determine the probability of structural failure under given random input parameters. With modern development of modeling techniques, computational models have achieved higher fidelity but at the increased cost of computational time, which poses a significant challenge for RA. Consequently, surrogate model-assisted RA has been explored as a means of improved efficiency and accuracy. This study proposes a novel learning function, Sample-based Expected Uncertainty Reduction (SEUR), for surrogate model-assisted RA. The SEUR function uses statistical information from the metamodeling with fixed hyper-parameters to construct expected failure probability bounds to sequentially update the design of experiment (DoE). The joint probability densities of input variables are accounted for through simulation methods, including Monte Carlo (MC) and subset simulation (SS). Furthermore, the discrete simulated annealing algorithm is used to search for the optimal design point. The performance of proposed AK-SEUR function is systematically evaluated using six examples of different dimensions, failure probability levels and nonlinearities. The AK-SEUR function is demonstrated to be more effective and efficient than other popular active learning methods in dealing with nonlinear performance functions, small probabilities, and complex limit states. The proposed SEUR function has the potential to improve the efficiency and accuracy of RA, particularly in situations where computational models are time-consuming and the search for the optimal solution is challenging.

Finite‐time optimal tracking control for a class of nonlinear systems with prescribed performance

AbstractIn this paper, a finite‐time optimal tracking control scheme based on integral reinforcement learning is developed for partially unknown nonlinear systems. In order to realize the prescribed performance, the original system is transformed into an equivalent unconstrained system so as to a composite system is constructed. Subsequently, a modified nonlinear quadratic performance function containing the auxiliary tracking error is designed. Furthermore, the technique of experience replay is used to update the critic neural network, which eliminates the persistent of excitation condition in traditional optimal methods. By combining the prescribed performance control with the finite‐time optimization control technique, the tracking error is driven to a desired performance in finite time. Consequently, it has been shown that all signals in the partially unknown nonlinear system are semiglobally practical finite‐time stable by stability analysis. Finally, the provided comparative simulation results verify the effectiveness of the developed control scheme.

Response Surface Method for Reliability Analysis Based on Iteratively-Reweighted-Least-Square Extreme Learning Machines

A response surface method for reliability analysis based on iteratively-reweighted-least-square extreme learning machines (IRLS-ELM) is explored in this paper, in which, highly nonlinear implicit performance functions of structures are approximated by the IRLS-ELM. Monte Carlo simulation is then carried out on the approximate IRLS-ELM for structural reliability analysis. Some numerical examples are given to illustrate the proposed method. The effects of parameters involved in the IRLS-ELM on accuracy in reliability analysis are respectively discussed. The results exhibit that a proper number of samples and neurons in hidden layer nodes, an appropriate regularization parameter, and the number of iterations for reweighting are of important assurance to obtain reasonable precision in estimating structural failure probability.

Time-Dependent Failure Possibility-Based Design Optimization by Using Kriging Model and Fuzzy Simulation

Time-dependent failure possibility-based design optimization (T-PBDO) can minimize the general cost while meeting the failure possibility requirement of aircraft structure in the service life. The accuracy of the T-PBDO solution obtained by existing efficient methods may be problematic in the case of nonlinear performance functions or multiple minimum performance target points. To overcome this limitation, this paper proposes a new double-loop method based on the adaptive kriging (AK) model and fuzzy simulation (FS), referred to as DL-AK-FS, is used to efficiently solve T-PBDO. In DL-AK-FS, to replace the real constraint performance function for dealing with the time-dependent failure possibility (TDFP) constraint, the inner loop is to adaptively construct a single-loop kriging model of the constraint performance function in the FS candidate sample pool. The outer loop is to search the optimal design parameters by optimization algorithm. The kriging model is first built in an augmented space that is spanned by design parameters and fuzzy inputs, and then it is adaptively and timely updated during the optimization iteration. Moreover, the strategy of reducing the size of the FS candidate sample pool is adopted to further improve the efficiency of analyzing the inner TDFP while ensuring the accuracy of the optimization solution. The strategy of combining FS with the AK model can extend the engineering applicability of the DL-AK-FS in estimating the inner TDFP, which is not limited by the complexity of the time-dependent performance function. The optimization results show that the proposed DL-AK-FS method in this paper is efficient and accurate for solving T-PBDO.

EMCS-SVR: Hybrid efficient and accurate enhanced simulation approach coupled with adaptive SVR for structural reliability analysis

In structural reliability analysis, robust and efficient sampling methods that address low failure probabilities are vital challenges. In this paper, a novel dynamical adaptive enhanced simulation method coupled with support vector regression (SVR) is proposed. Firstly, a more general and efficient approximation formula is proposed as an improved scheme. Furthermore, a dynamical adaptive simulation strategy for Monte Carlo simulation and an active training methodology basis SVR are developed. The dynamical active region for improving the efficiency and robustness of structural reliability analysis is applied for training the SVR models which are utilized for accurate estimating the failure probability by simulation methods. Analytical methods and crude Monte Carlo simulation are used for comparison, validation and discussion with the proposed hybrid simulation method using four numerical examples and four engineering problems. Through coupling SVR with dynamical active region, an accurate failure probability prediction with robust and low-computational cost is achieved. The proposed adaptive strategy applied in hybrid enhanced simulation approaches provided the accurate results with low-computational burden and these hybrid methods are robust than the analytical approaches. The proposed methods have shown strong​ capability for application in engineering problems with complex nonlinear performance functions.

Fuzzy First-Order and Second Moment Method for Failure Credibility Analysis in the Presence of Fuzzy Uncertainty

A fuzzy first-order and second moment (FFOSM) method is proposed to efficiently estimate time-independent and time-dependent failure credibility in the presence of fuzzy uncertainty. For the time-independent linear performance function with independently, regularly, and identically distributed fuzzy inputs, the analytical relationship is first derived between the failure credibility and fuzzy safety index, and the relation form is established between fuzzy safety index and fuzzy first two moments of the performance function. Second, in the standard space of fuzzy inputs, the minimum infinite norm distance from the coordinate origin to the linear limit state surface is analytically derived, and the point with the smallest infinite norm distance located on the limit state surface is defined as the fuzzy design point. Based on this definition, the geometric and physical meanings are explained for the fuzzy safety index and fuzzy design point, and the FFOSM method for time-independent failure credibility analysis is established for both linear and nonlinear performance function. Third, for time-dependent performance functions, the equivalence is proved between time-dependent fuzzy safety index and the minimum value of transient fuzzy safety indices, on which the iteration steps of the FFOSM method is designed for time-dependent fuzzy safety index and failure credibility estimation. Since the proposed FFOSM method is an approximately analytical approach, it has higher efficiency than the fuzzy simulation method and the cut level optimization method. Finally, the effectiveness and applicability of the proposed method are demonstrated by different examples.

Estimation of Small Failure Probability Based on Adaptive Subset Simulation and Deep Neural Network

Abstract The critical problem of reliability design is how to obtain a more accurate failure probability with a smaller number of evaluations of actual complex and nonlinear performance function. To achieve this objective, an adaptive subset simulation method with a deep neural network (DNN) is proposed for accurate estimation of small failure probability. A determinate criterion for threshold values is developed, and the subset number is adaptively quantified according to the initial estimated value of small failure probability. Therefore, the estimation of small failure probability is converted to estimation problem of multiple large conditional probabilities. An adaptive deep neural network model is constructed in every subset to predict the conditional probability with a smaller number of evaluations of the actual performance function. Furthermore, the sampling points for the next subset can be adaptively selected according to the constructed DNN model, which can decrease the number of invalid sampling points and evaluations of actual performance function, then the computational efficiency for estimating the conditional probability in every subset is increased. The sampling points with high probability density functions are recalculated with actual performance function values to replace the predicted values of the DNN model, which can verify the accuracy of DNN model and increase the estimation accuracy of small failure probability. By analyzing a nonlinear problem, a multiple failure domain problem and two engineering examples, the effectiveness and accuracy of the proposed methodology for estimating small failure probability are verified.

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.

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.

An equivalent expectation evaluation method for approximating the probability distribution of performance functions

This paper proposes an equivalent expectation evaluation method for approximating the probability distribution of performance functions. The innovation of the proposed method is to formulate the probability distribution of the performance function as an expectation regardless of the dimensionality and complexity of the problem. When a monotonic inverse function (MIF) of the performance function with respect to a certain random variable can be obtained, the probability distribution can be exactly formulated as an expectation related to the distribution of this variable and the MIF. While if such a MIF does not exist or cannot be easily obtained, by constructing two simple auxiliary functions whose MIF can be obtained, a generalized method is proposed to formulate the probability distribution of the performance function as an expectation related to the distribution of the auxiliary functions. Then, the Sobol sequence and point estimate method are employed to determine the expectation. The efficiency and accuracy of the proposed method are demonstrated through four numerical examples. The results show that the proposed method is applicable to approximating probability distributions of both static and dynamic reliability problems involving high-dimensional, nonlinear, or implicit performance functions.

An approximate single-loop chaos control method for reliability based design optimization using conjugate gradient search directions

Single-loop methods based on the Karush–Kuhn–Tucker conditions are considered to be efficient reliability based design optimization (RBDO) methods. The probabilistic performance functions are converted to deterministic functions for reducing the computational burden of reliability analysis. However, the most probable target point (MPTP) estimated using steepest descent search directions diverges or oscillates for highly nonlinear performance functions. Therefore, an approximate single-loop chaos control method is proposed to address this challenge by estimating MPTP using conjugate gradient search directions. An oscillation criterion is also proposed to track the oscillation of MPTP in every iteration. When this criterion is satisfied, the chaos control theory is used to update the current MPTP. The proposed method is tested on six mathematical and two engineering RBDO examples from the literature. Monte Carlo simulations are performed on the obtained solutions for estimating their reliability. The results demonstrate that the proposed method generates the best reliable solution and is also computationally efficient on the chosen set of examples over seven RBDO methods from the literature.

ILC-based two-layer strategy for economic performance improvement in industrial MPC systems

The conventional LQG based economic performance design has found its difficulty in industrial application and so far, there is still no systematic and effective way to improve economic performance. As learned from the LQG benchmark performance assessment method, the economic performance improvement in MPC systems can be realized through adjusting controller parameters in addition to the well-known setpoints change approach. Therefore, we take advantage of LQG and iterative learning control (ILC) to propose a new two-layer periodical economic performance improvement strategy applicable in industrial MPC systems. By dividing the whole time into multiple intervals called periods and optimize the performance periodically, the economics finally reach its optimal. Promoted twice in a certain period, the performance acquires its first promotion through fixed variance obtained from the lower MPC layer, which transforms the nonlinear economic performance function (EPF) of LQG into a linear one. The ILC-based weight coefficients adjustment algorithm then provides the parameters to the MPC controller in the next period with the updating principle based on the idea of minimizing the tracking error between the current controller economic performance and an optimal one, which realizes the second performance improvement. Room for the second promotion is analyzed and convergence of the algorithm is proved. Finally, the effectiveness and applicability of the strategy are verified via a typical industrial separation process.

AK-SESC: a novel reliability procedure based on the integration of active learning kriging and sequential space conversion method

To deal with evaluating small failure probabilities, AK–SESC: a novel approach integrating an active learning Kriging meta-model (AK-MCS) and the SESC, a sequential space conversion method, is suggested. The efficiency of the proposed approach relies on the advantages of the AK-MCS and its updating feature to evaluate the actual performance function and the superiority of SESC in estimating small failure probabilities. Although there are effective methods for small probabilities, the beauty of this approach is that it is derived from the probability integral with no simplifications while providing results of high accuracy.Different problems were solved to study the AK–SESC applicability. The main effort of this method is reducing the function call numbers of the original SESC while reaching the same accuracy as Monte Carlo Simulation (MCS). The reliability analysis results were compared with the main reliability methods of the Importance Sampling (IS), Subset Simulation (SubSim), Line Sampling (LS), First and also Second-Order Reliability Method (FORM and SORM). The solved problems indicate that the proposed approach provides accurate answers with much fewer function calls than SESC. So, it can be a promising method for reliability analyses involving nonlinear or high-dimensional performance functions with small failure probabilities.

MODELING AND PRODUCTIVITY PREDICTION OF THE COMPANIES-INTERNAL CROWDSOURCING-BASED IDEATION

AbstractThe internal crowdsourcing-based ideation within a company can be defined as an involvement of its staff, specialists, managers, and other employees, to propose solution ideas for a pre-defined problem. This paper addresses a question, how many participants of the company-internal ideation process are required to nearly reach the ideation limit for the problems with a finite number of workable solutions. To answer the research question, the author proposes a set of metrics and a non-linear ideation performance function with a positive decreasing slope and ideation limit for the closed-ended problems. Three series of experiments helped to explore relationships between the metric attributes and resulted in a mathematical model which allows companies to predict the productivity metrics of their crowdsourcing ideation activities such as quantity of different ideas and ideation limit as a function of the number of contributors, their average personal creativity and ideation efficiency of a contributors’ group.

Structural Reliability Analysis Using Group Teaching Optimization Algorithm

Conventional First order reliability method (FORM) usually encounters divergence and non-convergence for high nonlinear performance functions. This paper puts forward an improved FORM coupling with a newly raised metaheuristic optimization algorithm, that is, Group teaching optimization algorithm (GTO). The GTO algorithm is inspired by grouping teaching strategy, featured by no additional control parameters, and has shown excellent global search ability for high nonlinear problems. Thus, the GTO algorithm is utilized to solve the equivalent constrained optimization problem according to the first order reliability theory. The implementation procedure of the proposed method is then presented in this paper. A numerical example is employed to verify the performance of the proposed method. Traditional reliability methods and PSO-based FORM are also introduced to compared with the proposed approach. The simulation results prove that the proposed method provides good performance in terms of computational efficiency and accuracy.

Research on Bridge Structure Reliability Evaluation due to Vessels Collison Based on a Statistical Moment Method

Damage to bridge structures caused by vessel collision is a risk for bridges crossing water traffic routes. Therefore, safety around vessel collision of existing and planned bridges is one of the key technical problems that must be solved by engineering technicians and bridge managers. In the evaluation of the reliability of the bridge structure, the two aspects of vessel‐bridge collision force and structural resistance need to be considered. As there are many influencing parameters, the performance function is difficult to express by explicit function. This paper combines the moment method theory of structural reliability with finite element analysis and proposes a statistical moment method based on finite element analysis for the calculation of vessel‐bridge collision reliability, which solves the structural reliability problem with a nonlinear implicit performance function. According to the probability model based on current velocity, vessel velocity, and vessel collision tonnage, the estimate points in the standard normal space are converted into estimate points in the original state space through the Rosenblatt reverse transform. According to the estimate points in the original state space and the simplified dynamic load model of vessel‐bridge collision, the sample time‐history curve of random vessel‐bridge collision force is generated, the dynamic response of the bridge structure and the structural resistance of the bridge are calculated by establishing a finite element model, and the failure probability and reliability index of the bridge structure is calculated according to the fourth‐moment method. The statistical moment based on the finite element analysis is based on the finite element analysis and the moment method theory of structural reliability. The statistical moment of the limited performance function is calculated through a quite small amount of confirmatory finite element analysis, and the structural reliability index and failure probability are obtained. The method can be widely used in existing finite element analysis programs, greatly reducing the number of finite element analyses needed and improving the efficiency of structural reliability analysis.