Reliability-based Design Optimization Problems Research Articles

An approximate decoupled reliability-based design optimization method for efficient design exploration of linear structures under random loads

Reliability-based design optimization (RBDO) provides a promising approach for achieving effective structural designs while explicitly accounting for the effects of uncertainty. However, the computational demands associated with RBDO, often due to its nested loop nature, pose significant challenges, thereby impeding the application of RBDO for decision-making in real-world structural design. To alleviate this issue, an approximate decoupled approach is introduced for a class of RBDO problems involving linear truss structures subjected to random excitations, with the failure event defined by compliance. This contribution aims to provide an approximate but efficient way for design exploration to facilitate decision-making during the initial design phase. Specifically, the proposed approach converts the original RBDO problem into a deterministic optimization problem through a modest number of reliability analyses by the probability density evolution method (PDEM). Once the deterministic optimization problem is obtained, the solution of the whole RBDO problem can be obtained by solving this equivalent problem without further reliability analysis, resulting in notable enhancement in terms of computational efficiency. In this way, this contribution expands the frontier of application of the operator norm theory within the RBDO framework. Numerical examples are conducted to illustrate the effectiveness and capabilities of the proposed approach.

Quantile-based optimization under uncertainties for complex engineering structures using an active learning basis-adaptive PC-Kriging model

The Reliability-Based Design Optimization (RBDO) of complex engineering structures considering uncertainties has problems of being high-dimensional, highly nonlinear, and time-consuming, which requires a significant amount of sampling simulation computation. In this paper, a basis-adaptive Polynomial Chaos (PC)-Kriging surrogate model is proposed, in order to relieve the computational burden and enhance the predictive accuracy of a metamodel. The active learning basis-adaptive PC-Kriging model is combined with a quantile-based RBDO framework. Finally, five engineering cases have been implemented, including a benchmark RBDO problem, three high-dimensional explicit problems, and a high-dimensional implicit problem. Compared with Support Vector Regression (SVR), Kriging, and polynomial chaos expansion models, results show that the proposed basis-adaptive PC-Kriging model is more accurate and efficient for RBDO problems of complex engineering structures.

Accelerated PMA and its application in reliability-based design optimization

This paper proposes a novel approach based on the performance measure approach (PMA), which is termed accelerated PMA (aPMA) to overcome the non-convergence of the advanced mean value(AMV) and extend it to the reliability-based design optimization (RBDO). In aPMA, a threshold value by using inequality is integrated with the formula of AMV to make the efficiency and robustness of searching for the minimum performance target point (MPTP) better. The threshold value is able to eliminate the unnecessary probabilistic constraint in the inverse reliability analysis, thereby reducing the risk of numerical instability in AMV. Meanwhile, a judgment condition is defined according to the threshold value and reliability index in the design optimization, in which the active deterministic constraints can be quickly identified and updated. In addition, the recommended aPMA is merged into the double loop method (DLM) to consolidate the accuracy, efficiency and robustness of that for the high-dimensional RBDO problem. Since reliability analysis and structural optimization are carried out simultaneously, the convergence rate is enhanced. Besides, both the efficiency and robustness of the proposed aPMA are firstly trialed on six numerical/structural examples along with its performance on RBDO is tested through three RBDO problems quoted, showing significant merits.

A novel decoupled approach combining invertible cross-entropy method with Gaussian process modeling for reliability-based design and topology optimization

Design optimization considering the presence of uncertainties in parameters poses an extremely challenging problem. The source of difficulties comes with reliability-based formulations, where addressing the probabilistic problem exhausts the large computing efforts for failure estimations of the structure violating limit-state functions (LSFs). This paper proposes a novel decoupled approach for effectively solving reliability-based design optimization (RBDO) problems, namely an invertible cross-entropy (iCE) method advantageously combined with a Gaussian process regression (GPR) model, termed as iCE-GPR. The GPR model is applied to approximate the spectrum of LSFs under random parameters. Furthermore, to enhance the accurate prediction of the system failure probability, an active learning process is applied to systematically refine the GPR model by adding new learning points in the region with the largest uncertainty and high-reliability sensitivity through the maximization of an expected feasibility function (EFF). Based on the updated GPR model, the failure probability is estimated by a cost-effective cross-entropy (CE) method without any calls to the actual performance function. To perform the decoupling optimization process with the reliability analysis, the novel iCE, based on the CE method, is developed to update the most probable point (MPP) assigned for the next deterministic optimization process in determining the new optimal design. The method iteratively performs the deterministic optimization process based on the MPP underpinning LSFs sequentially updated by the active learning process. The proposed iCE-GPR method fast-converges the optimal design and significantly alleviates computational burdens associated with reliability analyses. The proposed method is also applied to solve a reliability-based topology optimization (RBTO) problem. Four numerical examples for both the RBDO and RBTO problems are provided to illustrate efficiency and robustness of the proposed iCE-GPR method.

Time-dependent reliability assessment and optimal design of corroded reinforced concrete beams

It remains a significant challenge to quantitatively describe the corrosion of reinforced concrete (RC) structures under chloride penetration. Moreover, when considering uncertainties throughout the life cycle of corroded RC structures for assessing their safety, reliability, and optimal design, the complexity of the problem intensifies. To address these issues, this paper studies the time-dependent reliability analysis and optimal design of corroded RC beams. At first, the time-dependent reliability of beams is investigated by considering both the serviceability limit state (SLS), which corresponds to the corrosion initiation of the reinforced steel, and the ultimate limit state (ULS), associated with the bending failure of the beam. This analysis takes into account the time-dependent chloride diffusion coefficient and incorporates a stochastic process. The reliability is evaluated using the Monte Carlo Simulation (MCS) method and the cumulative distribution function (CDF) method. Subsequently, a time-dependent reliability-based design optimization (TRBDO) problem is formulated, and the PSO-MCS, a methodology incorporating a particle swarm optimization (PSO) algorithm and MCS is adopted to solve it. After optimization, the initial cost of the specific RC beam is reduced from 1351.879€ to 1247.075€, while the time-dependent reliability within [0, 100] years is improved from 0.6057 to 0.6508. The effectiveness of the CDF, MCS and PSO-MCS methods are demonstrated through reliability analysis and design examples of corroded RC beams.

Augmented Space Integral Approach for Structural Reliability-Based Optimization

An efficient, fully decoupled approach is proposed to solve the structural reliability-based design optimization (RBDO) problem. The proposed approach utilizes augmented space integral and importance sampling (ASI-IS) to efficiently evaluate a functional relationship between the probability of failure and the design parameters, namely the so-called failure probability function. ASI-IS allows for the avoidance of both the intractable density fitting task associated with augmented space methods and the time-consuming repeated reliability evaluations associated with surrogate model approaches. The resulting functional relationship can be used to completely decouple the original RBDO problem into a deterministic one. Then, an iteration mechanism is constructed with gradually smaller design domains to enhance the efficiency of the optimization process. Furthermore, a sample reuse algorithm is proposed to improve the performance of the proposed approach by collecting the samples generated in previous iterations and reusing them in the current iteration in order to produce a better estimator of the failure probability function. Numerical and engineering examples, including a turbine blade and an aircraft inner flap, are given to demonstrate the efficiency and feasibility of the proposed approach.

A sequential sampling-based Bayesian numerical method for reliability-based design optimization

For efficiently solving the Reliability-Based Design Optimization (RBDO) problem with multi-modal, highly nonlinear and expensive-to-evaluate limit state functions (LSFs), a sequential sampling-based Bayesian active learning method is developed in this work. The penalty function method is embedded to transform the constrained optimization problem into a non-constrained one to reduce the model complexity. The proposed method for solving RBDO problems starts by training a Gaussian process (GP) model, in the augmented space of random and design variables. It is then based on an efficient sampling scheme for simulating the GP model, the adaptive Bayesian optimization (BO) and Bayesian reliability analysis (BRA) procedures are combined in a collaborative way for sequentially producing the joint training points. BO driven by expected improvement (EI) function is used for inferring the global optimum in the design space with global convergence, and the BRA equipped with U function is implemented for inferring the failure probabilities at the identified design points with the desired accuracy. The superiority of the proposed method is demonstrated with two numerical and two real-world engineering examples.

Reliability based design optimization of a two-stage wind turbine gearbox

This paper describes a multi-objective reliability-based design optimization (MORBDO) of a two-stage wind turbine gearbox. The optimization process incorporates the gear's reliability of accounting for the uncertainty of its internal geometric parameters. It also ensures that constraints relating to the gear's reliability index and efficiency are respected. The objective functions are to minimize both the total volume and the center distance. A specific reliability target is established, and to address the multi-objective reliability-based design optimization (MORBDO), the hybrid method (HM) in conjunction with the constrained non-dominated sorting genetic algorithm (C-NSGA-II) is employed. The outcomes demonstrate that applying C-NSGA-II to solve the multi-objective reliability-based design optimization problem yields dependable Pareto solutions that are well-distributed in relation to the desired reliability level. The optimization using C-NSGA-II with a population size of 300 particles and 1000 generations produced the most favorable outcomes. This research significantly contributes to the multi-objective design optimization of wind turbine gear while simultaneously considering their reliability.

Optimum-pursuing method for constrained optimization and reliability-based design optimization problems using Kriging model

This paper proposes a new active learning method named as optimum-pursuing method (OPM) from the viewpoint of optimization theory, which aims to provide an effective tool for solving constrained optimization and reliability-based design optimization (RBDO) problems with low computation cost. It uses the cheap Kriging metamodel to replace the expensive physical response. The novelty of the proposed OPM primarily lies in two aspects. First, the OPM utilizes the advantage of the optimization theory rather than sampling technology. By using the augmented Lagrangian approach, it comprehensively considers the objective, constraints, and their relations, thereby automatic identification of important region in the vicinity of the optimum. Second, the accordingly optimum-pursuing function consists of three parts: Kriging mean, Kriging standard deviation, and merit function. Also, the target reliability surface is further considered to enhance the local accuracy of the reliability analysis. The performance of OPM is tested for both deterministic optimization and problems, in which two mathematical and three real-world engineering examples are selected to showcase the feasibility and validity. The results demonstrate that OPM is promising for solving both deterministic optimization and RBDO problems by comparing with the well-known active learning methods.

An effective active learning strategy for reliability-based design optimization under multiple simulation models

This paper proposes an effective active learning strategy for reliability-based design optimization (RBDO) problems in which the constraint functions are acquired from multiple simulation models. To achieve this goal, a new active learning function (ALF) is derived by estimating the increased reliability of active constraint functions after adding one point to the train points of constraint functions in each simulation model. The proposed ALF distinguishes possibly active constraint functions that seem active near the current optimum and considers how the constraint functions are active. In the proposed RBDO method, a Kriging model is iteratively updated by adding the best point to the train points of constraint functions included in the crucial simulation model until the optimum converges and the Kriging model is sufficiently accurate. The best point and the crucial simulation model are obtained by comparing the proposed ALF. The ALF is further modified to apply to problems where the cost of each simulation model is different. To verify the effectiveness of the proposed method, two numerical and one engineering examples are analyzed. The results show that the proposed method efficiently and accurately obtains the RBDO optimum involving multiple simulation models, regardless of simulation cost.

Reliability-based design optimization of the spiral water jacket for motorized spindle

Spiral water jackets are essential refrigeration units in motorized spindles. The design of spiral water jackets directly affects the machining accuracy and reliability of motorized spindles. In this regard, a reliability-based design optimization (RBDO) method is presented to obtain the optimal configuration of the spiral water jacket. The proposed RBDO algorithm aims to minimize the volume of the spiral water jacket and power loss of the cooling equipment while ensuring machining reliability. The geometric correlation and machining reliability are selected as constraints. The Kriging surrogate model is established to approximate the time-consuming finite element model for improving computational efficiency. An efficient metaheuristic algorithm is introduced to perform optimal exploration for the proposed RBDO problem. The risk speed of the motorized spindle is investigated by the experiment and finite element analysis. Subsequently, the effects of various cooling configurations on thermal error are investigated thoroughly. Eventually, the proposed RBDO is conducted on a motorized spindle. Through probabilistic evaluation, machining reliability is guaranteed by the proposed RBDO compared to deterministic design optimization. This paper provides a new design paradigm for cooling configurations in motorized spindles from the perspective of machining reliability.

A reliability-based design and optimization strategy using a novel MPP searching method for maritime engineering structures

PurposeIn order to solve the problems faced by First Order Reliability Method (FORM) and First Order Saddlepoint Approximation (FOSA) in structural reliability optimization, this paper aims to propose a new Reliability-based Design Optimization (RBDO) strategy for offshore engineering structures based on Original Probabilistic Model (OPM) decoupling strategy. The application of this innovative technique to other maritime structures has the potential to substantially improve their design process by optimizing cost and enhancing structural reliability.Design/methodology/approachIn the strategy proposed by this paper, sequential optimization and reliability assessment method and surrogate model are used to improve the efficiency for solving RBDO. The strategy is applied to the analysis of two marine engineering structure cases of ship cargo hold structure and frame ring of underwater skirt pile gripper. The effectiveness of the method is proved by comparing the original design and the optimized results.FindingsIn this paper, the proposed new RBDO strategy is used to optimize the design of the ship cargo hold structure and the frame ring of the underwater skirt pile gripper. According to the results obtained, compared with the original design, the structure of optimization design has better reliability and stability, and reduces the risk of failure. This optimization can also better balance the relationship between performance and cost. Therefore, it is recommended for related RBDO problems in the field of marine engineering.Originality/valueIn view of the limitations of FORM and FOSA that may produce multiple MPPs for a single performance function, the new RBDO strategy proposed in this study provides valuable insights and robust methods for the optimization design of offshore engineering structures. It emphasizes the importance of combining advanced MPP search technology and integrating SORA and surrogate models to achieve more economical and reliable design.

Reliability-based design optimization using adaptive Kriging-A single-loop strategy and a double-loop one

With the development of the surrogate-assisted Reliability-Based Design Optimization (RBDO) methods in recent decade, efficiency has been continuously improved by various state-of-the-art learning schemes. However, ensuring sufficient accuracy and feasibility at the optimum is still challenging, especially for real scenarios with complex probabilistic constraints and high fidelity. In order to achieve a good balance between efficiency and accuracy and feasibility at the optimum, both a single-loop and a double-loop adaptive Kriging-based RBDO methods are proposed. By directly approximating the probabilistic constraints using Kriging models in the single-loop method, the original RBDO problem is converted into a trivial deterministic optimization, which can be solved by any type of optimization method. The real values of failure probabilities are estimated by Generalized Subset Simulation (GSS) at the initial construction or during each update of the Kriging models. To further raise efficiency in the double-loop adaptive Kriging-based method, GSS is substituted by an inner-loop Kriging-based failure probability estimation, which includes each local enrichment at the limit states corresponding to the active probabilistic constraints based on an initial global enrichment within the augmented reliability space. Once a certain Kriging model meets the requirement during the refining of all, only the others are kept updating until the accuracy of all are accepted. Four examples are used to demonstrate the performance of the proposed two methods.

Improved dynamic design method of ballasted high-speed railway bridges using surrogate-assisted reliability-based design optimization of dependent variables

Operating high-speed trains imposes excessive vibrations to bridges raising concerns about their safety. In this context, it was shown that some conventional design methods such as those related to the running safety suffer from a vague scientific background questioning their reliability or optimality. Therefore, the current article is devoted to updating the conventional design methodology, using Reliability-Based Design Optimization (RBDO) to propose the minimum allowable mass and stiffness which assures satisfying the target reliability. These proposed minimum design values can conceptually replace the conventional partial safety factor-based design method for running safety without the need for dynamic analysis. If the mass and stiffness resulting from the control of other limit states meet the proposed minimum values, the desired target reliability for running safety will be assured. This is achieved by adaptively training Kriging meta-models as a surrogate for the computational models decoupling the RBDO problem. In this regard, a new stopping criteria is proposed using mis-classification ratio of the cross-validated model; which reduces the generalization error of the trained meta-model and consequently the estimated failure probability. Moreover, due to the dependence of the design variables, the Copula concept is used to refine the augmented space and reformulate the RBDO problem.

A Kriging-assisted global reliability-based design optimization algorithm with a reliability-constrained expected improvement

Surrogate models have been extensively used in reliability-based design optimization (RBDO); however, few studies have focused on the global optimality of RBDO. This paper proposes a global RBDO framework that employs Kriging surrogate models to approximate both the objective function and performance functions. The proposed algorithm comprises two major modules: the global optimization module and the local optimization module. The former module aims to identify the interested region containing potential optima, while the latter module refines the local optima. To address the time-consuming acquisition of samples, two different infill strategies are implemented in these two modules. Furthermore, in addition to evaluating the optimal solution in the local optimization module for infill, a reliability-constrained expected improvement infill criterion is developed for the global optimization module. This criterion inherits the property of the expected improvement from the Kriging model, which balances the exploration and the exploitation of the objective space, while taking reliability into account by introducing the shifting vector into the calculation of the probability of feasibility. Numerical experiments indicate that the performance of the proposed infill criterion is significantly superior to others in searching for optima. Several examples verify the global optimization capability of the proposed algorithm and illustrate that it is more suitable for RBDO problems with multiple local optima.

An equivalent RBDO method under the hybrid of fuzzy and interval variables in the worst case

AbstractIn the reliability‐based design optimization (RBDO) problems of structures or systems, only a single type of variable involving is rare, and most of time are hybrid variables. Random variables are often used to describe uncertainty, and these random variables can usually be described by certain probability distributions. However, in practice, the probability distributions of some random variables are difficult or even impossible to be obtained, or the obtained probability distributions may be inaccurate, and can only be roughly described by fuzzy values or intervals. Therefore, this paper is committed to solving RBDO problems under the hybrid of fuzzy variables and interval variables. Firstly, the RBDO model of the structure under the hybrid variables is established. Secondly, the fuzzy variables are transformed to random variables according to entropy invariance. Finally, RBDO is carried out based on sequential single‐loop method in the worst case. A numerical example is presented to illustrate the feasibility of the proposed method.

A response band-based method for time-dependent reliability-based robust design optimization

In this paper, a response band-based method for time-dependent reliability-based robust design optimization is proposed. The proposed method provides a novel alternative framework, consist of a two-step transformation stage and a solving stage, to solve the time-dependent reliability-based robust design optimization problem. The original time-dependent reliability-based robust design optimization problem is transformed into an equivalent deterministic robust design optimization problem in the transformation stage, and the equivalent problem is settled in the solving stage. In the transformation stage, the dynamic modal decomposition technique and the kriging technique are combined to overcome the problem that there is no standard for both time division and observation sampling in the commonly used transformation methods. In the solving stage, an approach for constructing the response band of the objective function is presented, which significantly reduces the computational consumption of the variation evaluation of the objective function. Five cases are employed to verify the effectiveness of the proposed method.

Enriched global–local multi‑objective optimization scheme for fuzzy logic controller-driven magnetorheological damper-based structural system

To construct an optimal and reliable fuzzy logic controller (FLC) for magnetorheological (MR) damper-based structural systems, a reliability-based design optimization (RBDO) problem is formulated considering two competing objectives, namely, minimizing structural responses and control cost. This work incorporates a two-stage enrichment scheme with the non-dominated sorting genetic algorithm (NSGA-II)-based multi-objective optimization method to solve the RBDO problem. In the global stage of the proposed method (denoted as TS-MO method), an initial Kriging surrogate model is constructed in the augmented space and updated by sequentially adding enrichment points within the whole design space to obtain a coarse Kriging surrogate. In the local stage, within the multi-objective optimization iterations, the surrogate model is further refined using enrichment points selected in the vicinity of the current set of design points near the limit state function based on a distance-refined U learning function. Two convergence criteria with dynamic thresholds are employed to ensure the local precision of the trained surrogate model. The efficacy of the TS-MO method is demonstrated by conducting comparison works with two previous methods, i.e., the deterministic design optimization and global Kriging-based multi-objective optimization method (DDO-GK-MO) and the unified method.

Efficient decoupling approach for reliability-based optimization based on augmented Line Sampling and combination algorithm

This paper presents a novel decoupling approach to efficiently solve a class of reliability-based design optimization (RBDO) problems by means of augmented Line Sampling. The proposed approach can fully decouple the original RBDO by replacing the probabilistic constraint with the failure probability function (FPF), which is an explicit function of the design variables. One attractive feature is that the main numerical cost associated with this decoupling comes with only one implementation of augmented Line Sampling, which is actually highly efficient. And for the sake of accuracy, the proposed approach incorporates decoupling with the sequential optimization framework to solve the RBDO problem iteratively. On top of that, an optimal combination algorithm is proposed to reuse the information through aggregating the local estimates of FPF obtained in different iterations to produce an improved estimate, resulting in a more accurate and stable solution. Examples are given to show the effectiveness and efficiency of the proposed approach.

A Transformation-Based Improved Kriging Method for the Black Box Problem in Reliability-Based Design Optimization

In order to overcome the drawbacks of expensive function evaluation in the practical reliability-based design optimization (RBDO) problem, researchers have proposed the black box-based RBDO method. The algorithm flow of the commonly employed RBDO method for the black box problem consists of the outer construction loop of the surrogate model of the constraint function and the inner surrogate model-based solving loop. To improve the solving ability of the black box RBDO problem, this paper proposes a transformation-based improved kriging method to increase the effectiveness of the two loops identified above. For the outer loop, a sample distribution-based learning function is suggested to improve the construction efficiency of the surrogate model of the constraint function. For the inner loop, a paired incremental sample-based limit reliability boundary construction approach is suggested to transform the RBDO problem into an equivalent deterministic design optimization problem that can be efficiently solved by classical optimization algorithms. The test results of five cases demonstrate that the proposed method can accurately construct the surrogate model of the constraint function and efficiently solve the black box RBDO problem.