Efficient Reliability-based Design Optimization Research Articles

Sequential most probable point update combining Gaussian process and comprehensive learning PSO for structural reliability-based design optimization

This paper proposes an efficient reliability-based design optimization (RBDO) method that advantageously decouples comprehensive learning particle swarm optimization (CLPSO) algorithm with Gaussian process regression (GPR) model, termed as GPR-CLPSO. The method iteratively performs the CLPSO with deterministic parameters based on the most probable point (MPP) underpinning limit-state functions (LSFs) iteratively updated by the active learning reliability evaluation process. The GPR model approximates, from the design data given by CLPSO, the spectrum of LSFs under random parameters, and hence enables a significant computational reduction of Monte-Carlo simulations (MCSs) for failure probability approximation. The expected feasibility function is maximized using the CLPSO code to systematically refine the GPR model by adaptively adding new (intelligent) learning points in the region with high-reliability sensitivity leading to the more accurate prediction of failure probability. A novel inverse MCS constraint boundary method is developed to redefine the MPP assigned for the CLPSO algorithm in determining the new optimal design. The method efficiently leverages the decoupling approach, whilst significantly alleviating computing efforts, to quickly and accurately capture the optimal RBDO design. The resulting failure probability well satisfies the allowable limit. Four RBDO examples are provided to illustrate applications and robustness of the proposed decoupling GPR-CLPSO approach.

A novel reliability index approach and applied it to cushioning packaging design

This paper is aimed at proposing a new approach to Reliability-based Design Optimization (RBDO) and applying it to cushioning packaging design with a highly nonlinear system. The problem is formulated as an RBDO problem, which is included a minimizing cost function and probabilistic constraints. Here, the thickness of the cushion material is dealt with uncertainty and uncontrolled parameters. The traditional reliability index approach (RIA) has evolved as a powerful tool to solve the RBDO problem; however, due to its convergence problem, the modified reliability index approach (MRIA) is proposed. Although the MRIA method solves the problems of the traditional RIA, it inherits the low efficiency of searching for the most probability point (MPP). Thus, we developed a novel RIA based on MRIA to improve the efficiency and robustness during the RBDO process. The innovation active set strategy is developed in reliability assessment, which is a strict inequality to determine whether the current constraint is active or inactive. An application example is presented, and the results are compared with MRIA to assess cost-effectiveness and efficiency. Results indicate the proposed method is feasible to solve the uncertainty problem of packaging materials in the processing process and is also an efficient RBDO method.

A practical and efficient reliability-based design optimization method for rock tunnel support

Uncertainty is an intrinsic feature in rock mechanics and engineering, and uncertainty analysis is an important challenge to stability analysis, design, and construction in the context of rock engineering. In this study, a reliability-based design optimization (RBDO) approach is developed to determine the design parameters for engineering by combining the moment method and high-dimensional model representation (HDMR). Multiplicative dimensional reduction (MDRM) is used to approximate and determine the statistical moment of a rock tunnel response. The first-order second moment is used to estimate the reliability index based on the MDRM model. The complex, high-dimensional, and nonlinear response of the engineering structure is determined by HDMR. Microsoft Solver is used as optimization technology for the RBDO. The proposed approach was applied to a circular tunnel with rock bolts and the procedure was examined using a Monte Carlo simulation. The results show that the proposed method can determine the reliability index and design variables in the RBDO, and provide a simple and feasible way to consider uncertainties in design, stability analysis, and construction in rock engineering.

Efficient reliability-based design optimization for hydraulic pipeline with adaptive sampling region

In engineering practice, the active learning Kriging (AK) model is always embedded in the computation of reliability-based design optimization (RBDO) problem. Traditionally, with the iteration of the design points, the candidate sample pool for establishing the Kriging model must be constructed repeatedly. However, there are overlaps in the sample space, which affect the efficiency of the RBDO. In this study, a novel method combining the rejection sampling (RS) with AK is proposed to further ensure the accuracy of the Kriging model. The proposed method aims to reduce the number of candidate sample pools and improve the efficacy of sample utilization. In the optimization process, RS can capture the variation of the sampling region using the rejection algorithm, which avoids the need to update the Kriging model in the entire overlapping sample space, and focuses on improving the model accuracy in the additional sample space with the new design point. Thus, the Kriging model continuously updates with the iteration of the design variable in the optimization process, which significantly reduces the computational cost. Several examples were employed to validate the performance of the proposed method. Lastly, the proposed method was applied to the RBDO of a hydraulic pipeline system.

Reliability-Based Design Optimization of Structures Considering Uncertainties of Earthquakes Based on Efficient Gaussian Process Regression Metamodeling

The complexity of earthquakes and the nonlinearity of structures tend to increase the calculation cost of reliability-based design optimization (RBDO). To reduce computational burden and to effectively consider the uncertainties of ground motions and structural parameters, an efficient RBDO method for structures under stochastic earthquakes based on adaptive Gaussian process regression (GPR) metamodeling is proposed in this study. In this method, the uncertainties of ground motions are described by the record-to-record variation and the randomness of intensity measure (IM). A GPR model is constructed to obtain the approximations of the engineering demand parameter (EDP), and an active learning (AL) strategy is presented to adaptively update the design of experiments (DoE) of this metamodel. Based on the reliability of design variables calculated by Monte Carlo simulation (MCS), an optimal solution can be obtained by an efficient global optimization (EGO) algorithm. To validate the effectiveness and efficiency of the developed method, it is applied to the optimization problems of a steel frame and a reinforced concrete frame and compared with the existing methods. The results show that this method can provide accurate reliability information for seismic design and can deal with the problems of minimizing costs under the probabilistic constraint and problems of improving the seismic reliability under limited costs.

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.

Quantile surrogates and sensitivity by adaptive Gaussian process for efficient reliability-based design optimization

To obtain the optimal structural design satisfying probabilistic requirements, reliability-based design optimization (RBDO) has been widely studied and applied. However, its practical applications have been often hampered by huge computational costs. To address the challenge, the authors recently developed an RBDO method termed quantile surrogates by adaptive Gaussian process (QS-AGP), which approximates the quantiles of the performance functions adaptively using Gaussian process models to check whether the pre-generated design samples satisfy the reliability requirements. It has been shown that QS-AGP requires much fewer evaluations of performance functions than existing RBDO methods. However, the approach could be computationally expensive in high-dimensional applications since it may require an insurmountable memory to handle the pre-generated design samples. To alleviate this difficulty, a new quantile surrogate based RBDO framework is proposed in this paper. To this end, a non-sampling-based procedure is proposed for an efficient estimation of the quantile surrogates based on both input uncertainties and model error of surrogates. Moreover, to perform quantile-surrogate-based RBDO without relying on pre-generated design samples, the parameter sensitivity of the quantile surrogate is implemented. The computational efficiency of the proposed RBDO method, termed quantile surrogates and sensitivity by adaptive Gaussian process (QS2-AGP), is demonstrated by a variety of RBDO examples featuring up to 15 design parameters. The supporting source codes are available for download at https://github.com/Jungh0Kim/QS2-AGP.

Efficient reliability-based design optimization of composite structures via isogeometric analysis

Composite variable-stiffness (VS) panels with curvilinear fiber paths are very promising for aerospace structures. Due to the inherent complexity of VS laminates, buckling analysis and design optimization are extremely time-consuming and challenging, especially when uncertainties are considered, i.e. reliability-based design optimization (RBDO). In this study, an efficient bi-stage RBDO framework via isogeometric analysis (IGA) is established to release the tremendous computational burden. In Stage I, the layer thickness and lamination parameters are used as design variables to obtain an approximated layer number, which greatly reduces the design variable size. In Stage II, intermediate density variables are introduced, and IGA is employed for the buckling analysis and derivation of the analytical sensitivity. Furthermore, the augmented step size adjustment (ASSA) algorithm is used to enhance the efficiency and robustness of the RBDO process. Numerical results of VS panels are used to validate the performance of the proposed RBDO framework. The optimal results indicate that the proposed framework can find the optimal lightweight design that satisfies the manufacturing constraints in an efficient and accurate manner.

Reliability-based design optimization for the lattice boom of crawler crane

The reliability-based design optimization (RBDO) problem involving discrete design variables and multi-working constraints is tackled with the consideration of the uncertainty factors in the optimal design of crawler crane’s lattice boom. In the optimization model, the uncertainties of material property, geometric parameters and loads are represented by random variables. The geometrical dimensions, performance functions’ reliability indexes and the minimum weight of lattice boom are defined respectively as the random design variables, probabilistic constraints and objective function. Integrating the reliability analysis method based on response surface method (RSM) with multi-island genetic algorithm (MIGA), an efficient reliability-based design optimization process for the lattice boom is developed under the typical constrained working conditions. Several verification cases are presented to demonstrate the effectiveness of optimization results. It is shown that the reliability-based design optimization method can not only satisfy the reliability constraints of the lattice boom, but also ensure the safety and economy of crawler crane boom as well as minimize the structure weight. The developed method is capable to be extended to the reliability-based design optimization of other large and complex construction machinery structures.

Enriched single-loop approach for reliability-based design optimization of complex nonlinear problems

Single-loop approach (SLA) is an efficient reliability-based design optimization (RBDO) method, where the current most probable point (MPP) is located through the gradient information of previous MPP. However, the MPP obtained by SLA may not be accurate for complex nonlinear RBDO problems, which probably causes SLA to be inefficient, converge to the wrong optimal solution or even difficulty in convergence. In this study, an enriched single-loop approach based on enhanced advanced mean value (ESLA-EAMV) is proposed to improve convergence performance of the original SLA for complex nonlinear RBDO problems. First, an enhanced advanced mean value (EAMV) method is developed to find the MPP, where the negative gradient vector of the previous MPP in AMV is replaced by a vector with adaptive step size in EAMV. Then, the proposed EAMV method is integrated into the original SLA to improve the convergence ability of the original SLA. Finally, seven benchmark nonlinear problems are presented to verify the accuracy, efficiency and robustness of the proposed ESLA-MAMV compared with other existing RBDO methods. Comparison results show that the proposed ESLA-MAMV can improve the convergence performance of the original SLA.

An efficient reliability-based design optimization study for PCM-based heat-sink used for cooling electronic devices

Nowadays, development of new technologies requires efficient mechatronic components which needs more sophisticated cooling systems. Furthermore, phase change material (PCM) based heat sinks presents an appropriate technique for electronic devices. Then, the aim of this work is to propose an efficient methodology that determines an optimal design of such a PCM-based cooling system. Despite of the satisfaction of the optimal solution founded by the deterministic design optimization (DDO), the reliability level is not controlled. For this reason, reliability-based design optimization (RBDO) studies were performed. In fact, RBDO approaches aim to find the best compromise between safety and cost by taking into consideration uncertainties for the studied model. Therefore, several methods are studied, such as the optimum safety factor approach and the hybrid method. Yet, the application of these methods is limited only to linear materials. Hence, this study presents an extension in the case of nonlinear materials. In addition, to overcome problems of the classical hybrid approach, a Robust Hybrid Method (RHM) is proposed. A numerical application is used to study the different DDO and RBDO methods. Then, the efficiency of the RHM method for PCM-based heat sinks is verified.

Gaussian Surrogate Dimension Reduction for Efficient Reliability-Based Design Optimization

Reliability-based design optimization (RBDO) is commonly performed using surrogates to approximate the response due to deterministic and stochastic design variables, as well as random parameters, but is ultimately limited by the number of dimensions that can be accurately represented. In this paper, a new surrogate-centric RBDO formulation is developed to alleviate the computational expense of excessive random parameters. Nondeterministic kriging nonstationary variation estimation is used to characterize normally distributed random parameters and numerical noise rather than increasing the surrogate’s dimensionality. Simultaneously, surrogate-based uncertainty propagation is performed considering the nonnormally distributed or high nonlinear random parameters, and stochastic design variables. In this study, propagated uncertainties and the uncertainty captured by the kriging variation are recombined through the convolution of the two probability density functions. It was found that conglomerations of normally distributed random parameters allow this method to reduce the number of approximated spaces while maintaining accuracy compared to existing surrogate approaches. This paper contains a detailed description of the proposed methodology; foundational four-, five-, and six-dimensional RBDO examples with comparisons to previous methods from the literature; and the RBDO of a 10-dimensional nonlinear thermoelastic hat-stiffened panel under a frequency and stress constraint.

Efficient reliability-based robust design optimization of structures under extreme wind in dual response surface framework

This paper deals with an efficient reliability-based robust design optimization (RBRDO) of structures under extreme wind excitation. To solve such problems in the direct Monte Carlo simulation (MCS) framework, extensive computational time is required. To circumvent this, a new CDF-based RBRDO approach is proposed using the dual response surface method (RSM) framework. In the proposed approach, once the response surfaces are obtained by the dual RSM, the direct MCS can be fully avoided during optimization. Thereby, the approach evades several repetitive executions of finite element analyses code inside the optimization loop. Generally, RSM uses least squares method (LSM) which is reported to be a major source of error in few recent studies. Thus, the proposed approach utilizes the adaptive nature of moving least squares method (MLSM) for construction of response surfaces. The RBRDO is performed by using a two-criterion equivalent optimization problem, where the mean value of the cost of a structure and its standard deviation are optimized—ensuring constraint feasibility under parameter uncertainty. The dynamic response of structures necessary to define constraint functions is obtained using a random wind field model, which considers the effect of coherence and wind directionality. Two illustrative examples are presented to demonstrate the efficiency of the proposed RBRDO approach. When benchmarked through direct MCS results, the proposed MLSM-based RBRDO approach yields more accurate and robust solutions compared with that of the conventional LSM-based approach.

Efficient Reliability-Based Design Optimization of Degrading Systems Using a Meta-Model of the System Reliability

The application of reliability-based design optimization (RBDO) to degrading systems is challenging because of the continual interplay between calculating time-variant reliability (to ensure reliability policies are met) and moving the design point to optimize various objectives, such as cost, weight, size and so forth. The time needed for Monte Carlo Simulation (MCS) is lengthy when reliability calculations are required for each iteration of the design point. The common methods used to date to improve efficiency have some shortcomings: First, most approaches approximate probability via a method that invokes the most-likely failure point (MLFP), and second, tolerances are almost always excluded from the list of design parameters (hence only so-called parameter design is performed), and, without tolerances, true monetary costs cannot be determined, especially in manufactured systems. Herein, the efficiency of RBDO for degrading systems is greatly improved by essentially uncoupling the time-variant reliability problem from the optimization problem. First, a meta-model is built to relate time-variant reliability to the design space. Design of experiment techniques helps to select a few judicious training sets. Second, the meta-model is accessed to quickly evaluate objectives and reliability constraints in the optimization process. The set-theory approach (with MCS) is invoked to find the system reliability accurately and efficiently for multiple competing performance measures. For a case study, the seminal roller clutch with degradation due to wear is examined. The meta-model method, using both moving least-squares and kriging (using DACE in Matlab), is compared to the traditional approach whereby reliability is determined by MCS at each optimization iteration. The case study shows that both means and tolerances are found that correctly minimize a monetary cost objective and yet ensure that reliability policies are met. The meta-model approach is simple, accurate and very fast, suggesting an attractive means for RBDO of time-variant systems.

Integration of a Hybrid Method into an Efficient Reliability-Based Design Optimization Technique for Large Structures

The computational burden imposed by the repetitive and lengthy nature of the structural analyses of large structures has lead researches to continuously devise efficient methods for conducting their reliability analysis in addition to powerful methods for solving the design optimization problem. The presence of a considerably large number of random variables in the reliability analyses of large structure is another issue that tends to slow down the design optimization problem. In this paper, a hybrid method that was recently presented by the first author will be integrated into a structural reliability-based design optimization technique for large structures. In this technique, modified concepts of the weighted average simulation method are used to determine the most probable point of failure in a computationally efficient manner. The most probable point of failure is then transferred into the standard normal space, where the reliability index is calculated in closed-form. The firefly algorithm will be used for solving the optimization problem. It will be shown that the integrated approach significantly reduces the computational resources required for solving the design problem. The technique will be tested on a truss bridge. The results will shed light on the efficiency of the proposed technique in solving reliability-based design optimization problems.

Improved reliability-based design optimization of non-uniformly stiffened spherical dome

To resist external pressure, stiffened spherical domes have been widely used in underwater vehicles. In traditional design of these thin-walled structures, uniform stiffeners are usually utilized and overly conservative safety factor method is employed to envelop the multi-source uncertainties mainly caused by manufacturing. In this study, a concept of non-uniformly stiffened spherical dome (NUSPD) is developed to improve the structural buckling capacity. Furthermore, an efficient reliability-based design optimization (RBDO) framework is established for NUSPD, which includes a deterministic optimization loop to fix the stiffener layout, and subsequently followed by a RBDO loop to determine the detailed stiffener dimensions and decrease the structural weight under geometric and material property uncertainties. Since the RBDO of stiffened spherical domes is very complex, an improved enhanced chaos control method together with surrogate model is proposed to improve the convergence rate. Finally, the efficiency of the proposed RBDO framework is demonstrated by the NUSPD example.

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

ABSTRACTIn practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.

Iterative reliable design space approach for efficient reliability-based design optimization

Reliability-based design optimization has gained much attention in many engineering design problems with the consideration of uncertainties. Nevertheless, the application is limited by the huge computational cost for the repeated reliability analysis in the optimization process. To address this issue, an iterative reliable design space approach is proposed with less number of reliability analysis required. Specifically, a sequential optimization strategy is proposed to identify the iterative reliable design space and perform the equivalent deterministic optimization in a serial loop. Thus, the identification of the reliable design space, which is time-consuming, is eliminated from the equivalent probabilistic constraints. Furthermore, an improved shifting vector strategy is put forward for the identification. In this strategy, an approximate shifting vector is first constructed utilizing the partial derivatives at the current design point, and it is then modified based on an increment of the shifting vector and a correction provided by the true reliability analysis. Besides, several mathematical examples and engineering problems are tested to validate the performance of the proposed method. In conclusion, the proposed method is able to deal with different kinds of nonlinear problems with relatively high accuracy and efficiency.

Adaptive conjugate single-loop method for efficient reliability-based design and topology optimization

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.

An Adaptive Directional Boundary Sampling Method for Efficient Reliability-Based Design Optimization

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.