Reliability-based Design Optimization Framework 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.

A meta-heuristic approach for reliability-based design optimization of shell-and-tube heat exchangers

This study introduces a new framework for optimizing Shell-and-Tube Heat Exchanger (STHE) layouts using a reliability-based design optimization (RBDO) approach. The framework combines a control variate-based surrogate model and a hybrid metaheuristic algorithm. The proposed hybrid algorithm, k-means clustering and whale optimization algorithm (kWOA), was evaluated using CEC’2020 and a case study for optimizing STHE design under static conditions. kWOA showed superior performance in minimizing STHE's total annual cost and solving benchmark functions effectively. In our case study, the RBDO framework optimized the STHE design under two scenarios with target failure probabilities of 1% and 5%, resulting in cost increases of 112% and 82%, respectively, compared to deterministic optimization (DO). The integration of the RBDO approach with the STHE mathematical model, considering factors like inlet flow temperatures, mass flow rates, and fouling resistance, demonstrated the framework's ability to balance the trade-off between cost and reliability under uncertainty. Hybrid control variate radial basis function (RBF) models and Monte Carlo Simulation (MCS) were used to assess safety levels, showing the RBDO framework's superiority in improving safety and significantly reducing failure probability from 89% to 1% and 5%. The RBDO framework offers a robust approach for designing STHEs that achieve optimal performance while ensuring high reliability under uncertain conditions.

Active learning-based optimization of structures under stochastic excitations with first-passage probability constraints

In various efforts to manage the risk of structural failures caused by stochastic excitations such as wind and earthquake loads, the first-passage probability often needs to be calculated as a reliability constraint. Estimating the first-passage probability with low computational costs is essential, especially in structural optimization requiring iterative reliability calculations for design alternatives. This paper presents a new active learning-based framework for reliability-based design optimization (RBDO) of structures under stochastic excitations. A mixture-distribution-based formulation of the first-passage probability is utilized to handle the high-dimensional sequences of stochastic excitations during the optimization. The design parameter sensitivity of the first-passage probability is introduced to use a gradient-based optimizer in the RBDO iterations. These procedures employ heteroscedastic Gaussian process-based surrogates of the logarithmic responses. An active-learning scheme identifies the best training point to reduce the computational costs in the first-passage probability calculations and optimization. The numerical examples dealing with the optimal design of an eight-story building system and a tower structure subjected to stochastic wind loads demonstrate the accuracy and efficiency of the proposed method.

Multi-Task Learning for Design Under Uncertainty With Multi-Fidelity Partially Observed Information

Abstract The assessment of system performance and identification of failure mechanisms in complex engineering systems often requires the use of computation-intensive finite element software or physical experiments, which are both costly and time-consuming. Moreover, when accounting for uncertainties in the manufacturing process, material properties, and loading conditions, the process of reliability-based design optimization (RBDO) for complex engineering systems necessitates the repeated execution of expensive tasks throughout the optimization process. To address this problem, this paper proposes a novel methodology for RBDO. First, a multi-fidelity surrogate modeling strategy is presented, leveraging partially observed information (POI) from diverse sources with varying fidelity and dimensionality to reduce computational cost associated with evaluating expensive high-dimensional complex systems. Second, a multi-task surrogate modeling framework is proposed to address the concurrent evaluation of multiple constraints for each design point. The multi-task framework aids in the development of surrogate models and enhances the effectiveness of reliability analysis and design optimization. The proposed multi-fidelity multi-task machine learning model utilizes a Bayesian framework, which significantly improves the performance of the predictive model and provides uncertainty quantification of the prediction. Additionally, the model provides a highly accurate and efficient framework for reliability-based design optimization through knowledge sharing. The proposed method was applied to two design case studies. By incorporating POI from various sources, the proposed approach improves the accuracy and efficiency of system performance prediction, while simultaneously addressing the cost and complexity associated with the design of complex systems.

A two-level surrogate framework for demand-objective time-variant reliability-based design optimization

Complex engineering problems in the real world often involve uncertainties and require time-consuming simulations and experiments, hindering the efficiency of constraints processing. Additionally, practical engineering problems may have varying demands that pose new challenges for dealing with dynamic environments. However, most existing methods focus on immediate demands, making it inevitable to undergo tedious procedures to find feasible solutions. To address these issues, this paper proposes a demand-objective time-variant reliability-based design optimization framework to meet different demands in varying environments. Meanwhile, a corresponding two-level surrogate-based solving strategy is developed to reduce the computational resources required. The framework consists of two stages: time-variant reliability-based constraint handling and demand-objective optimization. An adaptive two-level surrogate method is proposed for time-variant reliability-based constraint handling by combining Kriging to reduce computational costs associated with evaluating constraints. This paper introduces moderate, conservative, and radical models for demand-objective optimization, combining the two-level surrogate method to deal with dynamic cost functions with different demands. Also, a new constrained minimax optimization method is developed for the radical model, which is the trickiest but very useful in practical engineering problems so that the algorithm can converge quickly. Finally, some examples are demonstrated to specify the proposed framework in applications.

Reduced-order model for RBDO of multiple TMDs on eccentric L-shaped buildings subjected to seismic excitations

L-shaped buildings subjected to ground accelerations undergo coupled lateral and torsional vibrations that can result in critical stress concentrations at their re-entrant corners. Even so, a comprehensive literature review indicates that the Reliability-Based Design Optimization (RBDO) of multiple Tuned Mass Dampers (TMDs) in L-shaped buildings has not been studied yet, to the best of the authors’ knowledge. Indeed, the more refined structural model required to capture the wings motion, rather than assuming shear-framed behavior as performed on asymmetric rectangular buildings, becomes critical in an RBDO framework. Thus, integrating a non-modal Model Order Reduction (MOR) strategy allows for a massive reduction in the computational burden. Emerging from control engineering, these techniques do not form their orthogonal subspace based on modal information. Furthermore, they dispense previous snapshots of the system solution while being suitable for non-classically damped systems. Within this context, this paper presents a novel MOR-based RBDO study of multiple TMDs in L-shaped buildings subjected to seismic excitation. The MOR algorithm connects second-order Krylov subspaces to build the reduced-order model at a specific expansion frequency with a windowing strategy for sweeping the complete frequency domain. Then, it is integrated into the RBDO, aiming to determine bi-directional TMD configurations that minimize the building’s annual failure rate for the serviceability limit state. The application case consists of a 4-story L-shaped building with 2880 active degrees of freedom (DOFs). The results show that a reduction in failure probability of up to 65% can be achieved while reaching a speed-up factor (SUF) of more than 35.

Optimization of module structure considering mechanical and thermal safety of pouch cell lithium-ion batteries using a reliability-based design optimization approach

Design optimization is an important method for improving the performance of lithium-ion batteries. However, the majority of earlier studies on battery optimization have generally concentrated on enhancing the performance of a single battery cell or focusing on particular objectives of the module and pack structures. Therefore, this study mainly focuses on developing a generalized optimization framework to increase the energy density of the module while satisfying the mechanical and thermal safety requirements. To the best of our knowledge, this work is the first to attempt to enhance the module performance while considering both mechanical and thermal safety. Moreover, this study develops a reliability-based design optimization (RBDO) framework for module structures that considers system uncertainties to mitigate the possibility of safety failures and obtain a reliable design. The mechanical behavior of the module shows how swelling, breathing, material types, and properties affect the final module length and stress. The effects of the C-rate and design parameters on the maximum temperature and temperature deviation are also investigated. All designs satisfy the target probability of failure, verifying that the RBDO framework of the module structure has been successfully established. The proposed RBDO framework provides guidance for the design and operation of battery modules.

Efficient adaptive reliability-based design optimization for geotechnical structures with multiple design parameters

Reliability-based design optimization (RBDO) aims to improve the reliability of geotechnical systems while minimizing the cost of design. However, the conventional RBDO method suffers from a large computational cost and intractable convergence problem due to the double-loop structure involving numerical optimization and repeated reliability analysis. To address this issue, this paper proposes an efficient adaptive RBDO framework for geotechnical structure with multiple design parameters. The proposed framework mainly includes two parts: a novel expanded inverse first-order reliability method (E-IFORM) for dealing with the inverse reliability problem with multiple design parameters, and an adaptive surrogate model to decouple the double-loop structure and improve the efficiency. The proposed methodology is applied to four geotechnical systems (i.e., a soil slope, an excavation pit, a strip footing and a retaining wall), and the results demonstrate the effectiveness and efficiency of the proposed framework.

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.

Stochastic Reliability-Based Design Optimization Framework for the Steel Plate Girder with Corrugated Web Subjected to Corrosion.

This paper proposes the framework for reliability-based design optimization (RBDO) of structural elements with an example based on the corrugated web I-girder. It tackles the problem of topological optimization of corroding structures with uncertainties. Engineering restrictions follow a concept of the limit states (LS) and extend it for stability and eigenfrequency assessment. The reliability constraints include all the LS; they are computed according to first- and second-order reliability methods. The RBDO example minimizes the bridge girder cross-section while satisfying the structural reliability level for the ultimate and the serviceability limit states, stability, and eigenfrequency. It takes into consideration two uncorrelated random effects, i.e., manufacturing imperfection and corrosion. They are both Gaussian; the first of them is applied at assembly time, while the second is applied according to the time series. The example confronts three independent FEM models with an increasing level of detailing, and compares RBDO results for three concurrent probabilistic methods, i.e., the iterative stochastic perturbation technique (ISPT), the semi-analytical method, and the Monte Carlo simulation. This study proves that the RBDO analysis is feasible even for computationally demanding structures, can support automation of structural design, and that the level of detailing in the FEM models influences its results. Finally, it exemplifies that reliability restrictions for LS are much more rigorous than for their deterministic counterparts, and that the fastest ISPT method is sufficiently accurate for probabilistic calculations in this RBDO.

Efficient decoupling-assisted evolutionary/metaheuristic framework for expensive reliability-based design optimization problems

Reliability-based design optimization (RBDO) algorithm is to minimize the objective under the probabilistic factors. While gradient-based and classical evolutionary RBDO algorithms provide promising performance on simple optimization problems, they are likely to perform poorly on challenging problems, including the multimodal functions, discrete design spaces, non-differential problems, etc. This paper proposes a unified framework to improve the performance of existing RBDO algorithms for complex RBDO problems. Our framework is based on three new strategies: generalized decoupling evolutionary and metaheuristic RBDO framework, particle’s memory saving strategy, and adaptive fractional-order equilibrium optimizer algorithm. The proposed algorithm is characterized by a decoupling strategy to enable the parallel operation of the inner reliability computation and outer deterministic optimization, a particle’s memory saving strategy to provide effective guidance from the previous iteration, and the adaptive fractional-order equilibrium optimizer algorithm to enhance the search efficiency and global convergence capacity. To evaluate the performance of the proposed algorithm, a wide range of experiments are conducted on different types of use cases. The experimental results demonstrate that our algorithm provides superior performance over other comparative algorithms.

A new framework for reliability-based design optimization using metaheuristic algorithms

This paper presents a new framework for reliability-based design optimization (RBDO) using metaheuristic algorithms based on decoupled methods. The sequential optimization and reliability assessment (SORA) is utilized as a decoupled method in this study. In the present framework, unlike the previous RBDO based on decoupled methods, a metaheuristic is employed on both reliability assessment and the optimization parts. Here, first a new reliability assessment method based on the metaheuristics is introduced. Then, a new termination condition for the metaheuristic inspired by the termination condition of the gradient-based method is presented. In order to investigate the efficiency of the proposed framework, an enhanced shuffled shepherd optimization algorithm (ESSOA) is used as an optimization approach. The efficiency of the proposed framework is evaluated by four well-known RBDO benchmarks investigated in the previous studies using the gradient-based method. Also, the two new RBDO problems are introduced in this study. The results show that the proposed framework can have better performance than the gradient-based method in the RBDO and can easily be utilized in a wide range of RBDO problems.

A general fidelity transformation framework for reliability-based design optimization with arbitrary precision

A general fidelity transformation framework for reliability-based design optimization with arbitrary precision

Optimasi Desain Penampang Struktur Rangka Batang Baja Berbasis Reliabilitas Menggunakan Symbiotic Organisms Search dan Artificial Neural Network

Safety and economic factors are the two main consideration in designing a structure. The structural engineer always try to find the optimal structure design with minimum cost that satisfy the safety requirement. This safety requirement can be expressed as structural reliability that associated to a certain failure probability threshold. An integrated Reliability-based Design Optimization (RBDO) framework usually employed to minimize the cost objective function subjected to the failure probability limit. Failure probability mostly computed by using a time-consuming Monte Carlo Simulation (MCS) method. This study develops two hybrid RBDO framework, SOS-ANN and PSO-ANN, which combine the metaheuristic method, Symbiotic Organisms Search (SOS) and Particle Swarm Optimization (PSO) with a machine learning method, Artificial Neural Network (ANN). The SOS and PSO method are used to solve the discrete optimization problem. The ANN method is adopted to replace the MCS method in predicting the reliability of every solution using binary classification. A practical RBDO case of steel truss structure is used to demonstrate the performance of both SOS-ANN and PSO-ANN method in finding the optimal structural design. The results show that the SOS-ANN method outperforms the PSO-ANN method in terms of solution quality, computational efficiency and consistency.

A unified fatigue reliability-based design optimization framework for aircraft turbine disk

To improve the modeling efficiency and optimization accuracy for fatigue reliability-based design optimization (FRBDO) of aircraft turbine disk, a unified framework integrating by hierarchical fuzzy-neuro (HFN) surrogate method and multi-level collaboration (MLC) optimization model are presented. The HFN method is developed with absorbing fuzzy-neuro surrogate into hierarchical modeling strategy. The MLC model is proposed by considering influencing factors and constraint conditions in multiple layers and multiple cycles. The presented framework was applied to the FRBDO of a high-pressure turbine disk. The optimization results show that the presented framework holds high computational efficiency and accuracy in FRBDO of turbine disk.

Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

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.

Reliability-based design optimization of a spar-type floating offshore wind turbine support structure

The application of reliability-based design optimization (RBDO) methods to offshore wind turbine systems is highly relevant regarding economic efficiency and for considering prevailing uncertainties within the design process. Furthermore, RBDO is a very promising approach in optimizing systems when classification and standardization are not fully available. The level of difficulty of design optimization already increases when including the reliability aspect, but becomes even more challenging when dealing with the highly complex system of floating wind turbines (FWTs), which has not yet been applied. Thus, this paper presents for the first time an integrated framework for RBDO of FWTs, combining concepts of optimization with reliability-based design and advanced modeling, requiring reasonable computational effort and time expenditure. In preprocessing, environmental conditions, limit states, and uncertainties are specified, an appropriate reliability assessment approach is elaborated, and response surfaces for various system geometries in the optimization design space are generated ahead of the RBDO execution. These are finally used by means of an interpolation approach for the reliability calculation integrated in the iterative design optimization. On the example of a spar-buoy FWT system, the application of the presented methodology and the feasibility of coupling FWT design optimization with reliability assessment are shown.

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.