Reliability-based Design Optimization Research Articles

Reliability-Based Design Optimization of Additive Manufacturing for Lithium Battery Silicon Anode

Abstract With the blooming of the electric vehicle market and the advancement in the lithium-ion battery industry, silicon anode has shown great potential for the next-generation battery. Using the state-of-the-art additive manufacturing technique (three-dimensional holographic lithography), researchers have demonstrated that silicon anode can be fabricated as a three-dimensional bicontinuous porous microstructure. However, the volume fluctuation of the silicon anode caused by lithiation during the discharging process causes continuous capacity decay and poor cycling life. Besides, uncertainties are inherent in the manufacturing and usage processes, making it crucial to systematically consider them in the silicon anode design to improve its performance and reliability. To fill the gap between current silicon anode research and future industrial need, this study established a digital twin to investigate the optimal design for silicon anode under the uncertainties of additive manufacturing and battery usage. This study started with developing multiphysics finite element models of the silicon anode lithiation process to investigate the volume fluctuation of silicon. Then, surrogate models were built based on the results from the finite element models to reduce computational cost. The reliability-based design optimization was employed to find the best design point for the silicon anode, in which an outer optimization loop maximized the objective function and an inner loop dedicated to reliability analysis. Finally, the Pareto optimal front of the silicon anode designs was obtained and validated, which shows over 10% improvements in the silicon anode's total capacity and rate capability.

A two-loop RBDO approach for steel frame structures using EVPS, GWO, and Monte Carlo simulation

This paper evaluates Enhanced Vibrating Particle System (EVPS) and Grey Wolf Optimizer (GWO) algorithms for Reliability-based Design Optimization (RBDO) of steel frames. The two-loop RBDO method minimizes frame weight while satisfying strength, drift, and reliability constraints, considering uncertainties in material properties and loads. Both EVPS and GWO effectively reduce structure weight and ensure required reliability levels, demonstrating their applicability in sustainable steel frame design.

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.

Quantile-based sequential optimization and reliability assessment method under random and interval hybrid uncertainty

Under random and interval hybrid uncertainties, solving hybrid reliability based design optimization (HRBDO) can acquire an optimal balance between structural performance and reliability. Since solving HRBDO includes a triple nested framework involving minimum analysis of performance function (PF), failure probability constraint analysis and design parameter optimization, the computational complexity of HRBDO is high, especially for dealing with complex structures. Therefore, a quantile-based sequential optimization and reliability assessment method (QSORA) is proposed for reducing the computational complexity of HRBDO. In the proposed QSORA for HRBDO, failure probability constraint is firstly transformed into minimum PF (MPF) quantile one corresponding to target failure probability. Then, approximating the difference between PF and its target quantile at current iteration by that at previous one, the failure probability constraint analysis is decoupled from the design parameter optimization. Moreover, by approximating the minimum point of the PF with respect to the interval input in the current iteration by that in the previous one, the minimum analysis of PF is separated from the design parameter optimization. By the separation of minimum analysis and failure probability constraint analysis from the design parameter optimization in the proposed QSORA, the triple nested framework of HRBDO is decoupled sequentially as the deterministic design optimization, the minimum analysis of the PF and the target MPF quantile estimation, and this way of reconstructing the HRBDO from the triple nested framework to three single-loop frameworks can significantly enhance the efficiency of solving HRBDO. Furthermore, the MPF quantile at the current design parameter is estimated by stochastic collocation based statistical moment method, in which the stochastic collocation method is employed to efficiently estimate the MPF moment to approximate the probability density function of MPF. The efficiency and accuracy of the QSORA are validated by four numerical and engineering examples finally.

Single-Loop Sampling for Estimating Failure-Probability Upper-Bound Function

Under random-interval mixed uncertainties of structures, failure-probability upper-bound function (FPUBF), which varies with the distribution parameters of random inputs, can not only provide the influence of distribution parameters on the failure-probability upper bound (FPUB), but also contribute to decoupling a reliability-based design optimization model. Although FPUBF can be estimated by repeatedly evaluating FPUBs at different distribution parameter realizations, it suffers from unaffordable computational cost resulting from this double-loop framework. To address this issue, this paper proposes a single-loop sampling strategy (SL) to estimate FPUBF at arbitrary realizations in the interested distribution parameter region. Instead of the huge computational cost of a double-loop framework, the SL estimates the entire FPUBF only by one simulation analysis. Moreover, importance sampling (IS) variance reduction technique is introduced, and a single-loop IS probability density function (PDF), or SL-IS-PDF, is constructed to more efficiently estimate FPUBF by reducing the required size of the candidate sample pool. For approximating the optimal SL-IS-PDF and identifying the states of candidate samples efficiently, the double-loop adaptive Kriging model of performance function is introduced to further reduce the number of performance function evaluations. A numerical example and two composite structure examples are employed to verify the accuracy, efficiency, and feasibility of the proposed methods.

Optimal path shape of friction-based Track-Nonlinear Energy Sinks to minimize lifecycle costs of buildings subjected to ground accelerations

Track Nonlinear Energy Sinks (T-NES) are Pendulum Tuned Mass Dampers (PTMDs) whose nonlinear restoring forces are obtained from curved path shapes. They combine the abilities of traditional spring-based NESs with the operational advantages of PTMDs. T-NES can resonate with broad frequency content, becoming effective even under damage to the host structure originating from strong ground motions. In the published literature, the integration of Friction Dampers (FDs) into T-NESs (so-called T-NES Type II) has only been studied on PTMDs with circular tracks, and in deterministic settings. Herein, a novel Reliability-Based Design Optimization (RBDO) procedure is presented to find the optimal track shape of T-NES Type II in buildings subjected to earthquakes. The objective is to minimize expected life-cycle damage costs corresponding to slight damage, moderate damage, and extensive damage limit states. A set of rational functions encompassing a wide range of generic curvatures is constructed to describe the T-NES track shape. The nonlinear description is used for restoring force and also in the dissipative mechanism. The case study is a medium-rise building located in Chile. The results indicate that, for the studied building, the optimal path is multimodal, and quite different track profiles can significantly reduce oscillation of the host structure.

Reliability-Based Design Optimization for a Vertical-Type Breakwater with an Emphasis on Sliding, Overturn, and Collapse Failure

To promote the application of reliability-based design within the Korean coastal engineering community, the author conducted reliability analyses and optimized the design of a vertical-type breakwater, considering multiple limit states in the seas off of Pusan and Gunsan – two representative ports in Korea. In this process, rather than relying on design waves of a specific return period, the author intentionally avoided such constraints. Instead, the author characterized the uncertainties associated with wave force, lift force, and overturning moment – key factors significantly influencing the integrity of a vertical-type breakwater. This characterization was achieved by employing a probabilistic model derived from the frequency analysis results of long-term in-situ wave data. The limit state of the vertical-type breakwater encompassed sliding, overturning, and collapse failure, with the close interrelation between wave force, lift force, and moment described using the Nataf joint probability distribution. Simulation results indicate, as expected, that considering only sliding failure underestimates the failure probability. Furthermore, it was shown that the failure probability of vertical-type breakwaters cannot be consistently secured using design waves with a specific return period. In contrast, breakwaters optimally designed to meet the reliability index requirement of <i>β</i>-3.5 to 4 consistently achieve a consistent failure probability across all sea areas.

Reliability-based design optimization for a vertical-type breakwater with multiple limit-state equations under Korean marine environments varying from sea to sea

In this study, as part of basic research aimed at enhancing the accuracy of load and resistance coefficients to ensure their suitability for practical design and promoting the application of underutilized reliability-based design in Korea, the author conducts optimal design based on the reliability analysis of a vertical-type breakwater in the seas off of Haeundae, Yeosu, Mokpo, Gunsan, and Incheon—representative ports in Korea. In doing so, the author utilized the double-loop approach, which simultaneously addresses a reliability problem nested within an optimization process, employing the Polak–He optimization algorithm. To mitigate the substantial numerical effort required by the double-loop approach based on the Polak–He optimization algorithm, which necessitates the gradients of both cost and constraint functions, the subset simulation method was employed. In this process, the author deliberately refrained from using design waves of a specific return period and linear probabilistic models such as the Gaussian distribution, especially concerning wave and lift forces, often viewed as barriers to the widespread application of reliability-based design in Korea. Instead, the author focused on characterizing the uncertainties associated with the wave force, lift force, and overturning moment—variables that significantly impact the integrity of vertical-type breakwaters—by developing probabilistic models for these random variables directly from long-term in situ wave data. These models capture the varied characteristics of the Korean marine environment from sea to sea. In this way, the need for additional assumptions concerning the interrelationship between significant wave and maximum wave heights, along with the wave period, can be eliminated. Following Occam's razor principle, which suggests that explanations constructed with the smallest possible set of assumptions are superior, the reliability-based design optimization of a vertical-type breakwater presented in this study demonstrates promise in terms of simplicity and practicality. The limit state of the vertical-type breakwater was defined to encompass sliding, overturning, and collapse failures, and the strong interrelations between the wave force, lift force, and overturning moment were described using the Nataf joint distribution. As anticipated, simulation results show that solely considering sliding failure, as in the current reliability-based design platform in Korea, leads to an underestimated failure probability. Furthermore, ensuring a consistent failure probability for vertical-type breakwaters using design waves with a specific return period, as in past studies, is not feasible. In contrast, this study demonstrated that breakwaters optimally designed to meet the reliability index requirement of β = 3.5–4 consistently maintain a target failure probability in all sea areas.

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.

Reliability-Based Design Optimization of the PEMFC Flow Field with Consideration of Statistical Uncertainty of Design Variables

Recently, with the fourth industrial revolution, the research cases that search for optimal design points based on neural networks or machine learning have rapidly increased. In addition, research on optimization is continuously reported in the field of fuel cell research using hydrogen as fuel. However, in the case of optimization research, it often requires a large amount of training data, which means that it is more suitable for numerical research such as CFD simulation rather than time-consuming research such as actual experiments. As is well known, the design range of fuel cell flow channels is extremely small, ranging from hundreds of microns to several millimeters, which means the small tolerance could cause fatal performance loss. In this study, the general optimization study was further improved in terms of reliability by considering stochastic tolerances that may occur in actual industry. The optimization problem was defined to maximize stack power, which is employed as objective function, under the constraints such as pressure drop and current density standard deviation; the performance of the optimal point through general optimization was about 3.252 kW/L. In the reliability-based optimization problem, the boundary condition for tolerance was set to 0.1 mm and tolerance was assumed to occur along a normal distribution. The optimal point to secure 99% reliability for the given constraints was 2.918 kW/L, showing significantly lower performance than the general optimal point.

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.

The comparison of robust and reliability based design optimization

The comparison of robust and reliability based design optimization

A novel and efficient stochastic collocation method for estimating failure probability function in one-dimensional reduced space

Failure probability function (FPF) is an important index when measuring the effect of random input distribution parameters on the safety of structures. It can decouple reliability-based design optimization. However, efficient estimation of the FPF is challenging. This study proposes a novel stochastic collocation method for estimating the FPF. The novelty of the proposed method lies in three aspects: First, an equivalent integral expression is proposed for the FPF in one-dimensional reduced space, enabling FPF estimation using an efficient stochastic collocation method. Secondly, a unified sampling density function is constructed to remove the coupling between the computational cost of FPF estimation and the number of distribution parameter realizations. Thirdly, a stochastic collocation point set is derived and shared to estimate the whole FPF using arbitrary distribution parameter realization. The proposed method is particularly suitable for the strength-stress and resistance-load models used in the field of engineering. The efficiency and accuracy of the proposed method are verified by several examples.

Reliability-Based Design Optimization Applied to a Rotor Supported by Hydrodynamic Bearings

Rotating machines are an important part of industrial equipment. It is essential to improve their performance while reducing the manufacturing, operating, and maintenance costs. Ensuring their reliability is also crucial because a machine breakdown can result in significant costs and potential environmental and safety damage. Reliability-based optimization is an approach that aims to find an optimal and robust design that guarantees a machine’s reliability. In this study, we focused on optimizing the shaft diameter and oil temperature of a rotor supported by hydrodynamic bearings. We considered the materials’ elastic moduli, density, and bearing clearance as uncertain parameters. Our goal was to ensure 99% reliability regarding both the vibration amplitude and stability threshold. To model the machine, we used the finite element method and represented the bearings using stiffness and damping coefficients, considering the linear short bearing model. Due to the complexity of the model, we employed surrogate models to solve the reliability-based optimization problem. Our results showed that the optimization problem could be solved successfully using Kriging, polynomial chaos expansion, and polynomial chaos Kriging.

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.

Meta model-based and cross entropy-based importance sampling algorithms for efficiently solving system failure probability function

The multi-mode system failure probability function (SFPF) can quantify how the distribution parameters of the random input vector affect the system safety and decouple the system reliability-based design optimization model. However, for a problem with a time-consuming implicit performance function and rare failure domain, efficiently solving the SFPF remains significantly challenging. Therefore, in this study, two efficient algorithms are proposed, namely, the meta model-based importance sampling and cross entropy-based importance sampling. The contributions of this study are twofold. The first is constructing a single-loop optimal importance sampling density (SL-OISD) method to decouple the double-loop framework for analyzing the SFPF. The second is establishing two methods to efficiently approximate the SL-OISD and complete the SFPF estimation. The first method is based on the meta model of the system performance function, which is abbreviated as SL-Meta-IS. The second method is based on minimizing the cross entropy between the Gaussian mixture density model and SL-OISD, which is abbreviated as SL-CE-IS. To reduce the number of evaluating the system performance function when approximating the SL-OISD, sampling the SL-OISD, and identifying the state of the samples for completing the SFPF estimation, an adaptive Kriging model of the system performance function is introduced into SL-Meta-IS and SL-CE-IS. Owing to decoupling the double-loop framework into a single-loop framework, replacing the time-consuming system performance function with the economic Kriging model, and employing importance sampling variance reduction techniques to address issues related to the rare failure domain, the proposed SL-Meta-IS and SL-CE-IS methods greatly enhance the efficiency of SFPF estimations. The numerical and practical examples demonstrate that the two proposed methods are superior to the existing algorithms; moreover, the efficiency of SL-CE-IS is higher than that of SL-Meta-IS.

Reliability-based optimization of supported pendulum TMDs’ nonlinear track shape using Padé approximants

Supported Pendulum Tuned Mass Dampers (PTMDs) are vibration absorbers whose mass is constrained to oscillate along a curved path, enabling to use gravity as a restoring force. Through a proper track shape selection, they can adapt to the so-called Nonlinear Energy Sinks (NES) concept, consolidated for spring–mass–damper TMDs. Accordingly, NESs are recognized for their ability to resonate in a wide range of frequencies due to their nonlinear nature, becoming more effective when variations on the host structure’s natural frequencies occur. However, regardless of the recent advances in the last few years, gaps remain to be explored in the track shape optimization of supported PTMDs. For instance, a comprehensive literature review shows that Reliability-Based Design Optimization (RBDO) studies to determine the track shape of supported PTMDs cannot be found to the best of the author’s knowledge. Within this context, this paper proposes an original RBDO approach to define the track shape of supported PTMDs in buildings subjected to seismic excitation. For this purpose, a generic rational function with an arbitrary curvature, determined from the Padé approximation of the circle equation, is constructed to allow the PTMD’s path shape search. This procedure allows optimizing fewer parameters than required in Taylor series expansions while providing a wider range of alternatives for numerical optimization. Finally, an active-learning Kriging-based Efficient Global Optimization (EGO) procedure is employed to find the optimum solutions with few objective function evaluations. The application case consists of a typical soft story building subjected to seismic loadings. The optimal track differs from the circle equation, with a higher slope for moderate displacements.

Life-cycle integrity design method considering load robustness of offshore oil and gas equipment: SCSSV as a case study

Components that are vulnerable to the external environment and harsh working conditions easily fail during their life cycle under variable loads of offshore oil and gas equipment. The reliability of the system in the life cycle is reduced, various risks are increased, and substantial maintenance costs are generated. However, the current reliability-based design optimization method has difficulty meeting the situation that the life cycle is taken as the time dimension of design considering reliability, risk, and maintenance. To solve the problem, a life cycle integrity design optimization method considering robustness is proposed. A simplified integrity model is established through random load sampling and global sensitivity analysis. A multiobjective particle swam hierarchical optimization algorithm is proposed, in which hierarchical optimization order is determined by the priority of objectives, and the optimization design with structural integrity as the optimization objective is completed. The method is applied in the design of pressure pipeline of all-electric surface-controlled subsurface safety valve system.

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.

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.