Reliability-based Design Optimization Research Articles

Reliability analysis and optimization design of hydrogen storage composite pressure vessel with hybrid random-fuzzy uncertainties

In this study, a progressive failure model for hydrogen storage composite pressure vessel (CPV) burst is established. The burst pressure and progressive damage behavior of the CPV are analyzed. Besides, kriging and response surface (RS) surrogate models for CPV burst are established. The verification and the comparative analysis of the surrogate models are carried out. Then, the reliability of the CPV is analyzed with hybrid random-fuzzy uncertainties, and the influence of longitudinal tensile strength, hoop layer thickness, and helical layer thickness on the hybrid reliability is explored. Finally, a unified reliability index is established with hybrid random-fuzzy uncertainties, and the modified particle swarm optimization (PSO) algorithm is applied to carry out reliability-based optimization design. The minimum weight of the composite shell is set as the objective, and the reliability of the CPV with hybrid random-fuzzy uncertainties is set as the constraint. The results show that the hoop layer of the CPV is the main load-bearing structure. The kriging and RS surrogate models are effectiveness to surrogate the burst pressure of the CPV, and the RS model is simpler and more computationally efficient than kriging model, which is more suitable for reliability analysis and optimization design. Under the condition of hybrid random-fuzzy uncertainties, the reliability of the hydrogen storage CPV exhibits fuzzy uncertainty, and there is a possibility of reliability equal to 0 under the low λ-cut level, making it difficult to meet reliability requirements. Increasing the longitudinal tensile strength, the hoop layer thickness and the helical layer thickness can improve the reliability of the hydrogen storage CPV. Furthermore, the optimization method proposed in this study improves its reliability from 0.7043 to 1. This study establishes reliability analysis and optimization design methods for hydrogen storage CPV with hybrid random-fuzzy uncertainties, which contributes to further improvement on the safety of CPV.

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 scheme of isolation capacity of nonlinear vibration isolators

Nonlinear vibration isolators (NVIs) are beneficial to isolate or suppress the adverse effects caused by harmful vibration. To a large extent, the isolation capacity of NVIs depends on the selection of design parameters (including structural, geometric, and excitation parameters) of vibration isolators. However, existent uncertainties of these parameters are unavoidable, which influence the reliability of the isolation capacity of NVIs. In this respect, an effective reliability-based design optimization (RBDO) scheme of isolation capacity of NVIs is presented by taking into account the parameter uncertainties of NVIs. In the proposed RBDO scheme, the force transmissibility of nonlinear isolators is regarded as the objective function, and the fundamental frequency, the dynamic displacement response amplitude, and reliability of vibration isolation performance are supplied as the constraints to search the optimal design parameters of NVIs. Considering the influence degree of design parameters uncertainties, the global reliability sensitivity analysis is employed for identifying the critical design parameters in NVIs. A surrogate model of the failure probability function is established by the adaptive learning Kriging approach to decrease the computational cost of the reliability evaluation nested in the RBDO scheme. An advanced meta-heuristic algorithm is executed to search for the design parameters for optimal isolation capacity of NVIs. The proposed scheme provides a potential direction for optimizing the isolation capacity of complicated vibration isolators.

Comparative analysis of deterministic and reliability-based structural optimization methods

Optimization is the act of obtaining the best possible result under established conditions. Usually, the optimization of a structural design is done considering the structure's dimensions, the materials' properties, and the loads as deterministic values. This way, the optimization process can lead to a more economical design without guaranteeing that this structure is safe. In practice, there are always uncertainties about the final dimensions of the built structure, material properties, and loads. Then, the need arises to use design optimization techniques based on reliability to guarantee a project that is both economical and safe. This objective is achieved by including uncertainties in the optimization process. This article evaluates the parameters that determine the global minimum of the optimization methods DDO (Deterministic Design Optimization) and RBDO (Reliability-Based Design Optimization). This work aims to compare the structural optimization methods of DDO and RBDO through an example. The results are obtained through the codes of the methods implemented in the Python language and show that when comparing the two optimization methods, the presence of uncertainties alters the optimal solution.

A Coupled Simulated Annealing and Particle Swarm Optimization Reliability-Based Design Optimization Strategy under Hybrid Uncertainties

As engineering systems become increasingly complex, reliability-based design optimization (RBDO) has been extensively studied in recent years and has made great progress. In order to achieve better optimization results, the mathematical model used needs to consider a large number of uncertain factors. Especially when considering mixed uncertainty factors, the contradiction between the large computational cost and the efficiency of the optimization algorithm becomes increasingly fierce. How to quickly find the optimal most probable point (MPP) will be an important research direction of RBDO. To solve this problem, this paper constructs a new RBDO method framework by combining an improved particle swarm algorithm (PSO) with excellent global optimization capabilities and a decoupling strategy using a simulated annealing algorithm (SA). This study improves the efficiency of the RBDO solution by quickly solving MPP points and decoupling optimization strategies. At the same time, the accuracy of RBDO results is ensured by enhancing global optimization capabilities. Finally, this article illustrates the superiority and feasibility of this method through three calculation examples.

Reliability-based design optimization for seismic structures considering randomness associated with ground motions

When addressing the dynamic reliability analysis of structures, it becomes necessary to account for multiple limit state functions or their combinations. In scenarios where structures are subjected to random excitation, this can lead to intricate inter-dependencies among different limit states, and the computational workload can pose a substantial challenge in ensuring sufficient precision. Code-based design primarily ensures safety at the member level, while deterministic optimization fails to accommodate the inherent uncertainties associated with external excitation or the system as a whole. Therefore, in such cases, to address both the uncertainties in excitations and the presence of multiple limit states while mitigating computational challenges, equivalent extreme-value criteria are employed within the framework of the probability density evolution method to calculate the global reliability of the structure subjected to stochastic ground motions generated from the physically motivated stochastic ground motion model. Numerical optimization is subsequently conducted using genetic algorithms, aiming to minimize the cost of the superstructure while adhering to the design performance criteria related to the inter-story drift ratio and considering global reliability. Additionally, multi-objective optimization is carried out using NSGA-II, permitting the generation of multiple solutions, from which one can select the most suitable solution as needed. The numerical results illustrate the effectiveness of this technique in achieving an optimal balance between the cost of the structure and the consideration of global reliability, providing a comprehensive solution for dynamic reliability analysis and design optimization of structures under random excitations.

Reliability-based design optimization of tuned mass-damper-inerter for mitigating structural vibration

The tuned mass-damper-inerter (TMDI), comprising mass blocks, springs, dampers, and two-terminal inertial elements, has garnered significant attention as an advanced iteration of the tuned mass damper (TMD) in recent years. However, previous studies of TMDI's design are primarily conducted according to the theoretical formulas based on deterministic analysis assuming harmonic excitations or stochastic analysis assuming Gaussian white noise type excitations. This treatment falls short of capturing the non-stationary and non-Gaussian characteristics exhibited by engineering excitations and thus cannot ensure the safety of controlled structure. A few studies considering non-stationary excitations and based on reliability metrics, in conjunction with stochastic simulation methods, often confront excessive computational overhead. Aiming at such a challenge, the reliability-based design optimization (RBDO) of TMDI system, combining the probability density evolution method (PDEM) and genetic algorithm (GA), for mitigating structural vibration under non-stationary excitations is performed in this study. For the purposes of illustration, a single-degree-of-freedom system attached the TMDI and subjected to stochastic ground motions is studied. Considering the cases of different levels of inertance-to-mass ratios, the RBDO of the TMDI is carried out. The solutions by the theoretical formulas and by the variance-based design optimization are compared to verify the superiority of the proposed method. For assessing the seismic mitigation performance of the optimized TMDI system, probability density evolution analysis of structure-TMDI system with three levels of inertance-to-mass ratios is performed. Comparison between the RBDO of TMDI and that of TMD under consistent mass ratio is further addressed. Numerical results reveal the technical advantages of TMDI in a probabilistic sense, i.e., high vibration mitigation performance, considerable mass reduction, and less stroke demand.

Time-dependent reliability-based design optimization of main shaft bearings in wind turbines involving mixed-integer variables

Main shaft bearings are essential transmission components in wind turbines. The failure of the main shaft bearings is inevitably accompanied by catastrophic consequences. In this regard, this paper presents a time-dependent reliability-based design optimization (TRBDO) approach for main shaft bearings involving mixed-integer variables. The maximization of fatigue life and the elastohydrodynamic film thickness under the load spectrum are selected as optimization objectives. The time-dependent reliability and geometric correlation are determined as nonlinear constraints. An efficient two-stage enrichment strategy is introduced to handle the time-dependent probabilistic constraint with mixed-integer design variables. A mixed-integer nonlinear optimization method is developed based on a meta-heuristic algorithm to solve the formulated TRBDO problem. Eventually, the effectiveness and robustness of the proposed approach are demonstrated by a real application in a 5 MW wind turbine.

A novel reliability updating based method for efficient estimation of failure-probability global sensitivity

Failure-probability (FP) global sensitivity (FP-GS) can measure the average effect of random input on FP, and it is significant in reliability-based design optimization. The key of FP-GS is estimating the conditional FPs on the different realizations of random inputs, which usually requires a time-demanding double-loop structure analysis. This paper originally discovers a reliability updating perspective to efficiently estimate FP-GS, in which all required conditional FPs can be approximated by the posterior FPs based on reliability updating strategy, and the double-loop structure is avoided in estimating the conditional FPs required by FP-GS. In the proposed novel reliability updating based FP-GS analysis method, all conditional FPs required by FP-GS are derived with the likelihood function on the given quasi observations, and they can be simultaneously estimated by a single random input sample set for analyzing the unconditional FP. To reduce the computational cost further, adaptive Kriging model is updated to replace the performance function for efficiently estimating the unconditional FP and all conditional FPs required by FP-GS. Several examples are presented to verify the efficiency and accuracy of the proposed novel reliability updating method for estimating the FP-GS.

Uncertainty-oriented optimal impedance control for EPS-human system with reliability evaluation

This study proposes an interval optimal impedance control method for the electric power steering- human (EPS-H) system with reliability constraints. Considering the relationship between the lateral control using electric power steering of the vehicle and the human, a position-based impedance control model for the integrated EPS-H system is first developed. Polynomial chaos expansion in the surrogate model is employed to establish the interval state-space equation to obtain the bounded responses of the impedance control system using the interval dimensional analysis method. Moreover, the influence of uncertain parameters on the overall control system is obtained by deriving the interval optimal control law consisting of the interval linear quadratic tracking (LQT) inner loop and the interval impedance control outer loop through the interval uncertainty analysis. Furthermore, a non-probabilistic time-varying reliability method is proposed to evaluate the safety of the whole impedance system using the partial derivatives with respect to the uncertain parameters derived from the interval state-space equation. Regarding the nominal cost function and steady-state error of nominal responses as two optimization objectives, the multi-objective reliability-based design optimization is built and solved. Finally, a numerical example of the impedance control model for the EPS-H system is applied to verify the effectiveness of the proposed method.

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.

A Review on Local Failure Probability Sensitivity Analysis

When assessing the reliability of a system, a mathematical model is often defined to replicate the system’s behavior. The inputs of the system are then gathered into two categories, random inputs and deterministic inputs. The failure of the system depends on both categories and here we focus on the influence of the deterministic inputs. Local failure probability sensitivity analysis consists in computing the derivatives of the failure probability with respect to these deterministic parameters and is a fundamental step in reliability-based design optimization. These sensitivities also provide valuable insights into how specific model parameters affect the failure probability, allowing engineers and designers to make informed decisions about adjusting those parameters to enhance reliability. This article explores various techniques developed in the literature for assessing the sensitivity of failure probability with respect to distribution or design parameters. Depending on the nature of the deterministic parameters and the selected input space, different methods are available. The statistical characteristics of the resulting estimates as well as their computational cost are discussed here, for comparison purpose.

Optimal sensor placement for permanent magnet synchronous motor condition monitoring using a digital twin-assisted fault diagnosis approach

Efficient health monitoring for identifying and quantifying damages can substantially improve the performance and structural integrity of engineered systems. Specifically, new advances in sensing technologies have pushed the research of large sensor networks to monitor complex mechanical structures. Given the need for health state monitoring, designing an optimal sensor framework with accurate detectability of failure modes has great significance. However, there is often little to no experimental data available for newly proposed mechanical systems; so a digital-twin method would make fault detection feasible for this applications. In this paper, a data-driven reliability-based design optimization (RBDO) approach is employed for sensor placement and fault detection of a permanent magnet synchronous motor (PMSM), which is a relatively new system for high power engineering applications. This system suffers from inter-turn and inter-phase short-winding faults, which can cause catastrophic failure of the whole structure. For PMSMs, current sensing and magnetic field sensing can be utilized for the detection of faults, but actual sensor placement has not been considered in recent literature. In this study, the first step is to create an FEA model of the PMSM for the simulation of faults, which serves as the digital twin. Next, a data-driven approach is implemented for sensor placement and classification of faults. The proposed method utilizes distance clustering for identification of various failure modes, which is suitable for many applications due to its high accuracy and computational efficiency. In addition, a genetic algorithm is implemented to determine the minimum number and optimal placement of sensors. This framework simultaneously searches for the optimal placement of sensors while training the classifier for detectability of system health states. Ultimately, the proposed methodology shows convergence to a solution with high accuracy for detection of faults, and is demonstrated on the novel system of a PMSM with magnetic field sensors.

Fatigue-sensitive feature extraction, failure prediction and reliability-based design optimization of the hyperloop tube

Hyperloop research focuses on investigating the design and limitations of its complex subsystems. However, the existing literature overlooks the fatigue failure characteristics of the Hyperloop tube, leaving future engineers without informed estimates of its reliability. To address this gap, this study examined the occurrence and prediction of fatigue failure in the Hyperloop system. The findings revealed that both the underground and above-ground configurations showed resistance to fatigue failure, with the underground system showing greater resilience. Additionally, sensitivity analysis highlighted support spacing, tube ultimate tensile strength, and tube radius as the most influential design variables affecting fatigue sensitivity. Moreover, a reliability-based design cost optimization was performed, taking into account demand uncertainty and utilizing insights from previous analyses to determine ideal design parameters. This research sheds light on critical design aspects of the Hyperloop tube that require intensified attention to effectively mitigate the risk of fatigue failure.

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.

Local Sensitivity of Failure Probability through Polynomial Regression and Importance Sampling

Evaluating the failure probability of a system is essential in order to assess its reliability. This probability may significantly depend on deterministic parameters such as distribution parameters or design parameters. The sensitivity of the failure probability with regard to these parameters is then critical for the reliability analysis of the system or in reliability-based design optimization. Here, we introduce a new approach to estimate the failure probability derivatives with respect to deterministic inputs, where the bias can be controlled and the simulation budget is kept low. The sensitivity estimate is obtained as a byproduct of a heteroscedastic polynomial regression with a database built with simulation methods. The polynomial comes from a Taylor series expansion of the approximated sensitivity domain integral obtained with the Weak approach. This new methodology is applied to two engineering use cases with the importance sampling strategy.

Fast solution of reliability-based robust design optimization by reducing the evaluation number for the performance functions

PurposeThe paper proposed a fast procedure for solving the reliability-based robust design optimization (RBRDO) by modifying the RBRDO formulation and transforming it into a series of RBRDO subproblems. Then for each subproblem, the objective function, constraint function and reliability index are approximated using Taylor series expansion, and their approximate forms depend on the deterministic design vector rather than the random vector and the uncertain estimation in the inner loop of RBRDO can be avoided. In this way, it can greatly reduce the evaluation number of performance function. Lastly, the trust region method is used to manage the above sequential RBRDO subproblems for convergence.Design/methodology/approachAs is known, RBRDO is nested optimization, where the outer loop updates the design vector and the inner loop estimate the uncertainties. When solving the RBRDO, a large evaluation number of performance functions are needed. Aiming at this issue, the paper proposed a fast integrated procedure for solving the RBRDO by reducing the evaluation number for the performance functions. First, it transforms the original RBRDO problem into a series of RBRDO subproblems. In each subproblem, the objective function, constraint function and reliability index caused are approximated using simple explicit functions that solely depend on the deterministic design vector rather than the random vector. In this way, the need for extensive sampling simulation in the inner loop is greatly reduced. As a result, the evaluation number for performance functions is significantly reduced, leading to a substantial reduction in computation cost. The trust region method is then employed to handle the sequential RBRDO subproblems, ensuring convergence to the optimal solutions. Finally, the engineering test and the application are presented to illustrate the effectiveness and efficiency of the proposed methods.FindingsThe paper proposes a fast procedure of solving the RBRDO can greatly reduce the evaluation number of performance function within the RBRDO and the computation cost can be saved greatly, which makes it suitable for engineering applications.Originality/valueThe standard deviation of the original objective function of the RBRDO is replaced by the mean and the reliability index of the original objective function, which are further approximated by using Taylor series expansion and their approximate forms depend on the deterministic design vector rather than the random vector. Moreover, the constraint functions are also approximated by using Taylor series expansion. In this way, the uncertainty estimation of the performance functions (i.e. the mean of the objective function, the constraint functions) and the reliability index of the objective function are avoided within the inner loop of the RBRDO.

A new solution framework for time-dependent reliability-based design optimization

Time-dependent reliability-based design optimization (TRBDO) has attracted intensive research attentions in recent years by virtue of its unique ability to allow consideration of dynamic uncertainties caused by stochastic processes and material property degradation. However, existing TRBDO methods are generally too intricate to be practically applicable for practical engineering application. On top of that, extremely high computational cost for complex TRBDO problems further hinders its practicability. To facilitate smooth implementation via enhancing computational efficiency in solving TRBDO problems, this study proposes an innovative and efficient solution framework. The strategy is that time-dependent performance function in each probabilistic constraint is discretized into a series of instantaneous performance functions to transform the original TRBDO problem into a RBDO problem. The reliability of each probabilistic constraint in the transformed RBDO problem is then considered under extreme value condition. With engagement of the first-order reliability method, the RBDO problem is transformed into two different triple-loop timeindependent RBDO problem. However, the issue of expensive computational cost still persists due to the triple-loop structure and identification of temporal variables under extreme value condition. To this gap, a decoupled strategy is adopted to resolve the triple-loop structure into a series of cycles of double-loop reliability analyses and deterministic optimization. Two numerical examples and three engineering applications are employed to demonstrate the supreme computational performance of the currently proposed solution framework. Results show that the proposed framework is capable of achieving a reliable optimal design at a fast convergence speed.

Structural reliability-based design optimization with non-probabilistic credibility level

With the increasing diversity of performance requirements for engineering structures, credibility has become an essential prerequisite for ensuring structural safety. This article introduces a reliability-based structural optimization with non-probabilistic credibility (NCRSO) method, to characterize the credibility of the optimization process. First the concept of uncertainty quantification for credible set is introduced. By utilizing a limited number of sample points, structural parameters with a high level of credibility are obtained and maintained throughout the optimization procedure. Additionally, a reliability index is established based on area ratio and credible feature distance, and a credible sequential optimization strategy is developed. Implementation of the proposed method significantly enhances the overall computational efficiency of structural optimization. The reliability index can directly utilized as the translation distance for deterministic constraints in the sequential strategy, minimizing unnecessary calculations and leading to improved convergence. Three engineering examples are eventually presented to effectively illustrate and emphasize the necessity of considering credibility when utilizing the structural optimization method under complex uncertainty from different perspectives.

Reliability-based design optimization of a pouch battery module using Gaussian process modeling in the presence of cell swelling

Reliability-based design optimization of a pouch battery module using Gaussian process modeling in the presence of cell swelling