Multi-objective Reliability-based Design Optimization Research Articles

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

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

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.

Multiobjective Reliability-Based Design Optimization of the Fuzzy Logic Controller for MR Damper-Based Structures

To devise an optimum and robust fuzzy logic controller for MR damper-based structures subjected to earthquake ground motions, the multiobjective reliability-based design optimization (RBDO) using the adaptive Kriging model is performed to determine the parameters of the fuzzy logic controller. The optimization problem is formulated with two objective functions, namely, the minimization of interstory drift and average control force of the concerned structure, and subjected to a probability constraint on structural dynamic responses under the effects of random structural stiffness and stochastic earthquake loadings. To reduce the computational cost of reliability assessment, a global Kriging model is constructed in an augmented space as a surrogate for computational evaluations. Subsequently, the trained metamodel combined with the nondominated sorting genetic algorithm (NSGA-II) is integrated into the framework of RBDO for solving the fuzzy logic control (FLC) optimization problem. The feasibility and effectiveness of the multiobjective RBDO in the FLC design are finally validated by conducting numerical simulations on both linear and nonlinear structures. As demonstrated in the linear case, the fuzzy logic controllers obtained from the multiobjective RBDO show more robustness than those derived from the multiobjective deterministic design optimization (DDO). In the nonlinear case, using the multiobjective DDO to prelocate a coarse safety domain can significantly reduce the number of samples for training the metamodel and facilitate the implementation of the multiobjective RBDO; in addition, the controlled structural performance with a specified fuzzy logic controller can be further improved by considering MR damper distribution optimization.

Chaos control assisted single-loop multi-objective reliability-based design optimization using differential evolution

Chaos control assisted single-loop multi-objective reliability-based design optimization using differential evolution

Single-Loop Multi-Objective Reliability-Based Design Optimization Using Chaos Control Theory and Shifting Vector with Differential Evolution

Multi-objective reliability-based design optimization (MORBDO) is an efficient tool for generating reliable Pareto-optimal (PO) solutions. However, generating such PO solutions requires many function evaluations for reliability analysis, thereby increasing the computational cost. In this paper, a single-loop multi-objective reliability-based design optimization formulation is proposed that approximates reliability analysis using Karush-Kuhn Tucker (KKT) optimality conditions. Further, chaos control theory is used for updating the point that is estimated through KKT conditions for avoiding any convergence issues. In order to generate the reliable point in the feasible region, the proposed formulation also incorporates the shifting vector approach. The proposed MORBDO formulation is solved using differential evolution (DE) that uses a heuristic convergence parameter based on hypervolume indicator for performing different mutation operators. DE incorporating the proposed formulation is tested on two mathematical and one engineering examples. The results demonstrate the generation of a better set of reliable PO solutions using the proposed method over the double-loop variant of multi-objective DE. Moreover, the proposed method requires 6×–377× less functional evaluations than the double-loop-based DE.

Two-Step Multi-Objective Reliability-Based Design Optimization of Aircraft Wing Structures

The multi-objective reliability-based design optimization (MORBDO) of an aircraft structure employing a non-probabilistic model, at present, still has a high level of analysis complexity while solving the possibility safety index (PSI) as they are a triple-loop nested problem. Many techniques have been proposed to expedite the process of solving their inner loop with a single objective function; however, research on applying multi-objective optimization to complete this task is required. This research paper aims to reduce the solution complexity in the MORBDO of an aircraft wing structure, which is a symmetrical part of the aircraft structure. The present framework is comprised of a two-step technique that begins with the multi-objective optimization (MODO) of the wing structure, followed by its reliability analysis. A non-probabilistic model is adopted for uncertainty consideration, contrary to frequently used probabilistic models. The reliability design problem has aircraft wing mass, flutter speed, and the possibility safety index as objective functions. According to the results, the proposed MORBDO technique is highly effective in reducing the complexity of aircraft wing structural design and can generate more conservative and feasible design solutions with various PSI values. Such a design can be achieved within a single run, which has not been done in previous studies. The results show that the highest reliability aircraft wing structure mass is 104.8504 kg at a flutter speed of 584.5670 m/s. Additionally, the developed framework explicitly states the relationship between MODO and MORBDO.

Grid Feature-Based Weighted Simulation Method for Multi-Objective Reliability-Based Design Optimization

Reliability analysis and reliability-based design have attracted huge attention since their origin, especially in the engineering design optimization field. Integrating a novel grid feature (GF) mechanism into the weighted simulation method (WSM), a grid feature-based weighted simulation method (GFWSM) is proposed in this paper. By sorting the samples in the random variable space, the GF mechanism serializes the failure probability calculation process, which can reduce the number of performance function evaluations required by the WSM. The proposed GFWSM is integrated into an efficient evolutionary multi-objective optimizer, disruption-based multi-objective equilibrium optimization algorithm (DMOEOA), for solving multi-objective reliability-based design optimization (MORBDO) problems. Different reliability analysis examples are employed to verify the advantages of the proposed GFWSM. Various constrained multi-objective reliability-based design optimization problems have been selected to test the effectiveness of the GFWSM-DMOEOA algorithm. The simulation results indicate that GFWSM and GFWSM-DMOEOA algorithm have achieved a good balance between accuracy and efficiency.

Geotechnical RBDO: Coupling the inverse reliability algorithm with multi-objective reliability-based design optimization of geotechnical systems

The analysis and design of geotechnical engineering inevitably deal with various sources of uncertainties. To further expand the application of reliability methods in geotechnical engineering design works, the inverse reliability algorithm is modified to incorporate unknown geotechnical design parameters in this paper. Based on the established HLRF algorithm for first order reliability analysis, the proposed approach can easily realize traditional single-objective inverse reliability-based design (RBD). Further, the multi-objective inverse reliability design is achieved by combining several single-objective inverse reliability analysis modules through a coupling matrix without seeking for any other surrogate functions. The Nataf transformation is integrated into the method to deal with the correlation of random variables. In addition, the algorithm can be combined with any nonlinear optimization method when the number of design parameters is greater than the number of performance functions, at which point the RBD is extended to reliability-based design optimization (RBDO) with an objective function related to the optimum design purpose. Verifications of the proposed approach are illustrated through several examples, which show that the proposed reliability design optimization framework is effective and can be used as a basic tool for RBDO problems in geotechnical engineering.

An efficient Bi-level hybrid multi-objective reliability-based design optimization of composite structures

This paper presents computationally an efficient multi-objective reliability-based design optimization (RBDO) method for composite structures with uncertain design parameters. The approach adopted is based on a novel hybrid decomposing-based multi-objective particle swarm optimization and sequential quadratic programming algorithms, which is coupled with bi-level modelling and optimization strategy of composite structures and RBDO using Cross entropy method. The reliability analysis considers the rare-event corresponding to the limit states, which is based on Tsai-Wu failure index. In addition, the growing requirements for composite structural design with a minimum weight, introduces significant research challenges in the RBDO approach as the weight and reliability are the two important opposing design requirements. To address this research challenges, the proposed approach involves both reliability and weight as objective functions. A laminated composite plate and benchmark problems were used to demonstrate the efficiency and accuracy of the proposed method. Results show that the proposed method provides computationally efficient and accurate evaluations of the performance of the design of composite structures in comparison to the conventional Monte Carlo simulation method and can provide a potential for solving large-scale multi-objective RBDO of composite structures design problems.

Multi-objective reliability-based seismic performance design optimization of SMRFs considering various sources of uncertainty

The purpose of this paper is to present an approach for multi-objective reliability-based seismic design optimization of Steel Moment Resisting Frames (SMRFs), considering weight and drift uniformity ratio (DUR) as objective functions plus deterministic and probabilistic criteria as constraints, while taking into account the effects of aleatory and epistemic uncertainties. To this end, structural reliability and performance-based seismic design criteria were used to preserve desired levels of structural safety during its life-time. The aleatory uncertainties were applied using estimated incremental dynamic analysis fractile curves, and epistemic uncertainties were applied by realization of stochastic variables. The mean annual frequency was used as a limit-state parameter to check reliability criteria. The novel technique of “shifting displacement-control node” was used to fix drawbacks of previous studies in SMRFs optimization, for non-convergence of solving finite element (FE) equations of different failure mechanisms, especially in the inelastic region and high drifts of pushover curve, as well as reaching the gravitational instability point for more reliable assessment of non-linear structural behavior. The proposed method, applied to 3 cases, ultimately led to the design of cost-effective structures with a desirable seismic performance. In some generated structures, the snap-back phenomenon occurred for inter-story as well as the overall drift of structures. Identification of this phenomenon in SMRFs has received less attention due to the difficulty of solving related FE problems. The effects of considering uncertainties and DUR in high drifts were studied, whereby unexpected results were achieved, such as occurrence and effects of the snap-back phenomenon, relation between DUR and reliability constraints, differences of structures satisfying deterministic and reliability criteria, the importance of selection of control node in pushover analysis, etc. The tools and algorithms developed through this research could help implement performance-based engineering in the design process of SMRFs, in a cost-effective, consistent, and repetitive manner. These algorithms help designers provide project stakeholders with decision-making tools suitable for risk-based structural engineering design for seismic hazard and methods to compare as well as evaluate alternative designs. These algorithms and methods also allow designers to easily compare and contrast structural designs concerning the expected performance.

Multiobjective reliability-based design optimization approach using the gray system and evidence theory

Multiobjective reliability-based design optimization approach using the gray system and evidence theory

Multiobjective Reliability-Based Design Optimization of Recycled Aggregates Used in Corrosive Environment Based on Response Surface Modelling

In this study, durability properties of concrete samples including rapid chloride migration and electrical resistance as corrosion evaluation indicators were tested. The effects of some environmental parameters including humidity percentage, temperature, and chloride concentration were also examined on concrete electrical resistance. The prediction models for mechanical and durability parameters of concrete were obtained using the response surface method. These models were then evaluated using a metaheuristic optimization algorithm coupled with a simple and fast usability new method of reliability evaluation. Eventually, the probabilistic optimal values of using recycled aggregates were calculated for achieving environmentally friendly concrete. Probabilistic multiobjective optimization results revealed that, in an environment with humidity of 70%, temperature of 23°C, and chloride ion concentrations of 3%, 5%, and 8%, use of recycled aggregates for above different chloride ion concentrations was limited to (33.56%, 13.23%), (32.14%, 5.56%), and (13.23%, 2.86%), recycled coarse and fine aggregates, respectively. Furthermore, optimization procedures were performed for the environment or precontaminant recycled aggregate with the chloride concentration of 10%, but the analysis procedure did not converge to an optimum design point.

An efficient multi-objective reliability-based design optimization method for structure based on probability and interval hybrid model

An efficient multi-objective reliability-based design optimization (MO-RBDO) method is proposed based on the probability and interval hybrid model. With probability variables coexist with interval variables in MO-RBDO problem, the original MO-RBDO problem could be considered as a three-layer nested multi-objective optimization problem. To reduce computing costs, firstly, an efficient decoupling strategy is developed to transform original three-layer MO-RBDO problem to single-layer optimization problem based on Karush–Kuhn–Tucker (KKT) necessary condition and the second-order fourth moment method. Secondly, a MO-RBDO method based on the adaptive approximation model is proposed for further improve computational efficiency. Then, the multi-objective genetic algorithm is employed to solve the MO-RBDO problem. Finally, the effectiveness and practicability of the proposed method are demonstrated by three numerical examples.

A Bi-stage Multi-objective Reliability-based Design Optimization Using Surrogate Model for Reusable Thrust Chambers

To reduce the computational burden of Multi-Objective Reliability-Based Design Optimization (MORBDO) and promote its application in complex practical systems, a bi-stage MORBDO procedure using surrogate models is proposed in this paper. The novel process consists of two stages: in the first stage, a narrowed solution space named as MORBDO design space is obtained by a pre-multi-objective deterministic design optimization in the whole original design space; In the second stage, the MORBDO design space is extended to form an augmented space by considering uncertainties, and then the MORBDO is executed with the double-loop strategy over the augmented space. Besides, the multiple response surface (MRS) is adopted in the bi-stage MORBDO procedure to replace the time-consuming finite element analysis. The proposed method is validated by conducting MORBDO on a lab-scale reusable LOX/H2 thrust chamber aiming at maximum residual strength and cyclic life. Compared with the traditional MORBDO implemented in the whole original design space, the bi-stage MORBDO can significantly reduce the computational burden of the surrogate model by 49.33% and only one-quarter of the optimization population is required in NSGA II to search for the optimal solution. Finally, a sensitivity analysis is performed to quantify the importance ranking of design variables of thrust chambers. The results indicate that the geometry of cooling channels near the throat plays the most important role in the thrust chamber's residual strength and cyclic life. The proposed method is easy to be conducted and holds great potentials in the MORBDO of other complex practical systems.

Implementation of a novel Six Sigma multi-objective robustness optimization method based on the improved response surface model for bumper system design

In the vehicle industry, a car’s weight and crashworthiness are two conflicting indicators. Owing to uncertainties in loads, geometries, material properties, and operational conditions, deterministic optimization could lead to unreliable or unstable designs and increase the risk of design failure. This study combined the non-dominated sorting genetic algorithm (NSGA-II), Monte Carlo simulation with descriptive sampling (MCSDS), Six Sigma robustness approach, and improved response surface model (IRSM) to develop a general Six Sigma multi-objective reliability-based design optimization method algorithm (6σ-MORO). The proposed method was applied to examine the crashworthiness of a bumper system. In comparison to the response surface model (RSM), the accuracy of the IRSM was improved by only 10 groups of sample points, proving that the IRSM could construct each response surface function efficiently and accurately. Finally, based on the IRSM, a Six Sigma multi-objective robustness optimization was conducted to explore optimal reliability solutions. The results revealed that the proposed method improved the crashworthiness of the bumper system and also ensured its reliability. Note that the optimal design could be fairly promising for engineering design problems with high dimensions.

Integration of multi-objective reliability-based design optimization into thermal energy management: Application on phase change material-based heat sinks

Phase change material-based heat sinks are widely used in several industrial applications such as in the field of industrial automation and mechatronics. Nowadays, several researchers aim to obtain an optimal design of these systems in order to ameliorate its performances. Nevertheless, uncertainties are not considered for the majority of these studies. Furthermore, it has been confirmed that deterministic multi-objective optimization (DMOO) may lead to an unreliable design. In this study, a new methodology that leads to perform the Multi-Objective Reliability-Based Design Optimization (MORBDO) for thermal management of a passive cooling system is proposed. It consists in coupling the Finite Element Model (FEM), MORBDO procedures and surrogate approaches. Kriging predictor is used to construct metamodels and then validated using Cross-Validation (CV) and error predictions. A numerical investigation is carried out to study the different DMOO and MORBDO approaches. In this study, a 3D PCM-based round pin-fin heat sink is detailed. The aim of this problem is to minimize two objective functions: the cost (total volume V) and the final cooling time tf by considering both thermal and physical constraints. For this purpose, geometrical uncertain parameters are considered such as length L, height H and ϕ pin diameter of the heat sink. This study leads to develop a well-distributed reliable Pareto solutions by combining the Robust Hybrid Method (RHM) and the Constrained Non-dominated Sorting Genetic Algorithm (C-NSGA-II). Then, the efficiency of the proposed MORBDO-RHM approach for PCM-based heat sink is verified.

Multi-objective reliability-based design optimization for the reducer housing of electric vehicles

In this study, a novel multi-objective reliability-based design optimization (MORBDO) method considering the maximum allowable deviation range of design variables is proposed for the reducer housing of electric vehicles. First, the numerical model of the reducer housing is established by ABAQUS and verified by experiments. A radial basis function (RBF) neural network model is used to construct the approximate finite element model. The structural parameters of the RBF are optimized using the heuristic global optimization ability of the particle swarm optimization (PSO) algorithm. Sequential quadratic programming (SQP) and non-dominated sorting genetic algorithm II (NSGA II) are used to perform the MORBDO. Finally, the technique for order preference by similarity to ideal solution, a multi-criteria decision-making (MCDM) method, is used to select the ideal design in multi-objective Pareto points. The optimization method generated a set of Pareto non-dominated solutions with three objectives, which can be selected for a more feasible scheme using MCDM. The proposed method comprehensively measures the requirements of manufacturing and performance criteria, and the optimization results provide a variety of optimization design schemes for the reducer housing of electric vehicles.

Reliability-based design optimization approach for compressor disc with multiple correlated failure modes

In this work, a reliability optimization method is proposed to resolve the multi-objective reliability-based design optimization (MORBDO) of the compressor disc with multiple correlated failure modes. The cost function-based reliability allocation model is firstly established to allocate the structural reliability into reliability under failure mode by minimizing the cost. During this process, the correlations among multi-failure modes are described by using the pair-copula function. Then, an improved MOPSO algorithm based on the niche sorting strategy and the dynamic inertia weight, is employed to solve the MORBDO problem while guaranteeing efficiency and accuracy. The proposed approach is verified by the MORBDO of the compressor disc considering low cycle fatigue (LCF), creep-fatigue (CF), and combined high and low cycle fatigue (H-LCF) failure modes. Strong non-linear positive correlations among the lifetimes of three failure modes are revealed. It is proved that ignoring the correlations among failure modes by assuming independent failure modes will result in an overly conservative reliability prediction, thus leading to redundancy of structural performance in MORBDO. Furthermore, the results proved that the improved MOPSO algorithm does not only provide a better Pareto front curve but also improves computational efficiency.

Multi-Objective, Reliability-Based Design Optimization of a Steering Linkage

Reliability-based design optimization (RBDO) of a mechanism is normally based on the non-probabilistic model, which is viewed as failure possibility constraints in each optimization loop. It leads to a double-loop nested problem that causes a computationally expensive evaluation. Several methods have been developed to solve the problem, which are expected to increase the realization of optimum results and computational efficiency. The purpose of this paper was to develop a new technique of RBDO that can reduce the complexity of the double-loop nested problem to a single-loop. This involves using a multi-objective evolutionary technique combined with the worst-case scenario and fuzzy sets, known as a multi-objective, reliability-based design optimization (MORBDO). The optimization test problem and a steering linkage design were used to validate the performance of the proposed technique. The proposed technique can reduce the complexity of the design problem, producing results that are more conservative and realizable.

Multi-objective reliability based design optimization using Kriging surrogate model for cementless hip prosthesis

Design optimization for cementless hip prosthesis signifies one of the key topics of research to improve its performances. However, majority of the studies have not considered the presence of uncertainties while it has been shown that a deterministic optimization leads to an unreliable design. In this paper, a multi-objective reliability-based design optimization (MORBDO) procedure is proposed for cementless hip prosthesis design. The proposed methodology consists in combining the finite element simulation (FES), surrogating techniques and optimization procedure. The constructed meta-models are validated and compared using different measures such as error predictions and cross-validation (CV). The results show that the constrained non-dominated sorting genetic algorithm (C-NSGA-II) coupled with the hybrid method (HM) was capable to generate well-distributed reliable Pareto solutions.