Reliability-based Robust Design Optimization Research Articles

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 response band-based method for time-dependent reliability-based robust design optimization

In this paper, a response band-based method for time-dependent reliability-based robust design optimization is proposed. The proposed method provides a novel alternative framework, consist of a two-step transformation stage and a solving stage, to solve the time-dependent reliability-based robust design optimization problem. The original time-dependent reliability-based robust design optimization problem is transformed into an equivalent deterministic robust design optimization problem in the transformation stage, and the equivalent problem is settled in the solving stage. In the transformation stage, the dynamic modal decomposition technique and the kriging technique are combined to overcome the problem that there is no standard for both time division and observation sampling in the commonly used transformation methods. In the solving stage, an approach for constructing the response band of the objective function is presented, which significantly reduces the computational consumption of the variation evaluation of the objective function. Five cases are employed to verify the effectiveness of the proposed method.

A multi-step approach for reliability-based robust design optimization of truss structures

There are two general methods of robust design optimization (RDO) and reliability-based robust design optimization (RBRDO) for the optimal design of structures with non-deterministic variables. In the problem-solving process of both RDO and RBRDO, assessing the probabilistic constraints of the problem and the standard deviation of the structural responses is time-consuming and costly. In this study, by presenting a multi-step approach, for a probable optimal solution, the deterministic constraints are first investigated. Then the limit state functions related to probabilistic constraints are computed based on the mean value of random variables (in the RBRDO problem) and in the step of uncertainties effects simulation on the response of the structure, probabilistic constraints are evaluated (in the RBRDO problem). Designs that not meeting the deterministic constraints or in the unsafe area for the mean values of the random variables are not included in calculating the first and second terms of the objective function and reliability assessment. This leads to reduce the volume of calculations. The Monte Carlo simulation method is used to assess the probabilistic constraints, and the EVPS metaheuristic algorithm is used for the optimization process. Three benchmark trusses of 10, 25, and 72 bars were studied to assess the efficiency of the proposed approach. The first objective function in these problems was the mean weight of trusses and the second objective function was the standard deviation of the displacement of a specific node in a certain direction.

A general methodology for reliability-based robust design optimization of computation-intensive engineering problems

Abstract As for complex engineering structures, their single deterministic solutions need to be calculated with the help of computation-intensive finite element software, and the corresponding computation time always lasts very long. When referring to robust design optimization of complex engineering structures, the above expensive job should be repeatedly carried out for a considerable amount of time during the full optimization process. Hence, such repetitive jobs may be difficult to complete for most engineering applications. Aiming at this problem, the paper proposes a general methodology for reliability-based robust design optimization (RBRDO). Firstly, the improved formulation of the RBRDO problem is proposed based on the conventional RBRDO form, and the original probabilistic constraint functions in the improved formulation are changed into the deterministic ones by means of employing the performance measure approach (PMA). Secondly, the above-mentioned improved RBRDO problem is approximately replaced by a new sequence of approximate suboptimizations. For each suboptimization, its original PMA functions are replaced by the approximate explicit form only concerning the deterministic design variables rather than random variables. In this way, each suboptimization is deterministic. Furthermore, the trust-region method is adopted to ensure that the new sequence of suboptimization converge to the original improved RBRDO problem. Lastly, a test problem and three applications are adopted to demonstrate the effectiveness and efficiency of the above-proposed methods. PACS(optional, as per journal): 75.40.-s; 71.20.LP

Reliability-Based Robust Design Optimization for Maximizing the Output Torque of Brushless Direct Current (BLDC) Motors Considering Manufacturing Uncertainty

In recent years, the deterministic design optimization method has been widely used to improve the output performance of brushless direct current (BLDC) motors. However, it does not contribute to reducing the failure rate and performance variation of products because it cannot determine the manufacturing uncertainty. In this study, we proposed reliability-based robust design optimization to improve the output torque of a BLDC motor while reducing the failure rate and performance variation. We calculated the output torque and vibration response of the BLDC motor using the electromagnetic–structural coupled analysis. We selected the tooth thickness, slot opening width, slot radius, slot depth, tooth width, magnet thickness, and magnet length as the design variables related to the shape of the stator and rotor that affect the output torque. We considered the distribution of design variables with manufacturing tolerances. We performed a reliability analysis of the BLDC motor considering the distribution of design variables with manufacturing tolerances. Using the reliability analysis results, we performed reliability-based robust design optimization (RBRDO) to maximize the output torque; consequently, the output torque increased by 8.8% compared to the initial BLDC motor, the standard deviation in output performance decreased by 46.9% with improved robustness, and the failure rate decreased by 99.2% with enhanced reliability. The proposed reliability-based robust design optimization is considered to be useful in the actual product design field because it can evaluate both the reliability and robustness of the product and improve its performance in the design stage.

Bayesian reliability-based robust design optimization of mechanical systems under both aleatory and epistemic uncertainties

Uncertainties can be divided into two general categories: aleatory and epistemic. Conventional reliability-based robust design optimization approaches, which disregard epistemic uncertainties due to lack of knowledge about the physical nature of systems, have previously been developed. To overcome this weakness, unlike previous methods, a Bayesian reliability-based robust design optimization method is proposed in the presence of both aleatory and epistemic uncertainties. The proposed formulation is presented as a multi-objective optimization problem. The univariate dimension reduction method is used to approximate the mean and variance of the design function. The non-dominated sorting genetic algorithm-II is used to solve the multi-objective optimization problem. To find the final optimum design from the Pareto front, Shannon’s entropy-based technique for order of preference by similarity to ideal solution (TOPSIS) algorithm is applied. Finally, to demonstrate the applicability of the proposed method, two case studies are considered and the results are compared and discussed.

An improved reliability-based robust design optimization method using Bayesian seemingly unrelated regression and multivariate loss function

An improved reliability-based robust design optimization method using Bayesian seemingly unrelated regression and multivariate loss function

Design optimization-under-uncertainty of a forward swept wing unmanned aerial vehicle using SAMURAI

In this paper the design optimization-under-uncertainty of a forward swept wing (FSW) blended wing body (BWB) unmanned aerial vehicle (UAV) is examined. Conventional BWBs are often tailless, which leads to a backward swept wing to ensure longitudinal static stability. This in turn can induce flow separation at the tip, leading to a loss of lift, controllability and the appearance of a nose-up pitching moment. A possible solution to this problem is a conceptual redesign by introducing a forward swept wing, which is inherently free of tip-stall, but needs a careful design in order to be controllable. However, fixed wing UAVs are often produced by means of direct injection expanded foam moulding, which is characterized by not negligible production tolerances. This lead to a reliability-based robust design optimization problem, for which a novel framework is employed: SAMURAI. Firstly, the method accounts for computational cost by means of surrogate modelling, an analytical treatment of the problem and an asynchronous updating scheme that balances design space exploration and objective exploitation. Secondly, the method treats the problem as a multi-objective problem, which leads to a Pareto front of robust and reliable designs. The result is a novel series of UAV designs that are inherently free of tip stall, perform robustly and meet the stability requirements with the target reliability obtained with a computationally feasible budget.

Reliability-Based Robust Design Optimization of Lithium-Ion Battery Cells for Maximizing the Energy Density by Increasing Reliability and Robustness

Lithium-ion batteries (LIBs) are increasingly employed in electric vehicles (EVs) owing to their advantages, such as low weight, and high energy and power densities. However, the uncertainty encountered in the manufacturing of LIB cells increases the failure rate and causes cell-to-cell variations, thereby degrading the battery capacity and lifetime. In this study, the reliability and robustness of LIB cells were improved using the design of experiments (DOE), and the reliability-based robust design optimization (RBRDO) approaches. First, design factors sensitive to the energy density and power density were selected as design variables through sensitivity analysis using the DOE. RBRDO was performed to maximize the energy density while reducing the failure rate and cell-to-cell variations. To verify the superiority of the reliability and robustness offered by RBRDO, the obtained results were compared with those from conventional deterministic design optimization (DDO), and reliability-based design optimization (RBDO). RBRDO increased the mean of the energy density by 33.5% compared to the initial value and reduced the failure rate by 98.9%, due to improved reliability, compared to DDO. Moreover, RBRDO reduced the standard deviation in the energy density (i.e., cell-to-cell variations) by 30.0% due to the improved robustness compared to RBDO.

A novel adaptive Kriging method: Time-dependent reliability-based robust design optimization and case study

The computational efficiency and accuracy of the time-dependent reliability-based robust design optimization (TRBRDO) directly rely on the capability to handle the time-dependent reliability analysis (TRA). Some TRA methods use ordinary efficient global optimization (EGO) to identify the extreme samples, and the Kriging model is utilized to approximate the implicit extreme value functions. However, the significant limitation of these methods lies in the unavailability for the parallelized reliability analysis, resulting from the point-to-point nature, which indicates the computational efficiency can be further improved. To construct a more efficient model for the TRA, this paper proposes an adaptive Kriging method, i.e., integrated parallel efficient global optimization (PEGO) and adaptive Kriging-Monte Carlo simulation (AK-MCS), which transforms the TRBRDO problem into an equivalent time-independent one. The proposed adaptive Kriging method was proven to be superior to existing TRBRDO methods in computing efficiency and accuracy, verified by the performance comparison via three cases, including a limit state function with only a time parameter, a two-dimensional function generator, and an engineering application.

Optimization of a higher-order sandwich composite beam under uncertainties

The present paper investigates the reliable and robust optimum design of a higher-order sandwich composite beam under the effect of uncertainty in material properties. The sandwich beam is modeled using the extended higher-order sandwich panel theory. The optimization approaches employed in this work are reliability-based design optimization, robust design optimization, and a hybrid reliability-based robust design optimization. The efficiency of the optimization process is enhanced by using a novel time domain spectral element method-based polynomial chaos surrogate model. The performance of the surrogate model is further improved by using Sobol indices based sensitivity analysis. The optimization procedure is performed using an accelerated particle swarm optimization. The numerical results of the reliable and robust optimal design are presented for both the soft and stiff core sandwich beam. Furthermore, the effect of load density and allowable deflection on the optimal design is also examined.

An effective approach for reliability-based robust design optimization of uncertain powertrain mounting systems involving imprecise information

The uncertain parameters of automotive powertrain mounting systems (PMSs) may involve imprecise information (e.g., incomplete, different and conflicting information) in engineering practice. An effective approach is proposed for the reliability-based robust design optimization (RBRDO) of uncertain PMSs involving imprecise information. In the proposed approach, the imprecise information of uncertain parameters is firstly addressed and combined based on evidence theory, and the uncertain parameters are treated as evidence variables. Then, an uncertainty analysis method named evidence perturbation-central difference method (EPCDM) is derived to fast estimate the mean intervals, standard deviation intervals, and the belief and plausibility measures related to system inherent characteristics. A reference method named evidence-Monte Carlo method (EMCM) is developed to verify the effectiveness of EPCDM. Next, to conduct robustness design, the weighted sum of the lower bounds of means and the upper bounds of standard deviations of system inherent characteristics are taken to construct optimization objective; while to perform reliability design, the belief measures related to system inherent characteristics are used to create reliability constraints. Afterwards, a nested RBRDO model is established to explore the optimum design of the PMS, which considers both reliability and robustness simultaneously. The nested PBRDO can be effectively simplified based on EPCDM. The effectiveness of the proposed approach is finally demonstrated by the application example.

Reliability-Based Robust Design Optimization in Consideration of Manufacturing Tolerance by Multi-Objective Evolutionary Optimization with Repair Algorithm

There are inherently various uncertainties in practical engineering, and reliability-based design optimization (RBDO) and robust design optimization (RDO) are two well-established methodologies when considering the uncertainties. Naturally, reliability-based robust design optimization (RBRDO) is a methodology developed recently by combining RBDO and RDO, in which the tolerances of random design variables are always assumed as constants. However, the tolerance of random design variables is a key factor for the objective robustness and manufacturing cost, and the tolerance allocation is the core problem in mechanical manufacturing. Inspired by the cost–tolerance relationship in mechanical manufacturing, this paper presents an integrated framework to simultaneously find the optimal design variable and the corresponding tolerance in the multi-objective RBRDO, with the trade-off between objective robustness and manufacturing cost. The failure mechanism of the constraint handling strategy of the constrained reference vector-guided evolutionary algorithm (C-RVEA) is discussed to solve the multi-objective optimization formulation. Then the robust repair operator and reliability-based repair operator are proposed to transform unfeasible solutions to the feasible ones under reliability constraints. Numerical results reveal that the proposed repair algorithm is effective. By solving the integrated multi-objective optimization problem, the Pareto front with the compromised solutions between the objective robustness and manufacturing cost could be obtained, from which decision makers can select the satisfying solution conveniently according to the preferred requirements.

Dimensional reduction applied to the reliability-based robust design optimization of composite structures

The need to quantify the uncertainty associated with the main design features and properties of composite laminates has been recognized by the scientific and industrial communities. Recently, robustness and reliability assessment have been combined in a structural design framework called Reliability-based Robust Design Optimization (RBRDO). However, reliability assessment is known to aggravate significantly the efficiency of evolutionary algorithms, in structural design optimization. The problem becomes particularly difficult in the design optimization of composite laminate structures. In this paper, it is proposed the application of an analytical dimensional reduction technique of the uncertainty space associated with reliability assessment, based on the approximate local solution of Sobol’ indices. This approach has a negligible computational cost. It is demonstrated that several random mechanical properties of the optimal design solutions are not important for reliability assessment and can be frozen. The proposed methodology allows the RBRDO problem to be solved much more efficiently.

An efficient Kriging based method for time-dependent reliability based robust design optimization via evolutionary algorithm

Uncertainty inadvertently exists in various stages of engineering system design, development, and operating conditions. During the system design and development stages, a design engineer encounters the reliability and robustness measures of a dynamic uncertain system. Due to the existence of dynamic uncertainties, incorporating the time-dependent reliability of an engineering system in reliability based robust design optimization (RBRDO) is crucial. However, the time-dependent and highly non-linear performance functions present a new challenge to the RBRDO problem. This paper presents a multiobjective integrated framework and corresponding algorithms to handle a time-dependent RBRDO problem. The mean and coefficient of variation of the cost function are taken as a multiobjective problem that needs to be optimized to maximize the robustness without destabilizing the system performance. An evolutionary algorithm is employed to find the optimal design points. The performance functions used to estimate the time-dependent reliability are taken as dynamic probabilistic constraints. The dynamic probabilistic constraints are then converted into deterministic constraints by predicting the corresponding time-dependent reliability. A transfer learning based method integrated with the Kriging surrogate models is proposed to predict the time-dependent reliability for a given time interval. Various examples are used to demonstrate the effectiveness of the proposed approach.

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

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

Reliability-based robust design optimization for torque ripple reduction considering manufacturing uncertainty of interior permanent magnet synchronous motor

This study performed reliability-based robust design optimization (RBRDO) of stator and rotor shape of the IPM type electric motor to reduce the torque ripple considering uncertainties. A metamodel based on the design of experiments results, which was calculated by using 2-D electromagnetic finite element analysis, was used to calculate manufacturing uncertainties and material property distribution. As a result, the mean torque ripple of the optimum electric motor design was reduced by 56.4 % and the robustness was increased by 61.8 % compared with the initial design. The RBRDO design was compared with the designs of deterministic design optimization and reliability-based design optimization, and it was shown that, by using the RBRDO method, the optimum IPMSM can be designed to satisfying not only the minimum torque ripple but also the robustness and the reliability of output torque and torque ripple.

Probability-space surrogate modeling for fast multidisciplinary optimization under uncertainty

This paper proposes a probability-space surrogate modeling approach for computationally efficient multidisciplinary design optimization under uncertainty. This paper uses a probability-space surrogate as opposed to an algebraic surrogate so that the probability distributions of the required outputs at a given design input can naturally be obtained without repeated Monte Carlo runs of an algebraic surrogate at different realizations of the uncertain variables. We consider three probability-space surrogates with analytical solutions for prediction and inference - Multivariate Gaussian, Gaussian Copula, and Gaussian Mixture Model, and investigate their applicability to perform multidisciplinary design optimization under uncertainty. All the input design and random variables, coupling variables, objective and constraint functions are incorporated within the probability-space surrogate, which helps analytically obtain the distributions of coupling variables, objective and constraint functions at desired design inputs while enforcing multidisciplinary compatibility. The training points for the probability-space surrogates are obtained by performing one-pass analysis through the disciplinary models at different realizations of the input variables. The proposed methodology is demonstrated for reliability-based design optimization (RBDO) and reliability-based robust design optimization (RBRDO) in an aircraft design example. The performance of the probability-space surrogates is compared against a Kriging algebraic surrogate and fully coupled Monte Carlo analysis.

Reliability-based robust design optimization of polymer nanocomposites to enhance percolated electrical conductivity considering correlated input variables using multivariate distributions

In this study, reliability-based robust design optimization (RBRDO) for polymer nanocomposites (PNCs) design is conducted to secure the reliability of design conditions concerning the electrical percolation threshold and carbon nanotube (CNT) aspect ratio as well as the robustness for electrical conductivity. CNT diameter and length are known as following the lognormal and Weibull distributions respectively. To reflect the different probability distributions of CNT geometry parameters and correlations between these random input variables, Nataf transformation is employed. By performing several case studies with the first-order reliability method (FORM)-based RBRDO approaches, the objective function exhibited a noteworthy change according to the correlation coefficients and the reliability and robustness for PNCs were satisfied concurrently. Furthermore, the highlight of this work is to provide a generic framework for practical PNC design and multi-objective optimization to enhance electrical performance efficiently.

Study of Uncertainties and Objective Function Modeling Effects on Probabilistic Optimization Results

Abstract In this work, we study the effect of uncertainties modeling and the choice of objective function on the results of optimization design problems in deterministic and probabilistic contexts. Uncertainties modeling are studied in two cases identified in the literature. The results show how the choice of two different objective functions, which lead to the same results in deterministic case, may lead to opposite results in probabilistic case. Also, the results show how the uncertainties modeling type can affect the antagonism between mean and standard deviation in the reliability-based robust design optimization (RBRDO) problems. Three mechanical applications chosen from the literature are used to illustrate these cases.