Dynamic reliability-based robust design optimization with time-variant probabilistic constraints

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.

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.

The methods of robust design optimization (RDO) and reliability-based robust design optimization (RBRDO) were implemented in the present study. RBRDO is an integrated method that accounts for the design robustness of an objective function and for the reliability of constraints. The objective function in RBRDO is expressed in terms of the mean and standard deviation of an original objective function. Thus, a multi-objective formulation is employed. The regressive approximate models are generated via the moving least squares method (MLSM) and constraint-feasible moving least squares method (CF-MLSM), which make it possible to realize the feasibility regardless of the multimodality/nonlinearity of the constraint function during the approximate optimization processes. The regression model based RBRDO is newly devised and its numerical characteristics are explored using the design of an actively controlled ten bar truss structure.

ABSTRACTProbabilistic and non-probabilistic methods have been proposed to deal with design problems under uncertainties. Reliability-based design and robust design are probabilistic strategies traditionally used for this purpose. In the present contribution, reliability-based robust design optimization (RBRDO) is formulated as a multi-objective problem considering the interaction of both approaches. The proposed methodology is based on the differential evolution algorithm associated with two strategies to deal with reliability and robustness, respectively, namely inverse reliability analysis and the effective mean concept. This multi-objective optimization problem considers the maximization of reliability and robustness coefficients as additional objective functions. The effectiveness of the methodology is illustrated by two classical test cases and a rotor-dynamics application. The results demonstrate that the proposed methodology is an alternative method to solve RBRDO problems.

This paper attempts to integrate a derivative-free probability analysis method to Reliability-Based Robust Design Optimization (RBRDO). The Eigenvector Dimension Reduction (EDR) method is used for the probability analysis method. It has been demonstrated that the EDR method is more accurate and efficient than the Second-Order Reliability Method (SORM) for reliability and quality assessment. Moreover, it can simultaneously evaluate both reliability and quality without any extra expense. Three practical engineering problems (vehicle side impact, layered bonding plates, and lower control arm) are used to demonstrate the effectiveness of the EDR method.

This paper describes the methodologies that have been developed by ESTECO during the first phase of UMRIDA European Project, in the field of Uncertainty Management and Robust Design Optimization, and that have been implemented in the software platform modeFRONTIER. In particular, in the first part there are proposed two methodologies, one based on SS-ANOVA regression applied directly to the uncertainties variables and one based on a stepwise regression methodology applied to the Polynomial Chaos terms used for the uncertainty quantification. Aeronautical test cases proposed by UMRIDA consortium are used to verify the validity of the methodologies. In the second part, the state of art methodologies for Robust Design Optimization are compared with a new proposed approach, based on a min-max definition of the objectives, and the application of Polynomial Chaos coefficients for an accurate definition of percentiles (reliability-based robust design optimization). Also in this case an Aeronautical CFD test case is proposed to validate the methodologies.

A primary concern in practical engineering design is ensuring high system reliability throughout a product life-cycle subject to time-variant operating conditions and component deteriorations. Thus, the capability to deal with time-dependent probabilistic constraints in reliability-based design optimization is of vital importance in practical engineering design applications. This paper presents a nested extreme response surface (NERS) approach to efficiently carry out time-dependent reliability analysis and determine the optimal designs. The NERS employs kriging model to build a nested response surface of time corresponding to the extreme value of the limit state function. The efficient global optimization technique is integrated with the NER S to extract the extreme time responses of the limit state function for any given system design. An adaptive response prediction and model maturation mechanism is developed based on mean square error (MSE) to concurrently improve the accuracy and computational efficiency of the proposed approach. With the nested response surface of time, the time-dependent reliability analysis can be converted into the time-independent reliability analysis and existing advanced reliability analysis and design methods can be used. The NERS is integrated with RBDO for the design of engineered systems with time-dependent performance deterioration. A design case study is used to demonstrate the efficacy of the proposed NERS approach.

Optimizing reliability-based robust design model using multi-objective genetic algorithm

Reliability-based robust design optimization: A comparative study

Dynamic reliability is defined as the probability that an engineered system successfully performs the predefined functionality over a certain period of time considering time-variant operation condition and component deterioration. In practice, it is still a major challenge to conduce dynamic reliability analysis due to the prohibitively high computational costs. In this study, a confidence-based meta-modeling approach is proposed for efficient sensitivity-free dynamic reliability analysis, referred to as double-loop adaptive sampling (DLAS). In DLAS a Gaussian process (GP) model is constructed to approximate extreme system responses over time, so that Monte Carlo simulation (MCS) can be employed directly to estimate dynamic reliability. A qualitative confidence measure is proposed to evaluate the accuracy of dynamic reliability estimation while using the MCS approach based on developed GP models. To improve the confidence, a double-loop adaptive sampling scheme is developed to efficiently update the GP model in a sequential manner, by considering system input variables and time concurrently in double sampling loops. The model updating process can be terminated once the user defined confidence target is satisfied. The DLAS approach does not require computationally expensive sensitivity analysis, thus substantially improves the efficiency of dynamic reliability assessment. Two case studies are used to demonstrate the effectiveness of DLAS for dynamic reliability analysis.

Based on the reliability-based optimization design theory, the reliability sensitivity technique and the robust design method, the reliability-based robust design of rubbing rotor system is extensively discussed and a numerical method for reliability-based robust design is proposed. The reliability sensitivity is added to the reliability-based optimization design model and the reliability-based robust design is described as a multi-objection optimization. On the condition of known first four moments of basic random variables, the respective program can be used to obtain the reliability-based robust design information of rubbing rotor system accurately and quickly using the fourth moment technique. According to the numerical results, the approach proposed is a convenient and practical reliability-based roust design method.

In this investigation, a decomposition approach is employed for reliability-based design optimization of multidisciplinary systems. In the last few years, a variety of different approaches for reliability-based design optimization have been proposed in the literature. Traditionally, reliability-based design optimization is formulated as a nested optimization problem, where the inner loop, generally involves the solution to an optimization problem for computing the probability of failure corresponding to the failure modes. Such a formulation is by nature computationally intensive, requiring a large numbers of function and constraint evaluations. To alleviate this problem, researchers have developed single level reliability-based optimization formulations. The single level approach is expensive for multidisciplinary systems because of the fact that the complex coupled nature of multidisciplinary systems are likewise computationally intensive requiring iterative solvers. To reduce the computational effort of performing reliability-based design optimization in application to multidisciplinary systems, a decomposition approach for optimization is employed. The method of Simultaneous ANalysis and Design (SAND) is employed as an optimization driver within a single level reliability-based design optimization strategy. The methodology is illustrated in application to multidisciplinary test problems.

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from un- known statistical distributions, known as epistemic uncer- tainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliabil- ity analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.

A novel reliability-based design optimization (RBDO) method using simulation-based techniques for reliability assessments and efficient optimization approach is presented in this paper. In RBDO, model-based reliability analysis needs to be performed to calculate the probability of not satisfying a reliability constraint and the gradient of this probability with respect to each design variable. Among model-based methods, the most widely used in RBDO is the first-order reliability method (FORM). However, FORM could be inaccurate for nonlinear problems and is not applicable for system reliability problems. This paper develops an efficient optimization methodology to perform RBDO using simulation-based techniques. By combining analytical and simulation-based reliability methods, accurate probability of failure and sensitivity information is obtained. The use of simulation also enables both component and system-level reliabilities to be included in RBDO formulation. Instead of using a traditional RBDO formulation in which optimization and reliability computations are nested, a sequential approach is developed to greatly reduce the computational cost. The efficiency of the proposed RBDO approach is enhanced by using a multi-modal adaptive importance sampling technique for simulation-based reliability assessment; and by treating the inactive reliability constraints properly in optimization. A vehicle side impact problem is used to demonstrate the capabilities of the proposed method.

The Study on Mechanical Reliability Design Method and Its Application