Incorporating Data Uncertainty in Reliability Based Design Optimization Using Inverse Reliability Measures

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.

To reduce the cost of prototype and physical test, CAE analysis has been widely used to evaluate the vehicle performance during product development process. Combining CAE analysis and optimization approach, vehicle design process can be implemented more efficiently with affordable cost. Reliability based design optimization (RBDO) formulation considers variations of input variables, such as component gauges and material properties. As a result, the design obtained by using RBDO is more reliable and robust compared to those by deterministic optimization. The RBDO process starts from running simulation at DOE sampling data points, generating surrogate models (response surface) and performing robust and reliability based design optimization on the surrogate models by using Monte Carlo simulation. This paper presents a RBDO framework in Excel enviroment. Within this framework, the engineer can perform DOE sampling, surrogate modeling, robustness assessment, Monte Carlo simulation, robust design and reliability-based optimization with any type of distribution.

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.

Reliability-based design optimization (RBDO) is a technique used for engineering design when uncertainty is being considered. A typical RBDO problem can be formulated as a stochastic optimization model where the performance of a system is optimized and the reliability requirements are treated as constraints. One major challenge of RBDO research has been the prohibitive computational expenses. In this research, a new approximation approach, termed the single-loop deterministic method for RBDO (SLDM_RBDO), is proposed to reduce the computational effort of RBDO without sacrificing much accuracy. Based on the first order reliability method, the SLDM_RBDO method converts the probabilistic constraints to approximate deterministic constraints so that the RBDO problems can be transformed to deterministic optimization problems in one step. Three comparison experiments are conducted to show the performance of the SLDM_RBDO. In addition, a reliable forearm crutch design is studied to demonstrate the applicability of SLDM_RBDO to a real industry case.

A new approach toward reliability based design optimization (RBDO) is proposed based on the response surface augmented moment method (RSMM). In RSMM, the reliability analysis procedure based on design of experiments (DOE) is combined with the response surface method (RSM) for numerical efficiency. It utilizes the Pearson system with four statistical moments to calculate the failure probability, and the progressive update of the response surface facilitates the calculation of these four statistical moments. In this study, a semi-analytic design sensitivity analysis is performed in connection with RSMM for an efficient implementation of RSMM in RBDO. The sensitivity of failure probability with respect to the design variables is calculated by direct differentiation and finite difference method with the Pearson system. It is integrated into a mathematical programming for RBDO and applied to several test examples. It was demonstrated that the proposed method of RBDO based on RSMM is efficient and robust.

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

<div class="htmlview paragraph">This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the hybrid mean value (HMV) first-order method for <i>ε</i>-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO while maintaining numerical accuracy and stability. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, several numerical examples are provided, including a large-scale multi-crash application.</div>

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.

Reliability-based design optimization (RBDO) of mechanical systems is computationally intensive due to the presence of two types of iterative procedures – design optimization and reliability estimation. Single-loop RBDO algorithms offer tremendous savings in computational effort, but they have so far only been able to consider individual component reliability constraints. This paper presents a single-loop RBDO formulation and an equivalent formulation that can also include system-level reliability constraints. The formulations allow the allocation of optimal reliability levels to individual component limit states in order to satisfy both system-level and component-level reliability requirements. Four solution algorithms to implement the second, more efficient formulation are developed. A key feature of these algorithms is to remove the most probable points from the decision space, thus avoiding the need to calculate Hessians or gradients of limit state gradients. It is shown that with the proposed methods, systems-level RBDO can be accomplished with computational expense equivalent to several cycles of computationally inexpensive singleloop RBDO based on second-moment methods. Examples of this new approach applied to series, parallel, and combined systems are provided. Nomenclature

Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization

This paper presents a reliability-based analysis of road vehicle accidents and the optimization of roadway radius and speed limit design based on vehicle dynamics, mainly focusing on windy environments. The performance functions are formulated as failure modes of vehicle rollover and sideslip and are defined on a finite set of basic variables with probabilistic characteristics, so-called random variables. The random variables are vehicle speed, steer angle, tire–road friction coefficient, road bank angle, and wind speed. The probability of accident was evaluated using the first-order reliability method (FORM) and numerical studies were conducted using a single-unit truck model. The analysis demonstrates that wind is a significant factor when assessing vehicle safety on roads, and probabilistic studies such as reliability-based design optimization (RBDO) are necessarily required to enhance vehicle safety in windy environments. Accordingly, design optimization of roadway radius and speed limit was conducted, and new designs were proposed satisfying the target reliability. This study suggests that probabilistic mechanics and theory can be of value for analysis and design of wind-related vehicle safety.

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.

Aircraft manufacturing companies have to consider multiple derivatives to satisfy various market requirements. They modify or extend an existing aircraft to meet new market demands while keeping the development time and cost to a minimum. Many researchers have studied the derivative design process, but these research efforts consider baseline and derivative designs together, while using the whole set of design variables. Therefore, an efficient process that can reduce cost and time for aircraft derivative design is needed. In this research, a more efficient design process is proposed which obtains global changes from local changes in aircraft design in order to develop aircraft derivatives efficiently. Sensitivity analysis was introduced to remove unnecessary design variables that have a low impact on the objective function. This prevented wasting computational effort and time on low priority variables for design requirements and objectives. Additionally, uncertainty from the fidelity of analysis tools was considered in design optimization to increase the probability of optimization results. The Reliability Based Design Optimization (RBDO) and Possibility Based Design Optimization (PBDO) methods were proposed to handle the uncertainty in aircraft conceptual design optimization. In this paper, Collaborative Optimization (CO) based framework with RBDO and PBDO was implemented to consider uncertainty. The proposed method was applied for civil jet aircraft derivative design that increases cruise range and the number of passengers. The proposed process provided deterministic design optimization, RBDO, and PBDO results for given requirements.

An efficient approach for reliability-based design optimization (RBDO) of series systems is presented. A modified formulation of the RBDO problem is employed in which the reliabilities of the failure modes of a system are included in the set of the design variables. This allows for an optimal apportionment of the reliability of a system among its failure modes. A sequential optimization and reliability assessment method is used to efficiently determine the optimum design. Here, the constraints on the reliabilities of the failure modes of the RBDO problem are replaced with approximate deterministic constraints. The proposed approach is demonstrated on two example problems that have been solved in previous studies without optimizing the required reliability levels of the failure modes. The first example performs RBDO to a cantilever beam with a rectangular cross section under lateral and vertical loads. The constraints are on the strength and the maximum allowable displacement. The second example performs RBDO to a cable-stayed box girder with five failure modes. Compared to the designs found by previous studies, the new approach finds designs with lower mass but without reducing the system reliability.

Traditional Reliability-based design optimization (RBDO) requires a double loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. It makes even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test the RBDO methods, including single and double loop methods. In addition its constraint size, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.