Liu Du

Enriched Performance Measure Approach for Reliability-Based Design Optimization.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, 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 reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

Integration of Possibility-Based Optimization and Robust Design for Epistemic Uncertainty

In practical engineering applications, there exist two different types of uncertainties: aleatory and epistemic uncertainties. This study attempts to develop a robust design optimization with epistemic uncertainty. For epistemic uncertainties, a possibility-based design optimization improves the failure rate, while a robust design optimization minimizes the product quality loss. In general, product quality loss is described using the first two statistical moments for aleatory uncertainty: mean and standard deviation. However, there is no metric for product quality loss defined when having epistemic uncertainty. This paper first proposes a new metric for product quality loss with epistemic uncertainty, and then a possibility-based robust design optimization. For numerical efficiency and stability, an enriched performance measure approach is employed for possibility-based robust design optimization, and the maximal possibility search is used for a possibility analysis. Three different types of robust objectives are considered for possibility-based robust design optimization: smaller-the-better type (S-Type), larger-the-better type (L-Type), and nominal-the-better type (N-Type). Examples are used to demonstrate the effectiveness of possibility-based robust design optimization using the proposed metric for product quality loss with epistemic uncertainty.

Possibility-Based Design Optimization Method for Design Problems With Both Statistical and Fuzzy Input Data

The reliability based design optimization (RBDO) method is prevailing in stochastic structural design optimization by assuming the amount of input data is sufficient enough to create accurate input statistical distribution. If the sufficient input data cannot be generated due to limitations in technical and/or facility resources, the possibility-based design optimization (PBDO) method can be used to obtain reliable designs by utilizing membership functions for epistemic uncertainties. For RBDO, the performance measure approach (PMA) is well established and accepted by many investigators. It is found that the same PMA is a very much desirable approach also for the PBDO problems. In many industry design problems, we have to deal with uncertainties with sufficient data and uncertainties with insufficient data simultaneously. For these design problems, it is not desirable to use RBDO since it could lead to an unreliable optimum design. This paper proposes to use PBDO for design optimization for such problems. In order to treat uncertainties as fuzzy variables, several methods for membership function generation are proposed. As less detailed information is available for the input data, the membership function that provides more conservative optimum design should be selected. For uncertainties with sufficient data, the membership function that yields the least conservative optimum design is proposed by using the possibility-probability consistency theory and the least conservative condition. The proposed approach for design problems with mixed type input uncertainties is applied to some example problems to demonstrate feasibility of the approach. It is shown that the proposed approach provides conservative optimum design.

Enriched Performance Measure Approach (PMA+) for Reliability-Based Design Optimization

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 enhanced hybrid mean value (HMV+) first-order method for e-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. 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, two numerical examples are provided, including a large-scale multi-crash application.

Inverse possibility analysis method for possibility-based design optimization

DOI: 10.2514/1.16546 Structural analysis and design optimization have recently been extended to the stochastic approach to consider variousuncertainties.However,inareaswhereitisnotpossibletoproduceaccurate statistical information forinput data, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, because improper modeling of uncertainty could cause a greater degree of statistical uncertainty than those of physical uncertainty. For systems with insufficient information for input data, possibility-based (or fuzzy set) methods have recently been introduced in structural analysis and design optimization. Using possibility methods, the extended fuzzy operations are much simpler than of random variables. Possibility-based design optimization will provide more conservative designs than those from probability methods, andwill provide a system level of failure possibility automatically. This paper proposes a new formulation of possibility-based design optimization using the performance measure approach. For the inverse possibility analysis, the maximal possibility search method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the multilevel-cut method. Two mathematical examples, including a nonmonotonic response and a physical example of vehicle side impact, are used to demonstrate the proposed maximal possibility search method and possibility-based design optimization.

A NEW FUZZY ANALYSIS METHOD FOR POSSIBILITY-BASED DESIGN OPTIMIZATION

ABSTRACT Structural analysis and design optimization have recently been extended to stochastic approach to take various uncertainties into account. However in areas whwew it is not possible to produce accurate statistical information, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty. For uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in stochastic structural analysis and design optimization. The main advantage of the fuzzy analysis is that it preserves the intrinsic random nature of physical variables through their membership functions and, when used for evaluation of designs, yields more conservative design than those from the probabilistic methods. There are two computational aspects in the fuzzy analysis compared to the probability analysis. First, the input fuzzy variables can be defined easier than the input random variables when no or very limited statistical data are available. Secondly, extended fuzzy operations are much simpler than those required to use probability, especially when a number of variables are involved. For possibility-based design optimization (PBDO), like for reliability-based design optimization (RBDO), the performance measure approach (PMA) with an inverse fuzzy analysis is more appropriate than other approaches, such as a possibility index approach. This paper proposes a new formulation of PBDO using PMA. It is also found that PBDO is more desirable than RBDO, since PBDO can inherently handle a system level possibility unlike RBDO. For the inverse fuzzy analysis, the maximal possibility search (MPS) method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the

Integration of Reliability- and Possibility-Based Design Optimizations Using Performance Measure Approach

Since deterministic optimum designs obtained without considering uncertainty lead to unreliable designs, it is vital to develop design methods that take account of the input uncertainty. When the input data contain sufficient information to characterize statistical distribution, the design optimization that incorporates the probability method is called a reliability-based design optimization (RBDO). It involves evaluation of probabilistic output performance measures. The enriched performance measure approach (PMA+) has been developed for efficient and robust design optimization process. This is integrated with the enhanced hybrid mean value (HMV+) method for effective evaluation of non-monotone and/or highly nonlinear probabilistic constraints. When sufficient information of input data cannot be obtained due to restrictions of budgets, facilities, human, time, etc., the input statistical distribution is not believable. In this case, the probability method cannot be used for reliability analysis and design optimization. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method should be used for structural analysis. A possibility-based design optimization (PBDO) method is proposed along with a new numerical method, called maximal possibility search (MPS), for fuzzy (or possibility) analysis and employing the performance measure approach (PMA) that improves numerical efficiency and stability in PBDO. The proposed RBDO and PBDO methods are applied to two examples to show their computational features. Also, RBDO and PBDO results are compared for implications of these methods in design optimization.

Alternative Methods for Reliability-Based Robust Design Optimization Including Dimension Reduction Method

The objective of reliability-based robust design optimization (RBRDO) is to minimize the product quality loss function subject to probabilistic constraints. Since the quality loss function is usually expressed in terms of the first two statistical moments, mean and variance, many methods have been proposed to accurately and efficiently estimate the moments. Among the methods, the univariate dimension reduction method (DRM), performance moment integration (PMI), and percentile difference method (PDM) are recently proposed methods. In this paper, estimation of statistical moments and their sensitivities are carried out using DRM and compared with results obtained using PMI and PDM. In addition, PMI and DRM are also compared in terms of how accurately and efficiently they estimate the statistical moments and their sensitivities of a performance function. In this comparison, PDM is excluded since PDM could not even accurately estimate the statistical moments of the performance function. Also, robust design optimization using DRM is developed and then compared with the results of RBRDO using PMI and PDM. Several numerical examples are used for the two comparisons. The comparisons show that DRM is efficient when the number of design variables is small and PMI is efficient when the number of design variables is relatively large. For the inverse reliability analysis of reliability-based design, the enriched performance measure approach (PMA+) is used.Copyright

Selection of Copula to Generate Input Joint CDF for RBDO

For RBDO problems with correlated input variables, it is necessary to obtain the input joint distribution (CDF, cumulative distribution function). Then Rosenblatt transformation is used to transform the correlated input variables into the independent standard normal variables for the purpose of inverse reliability analysis. However, in practical industry RBDO problems, often only the marginal CDFs and paired samples are available from limited experimental data. In this paper, a copula, which is a link between a joint CDF and marginal CDFs, is proposed to generate an input joint CDF from these marginal CDFs and paired samples. To identify the right copula from limited data, Bayesian method is proposed to use in this paper. Using Bayesian method, the number of samples required to properly identify the right copula is investigated for different types of copulas and for different correlation coefficients. A real industry problem is used to show how a copula can be identified from the limited experimental data.Copyright

A New Inverse Reliability Analysis Method Using MPP-Based Dimension Reduction Method (DRM)

There are two commonly used reliability analysis methods of analytical methods: linear approximation - First Order Reliability Method (FORM), and quadratic approximation - Second Order Reliability Method (SORM), of the performance functions. The reliability analysis using FORM could be acceptable for mildly nonlinear performance functions, whereas the reliability analysis using SORM is usually necessary for highly nonlinear performance functions of multi-variables. Even though the reliability analysis using SORM may be accurate, it is not desirable to use SORM for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is very difficult to develop. This paper proposes a method that can be used for multi-dimensional highly nonlinear systems to yield very accurate probability of failure calculation without requiring the second order sensitivities. For this purpose, the univariate dimension reduction method (DRM) is used. A three-step computational process is proposed to carry out the inverse reliability analysis: constraint shift, reliability index (β) update, and the most probable point (MPP) approximation method. Using the three steps, a new DRM-based MPP is obtained, which computes the probability of failure of the performance function more accurately than FORM and more efficiently than SORM.Copyright