An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches

Random Parameters ◽

Highly Nonlinear ◽

Deterministic optimum designs that are obtained without consideration of uncertainty could lead to unreliable designs, which call for a reliability approach to design optimization, using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and reliability analysis in an independent and standard normal random space (U-space). This process requires numerous nonlinear mapping between X- and U-spaces for a various probability distributions. Therefore, the nonlinearity of RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and U-spaces introduces additional nonlinearity to reliability-based performance measures evaluated during the RBDO process. Evaluation of probabilistic constraints in RBDO can be carried out in two different ways: the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity of RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of highly nonlinear transformations involved. However, PMA is rather independent of probability distributions because of little involvement of the nonlinear transformation.

An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches

10.1115/1.1701880 ◽

2003 ◽

Vol 126 (3) ◽

pp. 403-411 ◽

Cited By ~ 145

Author(s):

Byeng D. Youn ◽

Kyung K. Choi

Keyword(s):

Design Optimization ◽

Reliability Analysis ◽

Probability Distributions ◽

Performance Measure ◽

Random Parameters ◽

Highly Nonlinear ◽

Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and a reliability analysis in an independent and standard normal random space (U-space). This process requires numerous nonlinear mappings between X- and U-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and U-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.

Reliability-Based Design Optimization in X-Space Using Ensemble of Gaussian Reliability Analyses (EoGRA)

Volume 2B: 41st Design Automation Conference ◽

10.1115/detc2015-46130 ◽

2015 ◽

Cited By ~ 1

Author(s):

Po Ting Lin ◽

Shu-Ping Lin

Keyword(s):

Design Optimization ◽

Reliability Analysis ◽

Reliability Index ◽

Design Space ◽

Performance Measure ◽

Design Point ◽

Reliability-Based Design Optimization (RBDO) algorithms have been developed to solve design optimization problems with existence of uncertainties. Traditionally, the original random design space is transformed to the standard normal design space, where the reliability index can be measured in a standardized unit. In the standard normal design space, the Modified Reliability Index Approach (MRIA) measured the minimum distance from the design point to the failure region to represent the reliability index; on the other hand, the Performance Measure Approach (PMA) performed inverse reliability analysis to evaluate the target function performance in a distance of reliability index away from the design point. MRIA was able to provide stable and accurate reliability analysis while PMA showed greater efficiency and was widely used in various engineering applications. However, the existing methods cannot properly perform reliability analysis in the standard normal design space if the transformation to the standard normal space does not exist or is difficult to determine. To this end, a new algorithm, Ensemble of Gaussian Reliability Analyses (EoGRA), was developed to estimate the failure probability using Gaussian-based Kernel Density Estimation (KDE) in the original design space. The probabilistic constraints were formulated based on each kernel reliability analysis for the optimization processes. This paper proposed an efficient way to estimate the constraint gradient and linearly approximate the probabilistic constraints with fewer function evaluations. Some numerical examples with various random distributions are studied to investigate the numerical performances of the proposed method. The results showed EoGRA is capable of finding correct solutions in some problems that cannot be solved by traditional methods.

Dynamical accelerated performance measure approach for efficient reliability-based design optimization with highly nonlinear probabilistic constraints

Reliability Engineering & System Safety ◽

10.1016/j.ress.2018.05.015 ◽

2018 ◽

Vol 178 ◽

pp. 69-83 ◽

Cited By ~ 19

Author(s):

Behrooz Keshtegar ◽

Souvik Chakraborty

Keyword(s):

Design Optimization ◽

Performance Measure ◽

Performance Measure Approach ◽

Highly Nonlinear

A New Study on Reliability-Based Design Optimization

10.1115/1.2829499 ◽

1999 ◽

Vol 121 (4) ◽

pp. 557-564 ◽

Cited By ~ 591

Author(s):

J. Tu ◽

K. K. Choi ◽

Y. H. Park

Keyword(s):

Design Optimization ◽

Reliability Index ◽

Performance Measure ◽

Probabilistic Constraint ◽

Performance Measure Approach ◽

Index Approach ◽

Special Cases

This paper presents a general approach for probabilistic constraint evaluation in the reliability-based design optimization (RBDO). Different perspectives of the general approach are consistent in prescribing the probabilistic constraint, where the conventional reliability index approach (RIA) and the proposed performance measure approach (PMA) are identified as two special cases. PMA is shown to be inherently robust and more efficient in evaluating inactive probabilistic constraints, while RIA is more efficient for violated probabilistic constraints. Moreover, RBDO often yields a higher rate of convergence by using PMA, while RIA yields singularity in some cases.

A decoupling algorithm with first-order asymptotic integration for reliability-based design optimization

Advances in Mechanical Engineering ◽

10.1177/1687814018793336 ◽

2018 ◽

Vol 10 (9) ◽

pp. 168781401879333 ◽

Cited By ~ 1

Author(s):

Zhiliang Huang ◽

Tongguang Yang ◽

Fangyi Li

Keyword(s):

Design Optimization ◽

Reliability Analysis ◽

Optimization Problems ◽

Asymptotic Integration ◽

First Order Reliability Method ◽

First Order ◽

Reliability Method

Conventional decoupling approaches usually employ first-order reliability method to deal with probabilistic constraints in a reliability-based design optimization problem. In first-order reliability method, constraint functions are transformed into a standard normal space. Extra non-linearity introduced by the non-normal-to-normal transformation may increase the error in reliability analysis and then result in the reliability-based design optimization analysis with insufficient accuracy. In this article, a decoupling approach is proposed to provide an alternative tool for the reliability-based design optimization problems. To improve accuracy, the reliability analysis is performed by first-order asymptotic integration method without any extra non-linearity transformation. To achieve high efficiency, an approximate technique of reliability analysis is given to avoid calculating time-consuming performance function. Two numerical examples and an application of practical laptop structural design are presented to validate the effectiveness of the proposed approach.

Hybrid Analysis Method for Reliability-Based Design Optimization

10.1115/1.1561042 ◽

2003 ◽

Vol 125 (2) ◽

pp. 221-232 ◽

Cited By ~ 361

Author(s):

Byeng D. Youn ◽

Kyung K. Choi ◽

Young H. Park

Keyword(s):

Design Optimization ◽

Mean Value ◽

Performance Measure ◽

Equality Constraint ◽

Probabilistic Constraint ◽

Performance Function ◽

Advanced Mean Value

Reliability-based design optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the reliability index approach (RIA) and the performance measure approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the optimization process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the advanced mean value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the conjugate mean value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the hybrid mean value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.

An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses

10.1115/1.4033548 ◽

2016 ◽

Vol 138 (7) ◽

Cited By ~ 3

Author(s):

Po Ting Lin ◽

Shu-Ping Lin

Keyword(s):

Design Optimization ◽

Reliability Analysis ◽

Reliability Index ◽

Design Space ◽

Optimization Problems ◽

Performance Measure ◽

Original Design ◽

Design Point ◽

Reliability-based design optimization (RBDO) algorithms have been developed to solve design optimization problems with existence of uncertainties. Traditionally, the original random design space is transformed to the standard normal design space, where the reliability index can be measured in a standardized unit. In the standard normal design space, the modified reliability index approach (MRIA) measured the minimum distance from the design point to the failure region to represent the reliability index; on the other hand, the performance measure approach (PMA) performed inverse reliability analysis to evaluate the target function performance in a distance of reliability index away from the design point. MRIA was able to provide stable and accurate reliability analysis while PMA showed greater efficiency and was widely used in various engineering applications. However, the existing methods cannot properly perform reliability analysis in the standard normal design space if the transformation to the standard normal space does not exist or is difficult to determine. To this end, a new algorithm, ensemble of Gaussian reliability analyses (EoGRA), was developed to estimate the failure probability using Gaussian-based kernel density estimation (KDE) in the original design space. The probabilistic constraints were formulated based on each kernel reliability analysis for the optimization processes. This paper proposed an efficient way to estimate the constraint gradient and linearly approximate the probabilistic constraints with fewer function evaluations (FEs). Some numerical examples with various random distributions are studied to investigate the numerical performances of the proposed method. The results showed that EoGRA is capable of finding correct solutions in some problems that cannot be solved by traditional methods. Furthermore, experiments of image processing with arbitrarily distributed photo pixels are performed. The lighting of image pixels is maximized subject to the acceptable limit. Our implementation showed that the accuracy of the estimation of normal distribution is poor while the proposed method is capable of finding the optimal solution with acceptable accuracy.

Reliability-Based Design Optimization of Electromagnetic Devices by Evaluating Probabilistic Constraints Based on Performance Measure Approach

Journal of the Korean Magnetics Society ◽

10.4283/jkms.2013.23.2.062 ◽

2013 ◽

Vol 23 (2) ◽

pp. 62-67

Author(s):

Dong-Wook Kim ◽

Dong-Hun Kim

Keyword(s):

Design Optimization ◽

Performance Measure ◽

Performance Measure Approach ◽

Electromagnetic Devices

A Hybrid Loop Approach Using the Sufficient Descent Condition for Accurate, Robust, and Efficient Reliability-Based Design Optimization

10.1115/1.4034173 ◽

2016 ◽

Vol 138 (12) ◽

Cited By ~ 20

Author(s):

Behrooz Keshtegar ◽

Peng Hao

Keyword(s):

Design Optimization ◽

Nonlinear Problems ◽

Performance Measure ◽

Nonlinear Constraint ◽

Sufficient Descent Condition ◽

Highly Nonlinear ◽

Sufficient Descent ◽

Descent Condition

For reliability-based design optimization (RBDO) problems, single loop approaches (SLA) are very efficient but prone to converge to inappropriate point for highly nonlinear constraint functions, and double loop approaches (DLA) are proven to be accurate but require more iterations to achieve stable results. In this paper, an adjusted advanced mean value (AAMV) method is firstly proposed to improve the robustness and efficiency of performance measure approach. The global convergence of the AAMV is guaranteed using sufficient descent condition for the reliability loop in RBDO. Then, a hybrid RBDO method is developed to improve the efficiency of DLA and accuracy of SLA, on the basis of sufficient descent condition and AAMV method, named as hybrid single and double loops (HSD) method. Three nonlinear concave and convex performance functions are used to illustrate the efficiency and robustness of the AAMV method; then the accuracy, robustness, and efficiency of the proposed HSD method are compared to current SLA and DLA through another three benchmark nonlinear RBDO examples. Results show that the AAMV is more robust and efficient than the existing reliability analysis methods. The HSD is more accurate than the SLA for highly nonlinear problems, and also exhibits a better performance than the DLA from the point of view of both robustness and efficiency.

A Hybrid Reliability Approach for Reliability-Based Design Optimization

Volume 1: 36th Design Automation Conference, Parts A and B ◽

10.1115/detc2010-28871 ◽

2010 ◽

Cited By ~ 2

Author(s):

Po Ting Lin ◽

Yogesh Jaluria ◽

Hae Chang Gea

Keyword(s):

Design Optimization ◽

Reliability Index ◽

Optimization Problems ◽

Performance Measure ◽

Accurate Solution ◽

Index Approach ◽

Hybrid Reliability

Reliability-based Design Optimization problems have been solved by two well-known methods: Reliability Index Approach (RIA) and Performance Measure Approach (PMA). RIA generates first-order approximate probabilistic constraints using the measures of reliability indices. For infeasible design points, the traditional RIA method suffers from inaccurate evaluation of the reliability index. To overcome this problem, the Modified Reliability Index Approach (MRIA) has been proposed. The MRIA provides the accurate solution of the reliability index but also inherits some inefficiency characteristics from the Most Probable Failure Point (MPFP) search when nonlinear constraints are involved. In this paper, the benchmark examples have been utilized to examine the efficiency and stability of both PMA and MRIA. In our study, we found that the MRIA is capable of obtaining the correct optimal solutions regardless of the locations of design points but the PMA is much efficient in the inverse reliability analysis. To take advantages of the strengths of both methods, a Hybrid Reliability Approach (HRA) is proposed. The HRA uses a selection factor that can determine which method to use during optimization iterations. Numerical examples from the proposed method are presented and compared with the MRIA and the PMA.