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

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.

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.

An Investigation of Nonlinearity of Reliability-Based Design Optimization Approaches

Volume 2: 28th Design Automation Conference ◽

10.1115/detc2002/dac-34128 ◽

2002 ◽

Cited By ~ 10

Author(s):

Kyung K. Choi ◽

Byeng D. Youn

Keyword(s):

Design Optimization ◽

Reliability Analysis ◽

Probability Distributions ◽

Performance Measure ◽

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.

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 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.

Discussion of Mathematical Models of Probabilistic Constraints Calculation in Reliability-Based Design Optimization

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.243-249.5717 ◽

2011 ◽

Vol 243-249 ◽

pp. 5717-5726

Author(s):

Ping Yi

Keyword(s):

Mathematical Models ◽

Reliability Analysis ◽

Reliability Index ◽

Mean Value ◽

Performance Measure ◽

Probabilistic Constraint ◽

The Mean

In a reliability-based design optimization (RBDO) problem, most of the computations are used for probabilistic constraints assessment, i.e., reliability analysis. Therefore, the effectiveness, especially the correctness of the reliability analysis is very important. If the probabilistic constraint is misjudged, the optimization iteration would have convergence problems or arrive at erratic solutions. The probabilistic constraint assessment can be carried out using either the conventional reliability index approach (RIA) or the performance measure approach (PMA). In this paper, the mathematical models to calculate the reliability index in RIA and to calculate the probabilistic performance measure (PPM) in PMA are discussed. In RIA, through estimating whether the mean-value point in safe domain or not, we should use a positive or negative reliability index respectively. In PMA, one should always minimize the performance measure to compute PPM whether the performance measure at the mean-value point is positive or negative, which puts right the wrong mathematical model in some literatures and makes it possible to produce effective and efficient approach for RBDO.

Efficient optimum safety factor approach for system reliability-based design optimization with application to composite yarns

Vestnik of Don State Technical University ◽

10.23947/1992-5980-2019-19-3-221-230 ◽

2019 ◽

Vol 19 (3) ◽

pp. 221-230 ◽

Cited By ~ 1

Author(s):

Gh. Kharmanda ◽

I. R. Antypas

Keyword(s):

Design Optimization ◽

Safety Factor ◽

Reliability Index ◽

Performance Measure ◽

Alternative Methods ◽

Implementation Framework ◽

Reliability Level ◽

Convergence Stability

Introduction. The integration of reliability and optimization concepts seeks to design structures that should be both economic and reliable. This model is called Reliability-Based Design Optimization (RBDO). In fact, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational cost and weak convergence stability. Materials andMethods. Several methods have been developed to overcome these difficulties. The methods called Reliability Index Approach (RIA) and Performance Measure Approach (PMA) are two alternative methods. RIA describes the probabilistic constraint as a reliability index while PMA was proposed by converting the probability measure to a performance measure. An Optimum Safety Factor (OSF) method is proposed to compute safety factors satisfying a required reliability level without demanding additional computing cost for the reliability evaluation. The OSF equations are formulated considering RIA and PMA and extended to multiple failure case.Research Results. Several linear and nonlinear distribution laws are applied to composite yarns studies and then extended to multiple failure modes. It has been shown that the idea of the OSF method is to avoid the reliability constraint evaluation with a particular optimization process.Discussion and Conclusions. The simplified implementation framework of the OSF strategy consists of decoupling the optimization and the reliability analyses. It provides designers with efficient solutions that should be economic satisfying a required reliability level. It is demonstrated that the RBDO compared to OSF has several advantages: small number of optimization variables, good convergence stability, small computing time, satisfaction of the required reliability levels.

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.

Reliability Based Design Optimization of Reinforced Concrete Frames Using Genetic Algorithm

Periodica Polytechnica Civil Engineering ◽

10.3311/ppci.17150 ◽

2021 ◽

Author(s):

Mohammadreza Seify Asghshahr

Keyword(s):

Genetic Algorithm ◽

Reinforced Concrete ◽

Design Optimization ◽

Reliability Index ◽

Random Variables ◽

Rc Frame ◽

Form Analysis ◽

Reliability Based Design

This paper introduces a new framework for reliability based design optimization (RBDO) of the reinforced concrete (RC) frames. This framework is constructed based on the genetic algorithm (GA) and finite element reliability analysis (FERA) to optimize the frame weight by selecting appropriate sections for structural elements under deterministic and probabilistic constraints. Modulus of elasticity of the concrete and steel bar, dead load, live load, and earthquake equivalent load are considered as random variables. Deterministic constraints include the code design requirements that must be satisfied for all the frame elements according to the nominal values of the aforementioned random variables. On the other hand, this framework provides the minimum required reliability index as the probabilistic constraint. The first-order reliability method (FORM) using the Newton-type recursive relationship will be used to compute the reliability index. The maximum inter-story drift is considered as an engineering demand parameter to define the limit-state function in FORM analysis. To implement the proposed framework, a mid-rise five-story RC frame is selected as an example. Based on the analysis results, increasing the minimum reliability index from 6 to 7 causes an 11 % increase in the weight of the selected RC frame as an objective function. So, we can obtain a trade-off between the optimized frame weight and the required reliability index utilizing the developed framework. Furthermore, the high values of the reliability index for the frame demonstrate the conservative nature of code requirements for interstory drift limitations based on the linear static analysis method.