scholarly journals Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

2021 ◽  
Vol 11 (11) ◽  
pp. 5312
Author(s):  
Junho Chun
Keyword(s):  
Sensitivity Analysis ◽  
Design Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
Step Size ◽  
Reliability Based Design ◽  
Reliability Method ◽  
Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

scholarly journals Reliability-Based Design Optimization of Structures Using Complex-Step Approximation with Sensitivity Analysis

2021 ◽  
Vol 11 (10) ◽  
pp. 4708
Author(s):  
Junho Chun
Keyword(s):  
Sensitivity Analysis ◽  
Structural Optimization ◽  
Design Optimization ◽  
Optimization Problems ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
Step Size ◽  
First Derivative ◽  
Step Approximation ◽  
Reliability Based Design

Structural optimization aims to achieve a structural design that provides the best performance while satisfying the given design constraints. When uncertainties in design and conditions are taken into account, reliability-based design optimization (RBDO) is adopted to identify solutions with acceptable failure probabilities. This paper outlines a method for sensitivity analysis, reliability assessment, and RBDO for structures. Complex-step (CS) approximation and the first-order reliability method (FORM) are unified in the sensitivity analysis of a probabilistic constraint, which streamlines the setup of optimization problems and enhances their implementation in RBDO. Complex-step approximation utilizes an imaginary number as a step size to compute the first derivative without subtractive cancellations in the formula, which have been observed to significantly affect the accuracy of calculations in finite difference methods. Thus, the proposed method can select a very small step size for the first derivative to minimize truncation errors, while achieving accuracy within the machine precision. This approach integrates complex-step approximation into the FORM to compute sensitivity and assess reliability. The proposed method of RBDO is tested on structural optimization problems across a range of statistical variations, demonstrating that performance benefits can be achieved while satisfying precise probabilistic constraints.

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

2018 ◽  
Vol 10 (9) ◽  
pp. 168781401879333 ◽  
Author(s):  
Zhiliang Huang ◽  
Tongguang Yang ◽  
Fangyi Li
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Optimization Problems ◽  
Asymptotic Integration ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
First Order Reliability Method ◽  
First Order ◽  
Reliability Based Design ◽  
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.

Sensitivity Analyses of FORM-Based and DRM-Based Performance Measure Approach for Reliability-Based Design Optimization

2008 ◽  
Author(s):  
Ikjin Lee ◽  
Kyung K. Choi ◽  
Liu Du ◽  
David Gorsich
Keyword(s):  
Design Optimization ◽  
Performance Measure ◽  
Second Order ◽  
Probabilistic Constraints ◽  
Sensitivity Analyses ◽  
Probabilistic Constraint ◽  
Reliability Based Design Optimization ◽  
Performance Measure Approach ◽  
Reliability Based Design ◽  
Reliability Method

In a gradient-based design optimization, it is necessary to know sensitivities of the constraint with respect to the design variables. In a reliability-based design optimization (RBDO), the constraint is evaluated at the most probable point (MPP) and called the probabilistic constraint, thus it requires the sensitivities of the probabilistic constraints at MPP. This paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at MPP for both First Order Reliability Method (FORM)-based Performance Measure Approach (PMA) and Dimension Reduction Method (DRM)-based PMA. Numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method (FDM). However, since the sensitivity calculation at the true DRM-based MPP requires the second-order derivatives and additional MPP search, the sensitivity derivation at the approximated DRM-based MPP, which does not require the second-order derivatives and additional MPP search to find the DRM-based MPP, is proposed in this paper. A convergence study illustrates that the sensitivity at the approximated DRM-based MPP converges to the sensitivity at the true DRM-based MPP as the design approaches the optimum design. Hence, the sensitivity at the approximated DRM-based MPP is proposed to be used for the DRM-based RBDO to enhance the efficiency of the optimization.

Reliability-Based Design Optimization Methods

2004 ◽  
Author(s):  
Anukal Chiralaksanakul ◽  
Sankaran Mahadevan
Keyword(s):  
Design Optimization ◽  
Common Ground ◽  
Optimization Algorithms ◽  
Probabilistic Constraints ◽  
Objective Functions ◽  
Reliability Based Design Optimization ◽  
Reliability Based Design ◽  
Inverse Form ◽  
Reliability Method ◽  
Reliability Methods

Reliability-based design optimization (RBDO) methods are optimization algorithms that utilize reliability methods to evaluate probabilistic constraints and/or objective functions used to prescribe reliability. For practical applications, it is important that RBDO methods are efficient, i.e, they only require a manageable number of numerical evaluations of underlying functions since each one can be computationally expensive. The type of reliability methods and the manner in which they are used in conjunction with optimization algorithms strongly affect computational efficiency. The first order reliability method (FORM) and its inverse are proved to be efficient and widely accepted for reliability analysis. RBDO methods have therefore employed FORM or inverse FORM to numerically evaluate probabilistic constraints and objective functions. During the last decade, the efficiency of RBDO methods has been further improved through problem reformulation. Our goal is to present RBDO methods from a mathematical optimization perspective by formalizing FORM, inverse FORM, and associated RBDO formulations. This new perspective helps not only to clearly reveal their close relationships but also provides a common ground for understanding different types of RBDO methods. Using numerical studies reported in the literature, we indicate the numerical efficiency, convergence, and accuracy of existing RBDO methods.

Single-Loop System Reliability-Based Design Optimization Using Matrix-Based System Reliability Method: Theory and Applications

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Tam H. Nguyen ◽  
Junho Song ◽  
Glaucio H. Paulino
Keyword(s):  
Design Optimization ◽  
Failure Probability ◽  
System Reliability ◽  
Loop System ◽  
Probabilistic Constraints ◽  
System Failure ◽  
Reliability Based Design Optimization ◽  
Single Loop ◽  
Reliability Based Design ◽  
Reliability Method

This paper proposes a single-loop system reliability-based design optimization (SRBDO) approach using the recently developed matrix-based system reliability (MSR) method. A single-loop method was employed to eliminate the inner-loop of SRBDO that evaluates probabilistic constraints. The MSR method enables us to compute the system failure probability and its parameter sensitivities efficiently and accurately through convenient matrix calculations. The SRBDO/MSR approach proposed in this paper is applicable to general systems including series, parallel, cut-set, and link-set system events. After a brief overview on SRBDO algorithms and the MSR method, the SRBDO/MSR approach is introduced and demonstrated by three numerical examples. The first example deals with the optimal design of a combustion engine, in which the failure is described as a series system event. In the second example, the cross-sectional areas of the members of a statically indeterminate truss structure are determined for minimum total weight with a constraint on the probability of collapse. In the third example, the redistribution of the loads caused by member failures is considered for the truss system in the second example. The results based on different optimization approaches are compared for further investigation. Monte Carlo simulation is performed in each example to confirm the accuracy of the system failure probability computed by the MSR method.

A Hybrid Reliability Approach for Reliability-Based Design Optimization

2010 ◽  
Author(s):  
Po Ting Lin ◽  
Yogesh Jaluria ◽  
Hae Chang Gea
Keyword(s):  
Design Optimization ◽  
Reliability Index ◽  
Optimization Problems ◽  
Performance Measure ◽  
Accurate Solution ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
Reliability Based Design ◽  
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.

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