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

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.

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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 ◽

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.

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The Response Surface Single Loop Reliability-Based Design Optimization Method With Reliability Requirement on System Failure

Volume 2B: 42nd Design Automation Conference ◽

10.1115/detc2016-60505 ◽

2016 ◽

Author(s):

Rami Mansour ◽

Mårten Olsson

Keyword(s):

Design Optimization ◽

Response Surface ◽

Optimization Problem ◽

Optimization Method ◽

Reliability Based Design Optimization ◽

Single Loop ◽

Reliability Based Design ◽

Design Variables ◽

Reliability Method ◽

Deterministic Solution

In reliability-based design optimization (RBDO), an optimal design which minimizes an objective function while satisfying a number of probabilistic constraints is found. As opposed to deterministic optimization, statistical uncertainties in design variables and design parameters have to be taken into account in the design process in order to achieve a reliable design. In the most widely used RBDO approaches, the First-Order Reliability Method (FORM) is used in the probability assessment. This involves locating the Most Probable Point (MPP) of failure, or the inverse MPP, either exactly or approximately. If exact methods are used, an optimization problem has to be solved, typically resulting in computationally expensive double loop or decoupled loop RBDO methods. On the other hand, locating the MPP approximately typically results in highly efficient single loop RBDO methods since the optimization problem is not necessary in the probability assessment. However, since all these methods are based on FORM, which in turn is based on a linearization of the deterministic constraints at the MPP, they may suffer inaccuracies associated with neglecting the nonlinearity of deterministic constraints. In a previous paper presented by the authors, the Response Surface Single Loop (RSSL) Reliability-based design optimization method was proposed. The RSSL-method takes into account the non-linearity of the deterministic constraints in the computation of the probability of failure and was therefore shown to have higher accuracy than existing RBDO methods. The RSSL-method was also shown to have high efficiency since it bypasses the concept of an MPP. In RSSL, the deterministic solution is first found by neglecting uncertainties in design variables and parameters. Thereafter quadratic response surface models are fitted to the deterministic constraints around the deterministic solution using a single set of design of experiments. The RBDO problem is thereafter solved in a single loop using a closed-form second order reliability method (SORM) which takes into account all elements of the Hessian of the quadratic constraints. In this paper, the RSSL method is used to solve the more challenging system RBDO problems where all constraints are replaced by one constraint on the system probability of failure. The probabilities of failure for the constraints are assumed independent of each other. In general, system reliability problems may be more challenging to solve since replacing all constraints by one constraint may strongly increase the non-linearity in the optimization problem. The extensively studied reliability-based design for vehicle crash-worthiness, where the provided deterministic constraints are general quadratic models describing the system in the whole region of interest, is used to demonstrate the capabilities of the RSSL method for problems with system reliability constraints.

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System Reliability Based Design Optimization of Truss Structures with Interval Variables

Periodica Polytechnica Civil Engineering ◽

10.3311/ppci.14238 ◽

2019 ◽

Author(s):

Mohammad Zaeimi ◽

Ali Ghoddosain

Keyword(s):

Design Optimization ◽

System Reliability ◽

Truss Structures ◽

Equivalent Model ◽

Reliability Based Design Optimization ◽

Uncertain Variables ◽

Reliability Based Design ◽

Interval Variables ◽

Reliability Method ◽

Process Reliability

New products ranging from simple components to complex structures should be designed to be optimal and reliable. In this paper, for the first time, a hybrid uncertain model is applied to system reliability based design optimization (RBDO) of trusses. All uncertain variables are described by random distributions but those lack information are defined by variation intervals. For system RBDO of trusses, the first order reliability method, as well as an equivalent model and the branch and bound method, are utilized to determine the system failure probability; and Improved (μ + λ) constrained differential evolution (ICDE) is employed for the optimization process. Reliability assessment of some engineering examples is proposed to verify our results. Moreover, the effect interval variables on the optimum weight of the truss structures are investigated. The results indicate that the optimal weight depends not only on the uncertainty level but also on the equivalent standard deviation; and a falling-rising behavior is observed.

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Reliability-Based Design Optimization Methods

Volume 1: 30th Design Automation Conference ◽

10.1115/detc2004-57456 ◽

2004 ◽

Cited By ~ 3

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.

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System Reliability-Based Design Optimization Under Tradeoff Between Reduction of Sampling Uncertainty and Design Shift

Journal of Mechanical Design ◽

10.1115/1.4041859 ◽

2019 ◽

Vol 141 (4) ◽

Cited By ~ 3

Author(s):

Sangjune Bae ◽

Nam H. Kim ◽

Seung-gyo Jang

Keyword(s):

Sensitivity Analysis ◽

Bayesian Network ◽

Design Optimization ◽

Failure Probability ◽

System Reliability ◽

Global Sensitivity Analysis ◽

Conservative System ◽

Reliability Based Design Optimization ◽

Sampling Uncertainty ◽

Reliability Based Design

This paper presents a tradeoff between shifting design and controlling sampling uncertainty in system reliability-based design optimization (RBDO) using the Bayesian network. The sampling uncertainty is caused by a finite number of samples used in calculating the reliability of a component, and it propagates to the system reliability. A conservative failure probability is utilized to consider sampling uncertainty. In this paper, the sensitivity of a conservative system failure probability is derived with respect to the design change and the number of samples in a component using Bayesian network along with global sensitivity analysis (GSA). In the sensitivity analysis, GSA is used for local sensitivity calculation. The numerical results show that sampling uncertainty can significantly affect the conservative system reliability and needs to be controlled to achieve the desired level of system reliability. Numerical examples show that both shifting design and reducing sampling uncertainty are crucial in the system RBDO.

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

Applied Sciences ◽

10.3390/app11115312 ◽

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.

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

Volume 1: 34th Design Automation Conference, Parts A and B ◽

10.1115/detc2008-49493 ◽

2008 ◽

Cited By ~ 1

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.

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A Single-Loop Approach for System Reliability-Based Design Optimization

Volume 1: 32nd Design Automation Conference, Parts A and B ◽

10.1115/detc2006-99240 ◽

2006 ◽

Cited By ~ 1

Author(s):

Jinghong Liang ◽

Zissimos P. Mourelatos ◽

Efstratios Nikolaidis

Keyword(s):

Design Optimization ◽

System Reliability ◽

Failure Modes ◽

Optimization Problems ◽

Side Impact ◽

Sequential Optimization ◽

Reliability Based Design Optimization ◽

Single Loop ◽

Reliability Based Design ◽

Probable Failure

An efficient single-loop approach for series system reliability-based design optimization problems is presented in this paper. The approach enables the optimizer to apportion the system reliability among the failure modes in an optimal way by increasing the reliability of those failure modes whose reliability can be increased at low cost. Furthermore, it identifies the critical failure modes that contribute the most to the overall system reliability. A previously reported methodology uses a sequential optimization and reliability approach. It also uses a linear extrapolation to determine the coordinates of the most probable points of the failure modes as the design changes. As a result, the approach can be slow and may not converge if the location of the most probable failure point changes significantly. This paper proposes an alternative system RBDO approach that overcomes the above difficulties by using a single-loop approach where the searches for the optimum design and for the most probable failure points proceed simultaneously. An easy to implement active set strategy is used. The maximum allowable failure probabilities of the failure modes are considered as design variables. The efficiency and robustness of the method is demonstrated on three design examples involving a beam, an internal combustion engine and a vehicle side impact. The results are compared with deterministic optimization and the conventional component RBDO formulation.

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