Second‐order reliability method‐based inverse reliability analysis using Hessian update for accurate and efficient reliability‐based design optimization

First‐order reliability method (FORM) has been mostly utilized for solving reliability‐based design optimization (RBDO) problems efficiently. However, second‐order reliability method (SORM) is required in order to estimate a probability of failure accurately in highly nonlinear performance functions. Despite accuracy of SORM, its application to RBDO is quite challenging due to unaffordable numerical burden incurred by a Hessian calculation. For reducing the numerical efforts, a quasi‐Newton approach to approximate the Hessian is introduced in this study instead of calculating the true Hessian. The proposed SORM with the approximated Hessian requires computations only used in FORM, leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes a generalized chi‐squared distribution in order to achieve better accuracy. Furthermore, SORM‐based inverse reliability method is proposed in this study. An accurate reliability index corresponding to a target probability of failure is updated using the proposed SORM. Two approaches in terms of finding an accurate most probable point using the updated reliability index are proposed. The proposed SORM‐based inverse analysis is then extended to RBDO in order to obtain a reliability‐based optimum design satisfying probabilistic constraints more accurately even for a highly nonlinear system. The numerical study results show that the proposed reliability analysis and RBDO achieve efficiency of FORM and accuracy of SORM at the same time. Copyright © 2014 John Wiley & Sons, Ltd.     

Sensitivity analyses of FORM‐based and DRM‐based performance measure approach (PMA) for reliability‐based design optimization (RBDO)

In gradient‐based design optimization, the sensitivities of the constraint with respect to the design variables are required. In reliability‐based design optimization (RBDO), the probabilistic constraint is evaluated at the most probable point (MPP), and thus the sensitivities of the probabilistic constraints at MPP are required. 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, as 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. Copyright © 2009 John Wiley & Sons, Ltd.     

Reliability‐based design optimization with equality constraints

Equality constraints have been well studied and widely used in deterministic optimization, but they have rarely been addressed in reliability‐based design optimization (RBDO). The inclusion of an equality constraint in RBDO results in dependency among random variables. Theoretically, one random variable can be substituted in terms of remaining random variables given an equality constraint; and the equality constraint can then be eliminated. However, in practice, eliminating an equality constraint may be difficult or impossible because of complexities such as coupling, recursion, high dimensionality, non‐linearity, implicit formats, and high computational costs. The objective of this work is to develop a methodology to model equality constraints and a numerical procedure to solve a RBDO problem with equality constraints. Equality constraints are classified into demand‐based type and physics‐based type. A sequential optimization and reliability analysis strategy is used to solve RBDO with physics‐based equality constraints. The first‐order reliability method is employed for reliability analysis. The proposed method is illustrated by a mathematical example and a two‐member frame design problem. Copyright © 2007 John Wiley & Sons, Ltd.     

An integrated reliability-based design optimization of offshore towers

After recognizing the uncertainty in the parameters such as material, loading, geometry and so on in contrast with the conventional optimization, the reliability-based design optimization (RBDO) concept has become more meaningful to perform an economical design implementation, which includes a reliability analysis and an optimization algorithm. RBDO procedures include structural analysis, reliability analysis and sensitivity analysis both for optimization and for reliability. The efficiency of the RBDO system depends on the mentioned numerical algorithms. In this work, an integrated algorithms system is proposed to implement the RBDO of the offshore towers, which are subjected to the extreme wave loading. The numerical strategies interacting with each other to fulfill the RBDO of towers are as follows: (a) a structural analysis program, SAPOS, (b) an optimization program, SQP and (c) a reliability analysis program based on FORM. A demonstration of an example tripod tower under the reliability constraints based on limit states of the critical stress, buckling and the natural frequency is presented.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method

The first order reliability method has been extensively adopted for reliability-based design optimization (RBDO), but it shows inaccuracy in calculating the failure probability with highly nonlinear performance functions. Thus, the second order reliability method is required to evaluate the reliability accurately. However, its application for RBDO is quite challenge owing to the expensive computational cost incurred by the repeated reliability evaluation and Hessian calculation of probabilistic constraints. In this article, a new improved stability transformation method is proposed to search the most probable point efficiently, and the Hessian matrix is calculated by the symmetric rank-one update. The computational capability of the proposed method is illustrated and compared to the existing RBDO approaches through three mathematical and two engineering examples. The comparison results indicate that the proposed method is very efficient and accurate, providing an alternative tool for RBDO of engineering structures.     

Reliability analysis of metal‐composite adhesive joints under debonding modes I,II, and I/II using the results of experimental and FEM analyses

In the present study, the experimental and finite element (FE) analyses are first carried out to investigate the deboning behavior of metal‐composite adhesive joints under modes of I and mode II loading. To conduct an FE on the debonding propagation, cohesive zone method (CZM), as well as maximum nominal stress and energy criteria, is applied. In the reliability analysis, to achieve the probability of debonding growth (PODG), limit state functions are formulated based on the energy release rate. To that end, the first‐order reliability method (FORM), the second‐order reliability method (SORM), and the Monte Carlo simulation (MCS) are used to calculate the PODG. The effect of initial debonding length on the PODG in all mentioned modes is investigated. Results obtained from reliability analysis reveal that the random variables including the initial debonding length, width, and thickness are the most sensitive variables to ascertain the PODG.     

Probabilistic uncertainty analysis by mean-value first order Saddlepoint Approximation

Probabilistic uncertainty analysis quantifies the effect of input random variables on model outputs. It is an integral part of reliability-based design, robust design, and design for Six Sigma. The efficiency and accuracy of probabilistic uncertainty analysis is a trade-off issue in engineering applications. In this paper, an efficient and accurate mean-value first order Saddlepoint Approximation (MVFOSA) method is proposed. Similar to the mean-value first order Second Moment (MVFOSM) approach, a performance function is approximated with the first order Taylor expansion at the mean values of random input variables. Instead of simply using the first two moments of the random variables as in MVFOSM, MVFOSA estimates the probability density function and cumulative distribution function of the response by the accurate Saddlepoint Approximation. Because of the use of complete distribution information, MVFOSA is generally more accurate than MVFOSM with the same computational effort. Without the nonlinear transformation from non-normal variables to normal variables as required by the first order reliability method (FORM), MVFOSA is also more accurate than FORM in certain circumstances, especially when the transformation significantly increases the nonlinearity of a performance function. It is also more efficient than FORM because an iterative search process for the so-called Most Probable Point is not required. The features of the proposed method are demonstrated with four numerical examples.     

Discussion of “Reliability‐based design optimization with dependent interval variables” by Xiaoping Du,International Journal for Numerical Methods in Engineering 2012; 91:218–228

In the subject paper, a reliability‐based design optimization (RBDO) model with both random and dependent interval uncertainties was proposed based on the First Order Reliability Method. The lower bound of reliability defined in Equation (9) of the subject paper was utilized as the constraint in this RBDO model. The author claimed that it is the minimum reliability with both random and interval variables. However, we prove that it is not the minimum value. It is therefore suggested that the minimum reliability should be used in the RBDO model. Copyright © 2014 John Wiley & Sons, Ltd.     

Reliability-based design optimization of a car body using dimension- reduced Chebyshev polynomial

A reliability-based design optimization (RBDO) method of a car body is presented on basis of dimension-reduced Chebyshev polynomial method (DCM). To improve calculation efficiency and save computational time, complex models are often approximated by metamodels in reliability analysis. Traditional metamodels require a large number of sample points, which is time-consuming. To improve the efficiency, DCM is proposed to approximate the performance function of the car body. First, the performance function is decomposed by the dimension-reduction method into a sum of univariate functions, which are then fitted through Chebyshev polynomials. The reliability of the car body is predicted by the Taylor expansion method and the fourth-moment method. Finally, the result of RBDO is obtained using an improved adaptive genetic algorithm. The proposed method saves on the calculation time with high precision. Besides, the improved adaptive genetic algorithm reduces the number of iterations in the car body optimization and improves the efficiency.     

Reliability-based design optimization with frequency constraints using a new safest point approach

In the reliability-based design optimization (RBDO) model, the mean values of uncertain variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, an RBDO solution that reduces the structural weight in non-critical regions provides not only an improved design, but also a higher level of confidence in the design. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems that are likewise computationally intensive. This article focuses on the study of a particular problem representing the failure mode of structural vibration analysis. A new method is proposed, called safest point, that can efficiently give the reliability-based optimum solution under frequency constraints, and then several probability distributions are developed, which are mathematically nonlinear functions, for the proposed method. Finally, the efficiency of the extended approach is demonstrated for probability distributions such as log-normal and uniform distributions, and its applicability to the design of structures undergoing fluid–structure interaction phenomena, especially the design process of aeroelastic structures, is also demonstrated.     

Evolutionary topology optimization using design space adjustment based on fixed grid

Design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown. In the conventional topology optimization, the design domain is fixed. It is, however, desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables, which we call design space optimization. A breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background, but due to coupling effect and others, sensitivity results have not been satisfactory. Three innovative implementations are developed in this paper. Firstly, the proper characteristics of artificial material are defined. The second one is to decouple neighbouring elements for exact design space sensitivities. The previous design space optimization has been tedious because only one layer can be added. So, the third innovation is a new expansion strategy with multi‐layers based on design space sensitivities. As a result, the proposed evolutionary method can get an optimum much faster than ever before especially for large‐scale problems. It is also conjectured that this gives higher probability of getting the global optimum, as confirmed by numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Tolerance‐based reliability and optimization design of switched‐mode power supply

Due to its high efficiency and low power consumption, switched‐mode power supply (SMPS) represents the development trend of the stabilized voltage power supply. However, tolerance has become one of the key factors in the design of SMPS because of the process fluctuation of electronic components, unstable input parameters of the circuit system, influence of working conditions and environment, and the effect of aging. In order to improve the reliability of SMPS and reduce the manufacturing cost, this paper proposes a reliability analysis and optimization design method based on the tolerance and sensitivity analysis. Finally, this method is applied to the tolerance design for the positive switching power supply of the SMPS circuits, and the optimal tolerance design scheme is obtained. Furthermore, the reliability and probability density curves are evaluated.     

Reliability analysis of structures using neural network method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical procedure, such as finite element method. To reduce the computational effort required for reliability analysis, response surface method could be used. However the conventional response surface method is still time consuming especially when the number of random variables is large. In this paper, an artificial neural network (ANN)-based response surface method is proposed. In this method, the relation between the random variables (input) and structural responses is established using ANN models. ANN model is then connected to a reliability method, such as first order and second moment (FORM), or Monte Carlo simulation method (MCS), to predict the failure probability. The proposed method is applied to four examples to validate its accuracy and efficiency. The obtained results show that the ANN-based response surface method is more efficient and accurate than the conventional response surface method.     

Reliability-based design optimization using a moment method and a kriging metamodel

Reliability-based design optimization (RBDO) has been used for optimizing engineering systems with uncertainties in design variables and system parameters. RBDO involves reliability analysis, which requires a large amount of computational effort, so it is important to select an efficient method for reliability analysis. Of the many methods for reliability analysis, a moment method, which is called the fourth moment method, is known to be less expensive for moderate size problems and requires neither iteration nor the computation of derivatives. Despite these advantages, previous research on RBDO has been mainly based on the first-order reliability method and relatively little attention has been paid to moment-based RBDO. This article considers difficulties in implementing the moment method into RBDO; they are solved using a kriging metamodel with an active constraint strategy. Three numerical examples are tested and the results show that the proposed method is efficient and accurate.     

An integration design optimization framework of robust design,axiomatic design,and reliability‐based design

Robust design, axiomatic design, and reliability‐based design provide effective approaches to deal with quality problems, and their integration will achieve better quality improvement. An integration design optimization framework of robust design, axiomatic design, and reliability‐based design is proposed in this paper. First, the fitted response model of each quality characteristic is obtained by response surface methodology and the mean square error (MSE) estimation is given by a second‐order Taylor series approximation expansion. Then the multiple quality characteristics robust design model is developed by the MSE criteria. Finally, the independence axiom constraints for decoupling and reliability constraints are integrated into the multiple quality characteristics robust design model, and the integration design optimization framework is formulated, where the weighted Tchebycheff approach is adopted to solve the multiple objective programming. An illustrative example is presented at the end, and the results show that the proposed approach can obtain better trade‐offs among conflicting quality characteristics, variability, coupling degree and reliability requirements. Copyright © 2011 John Wiley & Sons, Ltd.     

An inverse reliability analysis method for reliability-based design optimization with random and dependent interval variables constrained within ellipsoids

In practical design problems, interval variables exist. Many existing methods can handle only independent interval variables. Some interval variables, however, are dependent. In this work, dependent interval variables constrained within a multi-ellipsoid convex set are considered and incorporated into reliability-based design optimization (RBDO). An efficient RBDO method is proposed by employing the sequential single-loop procedure, which separates the coupled reliability analysis procedure from the deterministic optimization procedure. In the reliability analysis procedure, a single-loop optimization for the inverse reliability analysis is performed, and an efficient inverse reliability analysis method for searching for the worst-case most probable point (WMPP) is developed. The search method contains two stages. The first stage deals the situation where the WMPP is on the boundary of the feasible region, while the second stage accommodates the situation where the WMPP is inside the feasible region by interpolation. Three examples are used for a demonstration.     

Enhanced modified reliability index approach for efficient and robust reliability-based design optimization

The reliability index approach (RIA) is one of the effective tools for solving the reliability-based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA-based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large-scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first-order reliability methods. Six RBDO problems quoted also indicate that DLS-based EMRIA has good performance to solve complex RBDO problems.     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.