Post optimization for accurate and efficient reliability‐based design optimization using second‐order reliability method based on importance sampling and its stochastic sensitivity analysis

In this study, a post optimization technique for a correction of inaccurate optimum obtained using first‐order reliability method (FORM) is proposed for accurate reliability‐based design optimization (RBDO). In the proposed method, RBDO using FORM is first performed, and then the proposed second‐order reliability method (SORM) is performed at the optimum obtained using FORM for more accurate reliability assessment and its sensitivity analysis. In the proposed SORM, the Hessian of a performance function is approximated by reusing derivatives information accumulated during previous RBDO iterations using FORM, indicating that additional functional evaluations are not required in the proposed SORM. The proposed SORM calculates a probability of failure and its first‐order and second‐order stochastic sensitivity by applying the importance sampling to a complete second‐order Taylor series of the performance function. The proposed post optimization constructs a second‐order Taylor expansion of the probability of failure using results of the proposed SORM. Because the constructed Taylor expansion is based on the reliability method more accurate than FORM, the corrected optimum using this Taylor expansion can satisfy the target reliability more accurately. In this way, the proposed method simultaneously achieves both efficiency of FORM and accuracy of SORM. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

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.     

基于HLRF法与修正对称秩1方法的改进可靠度方法EI北大核心CSCD

一次可靠度方法简单、高效,但在处理强非线性功能函数时存在较大误差;已有的二次可靠度方法在提高精度的同时往往降低了效率。为此,该文中在发展改进一次可靠度方法的同时提出了更好地兼顾精度与效率的改进二次可靠度方法。将修正对称秩1方法与HLRF法的步长确定策略相结合,提出了具有较好收敛性的改进一次可靠度方法,且在基本不增加计算量的前提下获得了功能函数的近似Hessian矩阵;结合坐标旋转、单变量降维近似和非中心卡方分布,提出了与改进一次可靠度方法同效率但具有更高精度的改进二次可靠度方法;通过数值算例和工程算例验证了建议方法的广泛适用性以及精度或效率上的优势。     

Application of the response surface methods to solve inverse reliability problems with implicit response functions

The inverse first-order reliability method (FORM) is considered to be one of the most widely used methods in inverse reliability analysis. It has been recognized that there are shortcomings of the inverse FORM in solving inverse reliability problems with implicit response functions, primarily inefficiency and difficulties involved in evaluating derivatives of the implicit response functions with respect to random variables. In order to apply the inverse FORM to structural inverse reliability analysis, response surface methods can be used to overcome the shortcomings. In the present paper, two different response surface methods, namely the polynomial-based response surface method and the artificial neural network-based response surface method, are developed to solve the inverse reliability problems with implicit response functions, and the accuracy and efficiency of the two response surface methods are demonstrated through two numerical examples of steel structures. It is found that the polynomial-based response surface method is more efficient and accurate than the artificial neural network-based response surface method. Recommendations are made regarding the suitability of the two response surface methods to solve the inverse reliability problems with implicit response functions.     

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.     

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.     

A new artificial neural network-based response surface method for structural reliability analysis

This paper presents a new artificial neural network-(ANN)based response surface method in conjunction with the uniform design method for predicting failure probability of structures. The method involves the selection of training datasets for establishing an ANN model by the uniform design method, approximation of the limit state function by the trained ANN model and estimation of the failure probability using first-order reliability method (FORM). In the proposed method, the use of the uniform design method can improve the quality of the selected training datasets, leading to a better performance of the ANN model. As a result, the ANN dramatically reduces the number of required trained datasets, and shows a good ability to approximate the limit state function and then provides a less rigorous formulation in the context of FORM. Results of three numerical examples involving both structural and non-structural problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure. Compared with the conventional ANN-based response surface method, the proposed method is much more economical to achieve reasonable accuracy when dealing with problems where closed-form failure functions are not available or the estimated failure probability is extremely small. Finally, several important parameters in the proposed method are discussed.     

Effect of blow-holes on reliability of cast component

This paper presents a study on the effect of blow-holes on the reliability of a cast component. The most probable point (MPP) based univariate response surface approximation is used for evaluating reliability. Crack geometry, blow-hole dimensions, external loads and material properties are treated as independent random variables. The methodology involves novel function decomposition at a most probable point that facilitates the MPP-based univariate response surface approximation of the original multivariate implicit limit state/performance function in the rotated Gaussian space. Once the approximate form of the original implicit limit state/performance function is defined, the failure probability can be obtained by Monte Carlo simulation (MCS), importance sampling technique, and first- and second-order reliability methods (FORM/SORM). FORTRAN code is developed to automate calls to ABAQUS for numerically simulating responses at sample points, to construct univariate response surface approximation, and to subsequently evaluate the failure probability by MCS, importance sampling technique, and FORM/SORM.     

Estimation of cable safety factors of suspension bridges using artificial neural network‐based inverse reliability method

The design of the main cables of suspension bridges is based on the verification of the rules defined by standard specifications, where cable safety factors are introduced to ensure safety. However, the current bridge design standards have been developed to ensure structural safety by defining a target reliability index. In other words, the structural reliability level is specified as a target to be satisfied by the designer. Thus, calibration of cable safety factors is needed to guarantee the specified reliability of main cables. This study proposes an efficient and accurate algorithm to solve the calibration problem of cable safety factors of suspension bridges. Uncertainties of the structure and load parameters are incorporated in the calculation model. The proposed algorithm integrates the concepts of the inverse reliability method, non‐linear finite element method, and artificial neural networks method. The accuracy and efficiency of this method with reference to an example long‐span suspension bridge are studied and numerical results have validated its superiority over the conventional deterministic method or inverse reliability method with Gimsing's simplified approach. Finally, some important parameters in the proposed method are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

A Structural Reliability Analysis Method Based on Radial Basis Function

The first-order reliability method (FORM) is one of the most widely used structural reliability analysis techniques due to its simplicity and efficiency. However, direct using FORM seems disability to work well for complex problems, especially related to high-dimensional variables and computation intensive numerical models. To expand the applicability of the FORM for more practical engineering problems, a response surface (RS) approach based FORM is proposed for structural reliability analysis. The radial basis function (RBF) is employed to approximate the implicit limit-state functions combined with Latin Hypercube Sampling (LHS) strategy. To guarantee the numerical stability, the improved HL-RF (iHL-RF) algorithm is used to assess the reliability index and corresponding probability of failure based on the constructed RS model. The effectiveness of the proposed method is demonstrated through five numerical examples.     

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.     

Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

Structural reliability assessment by salp swarm algorithm–based FORM

The first-order reliability method (FORM) is well recognized as an efficient approach for reliability analysis. Rooted in considering the reliability problem as a constrained optimization of a function, the traditional FORM makes use of gradient-based optimization techniques to solve it. However, the gradient-based optimization techniques may result in local convergence or even divergence for the highly nonlinear or high-dimensional performance function. In this paper, a hybrid method combining the Salp Swarm Algorithm (SSA) and FORM is presented. In the proposed method, a Lagrangian objective function is constructed by the exterior penalty function method to facilitate meta-heuristic optimization strategies. Then, SSA with strong global optimization ability for highly nonlinear and high-dimensional problems is utilized to solve the Lagrangian objective function. In this regard, the proposed SSA-FORM is able to overcome the limitations of FORM including local convergence and divergence. Finally, the accuracy and efficiency of the proposed SSA-FORM are compared with two gradient-based FORMs and several heuristic-based FORMs through eight numerical examples. The results show that the proposed SSA-FORM can be generally applied for reliability analysis involving low-dimensional, high-dimensional, and implicit performance functions.     

Enhanced high‐dimensional model representation for reliability analysis

This paper presents a new and alternative computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry based on high‐dimensional model representation (HDMR) generated from low‐order function components. HDMR is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher‐order variable correlations are weak, allowing the physical model to be captured by the lower‐order terms and facilitating lower‐dimensional approximation of the original high‐dimensional implicit limit state/performance function. When first‐order HDMR approximation of the original high‐dimensional implicit limit state/performance function is not adequate to provide the desired accuracy to the predicted failure probability, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of second‐ and higher‐order terms. The mathematical foundation of eHDMR is presented along with its applicability to approximate the original high‐dimensional implicit limit state/performance function for subsequent reliability analysis, given that conventional methods for reliability analysis are computationally demanding when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently the eHDMR approximation technique can capture complex model output uncertainty. The limit state/performance function surrogate is constructed using moving least‐squares interpolation formula by component functions of eHDMR expansion. Once the approximate form of implicit response function is defined, the failure probability can be obtained by statistical simulation. Results of five numerical examples involving elementary mathematical functions and structural/solid‐mechanics problems indicate that the failure probability obtained using the eHDMR approximation method for implicit limit state/performance function, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations. Copyright © 2008 John Wiley & Sons, Ltd.     

System‐level reliability sensitivity analysis by using weighted average simulation method

In addition to reliability analysis, investigation of the uncertainties' effect on the safety of structures can be regarded as one of the great concerns in engineering fields. The present study provides an efficient perturbation‐based reliability sensitivity analysis approach based on weighted average simulation method (WASM) to attain uncertainties effects on the structures safety. Without asking additional samplings and/or requiring to function derivation (that is necessary in score function method), the proposed approach simultaneously uses the finite difference and weight flexibility feature of WASM to estimate the parameter sensitivities of a reliability problem. The proposed method has also been successfully applied to the improved version of WASM to obtain reliability based sensitivity results with very few samples. The accuracy and efficiency of the proposed method is examined by solving five analytical and engineering examples with highly nonlinear performance functions and system‐level reliability problems. For each example, results are compared with those obtained by Monte Carlo simulation and common reliability methods. It is shown that the method is capable of solving real world system‐level engineering problems efficiently and accurately.     

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.     

40 years FORM: Some new aspects?

40 years ago Hasofer and Lind wrote their seminal paper [13] about FORM where they described an algorithm for finding the beta point. This algorithm, later in 1978 generalized by Rackwitz and Fiessler in [23] to include nonnormal random variables, is known as Hasofer-Lind–Rackwitz-Fiessler (HL–RF) algorithm and till now it is an important tool for reliability calculations. Here its relation with standard numerical optimization is explained. Further a simple method for computing the SORM factor is given and the connection of FORM/SORM with dimension reduction concepts is outlined.     

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.