Experience with approximate reliability-based optimization methods II: an exhaust system problem

Traditional reliability-based design optimization (RBDO) requires a double-loop iteration process. The inner optimization loop is to find the reliability and the outer is the regular optimization loop to optimize the RBDO problem with reliability objectives or constraints. It is known that the computation can be prohibitive when the associated function evaluation is expensive. This situation is even worse when a large number of reliability constraints are present. As a result, many approximate RBDO methods, which convert the double loop to a single loop, have been developed. In this research, an engineering problem with a large number of constraints (144) is designed to test RBDO methods based on the first-order reliability method (FORM), including single- and double-loop methods. In addition to the number of constraints, this problem possesses many local minimums. Some original authors of the RBDO methods are also asked to solve the same problem. The results and the efficiencies for different methods are published and discussed.     

Relaxed performance measure approach for reliability-based design optimization

The efficiency and robustness of reliability analysis methods are important factors to evaluate the probabilistic constraints in reliability-based design optimization (RBDO). In this paper, a relaxed mean value (RMV) approach is proposed in order to evaluate probabilistic constraints including convex and concave functions in RBDO using the performance measure approach (PMA). A relaxed factor is adaptively determined in the range from 0 to 2 using an inequality criterion to improve the efficiency and robustness of the inverse first-order reliability methods. The performance of the proposed RMV is compared with six existing reliability methods, including the advanced mean value (AMV), conjugate mean value (CMV), hybrid mean value (HMV), chaos control (CC), modified chaos control (MCC), and conjugate gradient analysis (CGA) methods, through four nonlinear concave and convex performance functions and three RBDO problems. The results demonstrate that the proposed RMV is more robust than the AMV, CMV, and HMV for highly concave problems, and slightly more efficient than the CC, MCC, and CGA methods. Furthermore, the proposed relaxed mean value guarantees robust and efficient convergence for RBDO problems with highly nonlinear performance functions.     

A general RBDO decoupling approach for different reliability analysis methods

There are available in the literature several papers on the development of methods to decouple the reliability analysis and the structural optimization to solve RBDO problems. Most of them focused on strategies that employ the First Order Reliability Method (FORM) to approximate the reliability constraints. Despite of all these developments, one limitation prevailed: the lack of accuracy in the approximation of the reliability constraints due to the use of FORM. Thus, in this paper, a novel approach for RBDO is presented in order to overcome such a limitation. In this approach, we use the concept of shifting vectors, originally developed in the context of the Sequential Optimization and Reliability Assessment (SORA). However, the shifting vectors are found and updated based on a novel strategy. The resulting framework is able to use any technique for the reliability analysis stage, such as Monte Carlo simulation, second order reliability methods, stochastic polynomials, among others. Thus, the proposed approach overcomes the aforementioned limitation of most of RBDO decoupling techniques, which required the use of FORM for reliability analysis. Several examples are analyzed in order to show the effectiveness of the methodology. Focus is given on examples that are poorly solved or even cannot be tackled by FORM based approaches, such as highly nonlinear limit state functions comprised by a maximum operator or problems with discrete random variables. It should be remarked that the proposed approach was not developed to be more computationally efficient than RBDO decoupling strategies based FORM, but to allow the utilization of any, including more accurate, reliability analysis method.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

A cell evolution method for reliability-based design optimization

The reliability-based design optimization (RBDO) presents to be a systematic and powerful approach for process designs under uncertainties. The traditional double-loop methods for solving RBDO problems can be computationally inefficient because the inner reliability analysis loop has to be iteratively performed for each probabilistic constraint. To solve RBDOs in an alternative and more effective way, Deb et al. [1] proposed recently the use of evolutionary algorithms with an incorporated fastPMA. Since the imbedded fastPMA needs the gradient calculations and the initial guesses of the most probable points (MPPs), their proposed algorithm would encounter difficulties in dealing with non-differentiable constraints and the effectiveness could be degraded significantly as the initial guesses are far from the true MPPs. In this paper, a novel population-based evolutionary algorithm, named cell evolution method, is proposed to improve the computational efficiency and effectiveness of solving the RBDO problems. By using the proposed cell evolution method, a family of test cells is generated based on the target reliability index and with these reliability test cells the determination of the MPPs for probabilistic constraints becomes a simple parallel calculation task, without the needs of gradient calculations and any initial guesses. Having determined the MPPs, a modified real-coded genetic algorithm is applied to evolve these cells into a final one that satisfies all the constraints and has the best objective function value for the RBDO. Especially, the nucleus of the final cell contains the reliable solution to the RBDO problem. Illustrative examples are provided to demonstrate the effectiveness and applicability of the proposed cell evolution method in solving RBDOs. Simulation results reveal that the proposed cell evolution method outperforms comparative methods in both the computational efficiency and solution accuracy, especially for multi-modal RBDO problems.     

An inverse-measure-based unilevel architecture for reliability-based design optimization

Reliability-based design optimization (RBDO) is a methodology for finding optimized designs that are characterized with a low probability of failure. Primarily, RBDO consists of optimizing a merit function while satisfying reliability constraints. The reliability constraints are constraints on the probability of failure corresponding to each of the failure modes of the system or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. During the last few years, a variety of different formulations have been developed for RBDO. Traditionally, these have been formulated as a double-loop (nested) optimization problem. The upper level optimization loop generally involves optimizing a merit function subject to reliability constraints, and the lower level optimization loop(s) compute(s) the probabilities of failure corresponding to the failure mode(s) that govern(s) the system failure. This formulation is, by nature, computationally intensive. Researchers have provided sequential strategies to address this issue, where the deterministic optimization and reliability analysis are decoupled, and the process is performed iteratively until convergence is achieved. These methods, though attractive in terms of obtaining a workable reliable design at considerably reduced computational costs, often lead to premature convergence and therefore yield spurious optimal designs. In this paper, a novel unilevel formulation for RBDO is developed. In the proposed formulation, the lower level optimization (evaluation of reliability constraints in the double-loop formulation) is replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper level optimization. Such a replacement is computationally equivalent to solving the original nested optimization if the lower level optimization problem is solved by numerically satisfying the KKT conditions (which is typically the case). It is shown through the use of test problems that the proposed formulation is numerically robust (stable) and computationally efficient compared to the existing approaches for RBDO.     

Reliability-based structural design optimization: hybridized conjugate mean value approach

The efficiency and robustness of reliability techniques are important in reliability-based design optimization (RBDO). Commonly, advanced mean value (AMV) is utilized in reliability loop of RBDO but unstable solutions using AMV may be obtained for highly concave performance functions. Owing to the challenges of commonly reliability methods, the conjugate gradient analysis (CGA) is proposed as a robust methodology but it shows inefficient results for convex constraints. In this research, hybrid conjugate mean value (HCMV) method is proposed using sufficient condition for the enhancement of efficiency and robustness of RBDO. The CGA and AMV are dynamically utilized for simple solution of convex/concave constraints using sufficient descent criterion in HCMV. The HCMV is used to evaluate the convergence performances and is compared with numerous existing reliability methods through three reliability problems (two concave/convex mathematical examples and one applicable structure) and four RBDO problems. From the numerical results, the HCMV exhibited the better efficiency, and robustness compared to other studied formulations in reliability and RBDO problems.

     

An optimal shifting vector approach for efficient probabilistic design

The application of reliability-based design optimization (RBDO) is hindered by the unbearable computational cost in the structure reliability evaluating process. This study proposes an optimal shifting vector (OSV) approach to enhance the efficiency of RBDO. In OSV, the idea of using an optimal shifting vector in the decoupled method and the notation of conducting reliability analysis in the super-sphere design space are proposed. The shifted limit state function, instead of the specific performance function, is used to identify the inverse most probable point (IMPP) and derive the optimal shifting vector for accelerating the optimization process. The super-sphere design space is applied to reduce the number of constraints and design variables for the novel reliability analysis model. OSV is very efficient for highly nonlinear problems, especially when the contour lines of the performance functions vary widely. The computation capability of the proposed method is demonstrated and compared to existing RBDO methods using four mathematical and engineering examples. The comparison results show that the proposed OSV approach is very efficient.     

Reliability-based design optimization using SORM and SQP

In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.     

A decoupling approach for evidence-theory-based reliability design optimization

For the problem of evidence-theory-based reliability design optimization (EBDO), this paper presents a decoupling approach which provides an effective tool for the reliability design of some complex structures with epistemic uncertainty. The approach converts the original nested optimization into a sequential iterative process including design optimization and reliability analysis. In each iteration step, through the uniformity algorithm, the original EBDO is firstly transformed to a conventional reliability-based design optimization (RBDO) and an optimal solution is obtained by solving it. At the solution, the first-order approximate reliability analysis method (FARM) is then used to perform the evidence-theory-based reliability analysis for each constraint. In addition, the RBDO solving and the evidence-theory-based reliability analysis are carried out alternately until reaching the convergence. Finally, two numerical examples and a practical engineering application show the effectiveness of the proposed method.     

Extension of optimum safety factor method to nonlinear reliability-based design optimization

In the reliability-based design optimization (RBDO) model, the mean values of uncertain system 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, a RBDO solution that reduces the structural weight in uncritical regions does not only provide an improved design but also a higher level of confidence in the design. In this paper, we present recent developments for the RBDO model relative to two points of view: reliability and optimization. Next, we develop several distributions for the hybrid method and the optimum safety factor methods (linear and nonlinear RBDO). Finally, we demonstrate the efficiency of our safety factor approach extended to nonlinear RBDO with application to a tri-material structure.     

Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from unknown statistical distributions, known as epistemic uncertainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliability analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.     

Performance assessment of a multi-objective parametric optimization algorithm with application to a multi-physical engineering system

The original problem of reliability-based design optimization (RBDO) is mathematically a nested two-level structure that is computationally time consuming for real engineering problems. In order to overcome the computational difficulties, many formulations have been proposed in the literature. These include SORA (sequential optimization and reliability assessment) that decouples the nested problems. SLA (single loop approach) further improves efficiency in that reliability analysis becomes an integrated part of the optimization problem. However, even SLA method can become computationally challenging for real engineering problems involving many reliability constraints. This paper presents an enhanced version of SLA where the first phase is based on approximation at nominal design point. After convergence of first iterative phase is reached the process transitions to a second phase where approximations of reliability constraints are carried out at their respective minimum performance target point (MPTP). The solution is implemented in Altair OptiStruct, where adaptive approximation and constraint screening strategies are utilized in the RBDO process. Examples show that the proposed two-phase approach leads to reduction in finite element analyses while preserving equal solution quality.     

An approximate sequential optimization and reliability assessment method for reliability-based design optimization

Sequential optimization and reliability assessment (SORA) is one of the most popular decoupled approaches to solve reliability-based design optimization (RBDO) problem because of its efficiency and robustness. In SORA, the double loop structure is decoupled through a serial of cycles of deterministic optimization and reliability assessment. In each cycle, the deterministic optimization and reliability assessment are performed sequentially and the boundaries of violated constraints are shifted to the feasible direction according to the reliability information obtained in the previous cycle. In this paper, based on the concept of SORA, approximate most probable target point (MPTP) and approximate probabilistic performance measure (PPM) are adopted in reliability assessment. In each cycle, the approximate MPTP needs to be reserved, which will be used to obtain new approximate MPTP in the next cycle. There is no need to evaluate the performance function in the deterministic optimization since the approximate PPM and its sensitivity are used to formulate the linear Taylor expansion of the constraint function. One example is used to illustrate that the approximate MPTP will approach the accurate MPTP with the iteration. The design variables and the approximate MPTP converge simultaneously. Numerical results of several examples indicate the proposed method is robust and more efficient than SORA and other common RBDO methods.     

Probabilistic sensitivity analysis for novel second-order reliability method (SORM) using generalized chi-squared distribution

Reliability-based design optimization (RBDO) requires evaluation of sensitivities of probabilistic constraints. To develop RBDO utilizing the recently proposed novel second-order reliability method (SORM) that improves conventional SORM approaches in terms of accuracy, the sensitivities of the probabilistic constraints at the most probable point (MPP) are required. Thus, this study presents sensitivity analysis of the novel SORM at MPP for more accurate RBDO. During analytic derivation in this study, it is assumed that the Hessian matrix does not change due to the small change of design variables. The calculation of the sensitivity based on the analytic derivation requires evaluation of probability density function (PDF) of a linear combination of non-central chi-square variables, which is obtained by utilizing general chi-squared distribution. In terms of accuracy, the proposed probabilistic sensitivity analysis is compared with the finite difference method (FDM) using the Monte Carlo simulation (MCS) through numerical examples. The numerical examples demonstrate that the analytic sensitivity of the novel SORM agrees very well with the sensitivity obtained by FDM using MCS when a performance function is quadratic in U-space and input variables are normally distributed. It is further shown that the proposed sensitivity is accurate enough compared with FDM results even for a higher order performance function.     

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis

The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) allows us to design structures which satisfy economical and safety requirements. However, in practical applications, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak convergence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and optimization iterations. This approach leads to low numerical efficiency, which is disadvantageous for engineering applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to parallel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed methodology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).     

A stability investigation of a simulation- and reliability-based optimization

This study developed a reliability-based design optimization (RBDO) algorithm focusing on the ability of solving problems with nonlinear constraints or system reliability. In this case, a sampling technique is often adopted to evaluate the reliability analyses. However, simulation with an insufficient sample size often possesses statistical randomness resulting in an inaccurate sensitivity calculation. This may cause an unstable RBDO solution. The proposed approach used a set of deterministic variables, called auxiliary design points, to replace the random parameters. Thus, an RBDO is converted into a deterministic optimization (DO, α-problem). The DO and the analysis of finding the auxiliary design points (β-problem) are conducted iteratively until the solution converges. To maintain the stability of the RBDO solution with less computational cost, the proposed approach calculated the sensitivity of reliability (in the β-problem) with respect to the mean value of the pseudo-random parameters rather than the design variables. The stability of the proposed method was compared to that of the double-loop approach, and many factors, such as sample size, starting point and the parameters used in the optimization, were considered. The accuracy of the proposed method was confirmed using Monte Carlo simulation (MCS) with several linear and nonlinear numerical problems.     

A hybrid sufficient performance measure approach to improve robustness and efficiency of reliability-based design optimization

The stable convergence and efficiency of reliability-based design optimization (RBDO) using performance measure approach (PMA) are the major issue to develop the reliability methods based on modified chaos control (MCC), hybrid chaos control (HCC) and finite-step length adjustment (FSL). However, these methods may be inefficient for RBDO problems with convex and concave probabilistic constraints. In this paper, an adaptive modified chaos control (AMC) is proposed to provide the robust and efficient results in RBDO. The proposed AMC is adjusted using dynamical chaos control factor, which is extracted using sufficient descent condition for PMA. Using sufficient criterion, the proposed AMC is adaptively combined with advanced mean value (AMV) to improve the performance of PMA, named as hybrid adaptive modified chaos control (HAMC). Considering the robustness and efficiency, the proposed HAMC is compared with several existing reliability methods by three nonlinear structural/mathematical performance functions and two RBDO problems. The results indicate that the proposed HAMC with sufficient descent condition provides superior convergences in terms of both robustness and efficiency, compared to existing PMA methods using AMV, MCC, HCC and FSL.