Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.     

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.     

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.

     

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.     

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.     

Convergence control of single loop approach for reliability-based design optimization

For solution of reliability-based design optimization (RBDO) problems, single loop approach (SLA) shows high efficiency. Thus SLA is extensively used in RBDO. However, the iteration solution procedure by SLA is often oscillatory and non-convergent for RBDO with nonlinear performance function. This prevents the application of SLA to engineering design problems. In this paper, the chaotic single loop approach (CLSA) is proposed to achieve the convergence control of original iterative algorithm in SLA. The modification involves automated selection of the chaos control factor by solving a novel one-dimensional optimization model. Additionally, a new oscillation-checking method is constructed to detect the oscillation of iterative process of design variables. The computational capability of CLSA is demonstrated through five benchmark examples and one stiffened shell application. The comparison of numerical results indicates that CSLA is more efficient than the double loop approach and the decoupled approach. CSLA also solves the RBDO problems with highly nonlinear performance function and non-normal random variables stably.     

A local adaptive sampling method for reliability-based design optimization using Kriging model

Reliability-based design optimization (RBDO) in practical applications is hindered by its huge computational cost during structure reliability evaluating process. Kriging-model-based RBDO is an effective method to overcome this difficulty. However, the accuracy of Kriging model depends directly on how to select the sample points. In this paper, the local adaptive sampling (LAS) is proposed to enhance the efficiency of constructing Kriging models for RBDO problems. In LAS, after initialization, new samples for probabilistic constraints are mainly selected within the local region around the current design point from each optimization iteration, and in the local sampling region, sample points are first considered to be located on the limit state constraint boundaries. The size of the LAS region is adaptively defined according to the nonlinearity of the performance functions. The computation capability of the proposed method is demonstrated using three mathematical RBDO problems and a honeycomb crash-worthiness design application. The comparison results show that the proposed method is very efficient.     

Single loop single vector approach using the conjugate gradient in reliability based design optimization

Conventional reliability-based design optimization (RBDO) approaches require high computing costs. Among the existing RBDO methods, the single loop single vector approach (SLSV) converts the RBDO problem into a single loop deterministic optimization. Hence, it can efficiently reduce the design cost compared to other methods. However, this method has a weakness in that instability or inaccuracy in convergence can be increased according to the problem characteristics. It often happens when the performance function is highly nonlinear or concave. In this study, a novel method is proposed to overcome the problems. It is an SLSV method using the conjugate gradient that is calculated with the gradient directions at the most probable points (MPP) of the previous cycles. Mathematical examples and structural applications are solved to verify the proposed method. The numerical performances of the proposed method are compared with other RBDO methods such as the RIA, PMA, SORA and SLSV approaches. It is shown that the SLSV method using the conjugate gradient (SLSVCG) is not greatly influenced by problem characteristics and the convergence capability is quite superior. Also, the computational cost of the proposed method is significantly reduced and an excellent solution satisfying the specified reliability is obtained.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

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.     

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.     

Application of reliability-based optimization to earth-moving machine: hydraulic cylinder components design process

Reliability-based design optimization (RBDO) incorporates probabilistic analysis into optimization process so that an optimum design has a great chance of staying in the feasible design space when the inevitable variability in design variables/parameters is considered. One of the biggest drawbacks of applying RBDO to practical problem is its high computational cost that is often impractical to industries. In search of the most suitable RBDO method for industrial applications, we first evaluated several existing RBDO approaches in details such as the double-loop RBDO, the sequential optimization and reliability assessment, and the response surface method. Then, based on industry needs, a platform incorporating/integrating the existing algorithm of optimization and reliability analysis is built for a practical RBDO problem. Effectiveness of the proposed RBDO approach is demonstrated using a simple cantilever beam problem and a more complicated industry problem.     

Reliability based design optimization of wire bonding in power microelectronic devices

This paper presents a numerical investigation of the probabilistic approach in estimating the reliability of wire bonding, and develops a reliability-based design optimization Methodology (RBDO) for microelectronic device structures. The objective of the RBDO method is to design structures which should be both economical and reliable where the solution reduces the structural weight in uncritical regions. It does not only provide an improved design, but also a higher level of confidence in the design. The Finite element simulation model intends to analyze the sequence of the failure events in power microelectronic devices. This numerical model is used to estimate the probability of failure of power module regarding the wire bonding connection. However, due to time-consuming of the multiphysics finite element simulation, a response surface method is used to approximate the response output of the limit state, in this way the reliability analysis is performed directly to the response surface by using the First and the Second Order Reliability Methods FORM/SORM. Subsequently the reliability analysis is integrated in the optimization process to improve the performance and reliability of structural design of wire bonding. The sequential RBDO algorithm is used to solve this problem and to find the best structural designs which realize the best compromise between cost and safety.     

An enhanced simulation-based design method coupled with meta-heuristic search algorithm for accurate reliability-based design optimization

The enhanced weighted simulation-based design method in conjunction with particle swarm optimization (PSO) is developed as a pseudo double-loop algorithm for accurate reliability-based design optimization (RBDO). According to this hybrid method, generated samples of weighed simulation method (WSM) are considered as initial population of the PSO. The proposed population is then employed to evaluate the safety level of each PSO swarm (design candidates) during movement. Using this strategy, there is no required to conduct new sampling for reliability assessment of design candidates (PSO swarms). Employing PSO as the search engine of RBDO and WSM as the reliability analyzer provide more accurate results with few samples and also increase the application range of traditional WSM. Besides, a shift strategy is also introduced to increase the capability of the WSM to investigate general RBDO problems including both deterministic and random design variables. Several examples are investigated to demonstrate the accuracy and robustness of the method. Results demonstrate the computational efficiency and superiority of the proposed method for practical engineering problems with highly nonlinear and implicit probabilistic constrains.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.     

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.     

Improved hybrid method as a robust tool for reliability-based design optimization

In the engineering problems, the randomness and the uncertainties of the distribution of the structural parameters are a crucial problem. In the case of reliability-based design optimization (RBDO), it is the objective to play a dominant role in the structural optimization problem introducing the reliability concept. The RBDO problem is often formulated as a minimization of the initial structural cost under constraints imposed on the values of elemental reliability indices corresponding to various limit states. The classical RBDO leads to high computing time and weak convergence, but a Hybrid Method (HM) has been proposed to overcome these two drawbacks. As the hybrid method successfully reduces the computing time, we can increase the number of variables by introducing the standard deviations as optimization variables to minimize the error values in the probabilistic model. The efficiency of the hybrid method has been demonstrated on static and dynamic cases with extension to the variability of the probabilistic model. In this paper, we propose a modification on the formulation of the hybrid method to improve the optimal solutions. The proposed method is called, Improved Hybrid Method (IHM). The main benefit of this method is to improve the structure performance by much more minimizing the objective function than the hybrid method. It is also shown to demonstrate the optimality conditions. The improved hybrid method is next applied to two numerical examples, with consideration of the standard deviations as optimization variables (for linear and nonlinear distributions). When integrating the improved hybrid method within the probabilistic model variability, we minimize the objective function more and more.     

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.

     

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.     

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.