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.     

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.     

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 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.     

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.     

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.     

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.     

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.     

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 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.     

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.     

Conservative reliability-based design optimization method with insufficient input data

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.     

Application of maximum entropy principle for reliability-based design optimization

The maximum entropy principle (MEP) is used to generate a natural probability distribution among the many possible that have the same moment conditions. The MEP can accommodate higher order moment information and therefore facilitate a higher quality PDF model. The performance of the MEP for PDF estimation is studied by using more than four moments. For the case with four moments, the results are compared with those by the Pearson system. It is observed that as accommodating higher order moment, the estimated PDF converges to the original one. A sensitivity analysis formulation of the failure probability based on the MEP is derived for reliability-based design optimization (RBDO) and the accuracy is compared with that by finite difference method (FDM). Two RBDO examples including a realistic three-dimensional wing design are solved by using the derived sensitivity formula and the MEP-based moment method. The results are compared with other methods such as TR-SQP, FAMM + Pearson system, FFMM + Pearson system in terms of accuracy and efficiency. It is also shown that an improvement in the accuracy by including more moment terms can increase numerical efficiency of optimization for the three-dimensional wing design. The moment method equipped with the MEP is found flexible and well adoptable for reliability analysis and design.     

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.     

Reliability-based and deterministic design optimization of a FSAE brake pedal: a risk allocation analysis

While probabilistic designs can translate into significant weight savings through better risk allocation, deterministic design optimization remains widely used in industry. To promote the use of probabilistic designs among engineering students and practitioners, this work solves reliability based design optimization (RBDO) and deterministic design optimization (DDO) models of a FSAE brake pedal with multiple failure modes (stress and buckling) with their relative performance evaluated through a risk allocation analysis. The problems of interest were systematically solved through the following steps: i) topology optimization to specify the brake-pedal shape, ii) numerical 3D brake-pedal modeling under uncertainty for stress and buckling analysis, iii) mass (M), maximum von Mises stress (S_max) and buckling load factor (f_buck) surrogate modeling, iv) global sensitivity analysis and surrogate model selection, and v) surrogate-based RBDO and DDO with risk allocation analysis. Results show that when compared to DDO with alternative safety factors, for the same probability of system failure, the RBDO brake pedal designs were significantly lighter and more robust (less mass variability).     

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.     

Reliability-based design optimization for crashworthiness of vehicle side impact

With the advent of powerful computers, vehicle safety issues have recently been addressed using computational methods of vehicle crashworthiness, resulting in reductions in cost and time for new vehicle development. Vehicle design demands multidisciplinary optimization coupled with a computational crashworthiness analysis. However, simulation-based optimization generates deterministic optimum designs, which are frequently pushed to the limits of design constraint boundaries, leaving little or no room for tolerances (uncertainty) in modeling, simulation uncertainties, and/or manufacturing imperfections. Consequently, deterministic optimum designs that are obtained without consideration of uncertainty may result in unreliable designs, indicating the need for Reliability-Based Design Optimization (RBDO).Recent development in RBDO allows evaluations of probabilistic constraints in two alternative ways: using the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). The PMA using the Hybrid Mean Value (HMV) method is shown to be robust and efficient in the RBDO process, whereas RIA yields instability for some problems. This paper presents an application of PMA and HMV for RBDO for the crashworthiness of a large-scale vehicle side impact. It is shown that the proposed RBDO approach is very effective in obtaining a reliability-based optimum design.     

Response surface single loop reliability-based design optimization with higher-order reliability assessment

Reliability-based design optimization (RBDO) aims at determination of the optimal design in the presence of uncertainty. The available Single-Loop approaches for RBDO are based on the First-Order Reliability Method (FORM) for the computation of the probability of failure, along with different approximations in order to avoid the expensive inner loop aiming at finding the Most Probable Point (MPP). However, the use of FORM in RBDO may not lead to sufficient accuracy depending on the degree of nonlinearity of the limit-state function. This is demonstrated for an extensively studied reliability-based design for vehicle crashworthiness problem solved in this paper, where all RBDO methods based on FORM strongly violates the probabilistic constraints. The Response Surface Single Loop (RSSL) method for RBDO is proposed based on the higher order probability computation for quadratic models previously presented by the authors. The RSSL-method bypasses the concept of an MPP and has high accuracy and efficiency. The method can solve problems with both constant and varying standard deviation of design variables and is particularly well suited for typical industrial applications where general quadratic response surface models can be used. If the quadratic response surface models of the deterministic constraints are valid in the whole region of interest, the method becomes a true single loop method with accuracy higher than traditional SORM. In other cases, quadratic response surface models are fitted to the deterministic constraints around the deterministic solution and the RBDO problem is solved using the proposed single loop method.     

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.     

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.