Efficient decoupling-assisted evolutionary/metaheuristic framework for expensive reliability-based design optimization problems

A Broyden–Fletcher–Goldfarb–Shanno algorithm for reliability-based design optimization

A novel reliability-based design optimization (RBDO) method using simulation-based techniques for reliability assessments and efficient optimization approach is presented in this paper. In RBDO, model-based reliability analysis needs to be performed to calculate the probability of not satisfying a reliability constraint and the gradient of this probability with respect to each design variable. Among model-based methods, the most widely used in RBDO is the first-order reliability method (FORM). However, FORM could be inaccurate for nonlinear problems and is not applicable for system reliability problems. This paper develops an efficient optimization methodology to perform RBDO using simulation-based techniques. By combining analytical and simulation-based reliability methods, accurate probability of failure and sensitivity information is obtained. The use of simulation also enables both component and system-level reliabilities to be included in RBDO formulation. Instead of using a traditional RBDO formulation in which optimization and reliability computations are nested, a sequential approach is developed to greatly reduce the computational cost. The efficiency of the proposed RBDO approach is enhanced by using a multi-modal adaptive importance sampling technique for simulation-based reliability assessment; and by treating the inactive reliability constraints properly in optimization. A vehicle side impact problem is used to demonstrate the capabilities of the proposed method.

<div class="htmlview paragraph">This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the hybrid mean value (HMV) first-order method for <i>ε</i>-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO while maintaining numerical accuracy and stability. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, several numerical examples are provided, including a large-scale multi-crash application.</div>

Reliability-based design optimization (RBDO) of mechanical systems is computationally intensive due to the presence of two types of iterative procedures – design optimization and reliability estimation. Single-loop RBDO algorithms offer tremendous savings in computational effort, but they have so far only been able to consider individual component reliability constraints. This paper presents a single-loop RBDO formulation and an equivalent formulation that can also include system-level reliability constraints. The formulations allow the allocation of optimal reliability levels to individual component limit states in order to satisfy both system-level and component-level reliability requirements. Four solution algorithms to implement the second, more efficient formulation are developed. A key feature of these algorithms is to remove the most probable points from the decision space, thus avoiding the need to calculate Hessians or gradients of limit state gradients. It is shown that with the proposed methods, systems-level RBDO can be accomplished with computational expense equivalent to several cycles of computationally inexpensive singleloop RBDO based on second-moment methods. Examples of this new approach applied to series, parallel, and combined systems are provided. Nomenclature

The application of reliability-based design optimization (RBDO) to degrading systems is challenging because of the continual interplay between calculating time-variant reliability (to ensure reliability policies are met) and moving the design point to optimize various objectives, such as cost, weight, size and so forth. The time needed for Monte Carlo Simulation (MCS) is lengthy when reliability calculations are required for each iteration of the design point. The common methods used to date to improve efficiency have some shortcomings: First, most approaches approximate probability via a method that invokes the most-likely failure point (MLFP), and second, tolerances are almost always excluded from the list of design parameters (hence only so-called parameter design is performed), and, without tolerances, true monetary costs cannot be determined, especially in manufactured systems. Herein, the efficiency of RBDO for degrading systems is greatly improved by essentially uncoupling the time-variant reliability problem from the optimization problem. First, a meta-model is built to relate time-variant reliability to the design space. Design of experiment techniques helps to select a few judicious training sets. Second, the meta-model is accessed to quickly evaluate objectives and reliability constraints in the optimization process. The set-theory approach (with MCS) is invoked to find the system reliability accurately and efficiently for multiple competing performance measures. For a case study, the seminal roller clutch with degradation due to wear is examined. The meta-model method, using both moving least-squares and kriging (using DACE in Matlab), is compared to the traditional approach whereby reliability is determined by MCS at each optimization iteration. The case study shows that both means and tolerances are found that correctly minimize a monetary cost objective and yet ensure that reliability policies are met. The meta-model approach is simple, accurate and very fast, suggesting an attractive means for RBDO of time-variant systems.

Reliability-based design optimization of volume fraction distribution in functionally graded composites

SummaryIn 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.

The increasing use of surrogate models has widened the range of application of classical reliability-based design optimization (RBDO) techniques to industrial problems. In this paper, we consider such an approach to the lightweight design of an automotive body structure. Solving this problem while approximating the complex models (nonlinear, noisy and high-dimensional) with a single metamodel would require a very large and non affordable design of experiments (DOE). We thus investigate and propose a methodology of adaptive Kriging based RBDO where an initial DOE is iteratively updated so as to improve the Kriging models only in regions that actually matter. The nested reliability analysis is expressed in terms of quantiles assessment. Two stages of enrichment are performed. The first one seeks to gradually improve the accuracy of the metamodels where the probabilistic constraints are likely to be violated. The second one is embedded in an evolution strategy optimization scheme where, at each iteration, the accuracy of the quantile estimation is improved if necessary. The methodology is applied on an analytical and crashworthiness design problems showing good performance by enhancing accuracy and efficiency with respect to a traditional approach.

An efficient, fully decoupled approach is proposed to solve the structural reliability-based design optimization (RBDO) problem. The proposed approach utilizes augmented space integral and importance sampling (ASI-IS) to efficiently evaluate a functional relationship between the probability of failure and the design parameters, namely the so-called failure probability function. ASI-IS allows for the avoidance of both the intractable density fitting task associated with augmented space methods and the time-consuming repeated reliability evaluations associated with surrogate model approaches. The resulting functional relationship can be used to completely decouple the original RBDO problem into a deterministic one. Then, an iteration mechanism is constructed with gradually smaller design domains to enhance the efficiency of the optimization process. Furthermore, a sample reuse algorithm is proposed to improve the performance of the proposed approach by collecting the samples generated in previous iterations and reusing them in the current iteration in order to produce a better estimator of the failure probability function. Numerical and engineering examples, including a turbine blade and an aircraft inner flap, are given to demonstrate the efficiency and feasibility of the proposed approach.

Reliability-based design optimisation of structural systems using high-order analytical moments

A stable Reliability-Based Design Optimization (RBDO) algorithm is proposed to produce an optimal design with a desired reliability. The main idea behind the proposed approach is to use a set of deterministic variables, called auxiliary design points, to replace the random parameters. Thus, the reliability analysis in the inner loop of an RBDO problem is relaxed. The auxiliary design points are found through an optimization procedure. The auxiliary design points are updated using the sums of the auxiliary design points and the differences between the mean values of the “pseudo” and actual random parameters in the previous step. The auxiliary design points can force the iteration of deterministic optimization to the vicinity of the probabilistic boundary. Note that in the proposed method, the coupled reliability analysis and deterministic optimization are decomposed to form a weakly coupled system. The stability and accuracy of the proposed method were investigated through linear and nonlinear numerical problems.

While reliability-based design optimization (RBDO) significantly improves engineering safety, computational expenses and accuracy concerns restrict its use in complex, high-dimensional, nonlinear, and black-box scenarios. To address this challenge, a Kriging-AMV-MCS reliability optimization method (KAMRO) is proposed in this paper. The algorithm integrates the Kriging surrogate model, an improved advanced mean value (AMV) method, and a Monte-Carlo simulation (MCS) validation mechanism, aiming to achieve efficient and robust design optimization. First, through an adaptive sampling strategy based on the expected feasibility function and space filling criteria, actively constrained regions are intelligently identified and sample distribution is optimized. Second, a two-stage optimization framework is adopted. In the first stage, the penalty function is used to quickly approximate the feasible domain, and in the second stage, explicit constraints are combined for precise optimization. Finally, an MCS-guided post-optimization process is proposed to further optimize and improve the design scheme when the reliability is insufficient. Through validation with engineering cases, including a two-dimensional analytical example, a seven-dimensional gear reducer structural optimization example, and a three-dimensional pressure control head structure design, this RBDO framework significantly reduces computational costs while ensuring design reliability, offering both optimization accuracy and engineering practicality.

A unified approach for time-invariant and time-variant reliability-based design optimization with multiple most probable points

<div class="htmlview paragraph">An efficient approach for series system reliability-based design optimization (RBDO) is presented. The key idea is to apportion optimally the system reliability among the failure modes by considering the target values of the failure probabilities of the modes as design variables. Critical failure modes that contribute the most to the overall system reliability are identified. This paper proposes a computationally efficient, system RBDO approach using a single-loop method where the searches for the optimum design and for the most probable failure points proceed simultaneously. Specifically, at each iteration the optimizer uses approximated most probable failure points from the previous iteration to search for the optimum. A second-order Ditlevsen upper bound is used for the joint failure probability of failure modes. Also, an easy to implement active strategy set is employed to improve algorithmic stability. The approach is demonstrated on two design examples involving a beam and, an internal combustion engine.</div>

To improve the efficiency and quality of solving expensive engineering design optimization problems, recently metamodel-based design and optimization technologies have been widely employed. In this paper, a novel adaptive metamodel-based optimization method called sequential radial basis function using support vector machine is proposed to solve the practical engineering optimization problems involving computationally expensive objective and constraints. The proposed metamodel-based optimization method uses the radial basis function metamodel to approximate the expensive simulations in the interesting sampling region, which is a subregion of the design space. To identify the interesting sampling region, a novel feature space fuzzy -means clustering method is proposed to obtain the cluster center of superior cheap points classified by support vector machine. The procedure of sequential radial basis function using support vector machine is presented first, and the interesting sampling region and feature space fuzzy -means clustering methods are discussed in detail. Several numerical benchmark problems are used to compare the proposed method with other well-known metamodel-based design and optimization methods including efficient global optimization, multiple surrogate efficient global optimization methodology, mode-pursuing sampling, constraint importance mode pursuing sampling, and constrained optimization by radial basis function interpolation. The comparison results show that the proposed method generally outperforms the competitive methods in terms of efficiency, convergence, and robustness. Finally, sequential radial basis function using support vector machine is applied in an Earth observation satellite multidisciplinary design optimization problem. The results illustrate that compared with genetic algorithm and sequential quadratic programming, the proposed method can find a better solution with fewer function evaluations, which demonstrates the practicality and effectiveness of the proposed method in solving real-world engineering design optimization problems.