Reliability-based design optimization using a moment method and a kriging metamodel

Reliability-based design optimization (RBDO) has been used for optimizing engineering systems with uncertainties in design variables and system parameters. RBDO involves reliability analysis, which requires a large amount of computational effort, so it is important to select an efficient method for reliability analysis. Of the many methods for reliability analysis, a moment method, which is called the fourth moment method, is known to be less expensive for moderate size problems and requires neither iteration nor the computation of derivatives. Despite these advantages, previous research on RBDO has been mainly based on the first-order reliability method and relatively little attention has been paid to moment-based RBDO. This article considers difficulties in implementing the moment method into RBDO; they are solved using a kriging metamodel with an active constraint strategy. Three numerical examples are tested and the results show that the proposed method is efficient and accurate.     

A global single-loop deterministic approach for reliability-based design optimization of truss structures with continuous and discrete design variables

A single-loop deterministic method (SLDM) has previously been proposed for solving reliability-based design optimization (RBDO) problems. In SLDM, probabilistic constraints are converted to approximate deterministic constraints. Consequently, RBDO problems can be transformed into approximate deterministic optimization problems, and hence the computational cost of solving such problems is reduced significantly. However, SLDM is limited to continuous design variables, and the obtained solutions are often trapped into local extrema. To overcome these two disadvantages, a global single-loop deterministic approach is developed in this article, and then it is applied to solve the RBDO problems of truss structures with both continuous and discrete design variables. The proposed approach is a combination of SLDM and improved differential evolution (IDE). The IDE algorithm is an improved version of the original differential evolution (DE) algorithm with two improvements: a roulette wheel selection with stochastic acceptance and an elitist selection technique. These improvements are applied to the mutation and selection phases of DE to enhance its convergence rate and accuracy. To demonstrate the reliability, efficiency and applicability of the proposed method, three numerical examples are executed, and the obtained results are compared with those available in the literature.     

Enhanced modified reliability index approach for efficient and robust reliability-based design optimization

The reliability index approach (RIA) is one of the effective tools for solving the reliability-based design optimization (RBDO) probabilistic model, which models the uncertainties with probability constraints. However, its wide application in engineering is limited due to low efficiency and convergence problems. The RIA-based modified reliability index approach (MRIA) appears to be very robust and accurate than RIA but yields inefficient for the most probable point (MPP) search with highly nonlinear probabilistic constraints. In this study, an enhanced modified reliability index approach (EMRIA) is developed to improve the efficiency and robustness of searching for MPP and is utilized for RBDO. In the EMRIA, an innovative active set using rigorous inequality is applied to construct the region of exploring for MPP, where the unnecessary probabilistic constraint could be eliminated adaptively during the iterative process. Moreover, the double loop strategy (DLS) is integrated into the EMRIA to strengthen the efficiency and robustness of large-scale RBDO problems. Two numerical examples demonstrated that the EMRIA is an efficient and robust method for MPP search in comparison with current first-order reliability methods. Six RBDO problems quoted also indicate that DLS-based EMRIA has good performance to solve complex RBDO problems.     

A multiple-design-point approach for reliability-based design optimization

The application of design-point-based reliability-based design optimization (RBDO) methods is hindered by the challenge of multiple-design-point problems. In this article, to improve the commonality of design-point-based RBDO methods, a novel multiple-design-point (MDP) approach is developed. The MDP approach uses the trace of the design points from consequent reliability analysis iterations to identify whether there are multiple design points, then all of the design points are used to calculate shifting vectors for the sequential optimization and reliability assessment method, and the corresponding probabilistic constraints are moved to the feasible region along these multiple shifting vectors at the same time. With multiple shifted probabilistic constraints, the design feasibility associated with this probabilistic constraint will be satisfied. Two mathematical examples, a speed reducer design and a honeycomb crashworthiness design, are presented to validate the effectiveness of the MDP method. The results show that the MDP approach is effective for handling multiple-design-point problems.     

Reliable design space and complete single-loop reliability-based design optimization

Reliability-based design optimization (RBDO) has been intensively studied due to its significance and its conceptual and mathematical complexity. This paper proposes a new method for RBDO on the basis of the concept of reliable design space (RDS), within which any design satisfies the reliability requirements. Therefore, a RBDO problem becomes a simple, deterministic optimization problem constrained by RDS rather than its deterministic feasible space. The RDS is found in this work by using the partial derivatives at the current design point as an approximation of the derivatives at its corresponding most probable point (MPP) on the limit state function. This work completely resolves the double loop in RBDO and turns RBDO into a simple optimization problem. Well-known problems from the literature are selected to illustrate the steps of the approach and for result comparison. Discussions will also be given on the limitation of the proposed method, which is shown to be a common limitation overlooked by the research community on RBDO.     

Reliability-based design optimization using a generalized subset simulation method and posterior approximation

The evaluation of the probabilistic constraints in reliability-based design optimization (RBDO) problems has always been significant and challenging work, which strongly affects the performance of RBDO methods. This article deals with RBDO problems using a recently developed generalized subset simulation (GSS) method and a posterior approximation approach. The posterior approximation approach is used to transform all the probabilistic constraints into ordinary constraints as in deterministic optimization. The assessment of multiple failure probabilities required by the posterior approximation approach is achieved by GSS in a single run at all supporting points, which are selected by a proper experimental design scheme combining Sobol’ sequences and Bucher’s design. Sequentially, the transformed deterministic design optimization problem can be solved by optimization algorithms, for example, the sequential quadratic programming method. Three optimization problems are used to demonstrate the efficiency and accuracy of the proposed method.     

Reliability‐based design optimization with equality constraints

Equality constraints have been well studied and widely used in deterministic optimization, but they have rarely been addressed in reliability‐based design optimization (RBDO). The inclusion of an equality constraint in RBDO results in dependency among random variables. Theoretically, one random variable can be substituted in terms of remaining random variables given an equality constraint; and the equality constraint can then be eliminated. However, in practice, eliminating an equality constraint may be difficult or impossible because of complexities such as coupling, recursion, high dimensionality, non‐linearity, implicit formats, and high computational costs. The objective of this work is to develop a methodology to model equality constraints and a numerical procedure to solve a RBDO problem with equality constraints. Equality constraints are classified into demand‐based type and physics‐based type. A sequential optimization and reliability analysis strategy is used to solve RBDO with physics‐based equality constraints. The first‐order reliability method is employed for reliability analysis. The proposed method is illustrated by a mathematical example and a two‐member frame design problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

An efficient decision-making framework for hybrid metamodelling

In engineering design optimization, the usage of hybrid metamodels (HMs) can take full advantage of the individual metamodels, and improve robustness of the predictions by reducing the impact of a poor metamodel. When there are plenty of candidates, it is difficult to make decisions on which metamodels to choose before building an HM. The decisions should simultaneously take into account of the number, accuracy and diversity of the selected metamodels. To address this problem, this research developed an efficient decision-making framework based on partial least squares for metamodel screening. A new significance index is firstly derived from the view of fitting error in a regression model. Then, a desirable metamodel combination which consist of only the significant ones is subsequently configured for further constructing the final HM. The effectiveness of the proposed framework is demonstrated through several benchmark problems.     

The Design Space Root Finding method for efficient risk optimization by simulation

Reliability-Based Design Optimization (RBDO) is computationally expensive due to the nested optimization and reliability loops. Several shortcuts have been proposed in the literature to solve RBDO problems. However, these shortcuts only apply when failure probability is a design constraint. When failure probabilities are incorporated in the objective function, such as in total life-cycle cost or risk optimization, no shortcuts were available to this date, to the best of the authors knowledge. In this paper, a novel method is proposed for the solution of risk optimization problems. Risk optimization allows one to address the apparently conflicting goals of safety and economy in structural design. In the conventional solution of risk optimization by Monte Carlo simulation, information concerning limit state function behavior over the design space is usually disregarded. The method proposed herein consists in finding the roots of the limit state function in the design space, for all Monte Carlo samples of random variables. The proposed method is compared to the usual method in application to one and n-dimensional optimization problems, considering various degrees of limit state and cost function nonlinearities. Results show that the proposed method is almost twenty times more efficient than the usual method, when applied to one-dimensional problems. Efficiency is reduced for higher dimensional problems, but the proposed method is still at least two times more efficient than the usual method for twenty design variables. As the efficiency of the proposed method for higher-dimensional problems is directly related to derivative evaluations, further investigation is necessary to improve its efficiency in application to multi-dimensional problems.     

An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method

The first order reliability method has been extensively adopted for reliability-based design optimization (RBDO), but it shows inaccuracy in calculating the failure probability with highly nonlinear performance functions. Thus, the second order reliability method is required to evaluate the reliability accurately. However, its application for RBDO is quite challenge owing to the expensive computational cost incurred by the repeated reliability evaluation and Hessian calculation of probabilistic constraints. In this article, a new improved stability transformation method is proposed to search the most probable point efficiently, and the Hessian matrix is calculated by the symmetric rank-one update. The computational capability of the proposed method is illustrated and compared to the existing RBDO approaches through three mathematical and two engineering examples. The comparison results indicate that the proposed method is very efficient and accurate, providing an alternative tool for RBDO of engineering structures.     

A system methodology for optimization design of the structural crashworthiness of a vehicle subjected to a high-speed frontal crash

The structural crashworthiness design of vehicles has become an important research direction to ensure the safety of the occupants. To effectively improve the structural safety of a vehicle in a frontal crash, a system methodology is presented in this study. The surrogate model of Online support vector regression (Online-SVR) is adopted to approximate crashworthiness criteria and different kernel functions are selected to enhance the accuracy of the model. The Online-SVR model is demonstrated to have the advantages of solving highly nonlinear problems and saving training costs, and can effectively be applied for vehicle structural crashworthiness design. By combining the non-dominated sorting genetic algorithm II and Monte Carlo simulation, both deterministic optimization and reliability-based design optimization (RBDO) are conducted. The optimization solutions are further validated by finite element analysis, which shows the effectiveness of the RBDO solution in the structural crashworthiness design process. The results demonstrate the advantages of using RBDO, resulting in not only increased energy absorption and decreased structural weight from a baseline design, but also a significant improvement in the reliability of the design.     

Sequential sampling designs based on space reduction

In the field of engineering design and optimization, metamodels are widely used to replace expensive simulation models in order to reduce computing costs. To improve the accuracy of metamodels effectively and efficiently, sequential sampling designs have been developed. In this article, a sequential sampling design using the Monte Carlo method and space reduction strategy (MCSR) is implemented and discussed in detail. The space reduction strategy not only maintains good sampling properties but also improves the efficiency of the sampling process. Furthermore, a local boundary search (LBS) algorithm is proposed to efficiently improve the performance of MCSR, which is called LBS-MCSR. Comparative results with several sequential sampling approaches from low to high dimensions indicate that the space reduction strategy generates samples with better sampling properties (and thus better metamodel accuracy) in less computing time.     

A Swarm Metaphor for Multiobjective Design Optimization

This paper presents a new optimization algorithm to solve multiobjective design optimization problems based on behavioral concepts similar to that of a real swarm. The individuals of a swarm update their flying direction through communication with their neighboring leaders with an aim to collectively attain a common goal. The success of the swarm is attributed to three fundamental processes: identification of a set of leaders, selection of a leader for information acquisition, and finally a meaningful information transfer scheme. The proposed algorithm mimics the above behavioral processes of a real swarm. The algorithm employs a multilevel sieve to generate a set of leaders, a probabilistic crowding radius-based strategy for leader selection and a simple generational operator for information transfer. Two test problems, one with a discontinuous Pareto front and the other with a multi-modal Pareto front is solved to illustrate the capabilities of the algorithm in handling mathematically complex problems. Three well-studied engineering design optimization problems (unconstrained and constrained problems with continuous and discrete variables) are solved to illustrate the efficiency and applicability of the algorithm for multiobjective design optimization. The results clearly indicate that the swarm algorithm is capable of generating an extended Pareto front, consisting of well spread Pareto points with significantly fewer function evaluations when compared to the nondominated sorting genetic algorithm (NSGA).     

Expected improvement based infill sampling for global robust optimization of constrained problems

A novel adaptive sampling scheme for efficient global robust optimization of constrained problems is proposed. The method addresses expensive to simulate black-box constrained problems affected by uncertainties for which only the bounds are known, while the probability distribution is not available. An iterative strategy for global robust optimization that adaptively samples the Kriging metamodel of the computationally expensive problem is proposed. The presented approach is tested on several benchmark problems and the average performance based on 100 runs is evaluated. The applicability of the method to engineering problems is also illustrated by applying robust optimization on an integrated photonic device affected by manufacturing uncertainties. The numerical results show consistent convergence to the global robust optimum using a limited number of expensive simulations.     

结构可靠性优化求解的解耦融合策略

在工程结构的可靠性优化过程中,求解的效率和精度是优化方法的关键。该文提出一种针对解耦优化的融合策略。所提方法在优化迭代解耦所用的失效概率函数为前几次迭代设计点构建的局部失效概率函数的加权融合形式。在对原可靠性优化问题进行解耦后,结合序列近似优化方法进行迭代求解。相比于常规的仅使用当次局部建立的失效概率函数而言,所提融合策略最大限度利用了各次迭代中产生的信息用于优化解耦求解,能够提高失效概率函数的近似精度,从而间接达到减少迭代次数和计算量的目的。最后给出了屋架和十杆结构的可靠性优化算例,验证该文方法的正确性和可行性。     

Hybrid metamodel-based design space management method for expensive problems

Metamodel-based global optimization methods have been extensively studied for their great potential in solving expensive problems. In this work, a design space management strategy is proposed to improve the accuracy and efficiency of metamodel-based optimization methods. In this strategy, the whole design space is divided into two parts: the important region constructed using several expensive points and the other region. Combined with a previously developed hybrid metamodel strategy, a hybrid metamodel-based design space management method (HMDSM) is developed. In this method, three representative metamodels are used simultaneously in the search of the global optimum in both the important region and the other region. In the search process, the important region is iteratively reduced and the global optimum is soon captured. Tests using a series of benchmark mathematical functions and a practical expensive problem demonstrate the excellent performance of the proposed method.     

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

A kriging metamodel-assisted robust optimization method based on a reverse model

The goal of robust optimization methods is to obtain a solution that is both optimum and relatively insensitive to uncertainty factors. Most existing robust optimization approaches use outer–inner nested optimization structures where a large amount of computational effort is required because the robustness of each candidate solution delivered from the outer level should be evaluated in the inner level. In this article, a kriging metamodel-assisted robust optimization method based on a reverse model (K-RMRO) is first proposed, in which the nested optimization structure is reduced into a single-loop optimization structure to ease the computational burden. Ignoring the interpolation uncertainties from kriging, K-RMRO may yield non-robust optima. Hence, an improved kriging-assisted robust optimization method based on a reverse model (IK-RMRO) is presented to take the interpolation uncertainty of kriging metamodel into consideration. In IK-RMRO, an objective switching criterion is introduced to determine whether the inner level robust optimization or the kriging metamodel replacement should be used to evaluate the robustness of design alternatives. The proposed criterion is developed according to whether or not the robust status of the individual can be changed because of the interpolation uncertainties from the kriging metamodel. Numerical and engineering cases are used to demonstrate the applicability and efficiency of the proposed approach.     

MULTIOBJECTIVE DESIGN OPTIMIZATION BY AN EVOLUTIONARY ALGORITHM

This paper presents an evolutionary algorithm for generic multiobjective design optimization problems. The algorithm is based on nondominance of solutions in the objective and constraint space and uses effective mating strategies to improve solutions that are weak in either. Since the methodology is based on nondominance, scaling and aggregation affecting conventional penalty function methods for constraint handling does not arise. The algorithm incorporates intelligent partner selection for cooperative mating. The diversification strategy is based on niching which results in a wide spread of solutions in the parametric space. Results of the algorithm for the design examples clearly illustrate the efficiency of the algorithm in solving multidisciplinary design optimization problems.