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Time-variant reliability-based design optimization (tRBDO) can rationally consider the time-variant uncertainties in engineering structures and find the optimal design that can keep reliable throughout its whole life cycle. However, solving the tRBDO involves a nested double-loop procedure and requires excessive computational cost. In this paper, a novel decoupled method called sequential approximate time-variant reliability analysis and optimization (SATO) is proposed to improve the efficiency of tRBDO. First, a two-step method is proposed to transform the original tRBDO problem into an equivalent deterministic optimization problem according to the results of time-variant reliability analysis (TRA). Second, a novel approximate TRA (ATRA) method based the least-square method is proposed to reduce the computational cost of TRA. Finally, the proposed SATO method decouples the original double-loop procedure in tRBDO into a sequential process of ATRA and deterministic optimization. Test results of a complicated welded beam problem verify that the proposed method can achieve similar accuracy and much higher efficiency than the compared methods. A rocket inter-stage structure problem demonstrates the capability of the proposed method in practical engineering applications.

Efficient aerodynamic design optimization method is of great value for improving the aerodynamic performance of little UAV's airfoil. Using engineering or semi-engineering estimation method to analyze aerodynamic forces in solving aerodynamic optimization problems costs little computational time, but the accuracy cannot be guaranteed. However, CFD method ensuring high accuracy needs much more computational cost, which is unfordable for optimization. Surrogate-based optimization can reduce the number of high-fidelity analyses to increase the optimization efficiency. However, the cost of CFD analyses is still huge for aerodynamic optimization due to multiple design variables, multi-optimal and strong nonlinearities. To solve this problem, a two-stage aerodynamic optimization method based on early termination of CFD convergence and variable-fidelity model is proposed. In the first optimization stage, the solutions by early termination CFD convergence and the convergenced CFD solutions are regarded as low-and high-fidelity data respectively for building variable-fidelity model. Then, the multi-island genetic algorithm is used in the global optimization based on the built variable-fidelity model. The modeling efficiency can be greatly improved due to many cheap low-fidelity data. In the second stage optimization, the global optimum from the first optimization stage is treated as the start of the Hooke-Jeeves algorithm to search locally based on convergenced CFD computations in order to acquire better-optimum. The proposed method is utilized in optimizing the aerodynamic performance of the airfoil of little UAV, and is compared with the EGO method based on single-fidelity Kriging surrogate model. The results show that the present two-level aerodynamic optimization method consumes less time.

Time-variant reliability analysis (TRA) is widely utilized to assess the performance of engineering structures under various time-variant uncertainties. Recently, the time discretization-based TRA (TDTRA) methods have been developed, which can achieve satisfactory accuracy but need to excessively perform most probable point (MPP) searches at many equidistant time instants. To improve the efficiency of TDTRA, this paper proposes a TRA method based on approximating the MPP trajectory, referred to as AMPPT. First, this paper introduces a new concept of the MPP trajectory (MPPT), which is defined as the moving path of the MPP in the U-space when time changes. Then, a one-dimensional Kriging model is constructed to approximate the MPPT by the adaptive sampling method, which only performs MPP searches at several critical time instants. To further improve the computational efficiency, a warm-starting strategy is proposed to accelerate the MPP search. Then, the approximated MPPT is employed to transform the time-variant response into an equivalent Gaussian process. Finally, the spectral decomposition method and Monte Carlo simulation are used to compute the time-variant reliability. Comparative studies on four numerical examples and one practical engineering example of the solid rocket engine shell verify that the proposed AMPPT outperforms TDTRA in terms of both accuracy and efficiency. Test results also indicate that the efficiency gain of the proposed AMPPT comes from not only the reduction in the number of MPP searches but also the acceleration of the MPP search itself.

The Radial Basis Function (RBF) network is one of the most widely used surrogate models in multidisciplinary design optimization. However, one of the challenges of applying the RBF network to engineering problems is how to efficiently optimize its width parameters. In this work, a novel space division–based width optimization (SDWO) method is proposed to decompose the complex multivariable width optimization into several small-scale single-variable width optimizations. The SDWO method consists of two main steps. First, a two-stage fuzzy clustering algorithm is carried out to group the samples and divide the input space into several overlapping local regions, and the overlapping degree is controlled by a dimensionless expansion factor. Second, in each local region, one small-scale local RBF network (LRBFN) is constructed by solving a single-variable optimization problem when the LRBFN is needed for prediction. All these LRBFNs are independent of each other and can be constructed in parallel. The proposed method is efficient and particularly suitable for large sample sets. Test results of four sample sets verify that the proposed SDWO method has better performance than the conventional width optimization method in terms of both training efficiency and approximation accuracy. An inter-stage structure optimization is carried out, which demonstrates the efficiency of the proposed method in practical engineering applications.

Surrogate-Based Optimization (SBO) is becoming increasingly popular since it can remarkably reduce the computational cost for design optimizations based on high-fidelity and expensive numerical analyses. However, for complicated optimization problems with a large design space, many design variables, and strong nonlinearity, SBO converges slowly and shows imperfection in local exploitation. This paper proposes a trust region method within the framework of an SBO process based on the Kriging model. In each refinement cycle, new samples are selected by a certain design of experiment method within a variable design space, which is sequentially updated by the trust region method. A multi-dimensional trust-region radius is proposed to improve the adaptability of the developed methodology. Further, the scale factor and the limit factor of the trust region are studied to evaluate their effects on the optimization process. Thereafter, different SBO methods using error-based exploration, prediction-based exploitation, refinement based on the expected improvement function, a hybrid refinement strategy, and the developed trust-region-based refinement are utilized in four analytical tests. Further, the developed optimization methodology is employed in the drag minimization of an RAE2822 airfoil. Results indicate that it has better robustness and local exploitation capability in comparison with those of other SBO methods. Keywords: Airfoil, Design optimization, Kriging model, Surrogate-based optimization, Trust-region method

In engineering design, the performance of the system and the budget of design uncertainty should be balanced, which means that it is best to optimize design variables and allocate the manufacturing uncertainty simultaneously. This work formulates this problem as an uncertainty optimization problem, where the input uncertainty is modeled by the probability method and both the design variables and the uncertainty magnitude are included in the optimization variables. A sequence optimization framework is proposed to solve the optimization problem. The Taylor-based first-order method is used to translate the probability constraint into a deterministic constraint. A correction coefficient is calculated by the dimensional adaptive polynomial chaos expansion method to improve the accuracy of the uncertainty analysis. The constraint translation and the correction coefficient calculation are executed sequentially. The accuracy and effectiveness of the proposed framework are validated by three benchmark problems, including a mathematical problem, a cantilever I-beam, and a ten-bar truss case.

Morphing aircraft can meet requirements of multi-mission during the whole flight due to changing the aerodynamic shape, so it is necessary to study its morphing rules along the trajectory. However, trajectory planning considering morphing variables requires a huge number of expensive CFD computations due to the morphing in view of aerodynamic performance. Under the given missions and trajectory, to alleviate computational cost and improve trajectory-planning efficiency for morphing aircraft, an offline optimization method is proposed based on Multi-Fidelity Kriging (MFK) modeling. The angle of attack, Mach number, sweep angle and axial position of the morphing wing are defined as variables for generating training data for building the MFK models, in which many inviscid aerodynamic solutions are used as low-fidelity data, while the less high-fidelity data are obtained by solving viscous flow. Then the built MFK models of the lift, drag and pressure centre at the different angles of attack and Mach numbers are used to predict the aerodynamic performance of the morphing aircraft, which keeps the optimal sweep angle and axial position of the wing during trajectory planning. Hence, the morphing rules can be correspondingly acquired along the trajectory, as well as keep the aircraft with the best aerodynamic performance during the whole task. The trajectory planning of a morphing aircraft was performed with the optimal aerodynamic performance based on the MFK models, built by only using 240 low-fidelity data and 110 high-fidelity data. The results indicate that a complex trajectory can take advantage of morphing rules in keeping good aerodynamic performance, and the proposed method is more efficient than trajectory optimization by reducing 86% of the computing time.

To improve the efficiency of solving uncertainty design optimization problems, a gradient-based optimization framework is herein proposed that combines the dimension adaptive polynomial chaos expansion (PCE) and sensitivity analysis. The dimension adaptive PCE is used to quantify the quantities of interest (e.g. reliability, robustness metrics) and sensitivity. The dimension adaptive property is inherited from the dimension adaptive sparse grid, which is used to evaluate the PCE coefficients. Robustness metrics, referred to as statistical moments, and their gradients with respect to design variables are easily derived from the PCE, whereas the evaluation of the reliability and its gradient require integrations. To quantify the reliability, the framework uses Heaviside step function to eliminate the failure domain and computes the integration by Monte Carlo Simulation with the function replaced by PCE. The PCE is further derived to compute the function gradient on the definition domain and is combined with Taylor’s expansion and the finite difference to compute the reliability sensitivity. Since the design vector may affect the sample set determined by dimension adaptive sparse grid, the update of the sample set is controlled by the norm variations of the design vector. The optimization framework is formed by combining reliability, robustness quantification and sensitivity analysis and the optimization module. The accuracy and efficiency of reliability quantification and the reliability sensitivity are verified through canonical examples of a mathematical example，a system of springs ,and a cantilever beam. The effectiveness of the framework for solving optimization problem is verified through a multiple limit states example, a truss optimization example with eight random variables and three design variables, and an airfoil problem with three random variables and eighteen design variables. The results demonstrate that the framework obtains accurate solutions at a manageable computational cost.

In modern engineering design optimization problems, high-fidelity analyses are always used for evaluating objectives and constraints, which might be quite expensive. Thus, efficient global optimization method should be developed to relieve the computational burden. This paper proposed a dynamic surrogate-based optimization (DSBO) using Kriging model, of which two criteria for selecting infill samples in refinement procedure are employed: maximizing expected improvement (EI) function and minimizing surrogate prediction. The DSBO are validated to be robust and efficient by six standard analytical tests. The inequality constraints are handled by three different means here: constraining EI function, penalizing surrogate prediction, and penalizing objective function. Analytical tests and an engineering optimization problem with inequality constraints are carried out. The results indicate that simultaneous constraining EI function and penalizing surrogate prediction is most efficient for DSBO, and there is no need of adjusting penalty factor.