ADAPTIVE MULTI-FIDELITY POLYNOMIAL CHAOS-KRIGING MODEL-BASED EFFICIENT AERODYNAMIC DESIGN OPTIMIZATION METHOD
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摘要: 多可信度代理模型已经成为提高基于代理模型的优化算法效率和可信度水平最有效的手段之一. 然而目前流行的co-Kriging和分层Kriging (HK)等多可信度代理模型泛化能力不足, 缺乏对高阶/高非线性建模问题的适应性, 难以广泛应用. 文章基于发展的自适应多可信度多项式混沌-Kriging (MF-PCK)代理模型, 在提高建模效率和对高阶/高非线性问题近似准确率的同时, 建立了基于该自适应MF-PCK模型的高效全局气动优化方法. 在发展的方法中, 提出了基于MF-PCK模型的新型变可信度期望改进加点方法, 使代理优化算法效率进一步提高. 为了验证发展方法的全面表现, 将其应用在经典的数值函数算例以及多个跨音速气动外形的确定性优化和稳健优化设计中, 并与基于Kriging和HK模型的代理优化算法进行了全面比较. 结果表明, 发展的新型多可信度全局气动优化方法其优化效率相对于基于Kriging和HK模型的优化效率显著提高, 结果更好也更加可靠, 并且稳健优化设计效率和结果也更符合工程应用需求, 证明了其相对于基于Kriging和HK模型的代理优化算法的显著优势.
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关键词:
- 多可信度代理模型 /
- 全局优化 /
- 跨音速气动优化 /
- 多可信度多项式混沌-克里金 /
- 稳健设计
Abstract: The multi-fidelity surrogate-assisted global aerodynamic optimization method has become one of the most efficient means to improve the efficiency and reliability of surrogate-based optimization algorithms. However, currently popular multi-fidelity surrogate models, e.g., co-Kriging and hierarchical Kriging (HK) models, lack the generalization ability and the adaptability to complex engineering applications, so that it is very difficult to promote them to be widely utilized. In this paper, based on the developed adaptive multi-fidelity polynomial chaos-Kriging (MF-PCK) surrogate model, an efficient global aerodynamic optimization framework based on the novel adaptive MF-PCK model is proposed while improving the modeling efficiency and the global approximation accuracy for highly-nonlinear and high-order response problems. In the development framework, we proposed a new adaptive multi-fidelity polynomial chaos-Kriging-based variable-fidelity expected improvement (MF-PCK-VF-EI) infilling strategy that can adaptively select new samples of both low- and high-fidelity, which further improves the efficiency and ability of this optimization framework. The developed methodology is verified by one analytical function example and demonstrated by three engineering application problems, including the deterministic optimization and the robust design optimization of several transonic aerodynamic shapes, compared with the popular Kriging-based and HK-based optimization algorithms. The results show that the global aerodynamic optimization efficiency of the developed method based on the novel adaptive MF-PCK model is higher than that based on the HK model and more than twice that based on the Kriging model, meantime with better and more reliable results. More importantly, the efficiency and performance of robust aerodynamic design optimization (RADO) for transonic aerodynamic shapes are prone to satisfy the requirements for complex engineering applications, which proves a significant advantage of the developed method over the popular Kriging-based and HK-based optimization algorithms. -
图 1 自适应选择最相关的低可信度多项式集合和最优的矫正多项式集合图示说明
Figure 1. Sketch map in selecting the optimal cardinalities of the low-fidelity PCE and corresponding correction expansion for the optimal MF-PCK model
图 2 基于自适应MF-PCK的多可信度气动优化流程图
Figure 2. The flow chart of adaptive MF-PCK-based multi-fidelity aerodynamic optimization algorithm
图 5 基于Kriging_EI加点过程
Figure 5. The infilling process of Kriging_EI optimization method
图 7 基于MF-PCK_VF-EI的加点过程
Figure 7. The infilling process of MF-PCK_VF-EI optimization method
图 9 代理模型预测误差收敛过程对比
Figure 9. The convergence history of the surrogate prediction errors
图 10 针对RAE2822翼型的不同可信度水平计算网格
Figure 10. Different levels of fidelity grids around RAE2822 airfoil
图 12 不同网格量计算压力分布与试验压力分布对比
Figure 12. Comparison of pressure distributions among computational and experimental results
图 15 阻力系数代理模型预测误差收敛过程
Figure 15. The convergence history of surrogate prediction errors of drag coefficients
图 17 优化翼型压力分布对比
Figure 17. Comparison of pressure distributions of optimized and initial airfoils
图 18 单点优化设计翼型在非设计点气动特性对比
Figure 18. Comparison of aerodynamic characteristics of optimized airfoil on off-design points
图 20 优化翼型阻力发散特性对比
Figure 20. Comparison of drag-divergence characteristics of optimized and initial airfoils
图 21 优化翼型与RAE2822翼型在不同状态下压力分布对比
Figure 21. Comparison of pressure distributions of optimized and initial airfoils under different Mach numbers
图 23 不同计算网格量下计算压力分布与试验压力分布对比
Figure 23. Comparison of computational and experimental pressure distributions under different number of grids
图 25 两种方法阻力收敛历程对比
Figure 25. The convergence history of different optimization algorithms
图 26 初始机翼和优化后机翼(MF-PCK_VF-EI)上表面压力云图对比
Figure 26. Comparison of pressure distribution contours of upper surface of M6 wing using the MF-PCK_VF-EI method
图 27 初始机翼和优化后的机翼(MF-PCK_VF-EI)各剖面压力分布和翼型对比
Figure 27. Comparison of pressure distributions and shapes of initial and optimized wing sections
表 1 3种优化方法结果对比
Table 1. Comparison of optimization results using three methods
Optimization method The optimal
locationsThe optimal values Prediction errors/10−4 Kriging_EI [−3.1702579,12.2190345] 0.4174474 8.953 HK_VF-EI [3.1912351, 2.3199426] 0.4166618 7.642 MF-PCK_VF-EI [9.4185951, 2.4763138] 0.3981135 3.692 
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表 2 RAE2822网格收敛结果对比
Table 2. Comparison of computational results of different levels of grids
Grid level Grid number $ {C}_{l} $ $ {C}_{d} $ $ {C}_{m} $ α/(°) $ {L}_{1} $ $4.0\times10^3$ 0.824 0.024297 −0.096164 3.022 $ {L}_{2} $ $ 2.1\times {10}^{4} $ 0.824 0.021261 −0.092284 2.973 $ {L}_{3} $ $ 8.1\times {10}^{4} $ 0.824 0.020112 −0.092868 2.917 $ {L}_{4} $ $1.27\times10^5$ 0.824 0.019960 −0.096062 2.869 $ {L}_{5} $ $1.98\times10^5$ 0.824 0.019729 −0.092778 2.806
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表 3 RAE2822翼型优化设计结果对比
Table 3. Comparison of optimization results of RAE2822 airfoil
Optimization methods $ {C}_{l} $ ${C}_{d}/10^{-3}$ Areas ${C}_{m}/10^{-3}$ Total time of CFD evaluations Kriging_EI 0.824 11.375 0.0779 −91.98 $ 70{N}_{h} + 40{N}_{h} = 110 t $ HK_VF-EI 0.824 11.282 0.0779 −91.75 $ 50{N}_{h} + 15{N}_{h} + 500{N}_{l}\approx 90 t $ MF-PCK_VF-EI 0.824 11.182 0.0779 −91.71 $ 30{\mathit{N}}_{\mathit{h}} + 15{\mathit{N}}_{\mathit{h}} + 500{\mathit{N}}_{\mathit{l}}\approx 70\mathit{t} $
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