﻿ 基于代理模型全局优化的自适应参数化方法
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Adaptive parameterization method for surrogate-based global optimization
ZHANG Wei, GAO Zhenghong, ZHOU Lin, XIA Lu
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Abstract: The results of the airfoil aerodynamic/stealth design with different numbers of design variables are compared and analyzed, revealing considerable impact of the design variable configuration on the results. Simple increase in the design variables cannot guarantee ideal results. This paper proposes an adaptive parameterization method for surrogate-based optimization. Using the global sensitivity analysis method and the element effect method, it obtains the sensitive information about the objectives in the design space to add design variables. The knot insertion algorithm is adopted to reconstruct samples in the high-dimensional space, avoiding the computational cost of resampling. Compared with the traditional fixed-dimensional method, the adaptive parameterization method expands the dimension in the sensitive area of the design space. The expanded design space can more accurately describe the shape and reflect the changing trend of the objective function. Therefore, the proposed method can significantly improve the quality and efficiency of optimization.
Keywords: aerodynamic/stealth design    surrogate-based global optimization    adaptive parameterization method    global sensitivity analysis    knot insertion algorithm

1 基本方法 1.1 基本效应法

 $\begin{array}{*{20}{l}} {{\rm{E}}{{\rm{E}}_i} = (f({X_1},{X_2}, \cdots ,{X_i} + {\varDelta _i}, \cdots ,{X_n}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} f({X_1},{X_2}, \cdots ,{X_i}, \cdots ,{X_n}))/{\varDelta _i}} \end{array}$ （1）

 ${\rm{S}}{{\rm{A}}_i} = \frac{{\sum\nolimits_1^r {{\rm{EE}}_i} }}{r}$ （2）

 ${\rm{S}}{{\rm{A}}_i} = \frac{{\sum\nolimits_1^r {\left| {{\rm{E}}{{\rm{E}}_i}} \right|} }}{r}$ （3）

1.2 Bspline曲线参数化方法

Bspline[22]方法凭借灵敏的局部扰动特性，广泛应用于工业设计中。通过一组基函数及其系数实现对外形的拟合和扰动，基函数的系数称为控制点，其位置由横、纵坐标共同决定。在翼型参数化中通常采用三次样条曲线。

Bspline通常由一组u∈[0, 1]的标量来组成。

 $G(u) = \sum\limits_{i = 0}^{n - 1} {{N_{i,p}}} (u){P_i}$ （4）
 ${N_{i,0}}(u) = \left\{ {\begin{array}{*{20}{c}} 1&{{r_i} \le u \le {r_{i + 1}}}\\ 0&{{\rm{ Otherwise }}} \end{array}} \right.$ （5）
 $\begin{array}{*{20}{c}} {{N_{i,k}}(u) = \frac{{u - {r_i}}}{{{r_{i + k}} - {r_i}}}{N_{i,k - 1}}(u) + }\\ {\frac{{{r_{i + k + 1}} - u}}{{{r_{i + k + 1}} - {r_{i + 1}}}}{N_{i + 1,k - 1}}(u)} \end{array}$ （6）

 $\begin{array}{*{20}{l}} {{P_{{\rm{ new }},j}} = }\\ {\qquad \left\{ {\begin{array}{*{20}{l}} {{P_j}}&{j = 0,1, \cdots ,i - k + 1}\\ {(1 - {\alpha _j}){P_{j - 1}} + {\alpha _j}{P_j}}&{j = i - k + 2, \cdots ,i - 1}\\ {{P_{j - 1}}}&{j = i - r + 1, \cdots ,n} \end{array}} \right.} \end{array}$ （7）

1.3 二维矩量法

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\hat n}} \times {\mathit{\boldsymbol{E}}^{\rm{i}}} = - \mathit{\boldsymbol{\hat n}} \times \oint\limits_S {\left[ {{\rm{j}}\omega \mu {\boldsymbol{J}}\psi - \frac{{\rm{j}}}{{\omega \varepsilon }}(\nabla {\boldsymbol{J}})\nabla \psi } \right]} {\rm{d}}S}\\ {\mathit{\boldsymbol{\hat n}} \times {\mathit{\boldsymbol{H}}^{\rm{i}}} = \frac{1}{2}\mathit{\boldsymbol{J}} - \mathit{\boldsymbol{\hat n}} \times \oint\limits_S \mathit{\boldsymbol{J}} \times \nabla \psi {\rm{d}}S} \end{array}} \right.$ （8）

 ${\sigma ^\prime } = \mathop {{\rm{lim}}}\limits_{r \to \infty } 2\pi r\frac{{{{\left| {{\mathit{\boldsymbol{E}}^{\rm{s}}}} \right|}^2}}}{{{{\left| {{\mathit{\boldsymbol{E}}^{\rm{i}}}} \right|}^2}}} = \mathop {\lim }\limits_{\rho \to \infty } 2\pi r\frac{{{{\left| {{\mathit{\boldsymbol{H}}^{\rm{s}}}} \right|}^2}}}{{{{\left| {{\mathit{\boldsymbol{H}}^{\rm{i}}}} \right|}^2}}}$ （9）

 $\sigma = \frac{{2{l^2}}}{\lambda }{\sigma ^\prime }\quad l \gg \lambda$ （10）

 图 1 FEKO三维计算模型 Fig. 1 3D computation model of FEKO

 图 2 FEKO与二维矩量法计算结果对比 Fig. 2 Results comparison of FEKO and 2D moment method
2 设计空间维度对翼型气动隐身特性影响 2.1 不同设计空间维度设计结果对比

 图 3 入射角示意图 Fig. 3 Sketch map of incident angle

 $\begin{array}{*{20}{l}} {{\rm{obj}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_D},{\rm{RCS}}}\\ {{\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}\left\{ {\begin{array}{*{20}{l}} {{C_L} = 0.25}\\ {{C_m} \ge 0.03}\\ {{\rm{ thic}}{{\rm{k}}_{{\rm{max}}}} \ge 0.18{\rm{ }}} \end{array}} \right.} \end{array}$ （11）

 翼型 阻力系数/count 力矩系数 RCS特性 CFD计算 基准 114.1 -0.01 0.0594 DV_12 88.2 0.0334 0.0087 100+500 DV_18 89.2 0.0321 0.0050 100+500 DV_24 89.7 0.0312 0.0049 100+500 DV_30 90.1 0.0355 0.0063 100+500
 图 4 设计翼型前向RCS特性对比 Fig. 4 Forward RCS comparison of optimized airfoils
 图 5 目标值收敛历程 Fig. 5 Convergence history of fitness

 图 6 设计翼型外形对比 Fig. 6 Shape comparison of optimized airfoils
 图 7 设计翼型压力分布对比 Fig. 7 Pressure distribution comparison of optimized airfoils
2.2 设计变量关于气动隐身特性敏感性分析

 图 8 设计变量关于阻力系数敏感性 Fig. 8 Sensitivity to CD of design variables
 图 9 设计变量关于前向RCS敏感性 Fig. 9 Sensitivity to forward RCS of design variables

3 自适应参数化方法

 图 10 自适应参数化方法流程 Fig. 10 Flowchart of adaptive parameterization method

 ${{\varDelta _t} = f{g_{i - t + 1}} - f{g_i}}$ （12）
 ${{f^*} = \sum\limits_{j = i - t + 1}^i {\frac{{{f_j} - f_j^{{\rm{ sur }}}}}{{{f_j}}}} }$ （13）
 ${ {\rm{Tr}} = {f^*}/t}$ （14）

Δt≤objtr，即目标函数优化效率不满足要求，以及Tr≤c，即代理模型精度满足要求的情况下，进行设计空间维度的扩展。

 $r = \frac{{{r_i} + {r_{i + 1}}}}{2}$ （15）

 $\begin{array}{l} {P_{{\rm{ new }}}} = \\ \left\{ {\begin{array}{*{20}{l}} {{P_{{\rm{ new }}}}}&{{\rm{ if }}(|{P_{{\rm{ new }}}} - {P_i}| > {\rm{d}}\rho ){\rm{\& }}(|{P_{{\rm{ new }}}} - {P_{i + 1}}| > {\rm{d}}\rho )}\\ {[{\kern 1pt} ]}&{{\rm{else}}} \end{array}} \right. \end{array}$ （16）

 ${\rm{V}}{{\rm{R}}_{{\rm{new}}}} = {\rm{V}}{{\rm{R}}_i}\frac{{|{P_{{\rm{new}}}} - {P_i}|}}{{|{P_{i + 1}} - {P_i}|}} + {\rm{V}}{{\rm{R}}_{i + 1}}\frac{{|{P_{{\rm{new}}}} - {P_{i + 1}}|}}{{|{P_{i + 1}} - {P_i}|}}$ （17）

4 NACA 65, 3-018翼型气动隐身设计

 $\begin{array}{l} {\rm{obj}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {C_D},{\rm{RCS}}\\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {{C_L} = 0.25}\\ {{C_m} \ge 0.03}\\ {{\rm{ thic}}{{\rm{k}}_{\max }}{\rm{ }} \ge 0.18} \end{array}} \right. \end{array}$ （18）

4.1 自适应参数化方法设计结果

 翼型 阻力系数/count 力矩系数 RCS特性 基准 114.1 -0.0098 0.0594 优化 87.9 0.0302 0.0045
 图 11 自适应参数化方法目标值收敛历程 Fig. 11 Fitness convergence history of adaptive parameterization method

 图 12 设计翼型外形 Fig. 12 Shape of optimized airfoil
 图 13 设计翼型压力分布 Fig. 13 Pressure distribution of optimized airfoil
 图 14 设计翼型前向RCS特性 Fig. 14 Forward RCS of optimized airfoil

 图 16 初始与最终阶段设计变量横坐标位置对比 Fig. 16 Design variable position comparison between initial and final stages
4.2 设计结果对比

 图 17 不同方法设计翼型外形对比 Fig. 17 Shape comparison of optimized airfoils of different methods
 图 18 不同方法优化收敛历程对比 Fig. 18 Comparison of fitness convergence histories in different methods

 翼型 阻力系数/count 力矩系数 RCS特性 CFD计算 自适应 87.9 0.0302 0.0045 100+200 DV_12 88.2 0.0334 0.0087 100+500 DV_18 89.2 0.0321 0.0050 100+500 DV_24 89.7 0.0312 0.0049 100+500 DV_30 90.1 0.0355 0.0063 100+500

5 结论

1) 基于设计空间维度对优化结果和效率的影响分析，提出一种适用于代理模型全局优化的自适应参数化方法。该方法在设计过程中从低维设计空间出发，结合设计变量的全局敏感性信息，在设计空间敏感性强的区域扩展维度。扩展后的高维设计空间能够更加精准地描述目标外形，反映目标变化的趋势，满足高效精细化设计需求。

2) 在设计空间维度扩张后，利用节点插入算法，将样本库内已有的低维样本在高维空间内重构，以重新训练代理模型，避免了在高维空间重新取样，实现了样本的高效配置。

3) 通过NACA 65, 3-018翼型气动隐身一体化设计算例，验证了本文提出的自适应参数化方法能够在优化过程中自适应配置设计变量，构造合理的设计空间，得到更加理想的设计结果。并与固定设计空间维度方法进行了对比，从设计质量和设计效率的角度分别验证了该方法的优越性。

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http://dx.doi.org/10.7527/S1000-6893.2020.23815

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文章信息

ZHANG Wei, GAO Zhenghong, ZHOU Lin, XIA Lu

Adaptive parameterization method for surrogate-based global optimization

Acta Aeronautica et Astronautica Sinica, 2020, 41(10): 123815.
http://dx.doi.org/10.7527/S1000-6893.2020.23815