﻿ 基于NFFD和高斯近似的螺旋桨多目标优化
 舰船科学技术  2022, Vol. 44 Issue (19): 46-51    DOI: 10.3404/j.issn.1672-7649.2022.19.010 PDF

1. 渤海船舶职业学院，辽宁 葫芦岛 125101;
2. 招商局重工(江苏)有限公司，江苏 南通 226000;
3. 华中科技大学，湖北 武汉 423201

Analysis of the influence of the three-propeller state of the heavy icebreaker on the performance of open water navigation
LIU Xu1, ZHOU Xi-ning2, ZHU Yao-long3
1. Bohai Shipbuilding Vocational College, Huludao 125101, China;
2. China Merchants Heavy Industry (Jiangsu) Co., Ltd., Nantong 226000, China;
3. Huazhong University of Science and Technology, Wuhan 423201, China
Abstract: In order to promote the efficient and optimal design of the propeller, combined with the NFFD technology, the multi-output Gaussian approximation model and the NSGA-II multi-objective optimization algorithm, a set of propeller high-efficiency propellers including propeller deformation reconstruction - rapid prediction of hydrodynamic performance - multi-objective optimization was constructed. Automatic optimization method. Firstly, based on NFFD technology, the geometric parameters of the propeller are used as design variables to realize the deformation and reconstruction of the three-dimensional model of the propeller; secondly, the hydrodynamic performance of the sample propeller is obtained by using finite element numerical simulation, and the propeller is established based on the multi-output Gaussian approximation theory. The approximate model between geometric parameters and hydrodynamic performance; finally, combined with Gaussian approximate model and NSGA-Ⅱ algorithm, the multi-objective optimization design of KP505 propeller is carried out, and the feasibility of applying the method to the optimal design of propeller is verified. The results show that the method achieves the multi-objective optimization of reducing the torque coefficient of KP505 propeller by 3.3% and increasing the efficiency by 2.6%, which verifies the feasibility of the method applied to the optimal design of the propeller.
Key words: propeller     multi-objective optimization     NFFD     Gaussian approximation     NSGA-II
0 引　言

1 NFFD方法及螺旋桨变形控制

 $Q = {Q_0} + tT + uU + vV。$ (1)

 ${P_{i,j,k}} = {Q_0} + {i / {l \cdot }}T + {j / {m \cdot }}U + {k / {n \cdot }}V 。$ (2)

 $Q\left( {t,u,v} \right) = \sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {\sum\limits_{k = 0}^n {{P_{i,j,k}}{B_{il}}} } } (t){B_{jm}}(u){B_{kn}}(v) 。$ (3)

 ${B_{il}}(t) = \frac{{l!}}{{i!(l - i)!}}{t^i}{(l - t)^{(l - i)}}。$ (4)

NFFD方法利用NURBS基函数替换Bernstein函数，螺旋桨任意一点Q坐标可表示为：

 $Q\left( {t,u,v} \right) = \frac{{\displaystyle\sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {\sum\limits_{k = 0}^n {{P_{i,j,k}}{W_{i,j,k}}{B_{il}}(t){B_{jm}}(u){B_{il}}(v)} } } }}{{\displaystyle\sum\limits_{i = 0}^l {\sum\limits_{j = 0}^m {\sum\limits_{k = 0}^n {{W_{i,j,k}}{B_{il}}(t){B_{jm}}(u){B_{il}}(v)} } } }}。$ (5)

 $Q(t,u,v) = \frac{{\hat {\boldsymbol{V}}{{\boldsymbol{M}}_k}(\hat {\boldsymbol{T}}{{\boldsymbol{M}}_i}{P_{i,j,k}}{W_{i,j,k}}{\boldsymbol{M}}_j^{\rm{T}}{{\hat {\boldsymbol{U}}}^{\rm{T}}})}}{{\hat {\boldsymbol{V}}{{\boldsymbol{M}}_k}(\hat {\boldsymbol{T}}{{\boldsymbol{M}}_i}{W_{i,j,k}}{\boldsymbol{M}}_j^{\rm{T}}{{\hat {\boldsymbol{U}}}^{\rm{T}}})}} 。$ (6)

 图 1 KP505桨形状控制效果和网格模型 Fig. 1 KP505 propeller shape control effect and mesh model
2 性能预测高斯近似模型 2.1 多输出高斯近似模型

 ${S_{learn}} = \left\{ {({x_i},{y_i}),i = 1, \cdots ,{N_{learn}}} \right\} ，$

 ${\hat y_i} \approx f({x_i}) ，$

 ${\hat y_i} = f\left( {{x_i}} \right) + \varepsilon ，$

 ${cov} \left( {\hat y} \right) = {K_n}\left( {X,X} \right) + \sigma _n^2I 。$

 ${K_n} = {K_f} \otimes {K_x} ，$

 $\left[ \begin{gathered} y \\ f\left( {{x_{test}}} \right) \\ \end{gathered} \right] \sim N\left( {\left[ \begin{gathered} 0 \\ 0 \\ \end{gathered} \right],\left[ \begin{gathered} K\left( {X,X} \right) + \sigma _n^2I{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k(X,{x_{_{test}}}){\kern 1pt} \\ k({x_{test}},X){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k({x_{_{test}}},{x_{_{test}}}) \\ \end{gathered} \right]} \right)。$

 $\left. {f\left( {{x_{test}}} \right)} \right|X,y,{x_{test}} \sim N\left( {\bar f({x_{test}}),{\sigma ^2}({x_{test}})} \right) ，$

 $\bar f({x_{test}}) = k{(X,{x_{test}})^{\rm{T}}}{\left[ {K\left( {X,X} \right) + \sigma _n^2I} \right]^{ - 1}}y ，$
 $\begin{split}{\sigma ^2}({x_{test}}) =& k({x_{test}},{x_{test}}) - k{(X,{x_{test}})^{\rm{T}}}\times \\ &{\left[ {K\left( {X,X} \right) + \sigma _n^2I} \right]^{ - 1}}k(X,{x_{test}}) 。\end{split}$

 $\left\{ \begin{gathered} X = \left\{ {x_i^a,x_i^b,x_i^c,x_i^d,x_i^e,x_i^f\cdots\cdots} \right\} ，\\ Y = \left\{ {{K_T},10{K_Q},{\eta _0},{P_{\min }}} \right\} 。\\ \end{gathered} \right.$

2.2 敞水性能数值仿真

 图 2 高雷诺数下螺旋桨的端涡和根涡现象 Fig. 2 End vortex and root vortex phenomenon of propeller at high Reynolds number

 图 3 螺旋桨仿真计算结果与试验结果对比 Fig. 3 Comparison between the simulation calculation results of the propeller and the test results

2.3 近似模型的验证

 图 4 高斯近似模型的交叉验证预测误差 Fig. 4 Cross-validation prediction error of Gaussian approximation model

3 螺旋桨多目标优化 3.1 NSGA-Ⅱ多目标优化

NSGA-Ⅱ是由Kalyanmoy等[23]针对NSGA提出的一种改进进化算法，该算法具有多目标适度简化、最优解拓展、求解精度高、高鲁棒性等优点，尤其针对少量样本、多变量多目标优化问题就较快的求解速度。

 $\left\{\begin{array}{l}目标：\left\{\begin{array}{l}\mathrm{max}{\eta }_{0}，\\ \mathrm{min}{K}_{Q}，\end{array}\right.\\ 约束：\left\{\begin{array}{l}{K}_{T}\geqslant {K}_{T}{}_{\text{initial}}，\\ {K}_{Q}\leqslant {K}_{Q}{}_{\text{initial}}，\\ {P}_{\mathrm{min}}\geqslant {P}_{\text{min-initial}}，\\ {x}_{\mathrm{min}}\leqslant {x}^{a/b/c/d/e/f}\leqslant {x}_{\mathrm{max}}，\\ 0.4\leqslant {A}_{e}/{A}_{d}\leqslant 1.4。\end{array}\right.\end{array}\right.$

 图 5 初始种群和Pareto解集扭矩系数和效率 Fig. 5 Torque coefficient and efficiency of initial population and Pareto solution set
3.2 优化设计结果及分析

 图 6 优化方案4桨叶r/R=0.7处剖面轮廓 Fig. 6 Section profile at r/R=0.7 of the optimized scheme 4 blade

 图 7 KP505模型桨叶宽度、厚度和倾斜角沿半径分布 Fig. 7 KP505 model blade width, thickness and inclination angle distribution along the radius

 图 8 KP505模型桨叶螺距比和曲率沿半径分布 Fig. 8 KP505 model blade pitch ratio and curvature distribution along the radius
4 结　语

1）以有限元仿真结果为样本，以螺旋桨参数沿半径的分布为输入，基于多输出高斯近似模型建立螺旋桨水动力性能预测模型。该模型与数值仿真结果得出的螺旋桨推力系数和效率误差在3%以内，可以较为准确预测螺旋桨水动力性能。

2）应用本文提出的螺旋桨高效自动优化方法，开展KP505桨的优化设计，实现了对KP505桨降低扭矩系数、提高效率的多目标优化，形成了Pareto解集，获得了理论上的最佳设计方案。

3）根据对优化结果的分析，0.65R～0.95R处螺距比和0.4R～0.6R处倾斜角对KP505桨的效率有较大影响。

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