﻿ 基于参数化的水下航行器外形稳健性优化
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (4): 622-628  DOI: 10.11990/jheu.201709086 0

### 引用本文

MIAO Yiran, GAO Liangtian, LIU Feng, et al. Robust optimization design of underwater vehicle shape based on parameterization[J]. Journal of Harbin Engineering University, 2018, 39(4), 622-628. DOI: 10.11990/jheu.201709086.

### 文章历史

Robust optimization design of underwater vehicle shape based on parameterization
MIAO Yiran, GAO Liangtian, LIU Feng, PENG Duojin
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In this study, to improve the robustness of a design scheme and the design efficiency, parametric design and robustness are optimized to design the shape of an underwater vehicle. Based on the resistance parameterization analysis of the underwater vehicle design, sample points were selected using the optimal Latin Hypercube method for resistance calculation and were fitted by a response surface model. An approximate hull resistance model that meets engineering demand was obtained. A multiobjective optimization model on the shape of underwater vehicle was established and the NSGA-Ⅲ multiobjective optimization algorithm was adopted to solve deterministic multiobjective optimization. Considering typical manufacturing errors in a shape design process, the robust optimization solution was performed. Compared with deterministic optimization, the hull resistance of robust optimization increased by 12.34%, and the total envelope volume increased by 51.98%. The robustness of the underwater vehicle was improved.
Key words: robustness    design efficiency    underwater vehicle    parameterization    shape optimization    NSGA-Ⅲ    optimal Latin hypercube method

1 外形参数化建模及直航阻力计算 1.1 水下航行器外形及设计参数

1.2 基于参数化的外形阻力计算

 $\begin{array}{l} \frac{{\partial (\rho {u_i})}}{{\partial t}} + \frac{\partial }{{{x_j}}}(\rho {u_i}{u_j}) = \frac{{-\partial p}}{{\partial {x_i}}} + \\ \frac{\partial }{{{x_j}}}\left( {\mu \frac{{\partial {u_i}}}{{\partial {x_j}}}-\rho u{\prime _i}u{\prime _j}} \right) + {S_i}\;\;\;\;\;i, j = 1, 2, 3 \end{array}$ (1)

 $\begin{array}{l} k = \frac{{\overline {u{\prime _i}u{\prime _j}} }}{2} = \frac{1}{2}(\overline {u{\prime _i}^2} + \overline {u{\prime _j}^2} + \overline {u{\prime _k}^2} )\\ \;\;\;\;\;\;\varepsilon = \frac{\mu }{\rho }\overline {\left( {\frac{{\partial u{\prime _i}}}{{\partial {x_k}}}} \right)\left( {\frac{{\partial u{\prime _i}}}{{\partial {x_k}}}} \right)} \end{array}$ (2)

 $\begin{array}{l} \frac{{\partial \left( {\rho k} \right)}}{{\partial t}} + \frac{{\partial (\rho k{u_i})}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_j}}}\left[{\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _k}}}} \right)\frac{{\partial k}}{{\partial {x_j}}}} \right] + \\ \;\;\;\;\;\;{G_k} + {G_b} - \rho \varepsilon - {Y_M} + {S_k}\\ \frac{{\partial \left( {\rho \varepsilon } \right)}}{{\partial t}} + \frac{{\partial (\rho \varepsilon {u_i})}}{{\partial {x_i}}} = \frac{\partial }{{\partial {x_j}}}\left[{\left( {\mu + \frac{{{\mu _t}}}{{{\sigma _\varepsilon }}}} \right)\frac{{\partial \varepsilon }}{{\partial {x_j}}}} \right] + \\ \;\;\;\;{C_{1\varepsilon }}\frac{\varepsilon }{k}({G_k} + {C_{3\varepsilon }}{G_b}) -{C_{2\varepsilon }}\rho \frac{{{\varepsilon ^2}}}{k} + {S_\varepsilon } \end{array}$ (3)

 Download: 图 3 参数化分析流程 Fig. 3 Flow chart of parametric analysis process
2 近似模型的建立 2.1 样本点的选取

2.2 响应面模型

 $\begin{array}{l} \hat F(x) = a_0 + \sum\limits_{i = 1}^N {{b_i}{x_i}} + \sum\limits_{i = 1}^N {{c_{ii}}x_i^2} + \sum\limits_{i = 1}^N {{d_i}x_i^3} + \\ \;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^N {{e_i}x_i^4} + \sum\limits_{1 \le i \le j \le N}^N {{f_{ij}}{x_i}{x_j}} \end{array}$ (4)

 $\begin{array}{l} {F_x} = 5\;685\;942-575{L_m}-80\;603\;801{L_r}-\\ 416\;473{L_t} + 979L_m^2 + 432\;955\;832L_r^2 + \\ \;1\;043\;632L_t^2 + 2\;276{L_m}{L_r} + 773{L_m}{L_t} - \\ \;115\;226{L_r}{L_t} - 214L_m^3 - 1\;031\;474\;381L_r^3 - \\ \;1\;078\;519L_t^3 + 14L_m^4 + 920\;889\;566L_r^4 + 414\;974L_t^4 \end{array}$ (5)

 $\begin{array}{l} V = 2.094\;304\;835\;9L_r^3 + 3.141\;592\;653\;8L_r^2{L_m} + \\ 1.047\;197\;551{L_t}(0.015\;625 + L_r^2 + 0.125{L_r}) \end{array}$ (6)

 ${R^2} = 1- \left[{\sum\limits_{i = 1}^n {{{({y_i}-{{\hat y}_i})}^2}} \sum\limits_{i = 1}^n {{{({y_i}-{{\bar y}_i})}^2}} } \right]$ (7)

R2是介于0~1实数，越接近1说明拟合效果越好，工程上要求其数值在0.9以上即可初步满足拟合要求。根据响应面模型，选择评估校核的验证样本为20组，其阻力值R2为0.969 32，满足工程需要。

3 多目标确定性优化 3.1 NSGA-Ⅲ算法

 $H = \left( {\begin{array}{*{20}{c}} {M - p{\rm{ }} + {\rm{ }}1}\\ p \end{array}} \right)$ (8)

 ${\rm{ASF}}\left( {X, W} \right) = {\rm{MA}}{{\rm{X}}_{i = 1:m}}\frac{{f{\prime _i}\left( x \right)}}{{{W_i}}}, f_i^n\left( x \right) = \frac{{f{\prime _i}\left( x \right)}}{{{a_i}}}$ (9)
 Download: 图 5 参考点选择示意图 Fig. 5 Schematic diagram of reference points selecting

 Download: 图 6 参考点关联个体示意图 Fig. 6 Schematic diagram of reference points correlate individuals

pj>0时选择距离最小个体进入pt+1，相比于上一代算法，NSGA-Ⅲ算法在选择子代时对标准超平面进行了有效分割的方法，代替了上一代算法中基于排挤机制的小生境技术，使子代更加均匀地分布在可能的寻优可行域空间内，提升了子代样本的多样性，并防止早熟。因此，更加适应于求解优化目标较多的多目标优化问题。

3.2 水下航行器外形多目标优化求解

 $\left\{ {\begin{array}{*{20}{l}} {{\rm{min}}:{{\{ {f_1}\left( x \right), {f_2}\left( x \right), \ldots {f_k}\left( x \right)\} }^{\rm{T}}}}\\ {{\rm{s}}.{\rm{t}}.}\\ {{g_j}\left( x \right) \le 0, j = 1, 2, \ldots, J}\\ {{x_L} \le x \le {x_U}} \end{array}} \right.$ (10)

 $\left\{ \begin{array}{l} {\rm{min}}:{F_x};{\rm{max}}:V\\ {\rm{s}}.{\rm{t}}.\\ {F_x} \le 18\;000\\ {\rm{D}}.{\rm{v}}:{L_r};{L_m};{L_t} \end{array} \right.$ (11)

 Download: 图 7 艇体阻力与设计变量Pareto解集 Fig. 7 Pareto solution set of hull resistance and design variable

4 多目标稳健性优化方法 4.1 稳健性优化

 Download: 图 8 确定性优化与稳健性优化比较 Fig. 8 Deterministic optimization compared with robust optimization

 $\begin{array}{l} {\rm{E}}\left( {L\left( y \right)} \right) = {\rm{E}}\{ {(y-{y_0})^2}\} = {\rm{E}}\{ {\left( {y-\bar y} \right)^2} + \;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\bar y-{y_0})2\} = Y_\mu ^2 + Y_\sigma ^2 \end{array}$ (11)

 $\begin{array}{l} {\rm{min}}:[{f_1}({Y_{\mu 1}}\left( x \right), {Y_{\sigma 1}}\left( x \right)), {f_2}({Y_{\mu 2}}\left( x \right), {Y_{\sigma 2}}\left( x \right)), \ldots \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f_k}({Y_{\mu k}}\left( x \right), {Y_{\sigma k}}\left( x \right)){]^{\rm{T}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{s}}.{\rm{t}}.\\ {g_{\mu j}}\left( x \right) + \eta {g_{\sigma j}}\left( x \right) \le 0, j = 1, 2, \ldots, J\\ \;\;\;\;\;\;\;\;{x_L} + \eta {x_\sigma } \le {x_\mu } \le {x_U} -\eta {x_\sigma } \end{array}$ (12)

 $\begin{array}{l} {\rm{min}}:\{ {F_{x\mu }};{F_{x\sigma }}\} ;{\rm{max}}:\{ {V_\mu };{V_\sigma }\} \\ {\rm{s}}.{\rm{t}}.\\ {F_{x\mu }} \le 16\;920\\ {\rm{D}}.{\rm{v}}:{L_r};{L_m};{L_t} \end{array}$ (13)

4.2 水下航行器外形多目标稳健性优化求解

 Download: 图 9 确定性优化与稳健性优化Pareto解集对比 Fig. 9 Pareto solution set′s comparison between of deterministic optimization and robustness optimization

5 结论

1) 参数分析的实现、响应面近似模型的建立，在保证计算精度的前提下，大幅减低了计算代价，提高了设计效率。

2) 在进行多个优化目标的全局Pareto解集求解过程中，相对于前一代非支配算法，NSGA-Ⅲ可提高计算效率，可有效避免早熟现象，适合作为稳健性优化的寻优方法。

3) 相比于初始方案，优化方案在提升艇体包络体积方面效果显著，虽然稳健性方案相对于确定性方案的降阻和增容效果稍差，但在一定程度上降低了外界因素对水下航行器性能的影响，更加满足工程需要。

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