﻿ 基于混合搜寻法的车用涡轮增压器多目标优化
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 同济大学学报(自然科学版)  2018, Vol. 46 Issue (6): 819-827.  DOI: 10.11908/j.issn.0253-374x.2018.06.015 0

### 引用本文

HAO Zhenzhen, NI Jimin, SHI Xiuyong, LI Dongdong. Multi-Objective Integrated Optimization of Vehicle Turbocharger Impeller Based on Hybrid Search Method[J]. Journal of Tongji University (Natural Science), 2018, 46(6): 819-827. DOI: 10.11908/j.issn.0253-374x.2018.06.015.

### 文章历史

Multi-Objective Integrated Optimization of Vehicle Turbocharger Impeller Based on Hybrid Search Method
HAO Zhenzhen, NI Jimin, SHI Xiuyong, LI Dongdong
School of Automotive Studies, Tongji University, Shanghai 201804, China
Abstract: Different from optimization study of the single performance of turbocharger impeller, this paper takes the compressor isentropic efficiency, the impeller maximum stress, and the impeller maximum deformation as the optimization objective. Based on ANSYS Workbench, the turbocharger compressor flow-thermal-solid multi-physics coupling simulation model was established. Using the orthogonal matrix method and the 2k method, the key design parameters of the compressor were studied. The radial basis function and non-dominated sorting genetic algorithm Ⅱ were applied to solve the global optimization problem. Finally, the optimal scheme of impeller structure design were obtained. The results show that the most significant decrease is the impeller maximum deformation of 28.09%, the impeller maximum stress is decreased by 16.34%, and the compressor isentropic efficiency is increased by 2.83%.
Key words: vehicle turbocharger    the structure optimization of impeller    hybrid Search method    isentropic efficiency    maximum stress    maximum deformation

1 仿真模型的建立与试验验证 1.1 强度分析理论

 图 1 节点三角形单元 Fig.1 Six-node triangle element

 ${\mathit{\boldsymbol{q}}_{\rm{e}}} = {\left[ {\begin{array}{*{20}{c}} {{u_1}}&{{v_1}}&{{u_2}}&{{v_2}}&{{u_3}}&{{v_3}} \end{array}} \right]^{\rm{T}}}$ (1)
 ${\mathit{\boldsymbol{F}}_{\rm{e}}} = {\left[ {\begin{array}{*{20}{c}} {{F_{{x_1}}}}&{{F_{{y_1}}}}&{{F_{{x_2}}}}&{{F_{{y_2}}}}&{{F_{{x_3}}}}&{{F_{{y_3}}}} \end{array}} \right]^{\rm{T}}}$ (2)
1.1.1 单元的位移

 $\left\{ \begin{array}{l} u\left( {x,y} \right) = {\alpha _0} + {\alpha _1}x + {\alpha _2}y\\ v\left( {x,y} \right) = {\beta _0} + {\beta _1}x + {\beta _2}y \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} u\left( {{x_i},{y_i}} \right) = {u_i},\\ v\left( {{x_i},{y_i}} \right) = {v_i}, \end{array} \right.\;\;\;\;\;i = 1,2,3$ (4)

 ${\alpha _0} = \frac{1}{{2A}}\left| {\begin{array}{*{20}{c}} {{u_1}}&{{x_1}}&{{y_1}}\\ {{u_2}}&{{x_2}}&{{y_2}}\\ {{u_2}}&{{x_3}}&{{y_3}} \end{array}} \right| = \frac{1}{{2A}}\left( {{a_1}{u_1} + {a_2}{u_2} + {a_3}{u_3}} \right)$ (5)
 ${\alpha _1} = \frac{1}{{2A}}\left| {\begin{array}{*{20}{c}} 1&{{u_1}}&{{y_1}}\\ 1&{{u_2}}&{{y_2}}\\ 1&{{u_3}}&{{y_3}} \end{array}} \right| = \frac{1}{{2A}}\left( {{b_1}{u_1} + {b_2}{u_2} + {b_3}{u_3}} \right)$ (6)
 ${\alpha _2} = \frac{1}{{2A}}\left| {\begin{array}{*{20}{c}} 1&{{x_1}}&{{u_1}}\\ 1&{{x_2}}&{{u_2}}\\ 1&{{x_3}}&{{u_3}} \end{array}} \right| = \frac{1}{{2A}}\left( {{c_1}{u_1} + {c_2}{u_2} + {c_3}{u_3}} \right)$ (7)
 ${\beta _0} = \frac{1}{{2A}}\left( {{a_1}{v_1} + {a_2}{v_2} + {a_3}{v_3}} \right)$ (8)
 ${\beta _1} = \frac{1}{{2A}}\left( {{b_1}{v_1} + {b_2}{v_2} + {b_3}{v_3}} \right)$ (9)
 ${\beta _2} = \frac{1}{{2A}}\left( {{c_1}{v_1} + {c_2}{v_2} + {c_3}{v_3}} \right)$ (10)

 $\begin{array}{l} A = \frac{1}{2}\left| {\begin{array}{*{20}{c}} 1&{{x_1}}&{{y_1}}\\ 1&{{x_2}}&{{y_2}}\\ 1&{{x_3}}&{{y_3}} \end{array}} \right| = \\ \;\;\;\;\frac{1}{2}\left( {{a_1} + {a_2} + {a_3}} \right) = \frac{1}{2}\left( {{b_1}{c_2} - {b_2}{c_1}} \right) \end{array}$ (11)
 $\left\{ \begin{array}{l} {a_1} = \left| {\begin{array}{*{20}{c}} {{x_2}}&{{y_2}}\\ {{x_3}}&{{y_3}} \end{array}} \right| = {x_2}{y_3} - {x_3}{y_2}\\ {b_1} = - \left| {\begin{array}{*{20}{c}} 1&{{y_2}}\\ 1&{{y_3}} \end{array}} \right| = {y_2} - {y_3}\\ {c_1} = \left| {\begin{array}{*{20}{c}} 1&{{x_2}}\\ 1&{{x_3}} \end{array}} \right| = - {x_2} + {x_3} \end{array} \right.$ (12)

 $\begin{array}{l} u\left( {x,y} \right) = {N_1}\left( {x,y} \right){u_1} + {N_2}\left( {x,y} \right){u_2} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{N_3}\left( {x,y} \right){u_3} \end{array}$ (13)
 $\begin{array}{l} v\left( {x,y} \right) = {N_1}\left( {x,y} \right){v_1} + {N_2}\left( {x,y} \right){v_2} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;{N_3}\left( {x,y} \right){v_3} \end{array}$ (14)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{u}}\left( {x,y} \right) = \left[ {\begin{array}{*{20}{c}} {u\left( {x,y} \right)}\\ {v\left( {x,y} \right)} \end{array}} \right] = }\\ {\left[ {\begin{array}{*{20}{c}} {{N_1}}&0&{{N_2}}&0&{{N_3}}&0\\ 0&{{N_1}}&0&{{N_2}}&0&{{N_3}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_1}}\\ {{v_1}}\\ {{u_2}}\\ {{v_2}}\\ {{u_3}}\\ {{v_3}} \end{array}} \right] = \mathit{\boldsymbol{N}}\left( {x,y} \right) \cdot {\mathit{\boldsymbol{q}}_{\rm{e}}}} \end{array}$ (15)

 $\mathit{\boldsymbol{N}}\left( {x,y} \right) = \left[ {\begin{array}{*{20}{c}} {{N_1}}&0&{{N_2}}&0&{{N_3}}&0\\ 0&{{N_1}}&0&{{N_2}}&0&{{N_3}} \end{array}} \right]$ (16)

 ${N_i} = \frac{1}{{2A}}\left( {{a_i} + {b_i}x + {c_i}y} \right),\;\;\;i = 1,2,3$ (17)
1.1.2 单元的应变场

 $\begin{array}{l} \mathit{\boldsymbol{\varepsilon }}\left( {x,y} \right) = \left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial x}}}\\ {\frac{{\partial u}}{{\partial y}}}\\ {\frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}} \end{array}} \right] = \\ \left[ {\begin{array}{*{20}{c}} {\frac{\partial }{{\partial x}}}&0\\ 0&{\frac{\partial }{{\partial y}}}\\ {\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial x}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {u\left( {x,y} \right)}\\ {v\left( {x,y} \right)} \end{array}} \right] = \partial \mathit{\boldsymbol{u}} \end{array}$ (18)

 $\partial = \left[ {\begin{array}{*{20}{c}} {\frac{\partial }{{\partial x}}}&0\\ 0&{\frac{\partial }{{\partial y}}}\\ {\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial x}}} \end{array}} \right]$ (19)

 $\mathit{\boldsymbol{\varepsilon }}\left( {x,y} \right) = \partial \mathit{\boldsymbol{N}}\left( {x,y} \right) \cdot {\mathit{\boldsymbol{q}}_{\rm{e}}} = \mathit{\boldsymbol{B}}\left( {x,y} \right) \cdot {\mathit{\boldsymbol{q}}_{\rm{e}}}$ (20)

 $\begin{array}{l} \mathit{\boldsymbol{B}}\left( {x,y} \right) = \partial \cdot \mathit{\boldsymbol{N = }}\\ \left[ {\begin{array}{*{20}{c}} {\frac{\partial }{{\partial x}}}&0\\ 0&{\frac{\partial }{{\partial y}}}\\ {\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial x}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{N_1}}&0&{{N_2}}&0&{{N_3}}&0\\ 0&{{N_1}}&0&{{N_2}}&0&{{N_3}} \end{array}} \right] \end{array}$ (21)

 $\begin{array}{l} \mathit{\boldsymbol{B}}\left( {x,y} \right) = \frac{1}{{2A}}\left[ {\begin{array}{*{20}{c}} {{b_1}}&0&{{b_2}}&0&{{b_3}}&0\\ 0&{{c_1}}&0&{{c_2}}&0&{{c_3}}\\ {{c_1}}&{{b_1}}&{{c_2}}&{{b_2}}&{{c_3}}&{{b_3}} \end{array}} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{B_1}}&{{B_2}}&{{B_3}} \end{array}} \right] \end{array}$ (22)

 ${\mathit{\boldsymbol{B}}_i} = \frac{1}{{2A}}\left[ {\begin{array}{*{20}{c}} {{b_i}}&0\\ 0&{{c_i}}\\ {{c_i}}&{{b_i}} \end{array}} \right],\;\;\;\;\;i = 1,2,3$ (23)
1.1.3 单元的应力场
 $\begin{array}{l} \mathit{\boldsymbol{\sigma }}\left( {x,y,z} \right) = \left[ {\begin{array}{*{20}{c}} {{\sigma _x}}\\ {{\sigma _y}}\\ {{\tau _{xy}}} \end{array}} \right] = \\ \;\;\;\;\;\;\;\;\frac{E}{{1 - {\mu ^2}}}\left[ {\begin{array}{*{20}{c}} 1&\mu &0\\ \mu &1&0\\ 0&0&{\frac{{1 - \mu }}{2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\varepsilon _x}}\\ {{\varepsilon _y}}\\ {{\gamma _{xy}}} \end{array}} \right] = \mathit{\boldsymbol{D}} \cdot \mathit{\boldsymbol{\varepsilon }} \end{array}$ (24)

 $\mathit{\boldsymbol{D}} = \frac{E}{{1 - {\mu ^2}}}\left[ {\begin{array}{*{20}{c}} 1&\mu &0\\ \mu &1&0\\ 0&0&{\frac{{1 - \mu }}{2}} \end{array}} \right]$ (25)

 $\mathit{\boldsymbol{\sigma }} = \mathit{\boldsymbol{D}} \cdot \mathit{\boldsymbol{B}} \cdot {\mathit{\boldsymbol{q}}_{\rm{e}}} = \mathit{\boldsymbol{S}} \cdot {\mathit{\boldsymbol{q}}_{\rm{e}}}$ (26)
 $\mathit{\boldsymbol{S}} = \mathit{\boldsymbol{D}} \cdot \mathit{\boldsymbol{B}}$ (27)
1.2 流-热-固耦合理论分析

 ${\rho _{\rm{s}}}{\mathit{\boldsymbol{d}}_{\rm{s}}} = \nabla {\mathit{\boldsymbol{\sigma }}_{\rm{s}}} + {\mathit{\boldsymbol{f}}_{\rm{s}}}$ (28)

 $\begin{array}{*{20}{c}} {\frac{{\partial \left( {\rho {h_{{\rm{tot}}}}} \right)}}{{\partial t}} - \frac{{\partial \rho }}{{\partial t}} + \nabla \left( {{\rho _f}v{h_{{\rm{tot}}}}} \right) = \nabla \left( {\lambda \nabla T} \right) + }\\ {\nabla \left( {v\tau } \right) + v\rho {f_{\rm{f}}} + {S_{\rm{E}}}} \end{array}$ (29)

 ${f_T} = {\alpha _T} \cdot \nabla T$ (30)

1.3 仿真模型的建立和试验验证

 图 2 压气机流量-等熵效率曲线 Fig.2 Traffic-isentropic efficiency of compressor
2 正交矩阵法和2k法混合搜寻设计变量

2.1 正交矩阵法析因分析

2.2 2 k法析因分析

(1) A组(叶轮轴与轴孔组)

(2) B组(叶根圆角和叶片厚度组)

(3) C组(叶轮出口组)

(4) D组(叶轮背盘组)

2.3 DoE混合搜寻结果

3 寻优计算及验证分析

3.1 帕雷托(Pareto)解集

3.2 权重分配

3.3 综合评价指标建立(选取最优帕雷托解)

 $Z = {Z_1} + {Z_2} + {Z_3}$ (31)

 ${Z_1} = {Z_4}\left( {{Z_5}/{Z_6}} \right) \times 100$ (32)

3.4 优化结果验证与分析 3.4.1 寻优计算结果

3.4.2 优化结果验证与分析

(1) 优化前后叶轮最大形变的变化

 图 3 原机叶轮尾缘最大形变 Fig.3 Maximum distortion in original impeller trailing edge
 图 4 优化后叶轮尾缘最大形变 Fig.4 Maximum distortion in impeller trailing edge after optimization

(2) 优化前后叶片根部应力的变化

 图 5 原机叶片根部最大应力 Fig.5 Maximum stress in original blade roots
 图 6 优化后叶片根部最大应力 Fig.6 Maximum stress in blade root after optimization

(3) 优化前后叶轮背盘应力的变化

 图 7 优化前后叶轮背盘最大应力 Fig.7 Maximum stress in impeller back plate after optimization

4 结论

(1) 优化后叶轮整体应力呈现下降趋势，叶轮最大形变量沿叶轮周向和径向的周期性分布也有所增强.

(2) 优化后叶轮背盘整体应力分布均匀性增强，而且由轴孔沿径向的应力分布阶梯更加均匀.

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