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 哈尔滨工程大学学报  2019, Vol. 40 Issue (8): 1502-1508  DOI: 10.11990/jheu.201805130 0

### 引用本文

HOU Zhenxiu, TIAN Shen, ZHUANG Ting, et al. PSO-based positioning method for the optimization of the supporting position of 3-D thin-walled pipe fittings[J]. Journal of Harbin Engineering University, 2019, 40(8), 1502-1508. DOI: 10.11990/jheu.201805130.

### 文章历史

PSO-based positioning method for the optimization of the supporting position of 3-D thin-walled pipe fittings
HOU Zhenxiu , TIAN Shen , ZHUANG Ting , XIE Liuwei , YAN Liang , YANG Ye , CHEN Yongkun
School of Mechanics Engineering, Harbin Institute of Technology, Harbin 150001, China
Abstract: The deformation of a three-dimensional (3-D) thin-walled tube will intensify if it is placed at a nonrational support position and support angle under gravity. The increment in the deformation of the tube will affect assembly precision during manufacturing and prolong operation and calculation when commercial software is used to identify the optimal support position. Thus, this work proposes a method based on particle swarm optimization (PSO) for optimizing the position of 3-D thin-walled pipe fittings. In this method, the axis of the pipe is extracted by the secondary development as the reference. Then, the finite element method is used to split the axis, and the finite element stiffness matrix of the pipe unit is rebuilt to calculate pipe deformation and to extract maximum deformation, which is considered as an index for the evaluation of the merits and demerits of the support position. Finally, the PSO algorithm is applied to identify the optimal combination of pipe support positions through the optimization iteration by using the maximum deformation as the evaluation function. Results show that the difference between the deformation calculated by the finite element model and that calculated by the commercial software is within 10%. The closeness between the calculated values indicates that the mathematical model is accurate. The finite element model combined with the PSO algorithm has the advantages of rapid iterative convergence. Specifically, the model can identify the optimal support position within 30 iterations. The time required by the proposed model to calculate 300 mathematical models is less than that required by commercial software.
Keywords: Three-dimensional thin-walled pipe fittings    Model establishment    Support position    PSO particle swarm optimization    Pipe fitting information extraction    Mechanical calculation analysis    Finite element simulation analysis    Optimization

1 空间管件数学模型的建立 1.1 三维管件轴线的提取

 Download: 图 1 ProE二次开发提取的轴线与模型对比 Fig. 1 Comparison of the axis and model of the secondary development of ProE
1.2 直杆单元力学模型的建立

 $\gamma\left[\begin{array}{cc}{1} & {-1} \\ {-1} & {1}\end{array}\right]\left[\begin{array}{l}{u_{1}} \\ {u_{2}}\end{array}\right]=\left[\begin{array}{l}{f_{x 1}} \\ {f_{x 2}}\end{array}\right]$ (1)

 $\delta\left[\begin{array}{cc}{1} & {-1} \\ {-1} & {1}\end{array}\right]\left[\begin{array}{c}{a_{x_{1}}} \\ {a_{x 2}}\end{array}\right]=\left[\begin{array}{c}{M_{x 1}} \\ {M_{x 2}}\end{array}\right]$ (2)

 $\alpha \left[ {\begin{array}{*{20}{c}} {12}&{6l}&{ - 12}&{6l}\\ {6l}&{4{l^2}}&{ - 6l}&{2{l^2}}\\ { - 12}&{ - 6l}&{12}&{ - 6l}\\ {6l}&{2{l^2}}&{ - 6l}&{4{l^2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V_1}}\\ {{v_{y1}}}\\ {{V_2}}\\ {{v_{y2}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{F_{y1}}}\\ {{M_{y1}}}\\ {{F_{y2}}}\\ {{M_{y2}}} \end{array}} \right]$ (3)

 $\beta \left[ {\begin{array}{*{20}{c}} {12}&{ - 6l}&{ - 12}&{6l}\\ { - 6l}&{4{l^2}}&{6l}&{2{l^2}}\\ { - 12}&{6l}&{12}&{6l}\\ {6l}&{2{l^2}}&{6l}&{4{l^2}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{W_1}}\\ {{v_{z1}}}\\ {{W_2}}\\ {{v_{z2}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{F_{z1}}}\\ {{M_{z1}}}\\ {{F_{z2}}}\\ {{M_{z2}}} \end{array}} \right]$ (4)

 $\mathit{\boldsymbol{k}} \times \mathit{\boldsymbol{d}} = \mathit{\boldsymbol{f}}$ (5)

 $\mathit{\boldsymbol{k}} = \left[ {\begin{array}{*{20}{c}} \gamma &0&0&0&0&0&{ - \gamma }&0&0&0&0&0\\ {}&{12\alpha }&0&0&0&{6l\alpha }&0&{ - 12\alpha }&0&0&0&{6l\alpha }\\ {}&{}&{12\beta }&0&{ - 6l\beta }&0&0&0&{ - 12\beta }&0&{ - 6l\beta }&0\\ {}&{}&{}&\delta &0&0&0&0&0&{ - \delta }&0&0\\ {}&{}&{}&{}&{4{l^2}\beta }&0&0&0&{6l\beta }&0&{2{l^2}\beta }&0\\ {}&{}&{}&{}&{}&{4{l^2}\alpha }&0&{ - 6l\alpha }&0&0&0&{2{l^2}\alpha }\\ {}&{}&{}&{}&{}&{}&\gamma &0&0&0&0&0\\ {}&{}&{}&{}&{}&{}&{}&{12\alpha }&0&0&0&{ - 6l\alpha }\\ {}&{}&{}&{}&{}&{}&{}&{}&{12\beta }&0&{6l\beta }&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&\delta &0&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{4{l^2}\beta }&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{4{l^2}\alpha } \end{array}} \right]$
 $\mathit{\boldsymbol{d}} = {\left[ {\begin{array}{*{20}{l}} {{\mathit{{u}}_1}}&{{\mathit{{V}}_1}}&{{\mathit{{W}}_1}}&{{a_{x1}}}&{{v_{y1}}}&{{v_{z1}}}&{{\mathit{{u}}_2}}&{{\mathit{{W}}_2}}&{{\mathit{{W}}_2}}&{{a_{x2}}}&{{v_{y2}}} \end{array}\;\;\;{v_{z2}}} \right]^{\rm{T}}}$
 $\mathit{\boldsymbol{f}} = {\left[ {\begin{array}{*{20}{c}} {{f_{x1}}}&{{F_{y1}}}&{{F_{z1}}}&{{M_{x1}}}&{{M_{y1}}}&{{M_{z1}}}&{{f_{x2}}}&{{F_{y2}}}&{{F_{z2}}}&{{M_{x2}}}&{{M_{y2}}}&{{M_{z2}}} \end{array}} \right]^{\rm{T}}}$

 $\mathit{\boldsymbol{k}} = \left[ {\begin{array}{*{20}{c}} {{h_1}}&0&0&0&0&0&{ - {h_1}}&0&0&0&0&0\\ {}&{{h_2}}&0&0&0&{{h_3}}&0&{ - {h_2}}&0&0&0&{{h_7}}\\ {}&{}&{{h_3}}&0&{ - {h_5}}&0&0&0&{ - {h_3}}&0&{ - {h_5}}&0\\ {}&{}&{}&{{h_4}}&0&0&0&0&0&{{h_4}}&0&0\\ {}&{}&{}&{}&{{h_6}}&0&0&0&{{h_1}}&0&{{h_9}}&0\\ {}&{}&{}&{}&{}&{{h_8}}&0&{{h_1}}&0&0&0&{{h_{10}}}\\ {}&{}&{}&{}&{}&{}&{{h_1}}&0&0&0&0&0\\ {}&{}&{}&{}&{}&{}&{}&{{h_2}}&0&0&0&{{h_7}}\\ {}&{}&{}&{}&{}&{}&{}&{}&{{h_3}}&0&{ - {h_5}}&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{{h_4}}&0&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{{h_6}}&0\\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{{h_8}} \end{array}} \right]$ (6)

$其中:{h_1} = EA/l;{h_2} = {\left( {\frac{{{l^3}}}{{12E{I_z}}} + \frac{{bl}}{{GA}}} \right)^{ - 1}};$ ${h_3} = {\left( {\frac{{{l^3}}}{{12E{I_y}}} + \frac{{bl}}{{GA}}} \right)^{ - 1}};{h_4} = GJ/l;$ $h_{5}=\left(\frac{l^{2}}{6 E I_{y}}+\frac{2 b}{G A}\right)^{-1} ; \quad h_{6}=\frac{4 E I_{y}\left(G A l^{2}+3 b E I_{y}\right)}{l\left(G A l^{2}+12 b E I_{y}\right)}$; $h_{7}=\left(\frac{l^{2}}{12 E I_{z}}+\frac{2 b}{G A}\right)^{-1} ; \quad h_{8}=\frac{4 E I_{z}\left(G A l^{2}+3 b E I_{z}\right)}{l\left(G A l^{2}+2 b E I_{z}\right)}$; ${h_9} = \frac{{2E{I_y}\left( {GA{l^2} - 6bE{I_y}} \right)}}{{l\left( {GA{l^2} + 12bE{I_y}} \right)}};\quad {h_{10}} = \frac{{2E{I_z}\left( {GA{l^2} - 6bE{I_z}} \right)}}{{l\left( {GA{l^2} + 12bE{I_z}} \right)}}。$

1.3 空间管件力学模型建立

 Download: 图 4 空间单元杆受力变形情况 Fig. 4 Space unit rod force deformation situation

 $\boldsymbol{F}_{0}=\lambda \boldsymbol{f}_{0}$ (7)

 $\boldsymbol{\lambda}=\left[\begin{array}{ccc}{\cos \langle x, X\rangle} & {\cos \langle y, X\rangle} & {\cos \langle z, X\rangle} \\ {\cos \langle x, Y\rangle} & {\cos \langle y, Y\rangle} & {\cos \langle z, Y\rangle} \\ {\cos \langle x, Z\rangle} & {\cos \langle y, Z\rangle} & {\cos \langle z, Z\rangle}\end{array}\right]$

 $\boldsymbol{d}=\boldsymbol{R}^{\mathrm{T}} \boldsymbol{D}$ (8)
 $\mathit{\boldsymbol{f}} = {\mathit{\boldsymbol{R}}^{\rm{T}}}\mathit{\boldsymbol{F}}$ (9)

 $\boldsymbol{K}=\boldsymbol{R} \boldsymbol{k} \boldsymbol{R}^{\mathrm{T}}$ (10)

 (11)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{F}}_1^{\left( i \right)} = {\left[ {\begin{array}{*{20}{c}} 0&0&{\frac{1}{2}{q_i}{l_i}}&0&0&{\frac{1}{{12}}{q_i}l_i^2} \end{array}} \right]^{\rm{T}}}\\ \mathit{\boldsymbol{F}}_2^{\left( i \right)} = {\left[ {\begin{array}{*{20}{c}} 0&0&{\frac{1}{2}{q_i}{l_i}}&0&0&{ - \frac{1}{{12}}{q_i}l_i^2} \end{array}} \right]^{\rm{T}}} \end{array} \right.$ (12)

 (13)
 ${\mathit{\boldsymbol{F}}_{{\rm{total}}}} = \left[ {\begin{array}{*{20}{c}} {F_1^{\left( 1 \right)}}\\ {F_2^{\left( 1 \right)} + F_1^{\left( 2 \right)}}\\ {F_2^{\left( 2 \right)} + F_1^{\left( 3 \right)}}\\ \vdots \\ {F_2^{\left( {i - 2} \right)} + F_1^{\left( {i - 1} \right)}}\\ {F_2^{i - 1} + F_1^{\left( i \right)}}\\ {F_2^{\left( i \right)} + F_1^{\left( {i + 1} \right)}}\\ \vdots \\ {F_2^{\left( {n - 2} \right)} + F_1^{\left( {n - 1} \right)}}\\ {F_2^{\left( {n - 1} \right)} + F_1^{\left( n \right)}}\\ {F_2^{\left( n \right)}} \end{array}} \right]$ (14)

1.4 有限元模型的精度仿真验证

 Download: 图 5 管件节点位置及旋转方向示意 Fig. 5 Tube node location and direction of rotation diagram

2 PSO优化算法的实现

2.1 粒子群优化算法基本原理

PSO粒子群算法是一种模拟鸟类觅食的多目标优化算法。每个可能解都被看做为一个粒子，群体所有粒子信息共享。在本文中，弯管的每一种支撑位姿都可以看作一个粒子，包含弯管的支撑位置和弯管的旋转角度。所有粒子都有一个由适应函数fitness()决定的适应度，在本文中适应度是指每种弯管支撑位置组合下的弯管最大变形量。PSO算法首先初始化一群随机粒子，所有粒子组成一个群体，再通过迭代更新每个粒子的速度和位置。粒子个体自身找到的最优解，称为个体最优pbest，这可以看做粒子自身的经验。整个群体到目前为止找到的最优解，称为群体最优gbest，这可以看做粒子群体的经验。在每一次更新中，粒子通过跟踪个体最优pbest和群体最优gbest来更新自己的状态[4-8]

 $\begin{array}{*{20}{c}} {v_i^{k + 1} = w \times v_i^k + {c_1} \times {\rm{rand}}\left( {\;\;} \right) \times \left( {{p_{{\rm{best}}}}_i^k - x_i^k} \right) + {c_2} \times }\\ {{\rm{rand}}\left( {\;\;} \right) \times \left( {{p_{{\rm{best}}}}^k - x_i^k} \right)x_i^{k + 1} = x_i^k + v_i^{k + 1}} \end{array}$ (14)

2.2 PSO算法应用实例

 $\boldsymbol{X}_{i}=\left[\begin{array}{llll}{n_{1}} & {n_{2}} & {n_{3}} & {a}\end{array}\right]$ (15)