﻿ 多工况拓扑优化算法在大型运输船高双层底实肋板结构设计中的应用
 舰船科学技术  2023, Vol. 45 Issue (23): 13-18    DOI: 10.3404/j.issn.1672-7649.2023.23.003 PDF

The application of multi-cases topology optimization algorithm in high plate floor design scheme of large transport vessel
TAO Peng, FAN Hui-li
Ship Development and Design Center, Wuhan 430064, China
Abstract: In order to solve the lack of refined degree in ship structural preliminary design process at the moment, this paper carries out the research and application exploration of normalization mathematical model based on the topology optimization algorithm. Considering the multi-cases topology optimization algorithm applied on ship structural preliminary design process, put forward the weight factor computing method based on the k-method. By the global stress restrain, the multi-objective topology optimization mathematical model involving the stress restrain function is established, and the research of multi-cases topology optimization of statics are conducted for the high plate floor design scheme. The results reveal that the final topology configuration of plate floor can satisfy the restrain conditions of the stress-strain, which has better manufacturability. The related achievements will provide a feasible way for ship structural preliminary design based on topology optimization method, which has normal academic significance and engineering application value.
Key words: topology optimization     stress restrain     ship structural design     weight factor
0 引　言

1 多工况下拓扑优化数学模型

1.1 多目标拓扑优化基本模型

 $\left\{ {\begin{array}{*{20}{l}} {\rm{Min}}\; C\left( {\rm{X}} \right) = \displaystyle\sum\limits_{j = 1}^m {{\omega _j}} {{{\boldsymbol{U}}}}_j^{\rm{T}}{{{{\boldsymbol{K}}}}_j}{{{{\boldsymbol{U}}}}_j} = \sum\limits_{j = 1}^m {{\omega _j}} {{{\boldsymbol{U}}}}_j^{\rm{T}}{{{\boldsymbol{K}}}}{{{{\boldsymbol{U}}}}_j}，\\ {\rm{s.t.}}\quad V - f{V_0} = \displaystyle\sum\limits_{i = 1}^n {{V_i}{x_i} - f{V_0} \leqslant 0}，\\ {{{{\boldsymbol{F}}}}_j} = {{{\boldsymbol{K}}}}{{{{\boldsymbol{U}}}}_j}，\;\; {j = 1,2,3...,m} ，\\ 0 < {x_{\min }} \leqslant {x_i} \leqslant 1，\;\; {i = 1,2,3...,n} ，\\ \displaystyle \sum\limits_{j = 1}^m {\omega _j} = 1, \;\;0 \leqslant {\omega _j} \leqslant 1 。\end{array}} \right.$ (1)

 $\left\{ {\begin{array}{*{20}{l}} {{\rm{Min}}\; f\left( {\boldsymbol{X}} \right) = {{\left\{ \displaystyle\sum\limits_{i = 1}^m {{\omega _i}^q{{\left[ \dfrac{{{C_i}\left( {\boldsymbol{X}} \right) - {C_i}^{\min }}}{{{C_i}^{\max } - {C_i}^{\min }}} \right]}^q}} \right\}}^{1/q}}}，\\ {\displaystyle\sum\limits_{i = 1}^m {{\omega _i} = 1，\;\; 0 \leqslant {\omega _i} \leqslant 1} }。\end{array}} \right.$ (2)

1.2 基于k方法的权因子确定方法

 $C_i^j = {C_i}\left( {{X_j}} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}，i = 1,2,3...,m；j = 1,2,3...,m，$ (3)

 $\sum\limits_{i = 1}^m {{\omega _i}\left| {\frac{{{C_i} - {C_i}^{\min }}}{{{C_i}^{\max } - {C_i}^{\min }}}} \right|} = {\omega _1}{f_1} + {\omega _2}{f_2} + ... + {\omega _m}{f_m} = k 。$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {{\omega _1}{f_1}^1 + {\omega _2}{f_2}^1 + ... + {\omega _m}f_m^1 = k}，\\ {{\omega _1}{f_1}^2 + {\omega _2}{f_2}^2 + ... + {\omega _m}f_m^2 = k} ，\\ \vdots \\ {{\omega _1}{f_1}^m + {\omega _2}{f_2}^m + ... + {\omega _m}f_m^m = k} ，\\ {{\omega _1} + {\omega _2} + ... + {\omega _m} = 1} 。\end{array}} \right.$ (5)

1.3 多工况拓扑优化模型迭代算法

 $\begin{split}L =& C\left( X \right) + {\lambda ^{\rm{T}}}\left( {{\boldsymbol{KU}} - {\boldsymbol{F}}} \right) + {\lambda _0}\left( {\sum\limits_{i = 0}^n {{V_i}{x_i} - f{V_0}} } \right) +\\ &\sum\limits_0^n {{\lambda _{1i}}} \left( {{x_{\min }} - {x_i}} \right) + \sum\limits_0^n {{\lambda _{2i}}} \left( {{x_i} - 1} \right)。\end{split}$ (6)

 $\frac{{\partial L}}{{\partial {x_i}}} = \frac{{\partial C}}{{\partial {x_i}}} + {\lambda ^T}\frac{{\partial KU}}{{\partial {x_i}}} + {\lambda _0}{V_i} - {\lambda _{1i}} + {\lambda _{2i}} = 0，$ (7)

 $\frac{{\partial L}}{{\partial {x_i}}} = \sum\limits_{j = 1}^m {\left( { - {U^T}\frac{{\partial {K_j}}}{{\partial {x_i}}}U} \right)} + {\lambda _0}{V_i} = - {U^T}\frac{{\partial K}}{{\partial {x_i}}}U + {\lambda _0}{V_i} = 0。$ (8)

 ${x_i}^{\left( {k + 1} \right)} = {B_i}^{\left( k \right)}x_i^{\left( k \right)}{\kern 1pt} {\kern 1pt} = \frac{{\displaystyle\sum\limits_{j = 1}^m {{\omega _j}{U_j}^{\rm{T}}\frac{{\partial K}}{{\partial {x_i}}}{U_j}} }}{{{\lambda _0}{V_i}}}x_i^{\left( k \right)}。$ (9)

1.4 全局化应力约束条件

 ${\hat \sigma _{\max }} \approx {\left( {\sum\limits_e {{\sigma _{veg}}^{{P_w}}} } \right)^{\frac{1}{{{P_w}}}}} \leqslant \bar \sigma 。$ (10)

Chau Le[10]在应力约束函数中引入了单元体积项Ve，即

 ${\hat \sigma _{\max }} \approx {\left( {\sum\limits_e {{V_e}{\sigma _{veg}}^{{P_w}}} } \right)^{\frac{1}{{{P_w}}}}} \leqslant \bar \sigma，$ (11)

Erik Holmberg[11]提出了成群应力约束的概念，对于第i个单元集群Φi来说，其整体应力约束为：

 ${\hat \sigma _{\max \_i}} \approx {\left( {\frac{1}{{{N_i}}}\sum\limits_{e \in {\phi _i}} {{\sigma _{veg}}^{{P_w}}} } \right)^{\frac{1}{{{P_w}}}}} \leqslant \bar \sigma ，$ (12)

${\sigma _{veg}}^{{P_w}}$ 值非常大时，目标函数会出现一定的振荡现象，难以收敛。于是需对 ${\sigma _{veg}}^{{P_w}}$ 进行标准化处理，令 ${S_{eg}} = {{{\sigma _{veg}}} /{\bar \sigma }}$ ，并给出了数学意义上等效的应力函数：

 ${S_{{P_w}}} = {\left( {\frac{1}{{{V_R}}}\sum\limits_e {{V_e}{{\left( {{S_{eg}}} \right)}^{{P_w}}}} } \right)^{\frac{1}{{{P_w}}}}}。$ (13)

 $\begin{split} \tilde T\left( X \right) = &\left\{ {\omega ^2}\left[ \sum\limits_{j = 0}^m {{\omega _j}\left[ {\frac{{{C_j}\left( X \right) - {C_j}^{\min }}}{{{C_j}^{\max } - {C_j}^{\min }}}} \right]} \right]^2 +\right. \\ &\left.{{\left( {1 - \omega } \right)}^2}{{\left[ {\sum\limits_{j = 0}^m {{\gamma _j}{S_{{P_w}j}}} } \right]}^2} \right\}^{1/2}\times \\ & \left( {\sum\limits_{j = 0}^m {{\omega _j} = 1} ,\sum\limits_{j = 0}^m {{\gamma _j} = 1} ,0 < \omega < 1} \right) 。\end{split}$ (14)

2 船舶高肋板结构拓扑优化设计实例 2.1 船舶高双层底肋板结构介绍

 图 1 双层底实肋板结构 Fig. 1 Double bottom plate floor structure

 图 2 实肋板结构有限元模型 Fig. 2 Plate floor FE model

 图 3 工况1应力云图 Fig. 3 Structure stress distribution
2.2 实肋板规范法设计

 图 4 实肋板规范法设计 Fig. 4 Plate floor standard design

 $\eta = \frac{{{V_0} - {V_r}}}{{{V_0}}} 。$ (15)

 图 5 工况1应力云图（规范法） Fig. 5 Structure stress distribution (Standard)
2.3 实肋板拓扑优化设计

 图 6 实肋板拓扑优化模型 Fig. 6 Topology optimization model of plate floor

 图 7 最终拓扑构型 Fig. 7 Final topology configuration

 图 8 结构应力云图 Fig. 8 Structure stress distribution

 图 9 体积分数迭代过程 Fig. 9 Iterations of volume fraction

 图 11 实肋板最终拓扑构型（设计） Fig. 11 Final topology configuration of plate floor (design)

 图 12 实肋板设计方案（一半） Fig. 12 Design scheme of plate floor（half）

 图 10 剩余体积分数曲线 Fig. 10 Residual volume fraction
3 结　语

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