﻿ 基于阶梯函数插值模型的复合材料层合板铺层优化设计
 舰船科学技术  2022, Vol. 44 Issue (10): 21-25    DOI: 10.3404/j.issn.1672-7649.2022.10.005 PDF

1. 北京理工大学 先进结构技术研究院，北京 100081;
2. 中国商用飞机有限责任公司北京民用飞机研究中心，北京 102211;
3. 北方车辆集团有限公司车辆研究院，北京 100072

Layer optimization design of composite laminates based on stepped function interpolation model
DING Wen-jie1, LI Meng-yu2, LIAO Hai-tao1, ZHAO Zhi-ying3, WANG Yun-kai3, LI Hua3, GENG Lin-lin3, XU Jing3
1. Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing 100081, China;
2. Beijing Aircraft Technology Research Institute of COMAC, Beijing 102211, China;
3. China North Vehicle Research Institute, Beijing 100072, China
Abstract: A stepped function interpolation model by making use of the penalty function idea of topology optimization is developed. Considering the discrete combinatorial optimization properties in the layer optimization of composite laminates. Then, the mapping relationship between composite continuous variables and discrete variables is established. The discrete variables can be applied in the optimized algorithm solution. The particle swarm algorithm based on the linear decreasing weight strategy is generalized. The discrete combination optimization design of the laminate angle is accomplished by taking into account the maximum basic frequency of the laminate. Moreover, different results obtained by the Evolution Algorithm (DE) and the Simulated annealing Algorithm (SA) are compared as well. The results reveal that the maximum fundamental frequencies optimized by the above three methods are identical, which proves the effectiveness of the proposed method.
Key words: composite material     laminates     stepped function     base frequency     layer angle
0 引　言

1 层合板基础频率分析

 图 1 复合材料层合板 Fig. 1 The structure of composite laminates

 $\begin{split}{D_{11}}\frac{{{\partial ^4}w}}{{\partial {x^4}}} +& 4{D_{16}}\frac{{{\partial ^4}w}}{{\partial {x^3}\partial y}} + 2\left( {{D_{12}} + 2{D_{66}}} \right)\frac{{{\partial ^4}w}}{{\partial {x^2}\partial {y^2}}} + \\ &4{D_{26}}\frac{{{\partial ^4}w}}{{\partial x\partial {y^3}}} + {D_{22}}\frac{{{\partial ^4}w}}{{\partial {y^4}}} = \rho h\frac{{{\partial ^2}w}}{{\partial {t^2}}} 。\end{split}$ (1)

 $\omega _1^2 = \frac{{{\text{π} ^4}}}{{\rho h}}\left[ {{D_{11}}{{\left( {\frac{1}{a}} \right)}^4} + 2\left( {{D_{12}} + 2{D_{66}}} \right)\left( {\frac{1}{a}} \right){{\left( {\frac{1}{b}} \right)}^2} + {D_{22}}{{\left( {\frac{1}{b}} \right)}^4}} \right]，$ (2)

 ${D_{ij}} = \sum\limits_{k = 1}^N {\int_{{Z_k}}^{{Z_{k + 1}}} {\overline Q _{ij}^{\left( k \right)}{z^2}{\rm{d}}z\quad i,j = 1,2,6} } 。$ (3)

 $\left\{ {\begin{array}{*{20}{l}} {{Q_{11}} = \dfrac{{{E_1}}}{{1 - {v_{12}}{v_{21}}}},{Q_{22}} = \dfrac{{{E_2}}}{{1 - {v_{12}}{v_{21}}}}} ，\\ {{Q_{12}} = \dfrac{{{v_{21}}{E_2}}}{{1 - {v_{12}}{v_{21}}}} = \dfrac{{{v_{12}}{E_1}}}{{1 - {v_{12}}{v_{21}}}},{Q_{66}} = {G_{12}}} ，\end{array}} \right.$ (4)
 $\left\{ {\begin{array}{*{20}{l}} {{{\overline Q }_{11}} = {Q_{11}}{{\cos }^4}\theta + 2\left( {{Q_{12}} + 2{Q_{66}}} \right){{\sin }^2}\theta {{\cos }^2}\theta + {Q_{22}}{{\sin }^4}\theta } ，\\ {{\overline Q }_{12}} = \left( {{Q_{11}} + {Q_{22}} - 4{Q_{66}}} \right){{\sin }^2}\theta {{\cos }^2}\theta + \\ \qquad\quad {Q_{12}}\left( {{{\sin }^4}\theta + {{\cos }^4}\theta } \right) ，\\ {{{\overline Q }_{22}} = {Q_{11}}{{\sin }^4}\theta + 2\left( {{Q_{12}} + 2{Q_{66}}} \right){{\sin }^2}\theta {{\cos }^2}\theta + {Q_{22}}{{\cos }^4}\theta }，\\ {{\overline Q }_{16}} = \left( {{Q_{11}} - {Q_{12}} - 2{Q_{66}}} \right)\sin \theta {{\cos }^3}\theta +\\ \qquad \quad\left( {{Q_{12}} - {Q_{22}} + 2{Q_{66}}} \right){{\sin }^3}\theta \cos \theta ，\\ {{\overline Q }_{26}} = \left( {{Q_{11}} - {Q_{12}} - 2{Q_{66}}} \right){{\sin }^3}\theta \cos \theta + \\ \qquad\quad \left( {{Q_{12}} - {Q_{22}} + 2{Q_{66}}} \right)\sin \theta {{\cos }^3}\theta ，\\ {{\overline Q }_{66}} = \left( {{Q_{11}} + {Q_{22}} - 2{Q_{12}} - 2{Q_{66}}} \right){{\sin }^2}\theta {{\cos }^2}\theta + \\ \qquad\quad {Q_{66}}\left( {{{\sin }^4}\theta + {{\cos }^4}\theta } \right)。\end{array}} \right.$ (5)

2 优化方案 2.1 阶梯插值模型

 $E(x|\xi ) = {E_0}{\text{ + }}\sum\limits_{i = 1}^N {{{\left\{ {H\left[ {f(x|\xi ) - u(i)} \right]} \right\}}^m}\left( {{E_i} - {E_{i - 1}}} \right)}，$ (6)
 $D(x|\xi ) = {D_0}{\text{ + }}\sum\limits_{i = 1}^N {{{\left\{ {H\left[ {f(x|\xi ) - u(i)} \right]} \right\}}^m}\left( {{D_i} - {D_{i - 1}}} \right)}。$ (7)

 $H(\Delta ) = \left\{ {\begin{array}{*{20}{c}} {1,\vartriangle \geqslant 0} ，\\ {0,\vartriangle < 0} 。\end{array}} \right.$ (8)

 $H\left( \Delta \right) = \frac{1}{{{\text{1 + }}{{\text{e}}^{ - \beta \cdot \Delta }}}}。$ (9)

 图 2 阶梯形插值模型函数关系 Fig. 2 The function relationship of stepped interpolation model

 图 3 复合材料结构阶梯形插值模型铺层力学设计研究方案 Fig. 3 The framwork of mechanical design for lamination of composite structure with stepped interpolation model

2.2 粒子群优化算法

 ${X_i} = \left( {{x_{i1}},{x_{i2}}, \cdot \cdot \cdot ,{x_{iD}}} \right),i = 1,2, \cdots N ，$ (10)

 ${V_i} = \left( {{v_{i1}},{v_{i2}}, \cdots {v_{iD}}} \right),i = 1,2, \cdots N ，$ (11)

i个粒子目前搜索的最优位置（个体极值）为：

 ${P_{besti}} = \left( {{p_{i1}},{p_{i2}}, \cdots {p_{iD}}} \right),i = 1,2, \cdots N，$ (12)

 ${g_{best}} = \left( {{g_1},{g_2}, \cdots {g_D}} \right)，$ (13)

 ${V_{i + 1}} = \omega {V_i} + {c_1}{r_1}\left( {{p_{besti}} - {x_i}} \right) + {c_2}{r_2}\left( {{g_{besti}} - {x_i}} \right)。$ (14)

$\omega$ 为惯性因子，其数值大小代表着算法的全局寻优能力。 $\omega$ 越大，全局寻优能力越强，局部寻优能力越弱。当 $\omega$ 在0.8~1.2之间时，综合收敛速度较为良好；当 $\omega > 1.2$ 时，容易陷入局部机制。动态 $\omega$ 比固定值有着更好的寻优效果，本文采取线性递减权值策略， $\omega$ 动态公式为：

 ${\omega ^{\left( i \right)}} = \left( {{\omega _{ini}} - {\omega _{end}}} \right)\left( {{G_k} - g} \right)/{G_k} + {\omega _{end}}。$ (15)

3 优化问题描述

 $\begin{split} &\min {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {w_1}(\theta ) \\ & {\text{s.t.}} - {\kern 1pt} \;{90^ \circ } \leqslant {\theta _i} \leqslant {90^ \circ }，\;\;i = 1,2, \cdots ,n ，\\ & \theta = {[{\theta _1}\;{\theta _2}\; \cdots {\theta _{i - 1}}\;{\theta _i}\;{\theta _{i + 1}}\; \cdots {\theta _n}]^{\text{T}}}。\end{split}$ (16)

1）材料选取玻璃纤维/环氧树脂，总层数n为16，纤维铺层角度在[−90,90]内，以45°为增量进行选择。

2）材料选取亚麻纤维/环氧树脂，总层数n为16，纤维铺层角度在[−90,90]内，以45°为增量进行选择。

3）材料选取碳纤维/环氧树脂，总层n为8，纤维铺层角度在[−90,90]内，以15°为增量进行选择。

 图 4 问题1和问题2阶梯插值模型 Fig. 4 Interpolation model of problem1 and problem2

 图 5 问题3阶梯插值模型 Fig. 5 Interpolation model of problem3
4 数值算例

5 结　语

1）提出了阶梯插值模型，可以将复合材料连续变量转化为离散物理量，进一步嵌入到优化算法中求解。

2）以铺层角度为设计变量，基于提出的插值模型和粒子群搜索算法建立了对称复合材料层合板频率特性优化的离散组合设计方法，实现了层合板频率优化问题的铺层设计，与文献[12]结果比较说明，基于阶梯插值模型的粒子群优化算法具有更好的全局搜索能力和计算稳定性，适合求解复合材料层合板的铺层优化问题。

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