﻿ 基于复合材料的水下耐压结构拓扑优化探究
 舰船科学技术  2017, Vol. 39 Issue (10): 14-21 PDF

1. 上海交通大学 海洋工程国家重点实验室，上海 200240;
2. 高新船舶与深海开发装备协同创新中心，上海 200240

Topology optimization design of underwater pressure structure with laminate composites
DAI Yang1, FENG Miao-lin1, ZHAO Min1,2
1. State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China;
2. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
Abstract: The paper studies the application of topology optimization with laminate composites in the design of underwater pressure structure. The methodology is based on the isoline method, the solid isotropic material with penalization (SIMP) model, and sensitivity filtering techniques. In addition, the equivalent element stiffness matrix for laminate composites is derived. By computing topology design cases of classical bridge-like structure, underwater structure subjected to hydrostatic pressure, and underwater pressure structure with initial void, the influence on the optimal result of structure under design-dependent loads taken by composites is analyzed. It is found that the structural optimization results of composite material and isotropic material are similar. The change of the angle and laminated type of composite layups may have a large influence on the optimal form of structure. There are similarities between the optimization results of pressure structure with initial void obtained in this paper and the concept put forward by MIT team. Therefore, the research of topology optimization with composite material will make contributions to the design of underwater pressure structure in the future.
Key words: pressure structure     topology optimization     composite material     design-dependent loads
0 引　言

 图 1 材料主向坐标系与自然坐标系 Fig. 1 Principle coordinate system of material and natural coordinate system

1 复合材料拓扑优化设计

2 拓扑优化数学模型

 ${E_e} = E\left( {{\rho _e}} \right) = \rho _e^p{E_0},0 \leqslant {\rho _e} \leqslant 1,e = 1, \cdots ,N\text{。}$ (1)

 \begin{aligned}\mathop {\min }\limits_\rho {\rm{: }}C\left( {\rho} \right) =& \displaystyle {{{U}}^{\rm T}}{{KU}} = \sum\limits_{e = 1}^N {{u}_e^{\rm T}{{k}_e}{{u}_e}}\text{，} \\& {\rm{s}}{\rm{.t}}{\rm{. : }}{{F = KU}}\text{，}\\& {\rm{: }}V\left( \rho \right)/{V_0} = volf\text{，}\\& {\rm{: 0 < }}{\rho _{\min }} \leqslant {\rho _e} \leqslant 1\text{。}\end{aligned} (2)

3 复合材料的单元刚度矩阵

 图 2 复合材料铺层模型 Fig. 2 A model laminated with composite material

 ${{D}} = \left[ {\begin{array}{*{20}{c}}{\displaystyle\frac{{{E_1}}}{{1 - {\nu _{12}}{\nu _{21}}}}}&{\displaystyle\frac{{{\nu _{12}}{E_2}}}{{1 - {\nu _{12}}{\nu _{21}}}}}&0\\[10pt]{\displaystyle\frac{{{\nu _{21}}{E_1}}}{{1 - {\nu _{12}}{\nu _{21}}}}}&{\displaystyle\frac{{{E_2}}}{{1 - {\nu _{12}}{\nu _{21}}}}}&0\\[8pt]0&0&{{G_{12}}}\end{array}} \right]\text{，}$ (3)

 $\bar { D} = { T D}{\left( { T} \right)^{\rm{T}}}\text{，}$ (4)

 ${{T}} = \left[ {\begin{array}{*{5}{c}}{{\rm {cos}}^2\theta }\quad\quad\quad {{\rm {sin}}^2\theta }\quad\quad\quad{2{\rm sin}\theta {\rm cos}\theta }\\{{\rm sin}^2\theta }\quad\quad\quad{{\rm cos}^2\theta }\quad\quad{ - 2{\rm sin}\theta {\rm cos}\theta }\\{ - {\rm sin}\theta {\rm cos}\theta }\quad\quad{{\rm sin}\theta {\rm cos}\theta }\quad\quad{{\rm cos}^2\theta - {\rm sin}^2\theta }\end{array}} \right]\text{。}$ (5)

 $\begin{array}{l}\left\{ \begin{array}{l}u\\v\end{array} \right\}{\rm{ = }}\left[ \begin{array}{l}{N_1}\;\;\;0\;\;\;{N_2}\;\;\;0\;\;\;{N_3}\;\;\;0\;\;\;\;{N_4}\;\;\;0\;\;\;\\\;0\;\;\;\;{N_1}\;\;\;0\;\;\;\;{N_2}\;\;\;0\;\;\;{N_3}\;\;\;0\;\;\;{N_4}\end{array} \right]\text{，}\\[10pt]\left\{ \begin{array}{l}{u_1}\\{v_1}\\{u_2}\\{v_2}\\{u_3}\\{v_3}\\{u_4}\\{v_4}\end{array} \right\}{\rm{ = }}\sum {{N_i}{d_i}} = Nd\text{。}\end{array}$ (6)

 ${N_i} = \left[ {\begin{array}{*{20}{c}}{{N_i}}&0\\0&{{N_i}}\end{array}} \right]\text{，}{\rm{ }}{d_i} = \left\{ {\begin{array}{*{20}{c}}{{u_i}}\\{{v_i}}\end{array}} \right\}\text{，}{\rm{ }}(i = 1,2,3,4)\text{，}$ (7)

 $\varepsilon = { L N}d = { B}d = \mathop \sum \nolimits^{ {L}} { N_i}{d_i}\text{，}$ (8)

 ${{L}} = \left[ {\begin{array}{*{20}{c}}{\displaystyle\frac{\partial }{{\partial {\rm{x}}}}}&0\\[10pt]0&{\displaystyle\frac{\partial }{{\partial {\rm{y}}}}}\\[10pt]{\displaystyle\frac{\partial }{{\partial {\rm{y}}}}}&{\displaystyle\frac{\partial }{{\partial {\rm{x}}}}}\end{array}} \right]\text{，}$ (9)

 ${\bf{\sigma }} = {{\bar { D}\varepsilon }}\text{，}$ (10)

 $U = \frac{1}{2}\int {\int_A {{\varepsilon ^{\rm{T}}}\sigma {\rm{d}}A = \frac{1}{2}d} } \left( {\int {\int_A {{{ B}^{\rm{T}}}\overline { D} {\rm{ }}{ B}dA} } } \right)d = \frac{1}{2}{d^{\rm{T}}}{ K}d\text{，}$ (11)

 $K = \int_{ - 1}^1 {\int_{ - 1}^1 {{{ B}^{\rm{T}}}{{\bar { D} B}}{\rm{det}}{{J}}{\rm{d}}\xi {\rm{d}}\eta } }\text{，}$ (12)

 ${\rm{det}}{ J} = \left| {\begin{array}{*{20}{c}}{\displaystyle\frac{{\partial {\rm{x}}}}{{\partial {\rm{\xi }}}}}&{\displaystyle\frac{{\partial {\rm{y}}}}{{\partial {\rm{\xi }}}}}\\[10pt] {\displaystyle\frac{{\partial {\rm{x}}}}{{\partial {\rm{\eta }}}}}&{\displaystyle\frac{{\partial {\rm{y}}}}{{\partial {\rm{\eta }}}}}\end{array}} \right|\text{。}$ (13)

 ${ K} = \frac{1}{n}{\rm{ }}\sum\limits_{i = 1}^n {{{ K}_i}} \text{。}$ (14)

4 灵敏度分析

 $\frac{{\rm{d}}}{{{\rm{d}}{\rho _e}}}\left[ {C\left( {{U}\left[ {{\rho _e}} \right],{\rho _e}} \right)} \right] = \frac{{\partial C}}{{\partial {\rho _e}}} + \frac{{\partial C}}{{\partial {U}}}\frac{{\partial {U}}}{{\partial {\rho _e}}}\text{。}$ (15)

 $\frac{{{\rm{d}}C}}{{{\rm{d}}{\rho _e}}} = - p{\rho ^{p - 1}}{E_0}{u}_e^{\rm T}{k}_e^0{{u}_e} + 2{{U}^{\rm T}}\frac{{\partial {F}}}{{\partial {\rho _e}}}\text{。}$ (16)

 $\frac{{\partial {F}}}{{\partial {\rho _e}}} = \frac{{\partial {F}}}{{\partial {{x}_f}}}\frac{{\partial {{x}_f}}}{{\partial {\rho _e}}} \approx \frac{{{\Delta _e}{F}}}{{\Delta {\rho _e}}}\text{，}$ (17)

 $\begin{split}\Delta eF = & F\left( {{x_f}\left( {{\rho _1},...,{\rho _e} + \Delta {\rho _e},...,{\rho _N}} \right)} \right) - \\& F\left( {{x_f}\left( {{\rho _1},...,{\rho _e},...,{\rho _N}} \right)} \right)\text{。}\end{split}$ (18)

5 数值算例

 图 3 各向同性情况下，外部受压的桥形结构计算模型及优化结果 Fig. 3 Bridge-like structure with isotropic material subjected to pressure loading
5.1 外部受压的桥形结构优化设计

 图 4 三种铺层形式下的桥形结构优化结果 Fig. 4 Optimal results of bridge-like structure with laminate composites

 图 5 静水压下耐压结构的计算模型 Fig. 5 Computing model for underwater structure subjected to hydrostatic pressure

5.2 静水压下耐压结构的优化设计

 图 6 静水压下的耐压结构优化结果 Fig. 6 Optimal results of underwater structure subjected to hydrostatic pressure

 图 7 内部铰支的空心耐压结构的计算模型 Fig. 7 Computing model for underwater structure with initial void and inside simply-supported constraints
5.3 内部铰支的空心耐压结构优化设计

 图 8 不同情形下内部铰支的耐压结构的优化结果 Fig. 8 Optimal results of underwater structure with initial void and inside simply-supported constraints

 图 9 材料各向同性多球铰支的耐压结构 Fig. 9 Optimal results of multiple intersecting spherical pressure hull with isotropic material

 图 10 MIT 提出的多球交接耐压结构模型 Fig. 10 Conceptual model of multiple intersecting spherical pressure hull

6 结　语

1）对于同一个优化设计问题，采用复合材料得到的优化结果与采用各向同性材料得到的结果在结构大体上相似，但局部结构会存在差异；

2）复合材料的铺层方式不同，优化结果也可能不同：如果是非对称铺层，优化结果很可能出现不对称的结构；如果 2 种铺层方式完全相反，两者的优化结果可能会是互成镜像；

3）对于有2个铺层的复合材料，铺层角度的变化，会改变实体材料的分布，对最优拓扑形式产生较大影响；

4）对于静水压作用下的耐压结构及空心耐压结构，复合材料的应用有可能消除或削弱原优化结果中的应力集中等不良现象。本文的研究只是对基于复合材料的拓扑优化在水下耐压结构设计上的初步应用，诸如基于复合材料的三维拓扑优化问题，以及更多形式的耐压结构设计问题等都是值得进一步探究的课题。

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