﻿ 三体船横舱壁拓扑优化设计及力学分析
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 哈尔滨工程大学学报  2020, Vol. 41 Issue (6): 805-811  DOI: 10.11990/jheu.201812031 0

### 引用本文

ZHANG Cong, JIA Dejun, LI Fanchun, et al. Topology optimization design and mechanical property analysis of the transverse bulkhead of a trimaran[J]. Journal of Harbin Engineering University, 2020, 41(6): 805-811. DOI: 10.11990/jheu.201812031.

### 文章历史

Topology optimization design and mechanical property analysis of the transverse bulkhead of a trimaran
ZHANG Cong , JIA Dejun , LI Fanchun , QIN Lunyang
Ship and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China
Keywords: trimaran    bulkhead    structural strength    topology optimization    lightweight    finite element method    real ships    arrangement of stiffeners

1 拓扑优化 1.1 构建优化模型

 ${\rho _i} = \left\{ \begin{array}{l} 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in {\mathit{\Omega} ^{\rm mat}}\\ 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i \in \mathit{\Omega} /{\mathit{\Omega} ^{\rm mat}} \end{array} \right.$ (1)

 $\int_\mathit{\Omega} {{\rho _i}{\rm d}\mathit{\Omega} \le {V_0},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le {\rho _i} \le 1,i \in \mathit{\Omega} }$ (2)

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {\rm Find} \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\rho _i},i = 1,2, \cdots ,m} \end{array}\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm min}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{C}} = {\mathit{\boldsymbol{F}}^{\rm T}}\mathit{\boldsymbol{u}}} \end{array}\\ \begin{array}{*{20}{l}} {\rm s.t.} \end{array}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} V/{V_0} - f \le 0\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\kern 1pt} \mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{F}}} \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 ＜ {\rho _{\rm min}} ＜ {\rho _i} \le 1{\kern 1pt} \end{array} \right.$ (3)

1.2 求解算法与敏度分析

 ${E_i} = \rho _i^pE_i^0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,2, \cdots ,m$ (4)

 $p \ge {p^ * } = {\rm max}\left\{ {\frac{2}{{1 - {\gamma _0}}},\frac{4}{{1 + {\gamma _0}}}} \right\},2D模型$ (5)
 $p \ge {p^ * } = {\rm max}\left\{ {15\frac{{1 - {\gamma _0}}}{{7 - 5{\gamma _0}}},\frac{3}{2}\frac{{1 - {\gamma _0}}}{{1 - 2{\gamma _0}}}} \right\},3D模型$ (6)

 $\mathit{\boldsymbol{C}} = {\mathit{\boldsymbol{F}}^{\rm T}}\mathit{\boldsymbol{u}} = {\mathit{\boldsymbol{F}}^{\rm T}}\mathit{\boldsymbol{u}} - {{\mathit{\boldsymbol{\hat u}}}^{\rm T}}(\mathit{\boldsymbol{Ku}} - \mathit{\boldsymbol{F}})$ (7)

 $\frac{{\partial \mathit{\boldsymbol{C}}}}{{\partial {\rho _i}}} = ({\mathit{\boldsymbol{F}}^{\rm T}} - - {{\mathit{\boldsymbol{\hat u}}}^{\rm T}}\mathit{\boldsymbol{K}})\frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial {\rho _i}}} - {{\mathit{\boldsymbol{\hat u}}}^{\rm T}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _i}}}\mathit{\boldsymbol{u}}$ (8)

K具有对称性，通过建立伴随方程K$\mathit{\boldsymbol{\hat u}}$=F，则：

 $\frac{{\partial \mathit{\boldsymbol{C}}}}{{\partial {\rho _i}}} = - {{\mathit{\boldsymbol{\hat u}}}^{\rm T}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _i}}}\mathit{\boldsymbol{u}}$ (9)

 $\frac{{\partial \mathit{\boldsymbol{C}}}}{{\partial {\rho _i}}} = - p\rho _i^{p - 1}\mathit{\boldsymbol{u}}_i^{\rm T}\mathit{\boldsymbol{k}}_i^0{\mathit{\boldsymbol{u}}_i}$ (10)

2 三体船舱壁结构拓扑优化设计 2.1 三体船舱壁结构有限元分析

2.2 舱壁结构的拓扑优化分析

2.3 其他工况校核

2.4 实船对比验证

 Download: 图 11 Stress distribution of bulkhead under CASE5 Fig. 11 Stress cloud map after optimization under different volume constraints

3 结论

1) 通过特定工况不同体积约束条件下优化后三体船舱壁的有限元结果可知，舱壁结构应力最大值与优化区域的体积减少比例之间不存在正相关关系，通过提高体积保形率降低舱壁应力的方法并不合理。

2) 通过变密度拓扑优化方法，对三体船非水密舱壁进行拓扑优化，得到了设计域内材料的最佳布局，能够在保证结构强度的前提下，减少舱壁优化区域内50%的结构重量，实现非水密舱壁的轻量化设计。

3) 通过与实船舱壁结构的对比，优化区域体积减少30%时，舱壁优化区域内的连接材料与实船舱壁中的加强筋布置相相似，充分验证了本文所用结构拓扑优化方法的合理性。

4) 通过3组水密舱壁的对比，在CASE5下按优化结果来布置加强筋后，舱壁结构强度更好，结果表明：合理的材料布置能够有效改善结构的强度，拓扑优化技术可在特定工况下为舱壁结构的加筋布置提供指导。

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