﻿ 考虑承载面最大位移约束的结构拓扑优化方法
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1. 华北电力大学 新能源电力系统国家重点实验室, 北京 102206;
2. 合肥工业大学 土木与水利工程学院, 合肥 230009

Structural topology optimization method with maximum displacement constraint on load-bearing surface
LONG Kai1, CHEN Zhuo1, GU Chunlu1, WANG Xuan2
1. State Key Laboratory for Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China;
2. College of Civil Engineering, Hefei University of Technology, Hefei 230009, China
Abstract: In practical engineering problems, there exists a class of engineering structures sustaining distributed force. To satisfy the requirement of the stiffness design for this kind of structure, a new topological design formulation is proposed to restrict the deflection on the load-bearing surface. The KS aggregation function is applied to integrate a mass of the displacement constraints on load-bearing surface into one single constraint. The corresponding adjoin equation and sensitivity expressions are conducted. Following the construction of the independent continuous mapping method, the explicit expressions of the objective function and constraint function are obtained by first-order and second-order Taylor expansion. Consequently, the optimization problem is transformed into a series of standard quadratic programs, which can be solved efficiently and robustly using the sequential quadratic programming. The feasibility and effectiveness of the proposed method are then verified by 2D and 3D numerical examples. The optimized results clearly demonstrate that the proposed formulation and corresponding optimization algorithm can effectively control the maximum deflection of local region.
Keywords: topology optimization    continuum structure    maximum displacement constraint    independent continuous mapping method    sequential quadratic programming

1 体积比约束下的拓扑优化列式

 $\begin{array}{l} {\rm{find}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\rho }}\\ {\rm{minimize}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} c = {\mathit{\boldsymbol{F}}^{\rm{T}}}\mathit{\boldsymbol{u}}\\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}\left\{ {\begin{array}{*{20}{l}} {V(\mathit{\boldsymbol{\rho }}) \le f{V_0}}\\ {\mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{F}}}\\ {0 < {\rho _{{\rm{min}}}} \le {\rho _e} \le 1\quad e = 1,2, \cdots ,{N_{\rm{E}}}} \end{array}} \right. \end{array}$ （1）

SIMP插值模型通过建立弹性模量(或刚度阵)与密度变量的假设关系，驱使在优化过程中密度变量向0或1靠近，其表达式为[1]

 ${\mathit{\boldsymbol{k}}_e} = \rho _e^p\mathit{\boldsymbol{k}}_e^0$ （2）

 $\frac{{\partial c}}{{\partial {\rho _e}}} = - {\mathit{\boldsymbol{U}}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _e}}}\mathit{\boldsymbol{U}} = - \mathit{\boldsymbol{u}}_e^{\rm{T}}\frac{{\partial {\mathit{\boldsymbol{k}}_e}}}{{\partial {\rho _e}}}{\mathit{\boldsymbol{u}}_e}$ （3）

2 考虑承载面最大位移约束的拓扑优化列式 2.1 拓扑优化列式

 $\begin{array}{l} {\rm{find}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\rho }}\\ {\rm{minimize}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} V/{V_0}\\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}\left\{ {\begin{array}{*{20}{l}} {{d_j} \le \bar d{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,J}\\ {\mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{F}}}\\ {0 < {\rho _{{\rm{min}}}} \le {\rho _e} \le 1\quad e = 1,2, \cdots ,{N_{\rm{E}}}} \end{array}} \right. \end{array}$ （4）

 $\begin{array}{l} {\rm{find}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathit{\boldsymbol{\rho }}\\ {\rm{minimize}}:{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} V/{V_0}\\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}\left\{ {\begin{array}{*{20}{l}} {{\rm{max}}\left( {{d_j}} \right) \le \bar d{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,J}\\ {\mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{F}}}\\ {0 < {\rho _{{\rm{min}}}} \le {\rho _e} \le 1\quad e = 1,2, \cdots ,{N_{\rm{E}}}} \end{array}} \right. \end{array}$ （5）

 ${d^{{\rm{KS}}}} = \frac{1}{\mu }{\rm{ln}}\left( {\sum\limits_j {{{\rm{e}}^{\mu \frac{{{d_j}}}{{\bar d}}}}} } \right) \le 1$ （6）

 $\mathop {{\rm{lim}}}\limits_{\mu \to \infty } {d^{{\rm{KS}}}} = \mathop {{\rm{lim}}}\limits_{\mu \to \infty } \frac{1}{\mu }{\rm{ln}}\left( {\sum\limits_j {{{\rm{e}}^{\mu \frac{{{d_j}}}{{\bar d}}}}} } \right) = 1$ （7）

 ${\tilde d^{{\rm{KS}}}} = {c^{{\rm{KS}}}} \cdot {d^{{\rm{KS}}}} \le 1$ （8）

 ${c^{{\rm{KS}}}} = \frac{{{\rm{max}}({d_j})}}{{\bar d \cdot {d^{{\rm{KS}}}}}}$ （9）
2.2 敏度分析

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}} = \frac{{{{\rm{e}}^{\mu \frac{{{d_j}}}{{\bar d}}}}}}{{\bar d\sum\limits_i {{{\rm{e}}^{\mu \frac{{{d_j}}}{{\bar d}}}}} }}$ （10）

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {\rho _e}}} = \sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot \frac{{\partial {d_j}}}{{\partial {\rho _e}}}$ （11）

 $\left\{ {\begin{array}{*{20}{l}} {{d_j} = \mathit{\boldsymbol{\alpha }}_j^{\rm{T}}\mathit{\boldsymbol{u}}}\\ {\mathit{\boldsymbol{\alpha }}_j^{\rm{T}} = \left[ {\begin{array}{*{20}{l}} 0&0& \cdots &1&0& \cdots \end{array}} \right]} \end{array}} \right.$ （12）

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {\rho _e}}} = \sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot \frac{{\partial \mathit{\boldsymbol{\alpha }}_j^{\rm{T}}\mathit{\boldsymbol{u}}}}{{\partial {\rho _e}}} = \sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot \mathit{\boldsymbol{\alpha }}_j^{\rm{T}}\frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial {\rho _e}}}$ （13）

 $\mathit{\boldsymbol{K}}\frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial {\rho _e}}} + \frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _e}}}\mathit{\boldsymbol{u}} = \frac{{\partial \mathit{\boldsymbol{F}}}}{{\partial {\rho _e}}}$ （14）

 $\frac{{\partial \mathit{\boldsymbol{u}}}}{{\partial {\rho _e}}} = - {\mathit{\boldsymbol{K}}^{ - 1}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _e}}}\mathit{\boldsymbol{u}}$ （15）

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {\rho _e}}} = \sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot \mathit{\boldsymbol{\alpha }}_j^{\rm{T}}\left( { - {\mathit{\boldsymbol{K}}^{ - 1}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _e}}}\mathit{\boldsymbol{u}}} \right)$ （16）

 $\mathit{\boldsymbol{K\lambda }} = {\left[ {\sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot \mathit{\boldsymbol{\alpha }}_j^{\rm{T}}} \right]^{\rm{T}}} = \sum\limits_j {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {d_j}}}} \cdot {\mathit{\boldsymbol{\alpha }}_j}$ （17）

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {\rho _e}}} = - {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {\rho _e}}}\mathit{\boldsymbol{u}}$ （18）

2.3 独立连续映射法显式化过程

 ${x_e} = \frac{1}{{\rho _e^p}}$ （19）

 ${\frac{{\partial V}}{{\partial {x_e}}} = - \frac{1}{p}x_e^{ - (1/p + 1)}{v_e}}$ （20a）
 ${\frac{{{\partial ^2}V}}{{\partial x_e^2}} = \frac{{p + 1}}{{{p^2}}}x_e^{ - (1/p + 2)}{v_e}}$ （20b）

 $\begin{array}{l} \mathit{\boldsymbol{H}} = \\ \frac{{p + 1}}{{{p^2}}}\left[ {\begin{array}{*{20}{c}} {x_1^ - (\frac{1}{p} + 2){v_1}}&0& \cdots &0\\ 0&{x_2^ - (\frac{1}{p} + 2){v_2}}& \cdots &0\\ \vdots & \vdots &{}& \vdots \\ 0&0& \cdots &{x_{{N_{\rm{E}}}}^{ - (1/\alpha + 2)}{v_{{N_{\rm{E}}}}}} \end{array}} \right] \end{array}$ （21）

 $\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {x_e}}} = - {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}\frac{{\partial \mathit{\boldsymbol{K}}}}{{\partial {x_e}}}\mathit{\boldsymbol{u}} = \mathit{\boldsymbol{\lambda }}_e^{\rm{T}}\frac{{{\mathit{\boldsymbol{k}}_e}}}{{{x_e}}}{\mathit{\boldsymbol{u}}_e}$ （22）

 ${c^{{\rm{KS}}}} \cdot \left[ {d_0^{{\rm{KS}}} + {{\sum\limits_{e = 1}^{{N_{\rm{E}}}} {\left. {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {x_e}}}} \right|} }_{\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{x}}^{(l)}}}}({x_e} - x_e^{(l)})} \right] \le 1$ （23）

 $\begin{array}{l} {\rm{find}}:\mathit{\boldsymbol{x}}\\ {\rm{minimize}}:{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{x}} + \frac{1}{2}{\mathit{\boldsymbol{x}}^{\rm{T}}}\mathit{\boldsymbol{Hx}}\\ {\rm{s}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{t}}{\rm{.}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{Ku}} = \mathit{\boldsymbol{F}}}\\ {{c^{{\rm{KS}}}} \cdot \left[ {d_0^{{\rm{KS}}} + {{\sum\limits_{e = 1}^{{N_{\rm{E}}}} {\left. {\frac{{\partial {d^{{\rm{KS}}}}}}{{\partial {x_e}}}} \right|} }_{\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{x}}^{(l)}}}}({x_e} - x_e^{(l)})} \right] \le 1}\\ {1 \le {x_e} \le \rho _{{\rm{min}}}^{ - \alpha }\quad e = 1,2, \cdots ,{N_{\rm{E}}}} \end{array}} \right. \end{array}$ （24）

 $|{V^{(l)}} - {V^{(l - 1)}}|/{V^{(l)}} \le \varepsilon$ （25）

3 数值算例与讨论

 图 1 长悬臂梁结构示意图 Fig. 1 Illustration of long cantilever structure
 图 2 所提方法的拓扑优化结果 Fig. 2 Optimized topology obtained from the proposed method

 图 3 采用拓扑优化列式(5)和MMA算法的拓扑结果 Fig. 3 Optimized topology obtained from Eq.(5) and MMA algorithm

 图 4 拓扑优化列式(1)下的拓扑优化结果 Fig. 4 Optimized topology using Eq.(1)

 图 5 平面结构示意图 Fig. 5 Illustration of plane structure
 图 6 所提列式下的拓扑优化结果与变形 Fig. 6 Optimized topology obtained from the proposed equation

 图 7 多点位移约束下的拓扑优化结果 Fig. 7 Optimized topology obtained from multiple nodal displacement constraints

 图 8 三维结构示意图 Fig. 8 Illustration of 3D structure
 图 9 不同列式下的拓扑优化结果 Fig. 9 Optimized topology obtained from different equations

4 结论

1) 与传统的体积比约束下柔顺度最小化列式对比，提出方法具有控制局部区域最大位移的效果。优化结果证明了提出方法的有效性。

2) 与多点位移约束对比，提出方法更精确、有效控制承载面的最大变形量。且将多点位移约束凝聚为单一约束，不仅减少了伴随工况计算工作量，且有利于优化求解。

3) 提出的优化列式和相应求解算法有望在几何非线性位移约束问题中进一步拓展。

 [1] MAUTE K, SIGMUND O. Topology optimization approaches[J]. Structural and Multidisciplinary Optimization, 2013, 48(6): 1-25. Click to display the text [2] DEATON J D, GRANDHI R V. A survey of structural and multidisciplinary continuum topology optimization:Post 2000[J]. Structural and Multidisciplinary Optimization, 2014, 49(1): 1-38. Click to display the text [3] ZHU J, ZHANG W, XIA L. Topology optimization in aircraft and aerospace structures[J]. Archives of Computational Methods in Engineering, 2016, 23(4): 595-622. Click to display the text [4] RONG J H, YI J H. A structural topological optimization method for multi-displacement constraints and any initial topology configuration[J]. Acta Mechanica Sinica, 2010, 26(5): 735-744. Click to display the text [5] 俞燎宏, 荣见华, 唐承铁, 等. 基于可行域调整的多相材料结构拓扑优化设计[J]. 航空学报, 2018, 39(9): 222023. YU L H, RONG J H, TANG C T, et al. Multi-phase material structural topology optimization design based on feasible domain adjustment[J]. Acta Aeronautica et Astronautica Sinica, 2018, 39(9): 222023. (in Chinese) Cited By in Cnki | Click to display the text [6] HUANG X, XIE Y M. Evolutionary topology optimization of continuum structures with an additional displacement constraint[J]. Structural and Multidisciplinary Optimization, 2010, 40(1-6): 409-416. Click to display the text [7] ZUO Z H, XIE Y M. Evolutionary topology optimization of continuum structures with a global displacement control[J]. Computer Aided Design, 2014, 56: 58-67. Click to display the text [8] QIAO H T, LIU S. Topology optimization by minimizing the geometric average displacement[J]. Engineering Optimization, 2013, 45(1): 1-18. Click to display the text [9] LI D, KIM I Y. Multi-material topology optimization for practical lightweight design[J]. Structural and Multidisciplinary Optimization, 2018, 58(3): 1081-1094. Click to display the text [10] LONG K, WANG X, GU X. Local optimum in multi-material topology optimization and solution by reciprocal variables[J]. Structural and Multidisciplinary Optimization, 2018, 57(3): 1283-1295. Click to display the text [11] ZHU J, LI Y, ZHANG W, et al. Shape preserving design with structural topology optimization[J]. Structural and Multidisciplinary Optimization, 2016, 53: 893-906. Click to display the text [12] LI Y, ZHU J, ZHANG W, et al. Structural topology optimization for directional deformation behavior design with orthotropic artificial weak element method[J]. Structural and Multidisciplinary Optimization, 2018, 57(3): 1251-1266. Click to display the text [13] LI Y, ZHU J, WANG F, et al. Shape preserving design of geometrically nonlinear structures using topology optimization[J]. Structural and Multidisciplinary Optimization, 2019, 59(4): 1033-1051. Click to display the text [14] CASTRO M S, SILVA O M, LENZI A, et al. Shape preserving design of vibrating structures using topology optimization[J]. Structural and Multidisciplinary Optimization, 2018, 58(3): 1109-1119. Click to display the text [15] 隋允康. 建模变换优化-结构综合方法新进展[M]. 大连: 大连理工大学出版社, 1996. SUI Y K. Modelling, transformation and optimization-new developments of structural synthesis method[M]. Dalian: Dalian University of Technology Press, 1996. (in Chinese) [16] 隋允康, 叶红玲. 连续体结构拓扑优化的ICM方法[M]. 北京: 科学出版社, 2013. SUI Y K, YE H L. Continuum topology optimization methods ICM[M]. Beijing: Science Press, 2013. (in Chinese) [17] SIGMUND O. A 99 line topology optimization code written in Matlab[J]. Structural and Multidisciplinary Optimization, 2001, 21(2): 120-127. Click to display the text [18] FLEURY C, BRAIBANT V. Structural optimization:A new dual method using mixed variables[J]. International Journal for Numerical Methods in Engineering, 1986, 23: 409-428. Click to display the text [19] SVANBERG K. The method of moving asymptotes-a new method for structural optimization[J]. International Journal for Numerical Methods in Engineering, 1987, 24: 359-373. Click to display the text [20] BRUYNEEL M, DUYSINX P, FLEURY C. A family of MMA approximations for structural optimization[J]. Structural and Multidisciplinary Optimization, 2002, 24(4): 263-276. Click to display the text [21] CHENG G D, GUO X. ε-relaxed approach in structural topology optimization[J]. Structural Optimization, 1997, 13(4): 258-266. Click to display the text [22] DUYSINX P, BENDSE M P. Topology optimization of continuum structures with local stress constraints[J]. International Journal for Numerical Methods in Engineering, 1998, 43(8): 1453-1478. Click to display the text [23] YANG D, LIU H, ZHANG W, et al. Stress-constrained topology optimization based on maximum stress measures[J]. Computers and Structures, 2018, 198: 23-29. Click to display the text [24] LONG K, WANG X, LIU H. Stress-constrained topol ogy optimization of continuum structures subjected to harmonic force excitation using sequential quadratic programming[J]. Structural and Multidisciplinary Optimization, 2019, 59(5): 1747-1759. Click to display the text [25] 龙凯, 谷先广, 王选. 基于多相材料的连续体结构动态轻量化设计方法[J]. 航空学报, 2017, 38(10): 221022. LONG K, GU X G, WANG X. Lightweight design method for continuum structure under vibration using multiphase materials[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(10): 221022. (in Chinese) Cited By in Cnki (4) | Click to display the text
http://dx.doi.org/10.7527/S1000-6893.2019.23577

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#### 文章信息

LONG Kai, CHEN Zhuo, GU Chunlu, WANG Xuan

Structural topology optimization method with maximum displacement constraint on load-bearing surface

Acta Aeronautica et Astronautica Sinica, 2020, 41(7): 223577.
http://dx.doi.org/10.7527/S1000-6893.2019.23577