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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。

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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