考虑可控性的压电作动器拓扑优化设计 1)

图1压电层合板示意图

Fig. 1Schematic diagram for piezoelectric laminated plate

由于压电层合板结构包含基体层和压电作动器层,因此结构整体的刚度阵和质量阵可进一步表达为${\pmb M} = {\pmb M}_{h} + {\pmb M}_{a} $和${\pmb K} = {\pmb K}_{h} + {\pmb K}_{a} $,其中下标h和a分别表示基体层和作动器层.

直接求解方程(1)的计算量很大,而对于一般的工程结构,振动能量主要集中在低阶振动模态中,并且由于主动控制主要应对低频振动,故引入常用的模态叠加法在保证精度的情况下降低计算量^[35]. 结构的位移可近似为

(2)$$ {\pmb x}(t) \approx {\pmb \varPsi} {\pmb \eta} (t) $$

式中,${\pmb \varPsi} = \left\{ { {\pmb\psi}_1 ,\;{\pmb\psi}_2 ,\; \cdots ,\;{\pmb\psi}_m } \right\}$和${\pmb\eta} = \left\{ {\eta _1 ,\;\eta _2 ,\; \cdots ,\;\eta _m } \right\}^{ T}$分别为对应的无阻尼系统的前$m$阶归一化模态矩阵和模态位移向量. 模态矩阵${\pmb \varPsi} $可通过求解如下的特征值问题获得

(3)$$ \left( { {\pmb K}-\omega _i^2 {\pmb M}} \right){\pmb\psi}_i = {\bf 0} $$

式中,$i = 1, 2, \cdots , m$;$\omega _i $为特征向量${\pmb\psi}_i $对应的圆频率.

对于低频振动为主的问题,也可采用模态加速法^[36-38]在一定程度上提高响应分析的精度.

假设结构的整体阻尼采用比例阻尼模型描述,则有

(4)$$ \tilde{\pmb C \ } = {\pmb \varPsi }^{T}{\pmb C}{\pmb \varPsi }= 2{\pmb Z}_{d} {\pmb \varOmega}_{ f} $$

式中,$\tilde{\pmb C \ } \in {\pmb R}^{m\times m}$为解耦的阻尼矩阵;${\pmb Z}_{d} \in {\pmb R}^{m\times m}$ 为阻尼系数组成的对角阵;${\pmb \varOmega}_{f} $为圆频率组成的对角矩阵,即${\pmb \varOmega} _{f} ={diag}\left( {\omega _1 , \omega _2 , \cdots , \omega _m } \right)$.

将方程(2)代入方程(1)并左乘${\pmb \varPsi}^{T}$,可得

(5)$$ \ddot {\pmb \eta }(t) + 2{\pmb Z}_{d} {\pmb \varOmega}_{f} \dot {\pmb \eta }(t) + {\pmb \varOmega}_{f}^2 {\pmb \eta} (t) = \\ \qquad {\pmb \varPsi}^{T}{\pmb f}_{a} (t) + {\pmb \varPsi}^{T}{\pmb f}(t) $$

压电控制力由施加在压电片上的电压产生,并由逆压电效应决定,满足如下关系式

(6)$$ {\pmb f}_{a} (t) =-{\pmb K}_{{u}\varphi } {\pmb \varPhi}_{a} \left( t \right) $$

式中,${\pmb \varPhi}_{a} \left( t \right)$为施加在压电作动器上的主动控制电压向量;${\pmb K}_{{u}\varphi }$是压电作动器的力电耦合矩阵;${\pmb K}_{{u}\varphi } $可表达为${\pmb K}_{{u}\varphi }= \int_{\varOmega _{a} } {\pmb B}_{u}^{T} e^{T}{\pmb B}_\varphi d\varOmega $,其中${\pmb B}_{u} $,${\pmb e}$,${\pmb B}_\varphi $和$\varOmega _{a} $分别是应变位移矩阵、压电系数矩阵、电场电压矩阵和作动器的体积域.

因此方程(5)可改写成如下振动控制方程

(7)$$ \ddot {\pmb \eta }(t) + 2{\pmb Z}_{d} {\pmb \varOmega}_{f} \dot {\pmb \eta }(t) + {\pmb \varOmega}_{f}^2 {\pmb \eta} (t) = {\pmb B}{\pmb \varPhi}_{a} (t) + {\pmb \varPsi}^{T}{\pmb f}(t) $$

式中,${\pmb B}$为控制矩阵并有${\pmb B} =-{\pmb \varPsi}^{T}{\pmb K}_{{u}\varphi } $.

2 可控性指标和拓扑优化模型

2.1 基于控制矩阵奇异值的可控性指标

压电控制力的内积可写为

(8)$$ {\pmb f}_{a} ^{T}{\pmb f}_{a} = {\pmb \varPhi}_{a} ^{T}{\pmb B}^{T}{\pmb B}{\pmb \varPhi }_{a} $$

对控制矩阵${\pmb B}$进行奇异值分解有

(9)$$ {\pmb B} ={\pmb M}{\pmb S}{\pmb N}^{T} $$

式中,${\pmb M}$和${\pmb N}$分别是控制矩阵的左右奇异向量;有${\pmb M} \in {\pmb R}^{m\times m}$,${\pmb M}^{T}{\pmb M} ={\pmb I}_m $,${\pmb N} \in {\pmb R}^{k\times k}$,${\pmb N}^{T}{\pmb N} = {\pmb I}_k $和

(10)$$ {\pmb S} = \left[ \!\! \begin{array}{cccc} {\sigma _1 } & { 0} & { 0} & {\bf 0} \\ { 0} & \ddots & { 0} & {\bf 0} \\ { 0} & { 0} & {\sigma _m } & {\bf 0} \end{array} \!\! \right] \in {\pmb R}^{m\times k} $$

式中,$\sigma _i \;\left( {i = 1,\;2,\; \cdots ,\;m} \right)$为矩阵${\pmb S}$的奇异值;$k$为作动器的数量. 这里假设作动器的数量比减缩的模态数量多,即$k > m$.

方程(8)可改写为

(11)$$ {\pmb f}_{a} ^{T}{\pmb f}_{a} ={\pmb \varPhi} _{a} ^{T}{\pmb N} {\pmb S}^2{\pmb N}^{ T}{\pmb \varPhi} _{a} $$

引入${\pmb \varPhi} = {\pmb N}^{T}{\pmb \varPhi}_{a} $可得(观察可知${\pmb \varPhi}^2 = {\pmb \varPhi}_{a}^2$)

(12)$$ {\pmb f}_{a}^{T}{\pmb f}_{a} = {\pmb \varPhi}^{T}{\pmb S}^2{\pmb \varPhi} $$

从能量的角度看,${\pmb \varPhi}$和${\pmb \varPhi}_{a} $是等价的,因此有

(13)$$ {\pmb f}_{a}^{T}{\pmb f}_{a}= \sum_{i = 1}^m \sigma _i^2 \phi _i^2 $$

式中,$\sigma _i $为矩阵${\pmb S}$的第$i$个奇异值;$\phi _i $为向量${\pmb \varPhi} $的第$i$个分量.

观察方程(13)可知,可通过提高矩阵${\pmb S}$的奇异值来提高系统的控制能量. 因此选取可控性指标(CI)为^[29]

(14)$$ W_{CI} = \prod\limits_{i = 1}^m {\sigma _i } $$

需要说明的是,以上的公式推导考虑了所有前$m$阶模态的可控性;若只考虑其中几阶模态的可控性,公式将有细微的改变,此处省略.

2.2 拓扑优化列式

本文以提高压电智能结构的可控性为目标设计压电作动器的布局,选取压电作动器层为设计区域,并假设基体层不变. 以压电材料的总用量为约束,作动器层单元的相对密度为设计变量. 由于迭代过程中可控性指标(CI)的数值将发生很大变化,故采用其对数形式为目标并最小化其负数. 优化列式为

(15)$$ \left.\!\!\begin{array}{ll} {\mathop {\min }\limits_\rho \;} & {J\left( \rho \right) =-\lg \prod\limits_{i = 1}^m {\sigma _i } } \\ {s.t.}\;\;\; & {\left( {K-\omega _i^2 M} \right)\psi _i = 0} , \ \ i = 1, 2, \cdots , m \\ & {\sum_{e = 1}^{N_{e} } {\rho _e V_e-f_{v} \sum_{e = 1}^{N_{e} } {V_e } } \leqslant 0} \\ & 0 < \rho _{\min } \leqslant \rho _e \leqslant 1 , \ \ e = 1, 2, \cdots , N_{e} \end{array}\!\!\right\} $$

式中,$\rho _e $为第$e$个压电单元的相对密度;$V_e $为第$e$个压电单元的体积;$f_{v} $为体积分数;$N_{e} $为单元的总数;$\rho _{\min } = 10^{-6}$为设计变量的下限.

采用SIMP模型,第$e$个压电单元的质量阵和刚度阵可表示为

(16)$$ \left.\begin{array} {\pmb M}_{a}^e = \rho _e \tilde {\pmb M \ }_{a}^e \\ {\pmb K}_{a}^e = (\rho _e )^{p_1 }\tilde{\pmb K \ }_{a}^e \end{array}\right\} $$

式中,$\tilde {\pmb M \ }_{a}^e $和$\tilde {\pmb K \ }_{a}^e $为相对密度为1时压电单元的质量阵和刚度阵;$p_{1} = 3$为惩罚系数. 由于基层板在优化过程中不变,低密度压电单元处将不会导致局部模态现象^[15].

类似于SIMP模型,引入惩罚模型对中间密度值的压电单元的压电系数进行惩罚,则有^[25]

(17)$$ {\pmb e} = \rho _e^{p_2 } \tilde {\pmb e} $$

式中,$\tilde {\pmb e}$为相对密度为1时压电材料的压电系数矩阵;$p_{2}$为压电系数的惩罚系数,本研究中取$p_{2} =1$.

3 灵敏度分析

由于优化模型的求解基于梯度优化算法,因此需要对目标函数关于设计变量做灵敏度分析. 为了提高计算效率,采用伴随变量法^[34,38-39]进行求解.

对方程(15)中的目标函数$J$关于设计变量$\rho _e $求导,可得

(18)$$ \begin{array} \dfrac{ dJ}{ d\rho _e } =-\dfrac{ d\lg \Big (\prod\limits_{i = 1}^m {\sigma _i }\Big) }{ d\rho _e } = \\ \qquad-\dfrac{1}{\prod\limits_{i = 1}^m {\sigma _i } }\left( {\sum_{i = 1}^m {\dfrac{\partial \sigma _i }{ d\rho _e }} \prod\limits_{j = 1}^m {\sigma _j } } \right) , \ \ j \ne i \end{array} $$

上式包含奇异值对设计变量的导数. 通过求解${\pmb W} = {\pmb B}{\pmb B}^{T}$矩阵的特征值$\lambda _i = \sigma _i^2 $的导数可得到奇异值$\sigma_i $对设计变量的导数. ${\pmb W}$矩阵的特征方程为

(19)$$ \left( {{\pmb W}-\lambda _i {\pmb I} } \right) {\pmb \varphi} _i = {\bf 0} $$

对方程(19)关于$\rho _e $求导,并左乘${\pmb \varphi}_i^{T} $可得

(20)$$ {\pmb \varphi}_i^{T} \left( {\dfrac{\partial {\pmb W}}{\partial \rho _e }-\dfrac{\partial \lambda _i }{\partial \rho _e }{\pmb I}} \right) {\pmb \varphi}_i + {\pmb \varphi}_i^{T} \left( {{\pmb W }- \lambda _i {\pmb I}} \right)\dfrac{\partial {\pmb \varphi}_i }{\partial \rho _e } = 0 $$

由于$\left( {{\pmb W}-\lambda _i {\pmb I}} \right)$是对称矩阵,因此方程(20)的第2项为0,可得

(21)$$ \dfrac{\partial \lambda _i }{\partial \rho _e } = {\pmb \varphi}_i^{T} \dfrac{\partial {\pmb W}}{\partial \rho _e }{\pmb \varphi}_i $$

对关系式$\sigma _i^ = \sqrt {\lambda _i } $求导,并引入方程(21),则有

(22)$$ \begin{array}\dfrac{\partial \sigma _i }{\partial \rho _e } = \dfrac{1}{2\sqrt {\lambda _i } }\dfrac{\partial \lambda _i }{\partial \rho _e } = \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \dfrac{\partial {\pmb W}}{\partial \rho _e }{\pmb \varphi} _i =\\ \qquad \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \dfrac{\partial \left( {{\pmb B}{\pmb B}^{T}} \right)}{\partial \rho _e }{\pmb \varphi}_i = \\ \qquad \dfrac{1}{2\sqrt {\lambda _i } }{\pmb \varphi}_i^{T} \left( {\dfrac{\partial B}{\partial \rho _e }{\pmb B}^{T} + {\pmb B}\dfrac{\partial {\pmb B}^{T}}{\partial \rho_e }} \right) {\pmb \varphi}_i \end{array} $$

其中

(23)$$ \begin{array} \dfrac{\partial {\pmb B}}{\partial \rho _e } = -{\pmb \varPsi} ^{T}\dfrac{\partial {\pmb K}_{{ u}\phi } }{\partial \rho _e } = \\ \qquad -{\pmb \varPsi}^{T}{\pmb G}^{T}p_2 \rho _e^{(p_2-1)} \int_{\varOmega _{a}^e } {{\pmb B}_{u}^{T} \tilde {e}^{T}{\pmb B}_\varphi } d\varOmega \cdot {\pmb G} \end{array} $$

式中,${\pmb G}$为自由度转换矩阵.

将方程(22)和(23)代入方程(18)就可得到目标函数对设计变量的敏度.

4 数值算例和讨论

4.1 灵敏度分析验证

算例1 如图2,左端固支悬臂板的长宽分别为$a = 1.6$ m,$b = 1.2 $ m. 下层基体板厚$t_{h} = 2\times 10^{-2}{m}$,密度$\rho _{h} = 2 700$ kg/m$^{3}$,杨氏模量$E_{h} = 6.9\times 10^{10}$ N/m$^2$,泊松比$\nu _{h} = 0.3$. 上层压电材料板厚$t_{a} = 5\times 10^{-4}$ m,其材料属性见表1. 假设阻尼系数矩阵为${\pmb Z}_{d} = \zeta _{d} {\pmb I}$,其中$\zeta _{d} = 1.0\times 10^{ -4}$. 优化模型中,目标函数考虑第1到第5阶模态的可控性,并取压电材料的体积分数为$f_{v} = 0.5$.

图2

图2左端固支悬臂板示意图

Fig. 2A cantilever plate with left end fixed

表1压电材料属性

Table 1 Material properties of piezoelectric material

新窗口打开|下载CSV

悬臂板划分成$N_{e} = 432(24\times 18)$个四节点Mindlin壳单元. 初始设计中,所有作动器单元的相对密度设为0.5. 采用本文所提灵敏度分析方法得到的目标函数关于设计变量的灵敏度结果如图3所示. 作为对比,分别采用有限差分法(finite difference method, FDM)计算了扰动为0.02%,0.1%和0.5%的灵敏度,得到的结果基本相同,并取扰动为0.1%的结果显示在图3中.

图3

图3本文所提灵敏度计算方法和有限差分法计算的灵敏度结果对比

Fig. 3Comparison of sensitives obtained by the proposed adjoint method and finite difference method

由图3可知,两种计算方法结果基本一致,这说明了本文灵敏度计算方法的有效性. 另外,本文灵敏度分析方法计算耗时2.147 s,有限差分法耗时342.148 s,可见本文方法更高效.

4.2 矩形悬臂板优化算例

算例2 考虑如图2所示左端固支矩形板作动器优化问题,板的长宽分别为 $a = 1.8$ m,$b = 1.2$ m. 材料属性、厚度、阻尼系数和体积分数与算例1相同. 在目标函数中考虑第1到第5阶模态的可控性. 将矩形板划分为$N_{ e} = 2 400$ $(60\times 40)$个四节点Mindlin壳单元,并设作动器单元初始的相对密度为0.5(初始设计). 采用移动渐近线算法(MMA)求解优化问题^[40].

优化经过49次迭代收敛,得到如图4所示的优化结果,结果显示灰色单元基本没有出现,因此本研究采用的压电材料人工模型表现出了一定的自惩罚性.图5为相应的目标函数和体积分数的优化迭代历史曲线;目标函数从36.539降低到33.986,即优化设计的可控性相对于初始设计的可控性提高了12.8倍.表2为初始设计和优化设计的前五阶固有频率,可以发现最大的频率变化只有3.26%.

图4

图4考虑1$\sim $5阶模态可控性的拓扑优化结果

Fig. 4Topology optimization result for the controllability considering first five modes

图5

图5可控性指标和体积分数的迭代历史曲线

Fig. 5Iteration history for controllability index and volume fraction

表2初始设计和优化设计的前5阶频率

Table 2 1st$\sim $5th order frequencies of the initial design and the optimized design

新窗口打开|下载CSV

为了进一步验证优化结果的有效性,这部分计算了优化设计与两个参考设计(如图6)在频率为80 Hz(悬臂板结构第五阶固有频率附近)的简谐外激励作用下等增益速度反馈(constant gain velocity feedback, CGVF)^[41]和线性二次型最优控制(linear quadratic regulator, LQR)^[33]的主动控制效果,取减缩阶数$m$为40 (由于结构的第40阶频率在820 Hz左右,因此认为对于80 Hz载荷的响应计算精度可以接受). 外激励作用在悬臂板的右端中部,方向垂直于板面向下,振幅为1$\times $10$^{3 }$ N;CGVF的反馈系数为$G_{ a}G_{c} = 1 \times 10^{6}$ V/A;LQR的权重矩阵${\pmb R}$取$m$阶单位阵,${\pmb Q}$取$\left[ \!\! \begin{array}{cc} {\bf 0} & {\bf 0} \\ {\bf 0} & {1\times 10^8 {\pmb I}^{m\times m}} \end{array} \!\! \right]$. CGVF和LQR的具体实施方法见文献^[33,41]. 选取结构的动柔度和控制电压向量幅值的内积^[33]作为衡量控制效果的指标,结果见表3. 由表3可知,同样条件下优化设计的动柔度(dynamic compliance)和能量消耗(energy consumption)都更小,可见优化设计在该结构CGVF和LQR控制中对能量的使用效率更高.

图6

图6两个参考设计

Fig. 6Two reference designs

表3优化设计,竖条纹参考设计与横条纹参考设计在CGVF和LQR控制下控制效果的对比

Table 3 Comparison of the control performance under CGVF and LQR for the optimized design, reference designs with vertical stripes and horizontal stripes

新窗口打开|下载CSV

为了研究体积分数对优化结果的影响,现考虑体积分数分别为0.4和0.6两种情况下的优化问题,优化结果如图7. 优化结果对应的可控性指标分别为34.332和33.724,相对于体积分数为0.5的优化结果的可控性分别降低了29.25%和增加了29.95%,可见增加材料的用量,可以提高所研究压电结构的可控性.

图7

图7体积分数为0.4和0.6下的拓扑优化结果

Fig. 7Optimization results with volume fraction of 0.4 and 0.6

图7

图7体积分数为0.4和0.6下的拓扑优化结果(续)

Fig. 7Optimization results with volume fraction of 0.4 and 0.6 (continued)

算例3 采用和算例2相同的模型,分别在目标函数中考虑第1, 2, 3, 4和5阶模态的可控性,体积分数都取为0.5,优化结果如图8. 由图可知,优化结果的拓扑构型发生了明显改变,即目标函数中所考虑的模态对优化结果影响很大.

图8

图8考虑不同模态可控性的拓扑优化结果

Fig. 8Optimization results for the controllability considering difference mode

5 结论

本文研究了考虑可控性指标的压电作动器拓扑优化设计问题. 首先推导了基于控制矩阵奇异值的可控性指标,构建了以此指标为目标函数的压电层合板结构的优化模型. 优化模型引入了对中间密度单元压电系数的惩罚. 进而,提出了可控性关于设计变量的灵敏度分析方法. 和差分法的对比验证了灵敏度分析方法的有效性. 数值算例得到了清晰的拓扑构型,优化设计相对于参考设计控制效果更好. 算例还讨论了体积分数和所选控制模态对优化结果的影响.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]

Gupta

, Sharma

, Thakur

Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: A technical review

Journal of Intelligent Material Systems and Structures, 2010 , 21 : 1227 - 1243

[2]

Salas

, Ramírez-Gil

, Montealegre-Rubio

, et al .

Optimized dynamic design of laminated piezocomposite multi-entry actuators considering fiber orientation

Computer Methods in Applied Mechanics and Engineering, 2018 , 335 : 223 - 254

DOIURL

[3]

, Li

Multi-parameter optimization of piezoelectric actuators for multi-mode active vibration control of cylindrical shells

Journal of Sound and Vibration, 2018 , 426 : 166 - 185

DOIURL

[4]

Yassin

, Lahcen

, Zeriab

ESM

Hybrid optimization procedure and application to location optimization of piezoelectric actuators and sensors for active vibration control

Applied Mathematical Modelling, 2018 , 62 : 701 - 716

[5]

, Ou

, Jiang

Integrated optimization of structural topology and control for piezoelectric smart plate based on genetic algorithm

Finite Elements in Analysis and Design, 2013 , 64 : 1 - 12

[6]

徐斌 , 赵普猛 , 黎莹 .

区间参数压电智能桁架结构/控制的多目标非概率可靠性拓扑优化

振动工程学报, 2013 , 26 ( 2 ): 169 - 177

( Xu

Bin

, Zhao

Pumeng

, Li

Ying

Multi-objective optimization for structural topology and control of piezoelectric smart truss with interval parameter considering non-probability reliability constraints

Journal of Vibration Engineering , 2013 , 26 ( 2 ): 169 - 177 (in Chinese))

[7]

Zorić

, Simonović

, Mitrović

, et al .

Optimal vibration control of smart composite beams with optimal size and location of piezoelectric sensing and actuation

Journal of Intelligent Material Systems and Structures, 2013 , 24 : 499 - 526

[8]

Bends?e

, Kikuchi

Generating optimal topologies in structural design using a homogenization method

Computer Methods in Applied Mechanics and Engineering, 1988 , 71 : 197 - 224

[9]

Bends?e

Optimal shape design as a material distribution problem

Structural Optimization, 1989 , 1 : 193 - 202

[10]

Wang

, Wang

, Guo

A level set method for structural topology optimization

Computer Methods in Applied Mechanics and Engineering, 2003 , 192 : 227 - 246

[11]

Allaire

, Jouve

, Toader

Structural optimization using sensitivity analysis and a level-set method

Journal of Computational Physics, 2004 , 194 : 363 - 393

[12]

Xie

, Steven

A simple evolutionary procedure for structural optimization

Computers & Structures, 1993 , 49 : 885 - 896

[13]

彭细荣 , 隋允康 .

频率响应位移幅值敏度分析的伴随法

应用力学学报, 2008 , 25 ( 2 ): 247 - 252, 357

[本文引用: 2]

( Peng

Xirong

, Sui

Yunkang

Sensitivity analysis for frequency response amplitude with adjoint method

Chinese Journal of Applied Mechanics , 2008 , 25 ( 2 ): 247 - 252, 357 (in Chinese))

[本文引用: 2]

[14]

彭细荣 , 隋允康 .

考虑破损——安全的连续体结构拓扑优化ICM方法

力学学报, 2018 , 50 ( 3 ): 611 - 621

( Peng

Xirong

, Sui

Yunkang

ICM method for fail-safe topology optimization of continuum structures

Chinese Journal of Theoretical and Applied Mechanics , 2018 , 50 ( 3 ): 611 - 621 (in Chinese))

[15]

Pedersen

Maximization of eigenvalues using topology optimization

Structural and Multidisciplinary Optimization, 2000 , 20 : 2 - 11

[16]

刘虎 , 张卫红 , 朱继宏 .

简谐力激励下结构拓扑优化与频率影响分析

力学学报, 2013 , 45 ( 3 ): 588 - 597

( Liu

, Zhang

Weihong

, Zhu

Jihong

Structural topology optimization and frequency influence analysis under harmonic force excitations

Chinese Journal of Theoretical and Applied Mechanics , 2013 , 45 ( 3 ): 588 - 597 (in Chinese))

[17]

, Sun

Reliability-based vibro-acoustic microstructural topology optimization

Structural and Multidisciplinary Optimization, 2017 , 55 : 1195 - 1215

[18]

, Olhoff

Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps

Structural and Multidisciplinary Optimization, 2007 , 34 : 91 - 110

[19]

Rong

, Tang

, Xie

, et al .

Topological optimization design of structures under random excitations using SQP method

Engineering Structures, 2013 , 56 : 2098 - 2106

[20]

Vicente

, Zuo

, Pavanello

, et al .

Concurrent topology optimization for minimizing frequency responses of two-level hierarchical structures

Computer Methods in Applied Mechanics and Engineering, 2016 , 301 : 116 - 136

[21]

Zhu

, He

, Liu

, et al .

Structural topology optimization under harmonic base acceleration excitations

Structural and Multidisciplinary Optimization, 2018 , 57 : 1061 - 1078

[22]

Gonçalves

, De Leon

, Perondi

Topology optimization of embedded piezoelectric actuators considering control spillover effects

Journal of Sound and Vibration, 2017 , 388 : 20 - 41

[23]

Larbi

, Deü

, da Silva

Design of shunted piezoelectric patches using topology optimization for noise and vibration attenuation

Advances in Acoustics and Vibration, 2017 : 23 - 33

[24]

Zhang

, Takezawa

, Kang

Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

Structural and Multidisciplinary Optimization, 2018 , 58 : 185 - 199

[25]

Kang

, Tong

Integrated optimization of material layout and control voltage for piezoelectric laminated plates

Journal of Intelligent Material Systems and Structures, 2008 , 19 : 889 - 904

[26]

Wang

, Luo

, Zhang

, et al .

Topological design of compliant smart structures with embedded movable actuators

Smart Materials and Structures, 2014 , 23 : 045024

[27]

Yang

, Zhu

, Wu

, et al .

Integrated optimization of actuators and structural topology of piezoelectric composite structures for static shape control

Computer Methods in Applied Mechanics and Engineering, 2018 , 334 : 440 - 469

[28]

Yoon

, Choi

, Hur

Multiphysics topology optimization for piezoelectric acoustic focuser

Computer Methods in Applied Mechanics and Engineering, 2018 , 332 : 600 - 623

[29]

Wang

, Wang

Optimal placement and size of piezoelectric patches on beams from the controllability perspective

Smart Materials and Structures, 2000 , 9 : 558

[30]

Wang

, Wang

A controllability index for optimal design of piezoelectric actuators in vibration control of beam structures

Journal of Sound and Vibration, 2001 , 242 : 507 - 518

[31]

Bruant

, Proslier

Optimal location of actuators and sensors in active vibration control

Journal of Intelligent Material Systems and Structures, 2005 , 16 : 197 - 206

[32]

Dhuri

, Seshu

Multi-objective optimization of piezo actuator placement and sizing using genetic algorithm

Journal of sound and vibration, 2009 , 323 : 495 - 514

[33]

, Zhang

, Kang

Layout design of piezoelectric patches in structural linear quadratic regulator optimal control using topology optimization

Journal of Intelligent Material Systems and Structures, 2018 , 29 ( 10 ): 2277 - 2294

DOIURL[本文引用: 5]

[34]

Zhang

, Kang

Dynamic topology optimization of piezoelectric structures with active control for reducing transient response

Computer Methods in Applied Mechanics and Engineering, 2014 , 281 : 200 - 219

[35]

Kang

, Park

, Arora

A review of optimization of structures subjected to transient loads

Structural and Multidisciplinary Optimization, 2006 , 31 : 81 - 95

[36]

Cheng

, Kang

, Wang

Dynamic optimization of a turbine foundation

Structural Optimization, 1997 , 13 : 244 - 249

[37]

Liu

, Zhang

, Gao

A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations

Structural and Multidisciplinary Optimization, 2015 , 51 : 1321 - 1333

DOIURL

[38]

Zhao

, Wang

Dynamic response topology optimization in the time domain using model reduction method

Structural and Multidisciplinary Optimization, 2016 , 53 : 101 - 114

[39]

Kang

, Zhang

, Jiang

, Cheng

On topology optimization of damping layer in shell structures under harmonic excitations

Structural and Multidisciplinary Optimization, 2012 , 46 : 51 - 67

[40]

Svanberg

A class of globally convergent optimization methods based on conservative convex separable approximations

SIAM journal on Optimization, 2002 , 12 : 555 - 573

[41]

Zhang

, Kang

Topology optimization of piezoelectric layers in plates with active vibration control

Journal of Intelligent Material Systems and Structures, 2014 , 25 : 697 - 71