BANDGAP OPTIMIZATION DESIGN OF PHONONIC CRYSTALS BASED ON SHAPE MEMORY ALLOY
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摘要: 声子晶体是一种具有声子带隙的周期性结构, 通过对其带隙的动态调控设计, 进而满足航空航天领域中重大装备对减振降噪性能的特定需求. 文章在声子晶体的带隙设计中, 引入智能材料, 采用拓扑优化方法, 开展多功能声子晶体带隙动态调控设计. 首先采用有限元方法分析声子晶体的带隙性能, 并建立形状记忆合金的温度本构关系模型; 其次基于变密度法的拓扑优化方法, 在满足特定体分比及强度约束、保证声子晶体单胞之间的连接性约束条件下, 以相对带隙最大化为目标函数建立声子晶体带隙设计优化模型; 最后根据改进的材料插值模型, 分析求解设计灵敏度, 采用移动渐进法开展多功能声子晶体带隙结构的拓扑优化设计. 优化结果表明: 在XY模式下形状记忆合金从马氏体转换到奥氏体带隙拓宽了103.9%, 在Z模式下带宽增大了3.75倍. 研究结果为声子晶体在复杂环境下实现更为主动带隙调控提供了一种有效的设计方法.Abstract: Phononic crystal is a kind of periodic structures with the phononic band gap. The dynamically controllable design of its band gap could improve the vibration and noise reduction performance of major equipment in the aerospace field. In the work, smart materials are introduced for band gap design of phononic crystals. And the topological optimization method is used to design the multifunctional phononic crystal with the dynamically controllable band gaps. Firstly, the band gap of phononic crystals are computed by finite element analysis. Simultaneously, the temperature constitutive model of shape memory alloy is established. Secondly, based on variable density method, topology optimization model is established with maximizing the relative band gap under the specific volume ratio and strength constraints. At the same time, the connectivity constraints among phononic crystal unit cells must be ensured. Lastly, the band gaps of multifunctional phononic crystals are optimized by using the moving asymptotic method. During the optimization process, the design sensitivities are calculated with the improved material interpolation model. The optimization results show that the band gap is widened by 103.9% in XY mode with the transformation of shape memory alloy from martensite to austenite. And the bandwidth is increased by 3.75 times in Z mode. This research provides an effective design way for more actively control the phononic crystals band gaps in the complex application environments. And the novel phononic crystals have a wider application prospect.
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引 言
声子晶体是一种具有弹性波带隙特性的人工周期性结构功能材料, 对带隙的设计和调控使其在航空航天等领域具有更为迫切的需求[1], 包括有效控制飞机舱室内低频噪声和振动[2]. 声子晶体带隙设计是为了实现在最低频率范围带宽最大化, 针对声子晶体的强可设计性[3], 通常有两种设计思路.
一种是基于带隙原理的正向设计. Poggetto等[4]通过设计蜘蛛网状声子晶体的几何尺寸和增加局部质量, 获得由布拉格散射和局部共振产生的宽完全带隙. Chen等[5]通过分析材料特性和几何参数对声子晶体宽完全带隙的影响设计带宽. 吴旭东等[6]设计出一种双侧振子布置形式的局域共振声子晶体梁结构, 实现范围更宽的多带隙特性, 也指出周期数和晶格常数对双振子梁带隙特性的影响. 陈长红等[7]通过将部分柱体嵌入到基底中增加了共振体的质量, 从而降低局域共振带隙, 以改变柱体的几何参数调控带隙特性. 孙向洋等[8]在分析经典单面柱声子晶体结构参数对带隙影响的基础上, 设计了两种具有更低频带隙的新型单面柱局域共振声子晶体结构. Panahi等[9]基于Maltese十字设计出两种新型结构声子晶体, 数值仿真和实验测试也验证其低频宽带隙特性. 基于带隙机理采用数值方法对各种不同形状的声子晶体进行分析计算, 能够得到低频范围宽带隙, 但设计目标单一, 且具有一定的盲目性.
另一种是基于拓扑优化的逆向设计. 作为一种挑战, 旨在生成功能性声子晶体结构设备的优化设计必将代替工程直观设计方法[10]. 拓扑优化是在满足一定约束条件下以特定性能最优为目标而获取创新构型的结构优化设计方法, 采用拓扑优化方法能够设计出具有多功能特性的低频宽带声子晶体. 自2003年Sigmund等[11]率先利用拓扑优化方法实现以相对带宽最大为目标的无限周期性声子带隙材料设计以来, 拓扑优化方法已广泛应用于声子晶体的带隙设计. Huang等[12]以相邻频带间带隙最大为优化目标设计了能阻隔特定方向弹性波传播的带隙材料微结构. He等[13]以偏向带隙最大化为目标函数设计了能促使弹性波定向传输的新型带隙材料微结构构型, 并考虑材料相设计域不确定性分布, 以带隙最大为目标实现了声子晶体的可靠性设计[14-15]. Han等[16]通过拓扑优化实现了三相声子晶体的带隙设计. Dong等 [17]以面内和面外模式相对带隙宽度最大化为目标, 基于遗传算法完成了二维声子晶体的构型设计, 并通过拓扑优化设计方法实现了双负声学超材料微结构构型的系统设计. Xie等[18]以最大带隙宽度为目标, 基于一种多项式方法对具有未知但有界参数的声子晶体能带结构进行分析, 采用遗传算法实现了带有不确定性参数声子晶体的拓扑优化设计. Li等[19]采用双向渐进结构优化方法(BESO)和有限元分析方法, 以特定带隙最大化为目标系统地开展了声子晶体的拓扑优化设计, 先后实现了带有六重对称六边形晶格和具有超宽全向带隙的声子晶体优化设计. Zhang等 [20]提出采用两步拓扑设计策略, 设计出精确控制的PnC谐振腔, 以确定禁带和缺陷带的频率范围. 为了满足航空航天领域对声子晶体应用的特殊需求, 在有效设计声子带隙的同时, 还需满足轻量化、高比刚强度要求. Dong等[21]以带隙宽度最大和质量最小为设计目标, 实现了二维多孔声子晶体的多目标拓扑优化设计. Xu等[22]以相对带隙宽度最大和质量最小为设计目标, 实现了三相周期性声子晶体多目标拓扑优化设计. Hedayatrasa等[23]以最大带隙宽度和面内刚度最大为目标, 实现了多孔声子晶体板的多目标拓扑优化设计. 吴明晨等[24]采用遗传算法实现了多相材料声子晶体的构型优化设计. 曹蕾蕾等[25]以特定频率段带隙最宽和结构质量最小为优化目标, 在对微结构进行连通性分析的基础上, 引入考虑可制造性因素的附加约束, 采用遗传算法实现了多相材料的声子晶体优化设计. 采用拓扑优化设计方法不仅能够获得低频宽带隙等特定性能的声子晶体创新构型设计, 也能够实现轻质和高比刚强度等性能的多功能声子晶体设计.
将智能材料引入声子晶体带隙特性设计中, 可进一步实现声子晶体带隙特性的动态调控设计. Hou等[26]指出声子晶体中压电材料的用量直接影响全带隙性能. Yeh [27]利用电流变弹性材料制备了一种声子晶体, 在外加电场的条件下实现了对带隙宽度和位置的调控. 廖涛等[28]将涂有硬质材料涂层的柱状压电散射体周期性连接在4个环氧树脂薄板, 设计出具有大带宽的新型二维压电声子晶体板, 通过在压电体表面施加不同的电边界条件, 实现多条完全带隙的动态调控. 刘书田等[29]将形状记忆合金与环氧树脂结合设计出一种2D穿孔智能结构, 通过布置缝隙与形状记忆合金相材料的位置, 实现声子晶体带隙性质的可调设计. 上述研究工作表明在声子晶体中引入智能材料, 可通过控制外部环境参数(如温度、磁场和电场等)实现对声子晶体的带隙大小和范围的动态调控.
综上所述, 国内外学者对声子晶体带隙正向设计到采用拓扑优化方法的逆向设计, 发展到将智能材料引入声子晶体中进行带隙正向调控设计, 为将智能材料和拓扑优化设计方法应用于声子晶体带隙调控的逆向设计提供了新思路. 在此基础上, 本工作在满足强度和质量约束的条件下, 以低频范围相对带宽最大为设计目标, 采用拓扑优化设计方法进一步开展基于形状记忆合金(shape memory alloy, SMA)的声子晶体带隙主动调控的逆向设计.
1. 理论分析模型
1.1 带隙分析
文中晶格对称性选择为方形对称, 其不可约布里渊区域如图1所示的阴影部分, kx和ky是波矢量k的分量.
由不同材料复合而成的弹性超材料晶体结构是一种非均匀各向同性弹性介质, 通过弹性动力学纳维(Navier)方程推广得到弹性波在无阻尼无源周期性结构中的本征波动方程为
$$ \rho \left( {\boldsymbol{r}} \right)\ddot {\boldsymbol{u}} = \nabla \left( {\lambda \left( {\boldsymbol{r}} \right) + 2\mu \left( {\boldsymbol{r}} \right)} \right)\left( {\nabla \cdot {\boldsymbol{u}}} \right) - \nabla \times \left( {\mu \left( {\boldsymbol{r}} \right)\nabla \times {\boldsymbol{u}}} \right) $$ (1) 式中, u = (ux, uy, uz)为位移矢量; r = (x, y, z)为空间位置矢量; λ和μ为材料的拉梅常数; ρ代表材料密度; $\nabla $为矢量Hamilton的微分算子.
在二维周期性无限介质内产生体弹性波, 可将波动方程解耦为XY平面内偏振的XY模式(面内模式)和垂直XY平面内偏振的Z模式(面外模式). 依据Fourier变换, 将时域方程转换为频域方程, 得到xOy平面内的位移矢量方程(横波方程)和z方向上的位移分量方程(纵波方程)
$$ \left.\begin{aligned} & \frac{\partial }{{\partial x}}\left[ {\left( {\lambda \left( {\boldsymbol{r}} \right) + 2\mu \left( {\boldsymbol{r}} \right)} \right)\frac{{\partial {u_x}}}{{\partial x}} + \lambda \left( {\boldsymbol{r}} \right)\frac{{\partial {u_y}}}{{\partial y}}} \right] +\\ &\qquad \frac{\partial }{{\partial y}}\left[ {\mu \left( {\boldsymbol{r}} \right)\left( {\frac{{\partial {u_x}}}{{\partial y}} + \frac{{\partial {u_y}}}{{\partial x}}} \right)} \right] = - \rho \left( {\boldsymbol{r}} \right){\omega ^2}{u_x} \\ & \frac{\partial }{{\partial y}}\left[ {\left( {\lambda \left( {\boldsymbol{r}} \right) + 2\mu \left( {\boldsymbol{r}} \right)} \right)\frac{{\partial {u_y}}}{{\partial y}} + \lambda \left( {\boldsymbol{r}} \right)\frac{{\partial {u_x}}}{{\partial x}}} \right] +\\ &\qquad \frac{\partial }{{\partial x}}\left[ {\mu \left( {\boldsymbol{r}} \right)\left( {\frac{{\partial {u_x}}}{{\partial y}} + \frac{{\partial {u_y}}}{{\partial x}}} \right)} \right] = - \rho \left( {\boldsymbol{r}} \right){\omega ^2}{u_y} \end{aligned} \right\} $$ (2) $$ \frac{\partial }{{\partial x}}\left( {\mu \left( {\boldsymbol{r}} \right)\frac{{\partial {u_z}}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {\mu \left( {\boldsymbol{r}} \right)\frac{{\partial {u_z}}}{{\partial y}}} \right) = - \rho \left( {\boldsymbol{r}} \right){\omega ^2}{u_z} $$ (3) 式中, ω为角频率; 式(2)和式(3)为以位移为未知函数的弹性超材料动力学波动方程.
采用有限单元法计算弹性超材料能带结构时, 根据声子晶体的周期性, 建立单胞结构的有限元模型, 利用Bloch-Floquet定理引入周期性边界条件
$$ {\boldsymbol{u}}\left( {\boldsymbol{r}} \right) = {{\rm{e}}^{{\rm{i}}k \cdot r}}{{\boldsymbol{u}}_k}\left( {\boldsymbol{r}} \right) $$ (4) 式中, r为单胞中局部坐标系的位置矢量; uk(r)为位置矢量的周期性函数; k = (kx, ky)为波矢量. 进而将弹性波传播的特征值求解问题转化为数值计算以下本征方程
$$ ({\boldsymbol{K}} - {\omega ^2}{\boldsymbol{M}}){\boldsymbol{U}} = {\boldsymbol{0}} $$ (5) 式中, ω为简谐振动角频率; K为周期性单胞的刚度矩阵; M为周期性单胞的质量矩阵; U为广义位移矢量. 分别表示为
$$ \left. \begin{split} & {\boldsymbol{K }}= \int {{{\boldsymbol{B}}^{\rm{T}}}{\boldsymbol{C}}({\boldsymbol{r}}){\boldsymbol{B}}} {\rm{d}}{V_\varOmega } \\ & {\boldsymbol{M}} = \int {{{\boldsymbol{N}}^{\rm{T}}}\rho ({\boldsymbol{r}}){\boldsymbol{N}}{\rm{d}}{V_\varOmega }} \\ & {\boldsymbol{U}} = \left[ {{{\boldsymbol{U}}_1}}\;{{{\boldsymbol{U}}_2}}\;{{{\boldsymbol{U}}_3}}\;\cdots {{\boldsymbol{U}}_n} \right]^{\rm{T}} \\ & {{\boldsymbol{U}}_i} = {\left[ {{u_i}}\;{{v_i}}\;{{w_i}} \right]^{\rm{T}}}\quad {\text{ (}}i = 1,2,3 \cdots n) \end{split} \right\} $$ (6) 式中, B为应变矩阵; N为形函数矩阵或插值函数矩阵; C为材料的弹性矩阵; $\rho $为单元等效密度; ${V_\varOmega }$为被积的单元区域; U为广义位移矢量; Ui为节点的位移矢量.
1.2 本构关系
形状记忆合金是一种智能材料, 描述其增率形式的本构方程由变量Green应变、温度T和马氏体体积分数ξ描述为
$$ \begin{split} & \dot \sigma {\text{ = }}\frac{{\partial \sigma }}{{\partial \varepsilon }}\dot \varepsilon + \frac{{\partial \sigma }}{{\partial \xi }}\dot \xi + \frac{{\partial \sigma }}{{\partial T}}\dot T =\\ &\qquad {{D}}(\varepsilon ,\xi ,T)\dot \varepsilon + {{\varOmega}} (\varepsilon ,\xi ,T)\dot \xi + {{\varTheta}} (\varepsilon ,\xi ,T)\dot T \end{split} $$ (7) 式中, ${{D}}(\varepsilon ,\xi ,T)$为弹性模量; ${{\varOmega }}(\varepsilon ,\xi ,T)$为相变模量; ${{\varTheta}} (\varepsilon ,\xi ,T)$为热弹性模量. 在由奥氏体向马氏体转变的相变过程中, 马氏体体积分数ξ变化规律的相变演化方程表示为
$$ \xi {\text{ = }}1 - \exp \left[ {{a^M}({M_s} - T) + {b^M}\sigma } \right] $$ (8) 相反, 表示为
$$ \xi {\text{ = }}\exp \left[ {{a^A}({A_s} - T) + {b^A}\sigma } \right] $$ (9) 式中, ${a}^{M}, {a}^{A}, {b}^{M}$和${b^A}$为积分常数; ${M_s}$为马氏体相变开始温度; ${A_s}$为奥氏体相变开始温度; ${A_f}$为奥氏体相变完成温度. 通过DSC测试相变温度实验所得Ni-Ti形状记忆合金的性能参数[30]如表1所示, 仿真分析后弹性模量随温度变化趋势由图2所示.
表 1 Ni-Ti形状记忆合金材料参数Table 1. Material parameters of Ni-Ti SMAMaterial parameters Martensite
(SMA-M)Austenitic
(SMA-A)elastic moduli E/GPa 25 81 density ρ/(kg·m−3) 6400 6400 Poisson ratio υ 0.378 0.412 longitudinal wave velocity/(m·s−1) 2764 5472 transverse wave velocity/(m·s−1) 1187 2117 采用3次样条分段曲线对图2中弹性模量随温度变化的数据进行拟合, 得到形状记忆合金的温度本构关系模型
$$ E(T) = \left\{ \begin{split} & 25,{\text{ }}T \lt 311 \\ & 4.94 \times {10^{ - 3}}{T^3} - 4.653\;4{T^2} + {\text{ }} \\ &\qquad 1461.894T - 153\;160.46,\quad T \in {\text{[311,333]}} \\ & 81,{\text{ }}T \gt 333 \end{split} \right. $$ (10) 式中, E (GPa)为形状记忆合金弹性模量; T (K)为相变过程中对应的温度.
2. 基于SMA的带隙调控拓扑优化
2.1 优化模型
采用基于变密度法的拓扑优化方法, 在特定体分比约束、单胞内部和单胞之间的连接性约束下, 以多孔弹性超材料单胞带隙最大化为目标函数对弹性超材料的单胞进行优化设计. 其优化模型可用以下数学模型描述
$$ \left. \begin{split} & {\rm{find}}{\text{ }}{\boldsymbol{\rho}} {\text{ = }}\left[ {{\rho _1}}\;{{\rho _2}}\;{\cdots}\;{{\rho _e}}\;{\cdots}\;{{\rho _N}} \right] \\ & \min {\text{ }} - f\left( {\boldsymbol{\rho}} \right) \\ & {\rm{s.t.}}{\text{ }}({\boldsymbol{K}}({{\boldsymbol{k}}}) - {\omega ^2}({{\boldsymbol{k}}}){\boldsymbol{M}}) \cdot {\boldsymbol{U}} = 0 \\ &\qquad {\text{ }}{V_{{\rm{th}}}} = \sum {{\rho _e} \cdot {v_e}} \leqslant {V_f} \cdot {V_0} \\ &\qquad {\text{ }}{G^{\text{*}}} \leqslant {G^{\rm{H}}} \\ &\qquad {\text{ }}{\sum {\left( {\rho _e^i - \rho _e^{i'}} \right)} ^2} \leqslant B \cdot {n_s} \\ &\qquad {\text{ }}0 \lt {\rho _{\min }} \leqslant {\rho _e} \leqslant 1,{\text{ }}\quad e = 1,2,\cdots,N{\text{ }} \end{split} \right\} $$ (11) 式中, N为设计变量的个数, 即设计域中有限单元的数目; ${\rho _e}$为取值范围为0 ~ 1的单元e的伪密度; ${\rho _{\min }}$为单元相对密度的下限值; f (ρ)为目标函数; GH是等效剪切模量; G*表示预设的单胞等效剪切模量下限值, 此约束能保证单胞内刚度满足一定的承载要求, 同时保证多孔弹性超材料胞内材料连接性; $ \rho _e^i $和$ \rho _e^{i'} $是多孔弹性超材料单胞对应边界上单元的伪密度值; B是边界连接性参数值, 预设其为足够小值用以要求单胞对应边界上有限单元伪密度值相等, 从而保证单胞间相连接; ${n_s}$为对应边界所取的单元格数目; ${V_{{\rm{th}}}}$表示单胞散射体的体分比; ${V_0}$表示设计域的总体积; ${V_f}$表示预设的单胞体分比上限值.
2.2 目标函数及灵敏度分析
采用移动渐进法(method of moving asymptotes)实施声子晶体带隙结构的拓扑优化设计, 需推导目标函数的灵敏度及各个约束函数对设计变量的导数. 采用形状记忆合金在不同温度下的带隙与带隙中心频率的比值来描述基于智能材料SMA带隙主动调控的声子晶体拓扑优化的目标函数, 即为相对带隙之差
$$ \begin{split} & \Delta {f_{{\rm{relative}}}} = \frac{{BG_\omega ^{{T_1}}}}{{\omega _c^{{T_1}}}} - \frac{{BG_\omega ^{{T_2}}}}{{\omega _c^{{T_2}}}} = \\ & \qquad 2\frac{{\min \omega _{n + 1}^{{T_1}}({\boldsymbol{k}}) - \max \omega _n^{{T_1}}({\boldsymbol{k}})}}{{\min \omega _{n + 1}^{{T_1}}({\boldsymbol{k}}) + \max \omega _n^{{T_1}}({\boldsymbol{k}})}}- \\ & \qquad 2\frac{{\min \omega _{n + 1}^{{T_2}}({\boldsymbol{k}}) - \max \omega _n^{{T_2}}({\boldsymbol{k}})}}{{\min \omega _{n + 1}^{{T_2}}({\boldsymbol{k}}) + \max \omega _n^{{T_2}}({\boldsymbol{k}})}} \end{split} $$ (12) 式中, k为波矢; $BG_\omega ^{{T_1}}$表示在温度T1时结构单胞的带隙宽度; $BG_\omega ^{{T_2}}$表示在温度T2时结构单胞的带隙宽度; $ \omega _c^{{T_1}} $和$ \omega _c^{{T_2}} $分别表示在温度T1和T2时带隙的中心频率. $\min \omega _{n + 1}^{{T_1}}({\boldsymbol{k}})$和$\max \omega _n^{{T_1}}({\boldsymbol{k}})$分别表示在温度T1时第n + 1阶特征频率的最小值和第n阶特征频率的最大值; $\min \omega _{n + 1}^{{T_2}}({\boldsymbol{k}})$和$\max \omega _n^{{T_2}}({\boldsymbol{k}})$分别表示在温度T2时第n + 1阶特征频率的最小值和第n阶特征频率的最大值, 且都是波矢k和设计变量的函数. 目标函数灵敏度计算如下
$$ \begin{split} & \frac{{\partial {f_{{\rm{relative}}}}}}{{\partial {\rho _e}}} = 4\frac{{\max \omega _n^{{T_1}}\dfrac{{\partial \min \omega _{n + 1}^{{T_1}}}}{{\partial {\rho _e}}}{{ - }}\min \omega _{n + 1}^{{T_1}}\dfrac{{\partial \max \omega _n^{{T_1}}}}{{\partial {\rho _e}}}}}{{{{\left( {\min \omega _{n + 1}^{{T_1}}{\text{ + }}\max \omega _n^{{T_1}}} \right)}^2}}} - \\ &\qquad 4\dfrac{{\max \omega _n^{{T_2}}\dfrac{{\partial \min \omega _{n + 1}^{{T_2}}}}{{\partial {\rho _e}}}{{ - }}\min \omega _{n + 1}^{{T_2}}\dfrac{{\partial \max \omega _n^{{T_2}}}}{{\partial {\rho _e}}}}}{{{{\left( {\min \omega _{n + 1}^{{T_2}}{\text{ + }}\max \omega _n^{{T_2}}} \right)}^2}}}\\[-12pt] \end{split} $$ (13) 式中, f是目标函数; ρe是取值范围为0 ~ 1的单元伪密度; ωn + 1与ωn分别为第n + 1和第n阶特征频率. 而特征频率对设计变量ρe的1阶导数可以表示为
$$ \frac{{\partial {\omega _n}}}{{\partial {\rho _e}}}{\text{ = }}\frac{1}{{2{\omega _n}}}{\boldsymbol{U}}_n^{\text{T}}\left( {\frac{{\partial {\boldsymbol{K}}}}{{\partial {\rho _e}}} - {\omega _n}^2\frac{{\partial {\boldsymbol{M}}}}{{\partial {\rho _e}}}} \right){{\boldsymbol{U}}_n} $$ (14) 式中, Un是ωn对应的特征向量; K是刚度矩阵; M为正交归一化的质量矩阵.
由于结构的对称性等原因可能导致重特征频率, 所相应的独立振型也不唯一, 以至于该频率所对应的振型不确定, 从而式(14)计算的导数值也不再唯一. 为解决这一问题, 首先假定具有2阶重频及两个相对应的振型U1和U2, 表示为
$$ {{\omega _r} = {\omega _1} = {\omega _2}} $$ (15) 显然, 这两个振型的任意线性组合仍然是特征频率ωr的振型, 即满足以下关系
$$\qquad\qquad\qquad {{\boldsymbol{U}}^*} = {c_1}{{\boldsymbol{U}}_1} + {c_2}{{\boldsymbol{U}}_2} $$ (16) $$ \qquad\qquad\qquad\left( {{\boldsymbol{K}}{{ - }}{\omega^2 _r}{\boldsymbol{M}}} \right){{\boldsymbol{U}}^*} = {\boldsymbol{0}} $$ (17) 将式(16)代入式(14), 可得
$$ \frac{{\partial {\omega _r}}}{{\partial {\rho _e}}}{\text{ = }}\frac{1}{{2{\omega _r}}}\left( {{c_1}^2{g_{11}} + {c_2}^2{g_{22}} + 2{c_1}{c_2}{g_{12}}} \right) $$ (18) $$ {g}_{ij}\text{ = }{{\boldsymbol{U}}}_{i}^{\text{T}}\left(\frac{\partial {\boldsymbol{K}}}{\partial {\rho }_{e}}-{\omega }_{r}^{2}\frac{\partial {\boldsymbol{M}}}{\partial {\rho }_{e}}\right){{\boldsymbol{U}}}_{j}\text{, }\quad i,j = 1,2 $$ (19) 由式(18)和式(19)可知, ωr的导数不仅与设计变量ρe有关, 而且还与振型Ui的选取有关, 即与c1和c2的取值有关. 基于∂ωr/∂ρe的计算结果与c1和c2的取值相关性, 改变c1和c2的取值可以使得∂ωr/∂ρe达到极值. 根据文献[31-32], ∂ωr/∂ρe的极值即为重频特征频率对设计变量的导数值.
为此, 构造拉格朗日函数
$$ L\left( {{c_1},{c_2},\lambda } \right) = \frac{{\partial {\omega _r}}}{{\partial {\rho _e}}} + \lambda \left( {1 - {c_1}^2 - {c_2}^2} \right) $$ (20) 分别对c1和c2求导, 取值为0, 得到
$$ \left. \begin{split} & {\frac{{\partial L}}{{\partial {c_1}}} = 2{g_{11}}{c_1} + 2{g_{12}}{c_2} - 2\lambda {c_1} = 0} \\ & {\frac{{\partial L}}{{\partial {c_2}}} = 2{g_{22}}{c_2} + 2{g_{12}}{c_1} - 2\lambda {c_2} = 0} \end{split} \right\} $$ (21) 简写为
$$ \left[ {\begin{array}{*{20}{c}} {{g_{11}}}&{{g_{12}}} \\ {{g_{21}}}&{{g_{22}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{c_1}} \\ {{c_2}} \end{array}} \right]{\text{ = }}\lambda \left[ {\begin{array}{*{20}{c}} {{c_1}} \\ {{c_2}} \end{array}} \right] $$ (22) 求解式(22)的特征值问题进而得到两组线性无关的c = [c1, c2]T, 代入式(18)便得到重频率的导数, 当有多个重频率合时, 可采用类似的方法计算其导数值.
采用均匀化理论计算等效弹性张量, 即
$$ E_{ij}^{\text{H}} = \frac{1}{S}\int_\varOmega { {{{\left( {{\boldsymbol{\varepsilon}} _0^i - {{\boldsymbol{\varepsilon}} ^i}} \right)}^{\text{T}}} {\boldsymbol{E}} \left( {{\boldsymbol{\varepsilon}} _0^i - {{\boldsymbol{\varepsilon}} ^i}} \right)} } {\rm{d}}\varOmega $$ (23) 式中, $E_{ij}^{\text{H}} $为等效弹性模量; E为给定点的弹性矩阵; S为单胞Ω的面积; ${\boldsymbol{\varepsilon}} _0^i $为3个线性无关的向量; εi为在满足周期性边界条件下数值计算所得应变场.
由等效弹性模量$E_{ij}^{\text{H}} $得到等效剪切模量GH, 满足刚度要求的约束函数对设计变量的导数为
$$ \frac{{\partial {G^{\text{H}}}}}{{\partial {\rho _e}}}{\text{ = }}\frac{{\partial E_{33}^{\text{H}}}}{{\partial {\rho _e}}} = \frac{1}{S}\int_\varOmega { {{{\left( {{\boldsymbol{\varepsilon}} _0^3 - {{\boldsymbol{\varepsilon}} ^3}} \right)}^{\text{T}}} {\frac{{\partial {\boldsymbol{E}}}}{{\partial {\rho _e}}}}\left( {{\boldsymbol{\varepsilon}} _0^3 - {{\boldsymbol{\varepsilon}} ^3}} \right)} } {\rm{d}}\varOmega $$ (24) 满足单胞间相连接的约束函数对设计变量的导数为
$$ \frac{{{{\displaystyle\sum {\left( {\rho _e^i - \rho _e^{i'}} \right)} }^2}}}{{\partial \rho _e^j}} = 2\left( {\rho _e^i - \rho _e^{i'}} \right){\delta _{ij}} $$ (25) 式中, i表示边界上的单元; $i' $表示相对应的单元; j表示所有单元.
3. 算例分析
在进行能带图后处理时, 为了消除晶格常数对带隙的影响, 在色散曲线图中, 采用归一化频率标识带隙, 记作$\varOmega {\text{ = }}\omega a/(2\text{π} C)$, 其中C为等效横波波速.
在优化设计过程中, 采用改进的MSIMP(modified solid isotropic material with penalization)[33]材料插值模型
$$ \left. \begin{split} & E = {\rho ^{penal}}{E_0} \\ & \rho = \left\{ \begin{split} & {{\rho _e}{\rho _0},\quad {\rho _e} \gt 0.1} \\ & {\left( {6 \times {{10}^5}\rho _e^6 - 5 \times {{10}^6}\rho _e^7} \right){\rho _0},\quad {\rho _e} \leqslant 0.1{\text{ }}} \end{split} \right. \end{split} \right\} $$ (26) 式中, ρe为伪密度值; E为单元弹性模量; E0为材料弹性模量; ρ0为材料密度; ρ为插值得到的单元密度; penal是惩罚因子, 一般取penal = 3. ${\rho _e} $>0.1时恢复成SIMP插值模型; ${\rho _e} $<0.1时, 插值得到的刚度和质量的比值将变成一个较大的数值; ${\rho _e} $=0时, MSIMP刚度与质量之比趋于正无穷, 可以有效地避免局部模态的产生.
由式(10)所得形状记忆合金弹性模量E与温度T的关系数学模型, 任取两个温度点T1和T2进行带隙调控优化. 为了展示典型调节效果, 算例中选取形状记忆合金处于马氏体和奥氏体两种状态下对应的极限温度点T1 = 311 K, T2 = 333 K进行单相蜂窝型单胞结构的带隙调控设计. 在马氏体(Martensite)状态下极限温度为T1时材料参数弹性模量E = 25 GPa, 泊松比v = 0.378, 密度ρ = 6400 kg/m3; 在奥氏体(Austenitic)状态下极限温度为T2时材料弹性模量E = 81 GPa, 泊松比v = 0.412, 密度ρ = 6400 kg/m3. 由式(11)拓扑优化数学模型, 其中体分比约束Vf=0.5, 边界连接性参数值B=$1.0\times {10^{ - 6}}$. 本工作分别对面内XY模式3阶带隙以及面外Z模式1 ~ 3阶带隙开展了基于SMA的带隙主动调控的拓扑优化设计.
3.1 面内XY模式带隙调控设计
由图3所示基于SMA单相蜂窝型声子晶体在面内XY模式下第3阶相对带隙调控优化设计构型及不同状态下色散曲线, 可知优化构型具有很好的对称性, 胞元内材料连续性分布, 单胞间对称性连接. 同时由优化所得声子晶体构型在两种温度下对应的色散曲线图3(b)和图3(c)可知: 马氏体状态下归一化带隙频率为0.2997, 奥氏体状态下归一化带隙频率为0.611, 带隙拓宽了103.9%. 根据式(27)可知两种状态下的共有带隙调节范围$\Delta {f_p}$=0.0945, 表明在主动调控过程中抑制弹性波传播的带隙频率范围, 从而能够实现特定频率隔振; 根据式(28)可知两种状态下全带隙调节范围$\Delta {f_t}$=0.8162, 显示了动态带隙调节的宽度和位置, 该值越大, 表明带隙调控能力越强
$$ \Delta {f_t} = \min {\omega ^{A*}_{n + 1}}({\boldsymbol{k}}) - \max {\omega ^{M*}_n({\boldsymbol{k}}}) $$ (27) $$ \Delta {f_p} = \min {\omega ^{M*}_{n + 1}({\boldsymbol{k}}}) - \max {\omega ^{A*}_n({\boldsymbol{k}}}) $$ (28) ![]()
图 3 SMA单相面内XY模式第3阶相对带隙调控优化构型及不同状态下色散曲线Figure 3. Optimal phononic crystal for third-order relative bandgap in the SMA single phase in-plane XY mode and the corresponding dispersion curves![]()
3 SMA单相面内XY模式第3阶相对带隙调控优化构型及不同状态下色散曲线 (续)3. Optimal phononic crystal for third-order relative bandgap in the SMA single phase in-plane XY mode and the corresponding dispersion curves (continued)3.2 面外Z模式带隙调控设计
如图4 ~ 图6所示基于SMA单相蜂窝型声子晶体在面外Z模式下不同阶数相对带隙调控优化设计构型及不同状态下色散曲线, 优化设计结果见表2.
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图 4 SMA单相面外Z模式第1阶带隙调控优化构型及不同状态下色散曲线Figure 4. Optimal phononic crystal for first-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves![]()
图 5 SMA单相面外Z模式第2阶带隙调控优化构型及不同状态下色散曲线Figure 5. Optimal phononic crystal for second-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves![]()
图 6 SMA单相面外Z模式第3阶带隙调控优化构型及不同状态下色散曲线Figure 6. Optimal phononic crystal for third-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves表 2 不同带隙阶数调控优化结果Table 2. Optimal results of different-order bandgapsWave order Martensite Austenitic Tunable range
$\Delta {f_t}$Shared range
$\Delta {f_p}$first 0.4384 0.8827 1.0207 0.3004 second 0.3444 0.6935 1.0240 0.0139 third 0.3256 1.2217 1.2273 0.3200 从图4 ~ 图5可以看出, Z模式优化结果单胞与XY模式优化结果相比具有很好的对称性, 所以波矢k扫描不可约布里渊区即可, 并且迭代曲线波动也较小, 其中3阶带隙调控优化的结果中在马氏体状态也存在2阶带隙. 结合表2和优化结果图可以明显看出, 3阶带隙优化的结果带隙总调节范围最大, 并且马氏体状态下带隙宽度是奥氏体状态的3.75倍, 表明带隙主动调控的能力较强, 其次是1阶带隙, 在两种状态下均有相对较宽的带隙, 2阶带隙的调控效果相对较低.
由图3(a) ~ 图6(a)所示, 优化设计声子晶体构型具有很好的对称性, 取相同的不可约布里渊区, 胞元内材料连续性分布, 单胞间对称性连接. 同时材料趋于单胞中心分布, 单胞之间由细韧带连接, 具有局域共振型声子晶体功能特征, 从而实现了基于形状记忆合金声子晶体的低频宽带隙设计.
4. 结 论
为了充分利用SMA的强度, 同时能够满足用于航空航天领域中带隙主动调控效果(也称为主动隔振)和刚度最大的要求, 并具有轻质特性, 建立了基于SMA二维单相蜂窝型弹性超材料带隙调控的拓扑优化模型. 基于形状记忆合金(SMA)的带隙调控拓扑优化, 是不同于以往的声子晶体带隙结构设计, 也不同于一般基于线弹性材料的静力学拓扑优化设计(设计材料的为线性材料), 首先智能材料形状记忆合属于热敏材料, 其材料属性能随温度发生实时的变化, 带隙的调控就是基于形状记忆合金这种特殊的材料属性实现的, 但这也是本文拓扑优化设计的难点, 所以为了降低了计算量和问题复杂度, 根据形状记忆合金的分析结果, 取两个特殊的温度点(即形状记忆合金处于马氏体状态和奥氏体状态下的温度点)分别对XY模式3阶带隙以及Z模式1阶、2阶和3阶带隙进行了带隙调控拓扑优化设计, 并创建了表征带隙调控效果的两个参数, 即共有带隙范围和带隙总调节范围, 优化结果数值分析表明, XY模式和Z模式3阶带隙的调控效果最好. 将智能材料与拓扑优化方法相结合进行带隙调控的结构设计, 基于智能材料拓扑优化设计的声子晶体带隙调控是一种有效的尝试, 从数据中可以看出优化后的形状记忆合金蜂窝单胞具有很好的带隙主动调控效果.
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图 3 SMA单相面内XY模式第3阶相对带隙调控优化构型及不同状态下色散曲线
Figure 3. Optimal phononic crystal for third-order relative bandgap in the SMA single phase in-plane XY mode and the corresponding dispersion curves
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3 SMA单相面内XY模式第3阶相对带隙调控优化构型及不同状态下色散曲线 (续)
3. Optimal phononic crystal for third-order relative bandgap in the SMA single phase in-plane XY mode and the corresponding dispersion curves (continued)
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图 4 SMA单相面外Z模式第1阶带隙调控优化构型及不同状态下色散曲线
Figure 4. Optimal phononic crystal for first-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves
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图 5 SMA单相面外Z模式第2阶带隙调控优化构型及不同状态下色散曲线
Figure 5. Optimal phononic crystal for second-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves
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图 6 SMA单相面外Z模式第3阶带隙调控优化构型及不同状态下色散曲线
Figure 6. Optimal phononic crystal for third-order relative bandgap in the SMA single phase out-plane Z mode and the corresponding dispersion curves
表 1 Ni-Ti形状记忆合金材料参数
Table 1 Material parameters of Ni-Ti SMA
Material parameters Martensite
(SMA-M)Austenitic
(SMA-A)elastic moduli E/GPa 25 81 density ρ/(kg·m−3) 6400 6400 Poisson ratio υ 0.378 0.412 longitudinal wave velocity/(m·s−1) 2764 5472 transverse wave velocity/(m·s−1) 1187 2117 
下载: 导出CSV
表 2 不同带隙阶数调控优化结果
Table 2 Optimal results of different-order bandgaps
Wave order Martensite Austenitic Tunable range
$\Delta {f_t}$Shared range
$\Delta {f_p}$first 0.4384 0.8827 1.0207 0.3004 second 0.3444 0.6935 1.0240 0.0139 third 0.3256 1.2217 1.2273 0.3200
下载: 导出CSV
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