﻿ 基于参数化水平集法的约束阻尼结构动力学拓扑优化
 上海理工大学学报  2021, Vol. 43 Issue (4): 349-359 PDF

1. 上海理工大学 机械工程学院，上海 200093;
2. 重庆大学 机械与运载工程学院，重庆 400044

Dynamic topology optimization for constrained layer damping structures using parametric level set method
WU Yonghui1, ZHANG Dongdong1, CHEN Jingyue1, ZHENG Ling2
1. School of Mechanical Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China;
2. College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, China
Abstract: Topology optimization of constrained layer damping (CLD) structure is an effective means to control vibration and noise under lightweight design. Combined with the finite element model of constrained layer damping structure, the first and second modal loss factor and its weighted value are defined as the objective functions. The expansion coefficients of the parametric level set function are chosen as the design variables, and the amount of CLD material is employed as the constraint condition. A topology optimization model based on parametric level set method (PLSM) is established. The sensitivity of objective function with respect to design variables is derived based on adjoint vector method and the design variables are updated by using optimization criterion method. Numerical results show that it is feasible and effective for topology optimization of the constrained layer damping structure using parametric level set method. In addition, the optimization efficiency can be improved by the initial configuration based on the strain energy distribution of the substrate structure. Furthermore, considering the additional mass kept unchanged, the influence of the thicknesses of constrained layer material and damping material on the modal loss factor and optimal configurations for constrained damping material is discussed.
Key words: parametric level set method     constrained layer damping structure     modal loss factor     optimality criteria

1 约束阻尼板有限元模型

 图 1 CLD有限元复合单元模型 Fig. 1 CLD finite composite element model

 $\left\{ {\begin{array}{*{20}{l}} {{\boldsymbol{M}} = {{\boldsymbol{M}}_{\rm{b}}} + \displaystyle\sum\limits_{i = 1}^n {\left( {{\boldsymbol{m}}_{{\rm{v}}i} + {\boldsymbol{m}}_{{\rm{c}}i}} \right)} } \\ {{{\boldsymbol{K}}_{\rm{R}}} = {{\boldsymbol{K}}_{\rm{b}}} + \displaystyle\sum\limits_{i = 1}^n {\left( {{\boldsymbol{k}}_{{\rm{v}}i} + {\boldsymbol{k}}_{{\rm{c}}i} + {\boldsymbol{k}}_{{\rm{\gamma v\_R}}i}} \right)} } \\ {{{\boldsymbol{K}}_{\rm{I}}} = \displaystyle\sum\limits_{i = 1}^n {{\boldsymbol{k}}_{{\rm{\gamma v\_I}}i}} } \end{array}} \right.$ (1)

 ${\boldsymbol{M}}{\ddot{\boldsymbol{ X}}} + ({{\boldsymbol{K}}_{\rm{R}}} +{\rm{i}}{{\boldsymbol{K}}_{\rm{I}}}){\boldsymbol{X}} = 0$ (2)

 $\left( { - \lambda _{r}^{\rm{*}}{\boldsymbol{M}} + ({{\boldsymbol{K}}_{\rm{R}}} + {\rm{i}}{{\boldsymbol{K}}_{\rm{I}}})} \right){\boldsymbol{\varPsi }}_{r}^{\rm{*}} = {\boldsymbol{0}}$ (3)

 ${\eta _r} = \frac{{{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}}}}{{{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{R}}}{{\boldsymbol{\varPsi }}_{r}}}}$ (4)

2 基于参数化水平集法的约束阻尼结构优化设计 2.1 参数化水平集法

 $\left\{ {\begin{array}{*{20}{l}} {{\boldsymbol{\varPhi }}({\boldsymbol{x}}){\rm{ > }}0,\;\;\;{\boldsymbol{x}} \in D{\rm{ }}} \\ {{\boldsymbol{\varPhi }}({\boldsymbol{x}}){\rm{ = }}0,\;\;\;{\boldsymbol{x}} \in \varGamma {\rm{ }}} \\ {{\boldsymbol{\varPhi }}({\boldsymbol{x}}){\rm{ < }}0,\;\;\;{\boldsymbol{x}} \in \varOmega /D{\rm{ }}} \end{array}} \right.\;\;\;\; \begin{array}{c}{(}{\text{材料}}{)}\\ {(}{\text{边界}}{)}\\ {(}{\text{孔洞}}{)}\end{array}$ (5)

 图 2 三维水平集函数与二维结构边界 Fig. 2 3D LSF and 2D structural boundary

 ${\boldsymbol{\varPhi }}({\boldsymbol{x}},t) = \sum\limits_{j = 1}^m {{g_{j}}\left( {\boldsymbol{x}} \right){\alpha _{j}}(t)}$ (6)

 ${\boldsymbol{\varPhi }}({\boldsymbol{x}},t) = {\boldsymbol{G}}{\rm{(}}{\boldsymbol{x}}{\rm{)}}{\boldsymbol{\alpha }}{\rm{(}}t{\rm{)}}$ (7)

 ${g_{j}}\left( {\boldsymbol{x}} \right)= \left\{ {\begin{array}{*{20}{l}} {\rm{0}}&{r_{j}}{\rm{(}}{\boldsymbol{x}}{\rm{)}} \geqslant 1 \\ {{\left( {{\rm{1 - }}{r_{j}}{\rm{(}}{\boldsymbol{x}}{\rm{)}}} \right)}^4}\left( {4{r_{j}}{\rm{(}}{\boldsymbol{x}}{\rm{)}} + 1}\right)&{r_{j}}{\rm{(}}{\boldsymbol{x}}{\rm{)}} < 1 \end{array}} \right.$ (8)

 ${r_{j}}{\rm{(}}{\boldsymbol{x}}{\rm{) = }}\frac{{\left\| {{\boldsymbol{x}} - {{\boldsymbol{x}}_{j}}} \right\|}}{R}$ (9)

2.2 优化模型

 $\left\{ {\begin{array}{*{20}{l}} {{\boldsymbol{M}}\left( {\boldsymbol{\varPhi }} \right) = {{\boldsymbol{M}}_{\rm{b}}} + {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}\left( {{\boldsymbol{m}}_{{\rm{v}}i} + {\boldsymbol{m}}_{{\rm{c}}i}} \right)} \\ {{{\boldsymbol{K}}_{\rm{R}}}\left( {\boldsymbol{\varPhi }} \right) = {{\boldsymbol{K}}_{\rm{b}}} + {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}\left( {{\boldsymbol{k}}_{{\rm{v}}i} + {\boldsymbol{k}}_{{\rm{c}}i} + {\boldsymbol{k}}_{{\rm{\gamma v\_R}}i}} \right)} \\ {{{\boldsymbol{K}}_{\rm{I}}}\left( {\boldsymbol{\varPhi }} \right) = {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}{\boldsymbol{k}}_{{\rm{\gamma v\_I}}i}} \end{array}} \right.$ (10)

 ${\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{) = }}\left\{ {\begin{array}{*{20}{l}} {a\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\boldsymbol{\varPhi }} \leqslant - \varDelta } \\ {\dfrac{{3(1 - a)}}{{4\varDelta }}\left( {\dfrac{{\boldsymbol{\varPhi }}}{\varDelta } - \dfrac{{{{\left( {\boldsymbol{\varPhi }} \right)}^3}}}{{3{\varDelta ^3}}}} \right) + \dfrac{{1 + a}}{2}\;\quad - \varDelta < {\boldsymbol{\varPhi }} < \varDelta } \\ {1\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\boldsymbol{\varPhi }} \geqslant \varDelta } \end{array}} \right.$ (11)

 $\begin{split} & find:\,\;{\alpha _j},\;\;j = 1,2, \cdots, m \\ &\max :\;\,J = \displaystyle\sum\limits_{r = 1}^k {{\beta _{r}}{\eta _{r}}} \\ & s.t.\quad {\rm{ }}\left( {{{\boldsymbol{K}}_{\rm{R}}} + {\rm{i}}{{\boldsymbol{K}}_{\rm{I}}}} \right){{\boldsymbol{\varPsi }}_{r}} = \lambda _{r}^{\rm{*}}{\boldsymbol{M}}{{\boldsymbol{\varPsi }}_{r}} \\ & \quad \;\quad {\rm{ }}{\boldsymbol{\varPsi }}_{e}^{\rm{T}}{\boldsymbol{M}}{{\boldsymbol{\varPsi }}_{f}} = {\delta _{{ef}}} \\ & \qquad \;{\alpha _{{j\rm{,min}}}} \leqslant {\alpha _{j}} \leqslant {\alpha _{{j\rm{,max}}}} \\ &\qquad\; V{\rm{ = }}{\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}\sum\limits_{i = 1}^n {{v_{i}}} \leqslant \bar V \end{split}$ (12)

2.3 灵敏度分析

 $\begin{split}{J_{r}} =& \frac{{{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{R}}}{{\boldsymbol{\varPsi }}_{r}}}}{{{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}}}}{\rm{ + }}{\rm{R_{\rm{e}}}}\left( {{\boldsymbol{\mu }}_{\rm{1}}^{\rm{H}}\left( {{\boldsymbol{K}}_{\rm{R}}+{\rm i}{\boldsymbol{K}}_{\rm{I}} - \lambda _{r}^{\rm{*}}{\boldsymbol{M}}} \right){{\boldsymbol{\varPsi }}_{r}}} \right) + \\ &{\mu _{\rm{2}}}\left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{\boldsymbol{M}}{{\boldsymbol{\varPsi }}_{r}} - 1} \right)\end{split}$ (13)

 $\begin{split}\dfrac{{\partial {J_{r}}}}{{\partial {\alpha _{j}}}} = &\dfrac{{\left\{ \begin{array}{l} \left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}}} \right)\left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}\dfrac{{\partial {{\boldsymbol{K}}_{\rm{R}}}}}{{\partial {\alpha _{j}}}}{{\boldsymbol{\varPsi }}_{r}}} \right) \\ - \left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{R}}}{{\boldsymbol{\varPsi }}_{r}}} \right)\left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}\dfrac{{\partial {{\boldsymbol{K}}_{\rm{I}}}}}{{\partial {\alpha _{j}}}}{{\boldsymbol{\varPsi }}_{r}}} \right) \end{array} \right\}}}{{{{\left( {\phi _{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{\phi _{r}}} \right)}^2}}}{\rm{ + }}\\ &{\rm{R_{\rm{e}}}}\left( {{\boldsymbol{\mu }}_{\rm{1}}^{\rm{H}}\left( {\dfrac{{\partial {\boldsymbol{K}}}}{{\partial {\alpha _j}}} - \lambda _{r}^{\rm{2}}\dfrac{{\partial {\boldsymbol{M}}}}{{\partial {\alpha _{j}}}}} \right){{\boldsymbol{\varPsi }}_{r}}} \right) + {\mu _2}\left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}\dfrac{{\partial {\boldsymbol{M}}}}{{\partial {\alpha _{i}}}}{{\boldsymbol{\varPsi }}_{r}}} \right)\end{split}$ (14)

 $\begin{split} &\left[ {\begin{array}{*{20}{c}} {\left( {{\boldsymbol{K}} - \lambda _{r}^{\rm{2}}{\boldsymbol{M}}} \right)}&{2{\boldsymbol{M}}{{\boldsymbol{\varPsi }}_{r}}} \\ {2{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{\boldsymbol{M}}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{\mu }}_1}} \\ {{\mu _2}} \end{array}} \right] = \\ &\qquad\left[{ \begin{array}{*{20}{c}} \dfrac{ - \left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}}} \right)\left( {2{{\boldsymbol{K}}_{\rm{R}}}{{\boldsymbol{\varPsi }}_{r}}} \right) + \left( {{\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}}} \right)\left( {2{{\boldsymbol{K}}_{\rm{R}}}{{\boldsymbol{\varPsi }}_{r}}} \right)}{\left( {\boldsymbol{\varPsi }}_{r}^{\rm{T}}{{\boldsymbol{K}}_{\rm{I}}}{{\boldsymbol{\varPsi }}_{r}} \right)^2 }\\ 0 \end{array}} \right] \end{split}$ (15)

 $\left\{ {\begin{array}{*{20}{l}} {\dfrac{{\partial {\boldsymbol{K}}\left( {\boldsymbol{\varPhi }} \right)}}{{\partial {\alpha _{j}}}}{\rm{ = }}\dfrac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}\dfrac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {\alpha _{j}}}}\left( {{\boldsymbol{k}}_{{\rm{c}}i} + {\boldsymbol{k}}_{{\rm{v}}i} + {\boldsymbol{k}}_{{\rm{\gamma v\_R}}i}{\rm{ + }}{\rm{i}}{\boldsymbol{k}}_{{\rm{\gamma v\_I}}i}} \right){\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}} \\ {\dfrac{{\partial {\boldsymbol{M}}\left( {\boldsymbol{\varPhi }} \right)}}{{\partial {\alpha _{j}}}}{\rm{ = }}\dfrac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}\dfrac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {\alpha _{j}}}}\left( {{\boldsymbol{m}}_{{\rm{v}}i} + {\boldsymbol{m}}_{{\rm{c}}i}} \right){\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}} \\ {\dfrac{{\partial {{\boldsymbol{K}}_{\rm{R}}}\left( {\boldsymbol{\varPhi }} \right)}}{{\partial {\alpha _{j}}}}{\rm{ = }}\dfrac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}\dfrac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {\alpha _{j}}}}\left( {{\boldsymbol{k}}_{{\rm{c}}i} + {\boldsymbol{k}}_{{\rm{v}}i} + {\boldsymbol{k}}_{{\rm{\gamma v\_R}}i}} \right){\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}} \\ {\dfrac{{\partial {{\boldsymbol{K}}_{\rm{I}}}\left( {\boldsymbol{\varPhi }} \right)}}{{\partial {\alpha _{j}}}}{\rm{ = }}\dfrac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}\dfrac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {\alpha _{j}}}}{\boldsymbol{k}}_{{\rm{\gamma v\_I}}i}{\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}} \end{array}} \right.$ (16)

Heaviside函数的偏导数由式（11）可得

 $\frac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}{\rm{ = }}\left\{ {\begin{array}{*{20}{c}} {\dfrac{{{\rm{3}}\left( {{\rm{1 - }}a} \right)}}{{{\rm{4}}\varDelta }}\left( {{\rm{1 - }}{{\left( {\dfrac{{\boldsymbol{\varPhi }}}{\varDelta }} \right)}^{\rm{2}}}} \right)\quad \quad \left| {\boldsymbol{\varPhi }} \right| \leqslant \varDelta } \\ {b\quad \quad \quad \quad \quad \quad \quad\quad \quad \left| {\boldsymbol{\varPhi }} \right| > \varDelta } \end{array}} \right.$ (17)

 $\frac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {\alpha _{j}}}}{\rm{ = }}{\left[ {\begin{array}{*{20}{c}} {g_1\left( {{{\boldsymbol{x}}}} \right)}& \cdots &{g_m\left( {{{\boldsymbol{x}}}} \right)} \end{array}} \right]^{\rm{T}}}$ (18)

 $\frac{{\partial V}}{{\partial {{\rm{\alpha }}_{j}}}} = \frac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {{\rm{\alpha }}_{j}}}}\sum\limits_{i = 1}^n {{v_{i}}} = \frac{{\partial {\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}}}{{\partial {\boldsymbol{\varPhi }}}}\frac{{\partial {\boldsymbol{\varPhi }}}}{{\partial {{\rm{\alpha }}_{j}}}}\sum\limits_{i = 1}^n {{v_i}}$ (19)
2.4 设计变量更新准则

 $\begin{split}L =& {J_{r}}\left( {{\alpha _{j}}} \right) + \lambda _{j}^{\rm{1}}\left({\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{)}}\sum\limits_{i = 1}^n {{v_{i}}} - \bar V\right) + \\ &\sum\limits_{j = 1}^m {\lambda _{j}^{\rm{2}}({\alpha _{j,{\rm{min}}}} - {\alpha _{j}})} + \sum\limits_{j = 1}^m {\lambda _{j}^{\rm{3}}({\alpha _{j}} - {\alpha _{j,{\rm{max}}}})} \end{split}$ (20)

 $\alpha _{j}^{{(q + 1)}}{\rm{ = }}\left\{ {\begin{array}{*{20}{l}} {{\rm{max(0,}}\;{\alpha _{j}} - p{\rm{), }}\;\;\;\;\;{\alpha _{j}} < {\alpha _{{\rm{min}}}}} \\ {{{\left( {D_{j}^{{(q)}}} \right)}^{\rm{\xi }}}\alpha _{j}^{{(k)}}{\rm{, }}\;\;\;\;\;\;\;{\alpha _{{\rm{min}}}} \leqslant {\alpha _{j}} \leqslant {\alpha _{{\rm{max}}}}} \\ {{\rm{min(1,}}\;{\alpha _{j}} + p{\rm{), }}\;\;\;\;{\alpha _{i}} > {\alpha _{{\rm{max}}}}} \end{array}} \right.$ (21)

3 优化流程

 图 3 参数化水平集法拓扑优化流程图 Fig. 3 Flowchart of topology optimization using PLSM

a. 初始化水平集函数 $\varPhi$ 、径向基函数 ${\boldsymbol G}\left( {\boldsymbol{x}} \right)$ ，并依据式（7）初始化设计变量（扩展系数） ${{{\alpha }}_{j}}$

b. 依据初始化的水平集函数确定约束阻尼材料的初始构型，并建立对应的约束阻尼板结构的有限元模型；求解约束阻尼结构的特征方程，得到初始结构的模态损耗因子；

c.采用伴随向量法分析目标函数对设计变量的灵敏度；

d. 依据优化准则法更新设计变量，即参数化水平函数的扩展系数，得到更新的水平集函数，更新约束阻尼板结构的有限元模型和特征方程；

e.求解特征方程得到更新的目标函数以及体积约束条件；

f. 判断收敛条件是否满足，若不满足，重复步骤c～e，直至满足迭代终止条件。本文中的收敛条件为：体积分数与目标体积之差、目标函数值连续20代的变化值同时小于0.001。

4 算例分析和讨论

4.1 优化结果与分析

 图 4 基板结构的模态应变能分布 Fig. 4 Modal strain energy distribution of base plate structure

 图 5 一阶模态损耗因子最大化时的两种初始构型 Fig. 5 Two initial configurations of CLD materials for maximizing first modal loss factor

 图 6 一阶模态损耗因子最大化的拓扑优化结果 Fig. 6 Topology optimization results for maximizing the first modal loss factor
 ${\boldsymbol{H}}{\rm{(}}{\boldsymbol{\varPhi }}{\rm{) = }}\left\{ {\begin{array}{*{20}{c}} {0,\;\;\;{\boldsymbol{\varPhi }} \leqslant {{d}}} \\ {1,\;\;\;{\boldsymbol{\varPhi }} > {{d}}} \end{array}} \right.$ (22)

 图 7 不同初始构型的模态损耗因子迭代曲线和体积分数收敛曲线 Fig. 7 Iteration history of modal loss factor and volume fraction under different initial configurations

 图 8 二阶模态损耗因子最大化的初始构型及优化结果 Fig. 8 Initial configuration and optimization results for maximizing second modal loss factor

 图 9 一阶和二阶加权初始构型及优化结果 Fig. 9 Initial configurations and optimization results for weighted first and second order modal loss factor

4.2 优化结果对厚度参数的依赖性分析

 图 10 约束层和阻尼层厚度变化时的优化结果 Fig. 10 Optimal results with varying thickness of constrained layer and damping layer

 图 11 改变约束阻尼层相对厚度的优化结果 Fig. 11 Optimization results of changing the relative thickness of constrained damping layer

5 结　论

 [1] 房占鹏, 郑玲. 约束阻尼结构的双向渐进拓扑优化[J]. 振动与冲击, 2014, 33(8): 165-170, 191. [2] LALL A K, NAKRA B C, ASNANI N T. Optimum design of viscoelastically damped sandwich panels[J]. Engineering Optimization, 1983, 6(4): 197-205. DOI:10.1080/03052158308902470 [3] ZHENG H, CAI C, PAU G S H, et al. Minimizing vibration response of cylindrical shells through layout optimization of passive constrained layer damping treatments[J]. Journal of Sound and Vibration, 2005, 279(3/5): 739-756. [4] SASIKUMAR K S K, ARULSHRI K P, SELVAKUMAR S. Optimization of constrained layer damping parameters in beam using Taguchi method[J]. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 2017, 41(3): 243-250. DOI:10.1007/s40997-016-0056-y [5] 王明旭, 陈国平. 基于均匀化方法的约束阻尼柱壳结构动力学性能优化[J]. 中国机械工程, 2011, 22(8): 892-897. [6] FANG Z P, ZHENG L. Topology optimization for minimizing the resonant response of plates with constrained layer damping treatment[J]. Shock and Vibration, 2015, 376854. [7] 窦松然, 桂洪斌, 李承豪, 等. 圆柱壳振动控制中约束阻尼拓扑优化研究[J]. 振动与冲击, 2015, 34(22): 149-153. [8] ARAÚJO A L, MADEIRA J F A, MOTA SOARES C M, et al. Optimal design for active damping in sandwich structures using the direct multisearch method[J]. Composite Structures, 2013, 105: 29-34. DOI:10.1016/j.compstruct.2013.04.044 [9] ZHANG D D, WU Y H, CHEN J Y, et al. Sound radiation analysis of constrained layer damping structures based on two-level optimization[J]. Materials, 2019, 12(19): 3053. DOI:10.3390/ma12193053 [10] OSHER S, SETHIAN J A. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988, 79(1): 12-49. DOI:10.1016/0021-9991(88)90002-2 [11] ALLAIRE G, JOUVE F, TOADER A M. Structural optimization using sensitivity analysis and a level-set method[J]. Journal of Computational Physics, 2004, 194(1): 363-393. DOI:10.1016/j.jcp.2003.09.032 [12] ANSARI M, KHAJEPOUR A, ESMAILZADEH E. Application of level set method to optimal vibration control of plate structures[J]. Journal of Sound and Vibration, 2013, 332(4): 687-700. DOI:10.1016/j.jsv.2012.09.006 [13] WANG S Y, WANG M Y. Radial basis functions and level set method for structural topology optimization[J]. International Journal for Numerical Methods in Engineering, 2006, 65(12): 2060-2090. DOI:10.1002/nme.1536 [14] WANG S Y, LIM K M, KHOO B C, et al. An extended level set method for shape and topology optimization[J]. Journal of Computational Physics, 2007, 221(1): 395-421. DOI:10.1016/j.jcp.2006.06.029 [15] PANG J, ZHENG W G, YANG L, et al. Topology optimization of free damping treatments on plates using level set method[J]. Shock and Vibration, 2020, 5084167. [16] CUI M T, LUO C C, LI G, et al. The parameterized level set method for structural topology optimization with shape sensitivity constraint factor[J]. Engineering with Computers, 2021, 37(2): 855-872. DOI:10.1007/s00366-019-00860-8 [17] ZHANG D D, QI T, ZHENG L. A hierarchical optimization strategy for position and thickness optimization of constrained layer damping/plate to minimize sound radiation power[J]. Advances in Mechanical Engineering, 2018, 10(10): 1-15. [18] OSHER S, FEDKIW R P. Level set methods: an overview and some recent results[J]. Journal of Computational Physics, 2001, 169(2): 463-502. DOI:10.1006/jcph.2000.6636 [19] LUO Z, WANG M Y, WANG S Y, et al. A level set-based parameterization method for structural shape and topology optimization[J]. International Journal for Numerical Methods in Engineering, 2008, 76(1): 1-26. DOI:10.1002/nme.2092 [20] VAN DER KOLK M, VAN DER VEEN G J, DE VREUGD J, et al. Multi-material topology optimization of viscoelastically damped structures using a parametric level set method[J]. Journal of Vibration and Control, 2017, 23(15): 2430-2443. DOI:10.1177/1077546315617333