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零泊松比超材料设计的多评价点功能基元拓扑优化方法

杨德庆 钟山

杨德庆, 钟山. 零泊松比超材料设计的多评价点功能基元拓扑优化方法[J]. 复合材料学报, 2020, 37(12): 3229-3241. doi: 10.13801/j.cnki.fhclxb.20200306.001
引用本文: 杨德庆, 钟山. 零泊松比超材料设计的多评价点功能基元拓扑优化方法[J]. 复合材料学报, 2020, 37(12): 3229-3241. doi: 10.13801/j.cnki.fhclxb.20200306.001
YANG Deqing, ZHONG Shan. Functional element topology optimization method based on multiple evaluation points for metamaterial design with zero Poisson’s ratio[J]. Acta Materiae Compositae Sinica, 2020, 37(12): 3229-3241. doi: 10.13801/j.cnki.fhclxb.20200306.001
Citation: YANG Deqing, ZHONG Shan. Functional element topology optimization method based on multiple evaluation points for metamaterial design with zero Poisson’s ratio[J]. Acta Materiae Compositae Sinica, 2020, 37(12): 3229-3241. doi: 10.13801/j.cnki.fhclxb.20200306.001

零泊松比超材料设计的多评价点功能基元拓扑优化方法

doi: 10.13801/j.cnki.fhclxb.20200306.001
基金项目: 国家自然科学基金(51479115);海洋工程国家重点实验室课题(GKZD010071);高技术船舶科研计划项目(科军评[2014]148号;联装函[2016]548号)
详细信息
    通讯作者:

    钟山,硕士研究生,研究方向为结构减振降噪控制理论与方法、优化设计方法 E-mail:zhongshan9988@163.com

  • 中图分类号: TB330.1

Functional element topology optimization method based on multiple evaluation points for metamaterial design with zero Poisson’s ratio

  • 摘要: 提出基于多评价点约束的零泊松比超材料功能基元拓扑优化设计方法。在同一功能基元拓扑基结构中,通过建立对于多个评价点的正、负泊松比约束,实现胞元零泊松比效应。分别采用最小质量和最大柔度目标函数拓扑优化模型优化设计出与半内六角蜂窝相似的零泊松比功能基元最优拓扑构型。提取功能基元最优构型并周期性序构了零泊松比超材料试件,通过有限元方法验证了该功能基元的零泊松比效应,并分析超材料试件的静、动力学特性。计算结果表明,最大柔度目标函数设计的功能基元构型的泊松比更接近于零,且具有更好的承载与隔振性能。设计了零泊松比超材料环肋双层圆柱壳结构,进行外壳静压和内部设备激振下壳体水下辐射噪声分析。研究表明,零泊松比超材料环肋可将外壳压缩变形转换为内外壳间环肋旋转,实现耐压壳内壳的保形,且具有较好的降噪性能。

     

  • 图  1  不同混合型零泊松比(ZPR)蜂窝及其周期微元

    Figure  1.  Different periodic elements of hybrid honeycombs with zero Poisson’s ratio (ZPR)

    图  2  超材料设计的功能基元拓扑优化(FETO)法示意图

    Figure  2.  Schematic of the functional element topology optimization (FETO) method of metamaterial design

    图  3  任意正/负泊松比超材料设计的矩形拓扑基结构

    Figure  3.  Rectangular topological ground structure of metamaterial with arbitrary positive/negative Poisson’s ratio

    图  4  ZPR超材料设计的拓扑基结构

    Figure  4.  Topological ground structure of metamaterial with ZPR

    图  5  不同目标函数下ZPR功能基元构型

    Figure  5.  Configurations of functional element with ZPR in different objective functions

    图  6  ZPR功能基元的有限元模型

    Figure  6.  FEM models of functional element with ZPR

    图  7  泊松比验证的评价点位置

    Figure  7.  Position of measuring points for Poisson’s ratio validation

    图  8  ZPR超材料结构轮廓尺寸参数

    Figure  8.  Dimension parameters of metamaterial structure with ZPR

    图  9  ZPR超材料结构振动评价点位置

    Figure  9.  Position of evaluation points for calculating vibration of metamaterial structure with ZPR

    图  10  不同ZPR超材料结构的频响曲线

    Figure  10.  Frequency response curves of metamaterial structure with ZPR in different finite element models

    图  11  不同ZPR超材料结构的加速度振级落差

    Figure  11.  Acceleration vibration level difference of metamaterial structure with ZPR in different finite element models

    图  12  ZPR超材料环肋圆柱壳

    Figure  12.  Ring rib of metamaterial structure with ZPR in cylindrical shell

    图  13  圆柱壳结构尺寸示意图

    Figure  13.  Dimension parameters of cylindrical shell structure

    图  14  常规圆柱壳与ZPR超材料圆柱壳

    Figure  14.  Conventional cylindrical shell and metamaterial cylindrical shell with ZPR

    图  15  双层圆柱壳静力学评价点位置

    Figure  15.  Position of evaluation points of double cylindrical shell in static analysis

    图  16  不同双层圆柱壳的应力分布

    Figure  16.  Stress distribution of different double cylindrical shells

    图  17  ZPR超材料圆柱壳静压变形示意图

    Figure  17.  Schematic of displacement in metamaterial cylindrical shell with ZPR under static pressure

    图  18  包含基座的双层圆柱壳FEM模型

    Figure  18.  FEM model of double cylindrical shell with base structure

    图  19  双层圆柱壳声学边界元模型

    Figure  19.  Acoustic boundary element model of double cylindrical shell

    图  20  不同圆柱壳的水下辐射声功率级

    Figure  20.  Underwater radiation noise power level of different cylindrical shells

    表  1  ZPR功能基元的验证

    Table  1.   Verification of functional element with ZPR

    FEM modelObjective functionOrientationStrainPoisson’s ratio
    Beam elementMinimal massX−1.17×10−40.091
    Y 1.06×10−5
    Maximal complianceX−7.84×10−5−0.0091
    Y−7.14×10−7
    Shell elementMinimal massX−1.07×10−50.095
    Y 1.01×10−6
    Maximal complianceX−6.95×10−6−0.012
    Y−8.53×10−8
    Note: FEM—Finite element model.
    下载: 导出CSV

    表  2  ZPR超材料结构宏观等效弹性模量和面内比模量

    Table  2.   Macroscopic equivalent elastic modulus and in-plane specific stiffness of metamaterial structures with ZPR

    FEM modelObjective functionP/NX/mmH/mmW/mmt1/mmA/mm2Ee/MPaKs/(m2·s−2)
    Beam elementMinimal mass1060.371682102420115.53146 117.42
    Maximal compliance1060.271682102420158.21220 660.68
    Shell elementMinimal mass1050.033168210204 200126.13159 517.11
    Maximal compliance1060.024168210204 200178.90249 129.51
    Notes:P—Load amplitude; △X—Overall deformation of metamaterial structure; H—Height of metamaterial structure; W—Width of metamaterial structure; t1—Thickness of metamaterial structure; A—Area of the plane perpendicular to the load in metamaterial structure; Ee and Ks—Macroscopic equivalent elastic modulus and in-plane specific stiffness of metamaterial structures.
    下载: 导出CSV

    表  3  ZPR超材料结构的总加速度振级落差

    Table  3.   Total acceleration VLD of metamaterial structure with ZPR

    FEM modelObjective functionEvaluation pointsAcceleration VLD/dB
    Beam elementMinimal massA-B1.60
    B-C2.21
    C-D3.09
    Maximal complianceA-B1.70
    B-C2.30
    C-D3.18
    Shell elementMinimal massA-B1.42
    B-C2.00
    C-D2.85
    Maximal complianceA-B1.74
    B-C2.60
    C-D3.66
    Note: VLD—Vibration level difference.
    下载: 导出CSV

    表  4  不同双层圆柱壳的径向位移

    Table  4.   Radial displacement of different double cylindrical shells

    Evaluation pointConventional cylindrical shellMetamaterial cylindrical shell
    Inner shell/mmOuter shell/mmIn-out ratioInner shell/mmOuter shell/mmIn-out ratio
    N1.581.730.9130.8852.690.329
    NE1.581.730.9130.8172.730.300
    E1.591.740.9140.8892.690.330
    SE1.561.710.9130.7402.690.275
    S1.571.720.9080.8412.650.318
    SW1.541.690.9130.7552.640.286
    W1.561.720.9080.8422.650.318
    NW1.561.700.9130.7822.690.291
    Average1.571.720.9120.8192.680.306
    下载: 导出CSV

    表  5  不同双层圆柱壳的内壳周向位移

    Table  5.   Circumferential displacement of inner shell in different double cylindrical shells

    Evaluation pointConventional cylindrical shell/mmMetamaterial cylindrical shell/mm
    N 4.38×10−3 −0.193
    NE 4.26×10−3 −0.218
    E 2.64×10−3 −0.241
    SE −1.12×10−3 −0.253
    S −9.50×10−4 −0.245
    SW −1.50×10−3 −0.225
    W 2.64×10−4 −0.197
    NW 4.69×10−3 −0.187
    Average 1.58×10−3 −0.220
    Ratio to radial displacement 1.01×10−3 0.269
    Note:Positive and negative values in the table mean counterclockwise and clockwise circumferential displacements respectively.
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-01-07
  • 录用日期:  2020-02-27
  • 网络出版日期:  2020-03-06
  • 刊出日期:  2020-12-15

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