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载荷方向不确定条件下结构动态稳健性拓扑优化设计

杜鼎新 王栋

杜鼎新, 王栋. 载荷方向不确定条件下结构动态稳健性拓扑优化设计. 力学学报, 待出版 doi: 10.6052/0459-1879-23-288
引用本文: 杜鼎新, 王栋. 载荷方向不确定条件下结构动态稳健性拓扑优化设计. 力学学报, 待出版 doi: 10.6052/0459-1879-23-288
Du Dingxin, Wang Dong. Robust dynamic topology optimization of continuum structure subjected to harmonic excitation with loading direction uncertainty. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-23-288
Citation: Du Dingxin, Wang Dong. Robust dynamic topology optimization of continuum structure subjected to harmonic excitation with loading direction uncertainty. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-23-288

载荷方向不确定条件下结构动态稳健性拓扑优化设计

doi: 10.6052/0459-1879-23-288
基金项目: 国家自然科学基金资助项目(51975470)
详细信息
    通讯作者:

    王栋, 教授, 主要研究方向为结构动力学行为分析与优化设计. E-mail: dwang@nwpu.edu.cn

  • 中图分类号: O327

ROBUST DYNAMIC TOPOLOGY OPTIMIZATION OF CONTINUUM STRUCTURE SUBJECTED TO HARMONIC EXCITATION WITH LOADING DIRECTION UNCERTAINTY

  • 摘要: 采用一种高效的方法开展了在外载荷作用方向不确定条件下, 连续体结构动态稳健性拓扑优化设计研究, 有效降低了结构的稳态动响应对简谐激励作用方向随机扰动的敏感性. 首先基于概率模型, 将动载荷作用方向的不确定性用正态分布函数表示. 其次通过二阶泰勒展开式, 高效地计算出在激励方向扰动情形下结构动柔顺度的均值和方差, 进而推导出了其对拓扑设计变量的一阶导数灵敏度显性表达式. 最后在材料体积约束下, 以动柔顺度概率特征指标的加权和为设计目标, 基于变密度方法, 对连续体结构进行动态稳健性拓扑优化设计, 并与传统载荷方向固定条件下的确定性优化结果进行对比, 充分展示了考虑外激励作用方向随机扰动对结构拓扑构型设计及其动柔顺度变化的影响. 对优化数值结果进一步分析表明, 采用文章提出的方法所得结构的动响应稳健性更高, 能有效抵抗外激励作用方向的随机扰动. 只需少许增加材料, 稳健性优化设计的动响应将在整个载荷扰动区域内优于确定性优化结果.

     

  • 图  1  结构受具有方向不确定外载荷作用的示意图

    Figure  1.  Schematic of a load with the direction uncertainty imposed to a structure

    图  2  简支平面矩形板结构

    Figure  2.  A simply supported rectangular panel

    图  3  确定性和稳健性设计策略结构拓扑构型对比

    Figure  3.  Comparison of the topology optimizations on two different design strategies of DTO and RTO

    图  4  结构动柔顺度随外激励作用方向变化情况

    Figure  4.  Variation of the structural dynamic compliance caused by the external loading direction disturbances

    图  5  两种设计策略结构拓扑优化在高频外激励条件下结果对比

    Figure  5.  Comparison of the topology optimizations on two different design strategies under an external load of a higher frequency

    图  6  MBB 梁设计区域及外激励

    Figure  6.  Design domain of the MBB beam and the external force

    图  7  两种设计策略结构拓扑优化结果对比

    Figure  7.  Comparison of the topology optimizations on two different design strategies

    图  8  结构动柔顺度随外激励作用方向变化情况

    Figure  8.  Variation of the structural dynamic compliance caused by the external loading direction disturbances

    图  9  两种设计策略拓扑优化收敛过程

    Figure  9.  Convergence curves of the dynamic compliance on the two design strategies

    表  1  两种方法所得结构动柔顺度概率特征指标对比

    Table  1.   Comparison of the obtained stochastic measurements of the structural dynamic compliance

    Method$ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J$ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/JCPU time/s
    MCS8.233 00.168 327.890 6
    proposed method8.224 50.170 80.015 6
    different ratio/%0.103 21.485 4−99.944 1
    下载: 导出CSV

    表  2  两种设计策略结构拓扑优化结果对比

    Table  2.   Comparison of the numerical results with the two optimization strategies

    Loading directionCd0/J$ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J$ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/JJ/Jcv
    fixed0.961 72.461 62.181 46.824 50.842 1
    uncertain1.108 91.170 30.016 01.202 30.013 7
    different ratio/%15.302 1−52.458 9−99.266 4−82.382 9−98.376 1
    下载: 导出CSV

    表  3  简支平板结构前3阶固有频率比较

    Table  3.   Comparison of the first three natural frequencies of the rectangular panel structure

    Design statusNatural frequency /Hz
    firstsecondthird
    initial265.64285.96468.53
    deterministic optimization483.12496.45613.87
    robust optimization482.73489.39595.12
    下载: 导出CSV

    表  4  两种设计策略结构拓扑在高频外激励条件下优化结果对比

    Table  4.   Comparison of the optimized results under an external load of a higher frequency

    LoadingdirectionCd0/J$ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J$ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/JJ/Jcv
    fixed0.816 41.090 40.079 11.248 60.072 5
    uncertain0.908 00.896 80.015 80.928 40.017 6
    different ratio/%11.220 0−17.755 0−80.025 3−25.644 7−75.724 1
    下载: 导出CSV

    表  5  结构拓扑优化结果对比

    Table  5.   Comparison of the optimized results

    Loading directionCd0/J$ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J$ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/Jcv (10−2)
    fixed2.967 73.001 20.047 51.581 7
    uncertain β = 13.085 42.989 70.006 60.220 8
    different ratio/%3.631 2−0.383 9−86.097 1−86.043 5
    uncertain β = 23.008 92.995 80.004 50.149 3
    different ratio/%1.389 2−0.180 7−90.577 0−90.559 9
    下载: 导出CSV
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