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基于参数化水平集法的材料非线性子结构拓扑优化

雷阳 封建湖

雷阳,封建湖. 基于参数化水平集法的材料非线性子结构拓扑优化 [J]. 应用数学和力学,2021,42(11):1150-1160 doi: 10.21656/1000-0887.420090
引用本文: 雷阳,封建湖. 基于参数化水平集法的材料非线性子结构拓扑优化 [J]. 应用数学和力学,2021,42(11):1150-1160 doi: 10.21656/1000-0887.420090
LEI Yang, FENG Jianhu. Topology Optimization of Nonlinear Material Structures Based on Parameterized Level Set and Substructure Methods[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1150-1160. doi: 10.21656/1000-0887.420090
Citation: LEI Yang, FENG Jianhu. Topology Optimization of Nonlinear Material Structures Based on Parameterized Level Set and Substructure Methods[J]. Applied Mathematics and Mechanics, 2021, 42(11): 1150-1160. doi: 10.21656/1000-0887.420090

基于参数化水平集法的材料非线性子结构拓扑优化

doi: 10.21656/1000-0887.420090
基金项目: 陕西省自然科学基金(2018JQ1027);中央高校基本科研业务费(300102120107)
详细信息
    作者简介:

    雷阳(1997—),女,硕士生(E-mail:leiyang_199703@163.com

    封建湖(1960—),男,教授,博士,博士生导师(通讯作者. E-mail:jhfeng@chd.edu.cn

  • 中图分类号: O302

Topology Optimization of Nonlinear Material Structures Based on Parameterized Level Set and Substructure Methods

  • 摘要: 针对利用传统水平集法进行非线性结构拓扑优化计算过程复杂及计算效率低等问题,将参数化水平集方法引入材料非线性结构拓扑优化中。通过全局径向基函数插值初始水平集函数,建立了以插值系数为设计变量、结构的应变能最小为目标函数、材料用量为约束条件的材料非线性结构拓扑优化模型,利用有限元分析对材料非线性结构建立平衡方程,并用迭代法求解。同时,采用子结构法划分设计区域为若干个子区域,将全自由度平衡方程的求解分解为缩减的平衡方程和多个子结构内部位移的求解,减小了计算成本。算例表明,这种处理非线性关系的方法可以在保证数值稳定的同时提高计算效率,得到边界清晰、结构合理的拓扑优化构形。
  • 图  1  二维结构边界及相应的水平集函数

    Figure  1.  The 2D structure boundary and the corresponding level set function

    图  2  L型设计域子结构划分

    Figure  2.  Division of substructures of the L-shaped design domain

    图  3  子结构法材料非线性结构拓扑优化流程图

    Figure  3.  The flowchart of nonlinear material structure topology optimization with the substructure method

    图  4  悬臂梁优化的设计域及边界条件

    Figure  4.  The design domain and boundary conditions for the optimization of a cantilever beam

    图  5  基于迭代法的材料非线性悬臂梁优化设计问题的优化历程

    Figure  5.  The evolution history of the nonlinear material cantilever beam optimized design based on the iterative method

    图  6  悬臂梁目标函数和体积分数的收敛曲线

    Figure  6.  Convergence curves of the objective function and the volume fraction of the cantilever beam problem

    图  7  简支梁优化的设计域和边界条件

    Figure  7.  The design domain and boundary conditions for the optimization of a freely supported beam

    图  8  基于迭代法的材料非线性简支梁优化设计问题的优化历程

    Figure  8.  The evolution history of the nonlinear material freely supported beam optimized design based on the iterative method

    图  9  简支梁目标函数和体积分数的收敛曲线

    Figure  9.  Convergence curves of the objective function and the volume fraction of the freely supported beam

    图  10  不同的初始设计和优化结果

    Figure  10.  Different initial designs and the optimized designs

    表  1  悬臂梁标准有限元法与子结构法计算结果对比

    Table  1.   The results of the cantilever beam standard finite element method and the substructure method

    $N_{\rm{CEX}} \times N_{\rm{CEY}}$$n_{\rm{elx}} \times n_{\rm{ely}}$objective function Jvolume fraction Vstep time t/soptimized design
    standard finite element method: 60×30 4.076 1E50.499 96.014 7
    $12 \times 6$$5 \times 5$4.072 7E50.500 34.937 9
    $6 \times 3$$10 \times 10$4.076 1E50.499 82.736 2
    $2 \times 1$$30 \times 30$4.066 5E50.500 31.336 4
    下载: 导出CSV

    表  2  简支梁标准有限元法与子结构法计算结果对比

    Table  2.   The results of the freely supported beam standard finite element method and the substructure method

    $N_{\rm{CEX}} \times N_{\rm{CEY}}$$n_{\rm{elx}} \times n_{\rm{ely}}$objective function Jvolume fraction Vstep time t/soptimized design
    standard finite element method: 120×30
    2.479 7E50.499 940.358 8
    $24 \times 6$$5 \times 5$2.475 5E50.499 947.779 5
    $12 \times 3$$10 \times 10$2.476 4E50.500 330.279 1
    $4 \times 1$$30 \times 30$2.476 3E50.501 016.701 1
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-04-07
  • 修回日期:  2021-05-19
  • 网络出版日期:  2021-12-07
  • 刊出日期:  2021-11-30

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