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基于PDE灵敏度滤波器的算法研究

孟换利 张岐良 王杰

孟换利,张岐良,王杰. 基于PDE灵敏度滤波器的算法研究 [J]. 应用数学和力学,2023,44(1):80-92 doi: 10.21656/1000-0887.430064
引用本文: 孟换利,张岐良,王杰. 基于PDE灵敏度滤波器的算法研究 [J]. 应用数学和力学,2023,44(1):80-92 doi: 10.21656/1000-0887.430064
MENG Huanli, ZHANG Qiliang, WANG Jie. Algorithm Research Based on the PDE Sensitivity Filter[J]. Applied Mathematics and Mechanics, 2023, 44(1): 80-92. doi: 10.21656/1000-0887.430064
Citation: MENG Huanli, ZHANG Qiliang, WANG Jie. Algorithm Research Based on the PDE Sensitivity Filter[J]. Applied Mathematics and Mechanics, 2023, 44(1): 80-92. doi: 10.21656/1000-0887.430064

基于PDE灵敏度滤波器的算法研究

doi: 10.21656/1000-0887.430064
基金项目: 中央高校基本科研业务费(76210-31610007)
详细信息
    作者简介:

    孟换利(1995—),女,硕士生(E-mail:menghli@mail2.sysu.edu.cn

    张岐良(1985—),男,助理教授,博士,硕士生导师(通讯作者. E-mail:zhangqliang@mail.sysu.edu.cn

  • 中图分类号: O241.82

Algorithm Research Based on the PDE Sensitivity Filter

  • 摘要:

    采用PDE灵敏度滤波器可以消除连续体结构拓扑优化结果存在的棋盘格现象、数值不稳定等问题,且PDE灵敏度滤波器的实质是具有Neumann边界条件的Helmholtz偏微分方程。针对大规模PDE灵敏度滤波器的求解问题,有限元分析得到其代数方程,分别采用共轭梯度算法、多重网格算法和多重网格预处理共轭梯度算法对代数方程进行求解,并且研究精度、过滤半径以及网格数量对拓扑优化效率的影响。结果表明:与共轭梯度算法和多重网格算法相比,多重网格预处理共轭梯度算法迭代次数最少,运行时间最短,极大地提高了拓扑优化效率。

  • 图  1  基于SIMP插值模型连续体结构拓扑优化实现流程图

    Figure  1.  The flow chart of continuum structure topology optimization based on the SIMP interpolation model

    图  2  悬臂梁

    Figure  2.  The cantilever beam

    图  3  Michell结构

    Figure  3.  The Michell structure

    图  4  不同精度下CG、MG和MGCG算法的悬臂梁的优化结构:(a) P=1E−2 (CG);(b) P=1E−2 (MG);(c) P=1E−2 (MGCG);(d) P=1E−3 (CG);(e) P=1E−3 (MG);(f) P=1E−3 (MGCG);(g) P=1E−4 (CG);(h) P=1E−4 (MG);(i) P=1E−4 (MGCG);(j) P=1E−5 (CG);(k) P=1E−5 (MG);(l) P=1E−5 (MGCG)

    Figure  4.  The optimized cantilever beams by CG, MG and MGCG algorithms with different degrees of precision: (a) P=1E−2 (CG); (b) P=1E−2 (MG); (c) P=1E−2 (MGCG); (d) P=1E−3 (CG); (e) P=1E−3 (MG); (f) P=1E−3 (MGCG); (g) P=1E−4 (CG); (h) P=1E−4 (MG); (i) P=1E−4 (MGCG); (j) P=1E−5 (CG); (k) P=1E−5 (MG); (l) P=1E−5 (MGCG)

    图  5  不同精度下CG、MG和MGCG算法的Michell结构的优化结构:(a) P=1E−2 (CG);(b) P=1E−2 (MG);(c) P=1E−2 (MGCG);(d) P=1E−3 (CG);(e) P=1E−3 (MG);(f) P=1E−3 (MGCG);(g) P=1E−4 (CG);(h) P=1E−4 (MG);(i) P=1E−4 (MGCG);(j) P=1E−5 (CG);(k) P=1E−5 (MG); (l) P=1E−5 (MGCG)

    Figure  5.  The optimized Michell structures by CG, MG and MGCG algorithms with different degrees of precision: (a) P=1E−2 (CG); (b) P=1E−2 (MG); (c) P=1E−2 (MGCG); (d) P=1E−3 (CG); (e) P=1E−3 (MG); (f) P=1E−3 (MGCG); (g) P=1E−4 (CG); (h) P=1E−4 (MG); (i) P=1E−4 (MGCG); (j) P=1E−5 (CG); (k) P=1E−5 (MG); (l) P=1E−5 (MGCG)

    图  6  悬臂梁结构优化时间分布图

    Figure  6.  The optimized time distribution of the cantilever beam

    图  7  Michell结构优化时间分布图

    Figure  7.  The optimized time distribution of the Michell structure

    图  8  不同过滤半径下CG、MG和MGCG算法的悬臂梁的优化结构:(a) R=3 (CG);(b) R=3 (MG);(c) R=3 (MGCG);(d) R=6 (CG);(e) R=6 (MG);(f) R=6 (MGCG);(g) R=9 (CG);(h) R=9 (MG);(i) R=9 (MGCG);(j) R=12 (CG);(k) R=12 (MG);(l) R=12 (MGCG)

    Figure  8.  The optimized cantilever beams by CG, MG and MGCG algorithms with different filter radius: (a) R=3 (CG); (b) R=3 (MG); (c) R=3 (MGCG); (d) R=6 (CG); (e) R=6 (MG); (f) R=6 (MGCG); (g) R=9 (CG); (h) R=9 (MG); (i) R=9 (MGCG); (j) R=12 (CG); (k) R=12 (MG); (l) R=12 (MGCG)

    图  9  不同过滤半径下CG、MG和MGCG算法的Michell结构的优化结构:(a) R=3 (CG);(b) R=3 (MG);(c) R=3 (MGCG);(d) R=6 (CG);(e) R=6 (MG);(f) R=6 (MGCG);(g) R=9 (CG);(h) R=9 (MG);(i) R=9 (MGCG);(j) R=12 (CG);(k) R=12 (MG);(l) R=12 (MGCG)

    Figure  9.  The optimized Michell structures by CG, MG and MGCG algorithms with different filter radius: (a) R=3 (CG); (b) R=3 (MG); (c) R=3 (MGCG); (d) R=6 (CG); (e) R=6 (MG); (f) R=6 (MGCG); (g) R=9 (CG); (h) R=9 (MG); (i) R=9 (MGCG); (j) R=12 (CG); (k) R=12 (MG); (l) R=12 (MGCG)

    图  10  悬臂梁结构优化时间分布图(不同过滤半径)

    Figure  10.  The optimized time distribution of the cantilever beam (different filter radius)

    图  11  Michell结构优化时间分布图(不同过滤半径)

    Figure  11.  The optimized time distribution of the Michell structure (different filter radius)

    图  12  不同网格数量下CG、MG和MGCG算法的悬臂梁的优化结构:(a) 网格为240 × 80 (CG);(b) 网格为240 × 80 (MG);(c) 网格为240 × 80 (MGCG);(d) 网格为480 × 160 (CG);(e) 网格为480 × 160 (MG);(f) 网格为480 × 160 (MGCG);(g) 网格为960 × 320 (CG);(h) 网格为960 × 320 (MG);(i) 网格为960 × 320 (MGCG)

    Figure  12.  The optimized cantilever beams by CG, MG and MGCG algorithms with different grids: (a) grid 240 × 80 (CG); (b) grid 240 × 80 (MG); (c) grid 240 × 80 (MGCG); (d) grid 480 × 160 (CG); (e) grid 480 × 160 (MG); (f) grid 480 × 160 (MGCG); (g) grid 960 × 320 (CG); (h) grid 960 × 320 (MG); (i) grid 960 × 320 (MGCG)

    图  13  不同网格数量下CG、MG和MGCG算法的Michell结构的优化结构:(a) 网格为240 × 120 (CG);(b) 网格为240 × 120 (MG);(c) 网格为240 × 120 (MGCG);(d) 网格为480 × 240 (CG);(e) 网格为480 × 240 (MG);(f) 网格为480 × 240 (MGCG);(g) 网格为960 × 480 (CG);(h) 网格为960 × 480 (MG);(i) 网格为960 × 480 (MGCG)

    Figure  13.  The optimized Michell structures by CG, MG and MGCG algorithms with different grids: (a) grid 240 × 120 (CG); (b) grid 240 × 120 (MG); (c) grid 240 × 120 (MGCG); (d) grid 480 × 240 (CG); (e) grid 480 × 240 (MG); (f) grid 480 × 240 (MGCG); (g) grid 960 × 480 (CG); (h) grid 960 × 480 (MG); (i) grid 960 × 480 (MGCG)

    图  14  悬臂梁结构优化时间分布图(不同网格数量)

    Figure  14.  The optimized time distribution of the cantilever beam (different grids)

    图  15  Michell结构优化时间分布图(不同网格数量)

    Figure  15.  The optimized time distribution of the Michell structure (different grids)

      算法1
      1 input:
      2 level k;
      3 initial guess $ {\boldsymbol{u}}_k^m $;
      4 right hand side vector $ {{\boldsymbol{f}}_k} $;
      5 coefficient matrix under level $ k $ grid $ {{\boldsymbol{A}}_k} $;
      6 output:
      7 updated solution $ {\boldsymbol{u}}_k^{m + 1} $
      8 pre-smooth: $ {\boldsymbol{u}}_k^{m + 1} = {{\boldsymbol{S}}^{{\mu _1}}}({{\boldsymbol{A}}_k},{{\boldsymbol{f}}_k},{\boldsymbol{u}}_k^m) $
      9 coarse grid correction:
      10 compute residual: $ {\boldsymbol{r}}_k^m = {{\boldsymbol{f}}_k} - {{\boldsymbol{A}}_k}{\boldsymbol{u}}_k^m $
      11 restrict the residual to coarse-grid: $ {\boldsymbol{r}}_{k + 1}^m = {\boldsymbol{R}}_k^{k + 1}{\boldsymbol{r}}_k^m $
      12 if k + 1=L then
      13  solve approximate error solution $ \tilde {\boldsymbol{e}}_{k + 1}^m = {\boldsymbol{A}}_{k + 1}^{ - 1}{\boldsymbol{r}}_{k + 1}^m $
      14 else
      15  $ \tilde {\boldsymbol{e}}_{k + 1}^m{\text{ = }}V \cdot {\text{cycle(}}{{\boldsymbol{A}}_{k + 1}},{\boldsymbol{r}}_{k + 1}^m,0,k + 1) $
      16 end
      17 interpolate:
      18 $ \tilde {\boldsymbol{e}}_k^m = P_{k + 1}^k\tilde {\boldsymbol{e}}_{k + 1}^m $
      19 $ {\boldsymbol{u}}_k^{m + 1} = {\boldsymbol{u}}_k^{m + 1} + \tilde {\boldsymbol{e}}_k^m $
      20 post-smooth:
      21 $ {\boldsymbol{u}}_k^{m + 1} = {{\boldsymbol{S}}^{{\mu _2}}}({{\boldsymbol{A}}_k},{{\boldsymbol{f}}_k},{\boldsymbol{u}}_k^{m + 1}) $。
    下载: 导出CSV
      算法2
      input:
       symmetric positive definite coefficient matrix $ {\boldsymbol{A}} $;
       right hand side vector $ {\boldsymbol{b}} $;
       initiate a guess for $ {{\boldsymbol{x}}_0} \in {\mathbb{R}^n} $
      output:
       solution $ {\boldsymbol{x}} $;
      1 compute the residual on the finest grid $ {{\boldsymbol{r}}_0} \mathop {\rm{ = }}\limits^\Delta {\boldsymbol{b}} - {\boldsymbol{A}}{{\boldsymbol{x}}_0} $;
      2 for $ i = 1,2,3, \cdots $ do
      3 solve $ {\boldsymbol{A}}{{\boldsymbol{z}}_0} = {{\boldsymbol{r}}_0} $ by the multigrid V-cycle algorithm
      4 initialize $ {{\boldsymbol{p}}_0} = {{\boldsymbol{z}}_0} $
      5 compute a step length the $ {\alpha _i} = ({\boldsymbol{r}}_{i - 1}^{\text{T}}{{\boldsymbol{z}}_{i - 1}})/({\boldsymbol{p}}_{i - 1}^{\text{T}}{\boldsymbol{A}}{{\boldsymbol{p}}_{i - 1}}) $
         (where $ {{\boldsymbol{z}}_{i - 1}} $ can be solved by the multigrid V-cycle algorithm)
      6 update the approximate solution $ {{\boldsymbol{x}}_i} = {{\boldsymbol{x}}_{i - 1}} + {\alpha _i}{{\boldsymbol{p}}_{i - 1}} $
      7 update the residual $ {{\boldsymbol{r}}_i} = {{\boldsymbol{r}}_{i - 1}} - {\alpha _i}{\boldsymbol{A}}{{\boldsymbol{p}}_{i - 1}} $
      8 solve $ {\boldsymbol{A}}{{\boldsymbol{z}}_i} = {{\boldsymbol{r}}_i} $ by the multigrid V-cycle algorithm
      9 compute a gradient correction factor $ {\beta _i} = ({\boldsymbol{r}}_i^{\rm{T}}{{\boldsymbol{z}}_i})/({\boldsymbol{r}}_{i - 1}^{\rm{T}}{{\boldsymbol{z}}_{i - 1}}) $
      10 set the new search direction $ {{\boldsymbol{p}}_i} = {{\boldsymbol{z}}_i} + {\beta _i}{{\boldsymbol{p}}_{i - 1}} $
      11 end
      12 until convergence。
    下载: 导出CSV

    表  1  两种结构的拓扑优化迭代次数和柔度值

    Table  1.   Total numbers of iterations and compliance for topology optimization of 2 structures

    structureprecision PCG algorithmMG algorithmMGCG algorithm
    compliance Ctime T/siterations Icompliance Ctime T/siterations Icompliance Ctime T/siterations I
    cantilever beam1E−2186.885 287170187.306 996167184.738 364114
    1E−3186.883 482155186.939 3100160184.414 754100
    1E−4186.884 284155186.939 2105160184.014 075133
    1E−5186.883 984155186.939 1111160185.215 564113
    Michell structure1E−217.125 233138716.909 5707816.923 86878
    1E−317.123 423127117.119 926027716.923 96778
    1E−417.123 123027117.119 926927716.952 68093
    1E−517.123 123127117.074 726726716.943 66271
    下载: 导出CSV

    表  2  两种结构的拓扑优化迭代次数和柔度值(不同过滤半径)

    Table  2.   Total numbers of iterations and compliance for topology optimization of 2 structures (different filter radius)

    structurefilter radius RCG algorithmMG algorithmMGCG algorithm
    compliance Ctime T/siterations Icompliance Ctime T/siterations Icompliance Ctime T/siterations I
    cantilever beam3180.759 1505917182.218 3147225182.352 058103
    6186.884 282155186.939 2101160184.014 073133
    9192.522 998183193.053 764100193.096 75397
    12197.418 773136197.669 567105197.641 962115
    Michell structure316.738 584092616.864 516516816.894 18090
    617.123 123927117.119 928027716.952 68493
    917.382 911713317.322 611511717.168 65967
    1217.696 49911417.696 510711417.686 4100114
    下载: 导出CSV

    表  3  两种结构的拓扑优化迭代次数和柔度值(不同网格数量)

    Table  3.   Total numbers of iterations and compliance for topology optimization of 2 structures (different grids)

    structuregrid sizeCG algorithmMG algorithmMGCG algorithm
    compliance Ctime T/siterations Icompliance Ctime T/siterations Icompliance Ctime T/siterations I
    cantilever beam240 × 80186.484 016133186.315 319133183.148 41296
    480 × 160186.884 282155186.939 2101160184.014 073133
    960 × 320188.099 5430167188.513 5495173185.695 5419155
    Michell structure240 × 12015.995 44223315.867 0209115.816 01583
    480 × 24017.123 123527117.119 927427716.952 68093
    960 × 48018.293 9131929218.288 6148 130618.095 936191
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-03-01
  • 修回日期:  2022-12-15
  • 网络出版日期:  2023-01-06
  • 刊出日期:  2023-01-15

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