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线法二阶常微分方程组有限元分析的结点精度修正及其超收敛计算

黄泽敏 袁驷

黄泽敏, 袁驷. 线法二阶常微分方程组有限元分析的结点精度修正及其超收敛计算[J]. 工程力学, 2022, 39(S): 9-14, 34. doi: 10.6052/j.issn.1000-4750.2021.06.S002
引用本文: 黄泽敏, 袁驷. 线法二阶常微分方程组有限元分析的结点精度修正及其超收敛计算[J]. 工程力学, 2022, 39(S): 9-14, 34. doi: 10.6052/j.issn.1000-4750.2021.06.S002
HUANG Ze-min, YUAN Si. NODAL ACCURACY IMPROVEMENT AND SUPER-CONVERGENT COMPUTATION IN FEM ANALYSIS OF FEMOL SECOND ORDER ODES[J]. Engineering Mechanics, 2022, 39(S): 9-14, 34. doi: 10.6052/j.issn.1000-4750.2021.06.S002
Citation: HUANG Ze-min, YUAN Si. NODAL ACCURACY IMPROVEMENT AND SUPER-CONVERGENT COMPUTATION IN FEM ANALYSIS OF FEMOL SECOND ORDER ODES[J]. Engineering Mechanics, 2022, 39(S): 9-14, 34. doi: 10.6052/j.issn.1000-4750.2021.06.S002

线法二阶常微分方程组有限元分析的结点精度修正及其超收敛计算

doi: 10.6052/j.issn.1000-4750.2021.06.S002
基金项目: 国家自然科学基金项目(51878383,51378293)
详细信息
    作者简介:

    袁 驷(1953−),男,北京人,教授,博士,主要从事结构工程研究(E-mail: yuans@tsinghua.edu.cn)

    通讯作者:

    黄泽敏(1996−),男,福建人,博士生,主要从事结构工程研究(E-mail: hzm18@mails.tsinghua.edu.cn)

  • 中图分类号: O241.81

NODAL ACCURACY IMPROVEMENT AND SUPER-CONVERGENT COMPUTATION IN FEM ANALYSIS OF FEMOL SECOND ORDER ODES

  • 摘要: 采用$ m $次单元对线法二阶常微分方程组(ODEs)进行有限元(FEM)求解,其单元内部位移为$ m + 1 $阶收敛,而端结点位移收敛阶可达$ 2m $阶。单元能量投影(EEP)超收敛计算恢复的单元内部位移精度一般为$\min $$ (m + 2,2m)$阶,此收敛阶既受益于也受限于有限元端结点位移的精度。该文提出了一种修正EEP法(M-EEP),利用EEP超收敛解,先对端结点位移进行修正,再用其恢复单元内部位移。广泛的数值试验表明:对端结点位移修正后的收敛阶可达$ 2m + 2 $阶,再次修复的单元内部位移始终可达$ m + 2 $阶收敛,摆脱了$ 2m $阶收敛精度的限制。对于线性元,修正后结点位移的精度翻倍,单元内部M-EEP位移亦摆脱了原FEM解2阶收敛精度的限制,升到3阶收敛,基本达到二次元的收敛精度,效果显著。
  • 图  1  求解域和FEMOL网格

    Figure  1.  Solution domain and FEMOL meshes

    图  2  求解域

    Figure  2.  Solution domain

    图  3  FEMOL网格

    Figure  3.  FEMOL mesh

    表  1  常系数问题端结点位移收敛阶

    Table  1.   Convergence orders of nodal displacement of constant coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解修正解
    误差$ e_{i,{\kern 1pt} \max }^h $收敛阶ρ误差$ \tilde e_{i,{\kern 1pt} \max }^h $收敛阶ρ
    1 4 3.68×10−3 2.38×10−4
    8 9.10×10−4 2.01 2.11×10−5 3.50
    16 2.27×10−4 2.00 1.40×10−6 3.91
    32 5.67×10−5 2.00 7.82×10−8 4.16
    64 1.42×10−5 2.00 4.76×10−9 4.04
    2 4 5.62×10−5 1.03×10−5
    8 5.17×10−6 3.44 3.16×10−7 5.03
    16 3.36×10−7 3.94 5.34×10−9 5.89
    32 2.05×10−8 4.04 8.25×10−11 6.01
    64 1.29×10−9 3.99 1.31×10−12 5.98
    3 4 3.18×10−6 3.54×10−7
    8 9.71×10−8 5.03 3.37×10−9 6.71
    16 1.62×10−9 5.91 1.54×10−11 7.78
    32 2.48×10−11 6.03 5.92×10−14 8.02
    64 3.92×10−13 5.98 2.30×10−16 8.01
    收敛阶 $ {{2}}{m} $ $ {{2}}{m}{{ + 2}} $
    下载: 导出CSV

    表  2  常系数问题内部结点位移收敛阶

    Table  2.   Convergence orders of interior nodal displacement of constant coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解EPP解M-EEP解
    误差$ e_{\max }^h $收敛阶ρ误差$ e_{\max }^{\text{*}} $收敛阶ρ误差$ \tilde e_{\max }^* $收敛阶ρ
    1 4 1.30×10−2 3.68×10−3 1.08×10−3
    8 3.54×10−3 1.88 9.11×10−4 2.02 1.08×10−4 3.31
    16 9.47×10−4 1.90 2.27×10−4 2.00 9.24×10−6 3.55
    32 2.57×10−4 1.88 5.67×10−5 2.00 7.07×10−7 3.71
    64 6.69×10−5 1.94 1.42×10−5 2.00 5.21×10−8 3.76
    2 4 6.07×10−4 7.39×10−5 4.61×10−5
    8 1.02×10−4 2.58 6.63×10−6 3.48 3.04×10−6 3.92
    16 1.58×10−5 2.69 4.04×10−7 4.04 1.32×10−7 4.52
    32 2.16×10−6 2.87 2.28×10−8 4.15 5.02×10−9 4.72
    64 2.82×10−7 2.93 1.36×10−9 4.06 1.72×10−10 4.87
    3 4 4.72×10−5 1.02×10−5 4.12×10−6
    8 4.65×10−6 3.34 4.21×10−7 4.60 1.44×10−7 4.84
    16 3.73×10−7 3.64 1.09×10−8 5.27 3.42×10−9 5.39
    32 2.83×10−8 3.72 2.20×10−10 5.64 6.58×10−11 5.70
    64 1.97×10−9 3.85 3.90×10−12 5.82 1.14×10−12 5.85
    收敛阶 $ {m}{{ + 1}} $ ${ {{\rm{min}}(2} }{m},{m}{ { + 3)} }$ $ {m}{{ + 3}} $
    下载: 导出CSV

    表  3  变系数问题端结点位移收敛阶

    Table  3.   Convergence orders of nodal displacement of variable coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解修正解
    误差$ e_{i,{\kern 1pt} \max }^h $收敛阶ρ误差$ \tilde e_{i,{\kern 1pt} \max }^h $收敛阶ρ
    1 32 1.54×10−4 8.76×10−7
    64 3.86×10−5 2.00 5.37×10−8 4.03
    128 9.66×10−6 2.00 3.32×10−9 4.01
    256 2.42×10−6 2.00 2.08×10−10 4.00
    512 6.04×10−7 2.00 1.30×10−11 4.00
    2 32 2.04×10−7 1.59×10−9
    64 1.26×10−8 4.02 2.46×10−11 6.01
    128 7.83×10−10 4.00 3.84×10−13 6.00
    256 4.90×10−11 4.00 6.01×10−15 6.00
    512 3.06×10−12 4.00 9.39×10−17 6.00
    3 32 4.69×10−10 1.71×10−12
    64 7.28×10−12 6.01 6.64×10−15 8.01
    128 1.14×10−13 5.99 2.62×10−17 7.99
    256 1.78×10−15 6.00 1.02×10−19 8.00
    512 2.79×10−17 6.00 3.99×10−22 8.00
    收敛阶 $ {{2}}{m} $ $ {{2}}{m}{{ + 2}} $
    下载: 导出CSV

    表  4  变系数问题内部结点位移收敛阶

    Table  4.   Convergence orders of interior nodal displacement of variable coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解EEP解M-EEP解
    误差$ e_{\max }^h $收敛阶ρ误差$ e_{\max }^{\text{*}} $收敛阶ρ误差$ \tilde e_{\max }^*$收敛阶ρ
    1 32 7.45×10−4 1.54×10−4 7.27×10−6
    64 1.98×10−4 1.91 3.86×10−5 2.00 5.14×10−7 3.82
    128 5.14×10−5 1.95 9.66×10−6 2.00 4.88×10−8 3.40
    256 1.31×10−5 1.97 2.42×10−6 2.00 5.97×10−9 3.03
    512 3.31×10−6 1.99 6.04×10−7 2.00 7.38×10−10 3.02
    2 32 1.12×10−5 2.10×10−7 5.68×10−8
    64 1.51×10−6 2.89 1.26×10−8 4.06 2.06×10−9 4.78
    128 1.96×10−7 2.94 7.83×10−10 4.00 6.97×10−11 4.89
    256 2.36×10−8 3.05 4.90×10−11 4.00 3.89×10−12 4.16
    512 2.98×10−9 2.99 3.06×10−12 4.00 2.44×10−13 4.00
    3 32 2.58×10−7 3.51×10−9 1.10×10−9
    64 1.80×10−8 3.84 6.51×10−11 5.75 1.97×10−11 5.80
    128 1.19×10−9 3.92 1.11×10−12 5.88 3.31×10−13 5.90
    256 6.73×10−11 4.14 2.32×10−14 5.58 6.95×10−15 5.57
    512 4.26×10−12 3.98 7.32×10−16 4.99 2.19×10−16 4.98
    收敛阶 $ {m}{{ + 1}} $ $ {{{\rm{min}}(2}}{m},{m}{{ + 2)}} $ $ {m}{{ + 2}} $
    下载: 导出CSV

    表  5  变系数问题端结点应力函数收敛阶

    Table  5.   Convergence orders of nodal stress function of variable coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解修正解
    误差$ e_{i,{\kern 1pt} \max }^h $收敛阶ρ误差$ \tilde e_{i,{\kern 1pt} \max }^h $收敛阶ρ
    141.44×10−26.21×10−3
    83.50×10−32.041.37×10−32.18
    161.22×10−31.522.71×10−42.33
    323.87×10−41.663.37×10−53.01
    649.27×10−52.062.58×10−63.70
    下载: 导出CSV

    表  6  变系数问题内部结点应力函数收敛阶

    Table  6.   Convergence orders of interior nodal stress function of variable coefficient problem

    单元次数$ m $单元数$ {N_e} $有限元解EEP解M-EEP解
    误差$ e_{\max }^h $收敛阶ρ误差$ e_{\max }^{\text{*}} $收敛阶ρ误差$ \tilde e_{\max }^* $收敛阶ρ
    144.44×10−23.59×10−23.07×10−2
    82.17×10−21.038.81×10−32.037.60×10−32.01
    161.04×10−21.071.88×10−32.231.66×10−32.19
    324.19×10−31.314.33×10−42.122.85×10−42.54
    641.42×10−31.569.83×10−52.143.85×10−52.89
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-06-23
  • 修回日期:  2022-02-16
  • 网络出版日期:  2022-04-07
  • 刊出日期:  2022-06-06

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