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Gappy POD方法重构湍流数据的研究

李天一 BuzzicottiMichele BiferaleLuca 万敏平 陈十一

李天一, Buzzicotti Michele, Biferale Luca, 万敏平, 陈十一. Gappy POD方法重构湍流数据的研究. 力学学报, 2021, 53(10): 2703-2711 doi: 10.6052/0459-1879-21-464
引用本文: 李天一, Buzzicotti Michele, Biferale Luca, 万敏平, 陈十一. Gappy POD方法重构湍流数据的研究. 力学学报, 2021, 53(10): 2703-2711 doi: 10.6052/0459-1879-21-464
Li Tianyi, Buzzicotti Michele, Biferale Luca, Wan Minping, Chen Shiyi. Reconstruction of turbulent data with gappy POD method. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2703-2711 doi: 10.6052/0459-1879-21-464
Citation: Li Tianyi, Buzzicotti Michele, Biferale Luca, Wan Minping, Chen Shiyi. Reconstruction of turbulent data with gappy POD method. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2703-2711 doi: 10.6052/0459-1879-21-464

Gappy POD方法重构湍流数据的研究

doi: 10.6052/0459-1879-21-464
基金项目: 国家数值风洞工程(NNW2019ZT1-A01)和深圳市科技计划(KQTD20180411143441009)资助项目
详细信息
    作者简介:

    万敏平, 教授, 主要研究方向: 湍流数值模拟和理论, 磁流体和太阳风. E-mail: wanmp@sustech.edu.cn

  • 中图分类号: O357.5

RECONSTRUCTION OF TURBULENT DATA WITH GAPPY POD METHOD

  • 摘要: Gappy POD 是一种基于本征正交分解(proper orthogonal decomposition, POD)的数据重构方法. 本文研究了gappy POD在湍流数据重构中的应用, 主要关注了以下两个因素的影响: 第一, 数据本身的复杂程度, 即构成流场的POD模态数量; 第二, 破损区域的面积大小和几何形状. 考虑到上述因素, 本文重新严格地表述了gappy POD的重构过程, 并推导出gappy POD重构误差的公式. 论文选取旋转湍流数据为案例进行了gappy POD重构的研究, 并解释了构成gappy POD重构误差的两个部分. 第一部分来自流场POD展开的截断误差, 该截断误差会被POD基函数在已知点上的值组成的矩阵的最小特征值放大. 这部分误差主要取决于流场的复杂程度, 当流场复杂程度较低时, 相应误差随采用的POD模态数目增大而减小. 当流场复杂程度较高时, 很小的POD截断误差也会导致很大的重构误差, 此时需要采用流场所有的POD模态进行重构以消除截断误差. 重构误差的第二部分来自POD基函数在已知点上的值组成的矩阵的非列满秩性, 它主要取决于破损区域的面积大小和几何形状. 破损区域的面积越大, 或者破损面积相同时, 破损区域内信息所包含的相关性越大, 第二部分的重构误差越大.

     

  • 图  1  POD模态的特征值曲线

    Figure  1.  Eigenvalues of the POD modes

    2  (a)-(e) 破损面积相同但破损区域几何不同的部分缺失流场, (f) 真实的完整流场

    2.  (a)-(e) Damaged flow fields with gaps of the same area but different geometries. (f) The complete flow field

    图  2  (a)-(e) 破损面积相同但破损区域几何不同的部分缺失流场, (f) 真实的完整流场 (续)

    Figure  2.  (a)-(e) Damaged flow fields with gaps of the same area but different geometries. (f) The complete flow field (continued)

    图  3  破损区域中的均方重构误差$ {\rm{MS}}{{\rm{E}}_{{\rm{gap}}}} $关于${{{N_{{\rm{comp}}}}} \mathord{\left/ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} \right. } {{N_{{\rm{flow}}}}}}$的变化曲线

    Figure  3.  Normalized mean square error in the gap, $ {{MS}}{{{E}}_{{\rm{gap}}}} $, as a function of ${{{N_{{\rm{comp}}}}} \mathord{\left/ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} \right. } {{N_{{\rm{flow}}}}}}$

    图  4  $\boldsymbol{\tilde X}$的最小奇异值$ {\sigma _{\min }} $关于${{N_{{\rm{comp}}}}}$/$ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} {{N_{{\rm{flow}}}}} $的变化曲线

    Figure  4.  The minimum singular value of $ \boldsymbol{\tilde X} $, $ {\sigma _{\min }} $, as a function of ${{N_{{\rm{comp}}}}}$/${ {\vphantom {{{N_{{\rm{comp}}}}} {{N_{{\rm{flow}}}}}}} } {{N_{{\rm{flow}}}}} $

    图  5  对于不同几何形状的破损, 破损区域中的均方重构误差$ {{MS}}{{{E}}_{{\rm{gap}}}} $关于破损尺寸的变化曲线

    Figure  5.  Normalized mean square error in the gap, $ {{MS}}{{{E}}_{{\rm{gap}}}} $, as a function of the gap size for different gap geometries

    图  6  对于不同几何形状的破损, ${N_{{\rm{flow}}}} - r\left( {\boldsymbol{\tilde X}} \right)$关于破损尺寸的变化曲线

    Figure  6.  ${N_{{\rm{flow}}}} - r\left( {\boldsymbol{\tilde X}} \right)$ as a function of the gap size for different gap geometries

    图  7  相同破损面积, 不同破损几何的gappy POD重构结果(对应不同行). 第1列: 破损的流场; 第2列: 重构的流场; 第3列: 原始流场; 第4列: 第一列中红色参考线上重构结果(虚线)与真实结果(实线)的分布; 第5列: 平均重构误差(红色), $ {\varDelta _u}\left( {{x_1}} \right) $, 及其在测试集上的平均(黑色), $ \left\langle {{\varDelta _u}\left( {{x_1}} \right)} \right\rangle $. ${x_1}$${x_2}$分别代表水平和竖直方向

    Figure  7.  Gappy POD reconstruction results for the same gap area and different gap geometries (one for each row). 1st column: damaged image in input. 2nd column: image generated in output. 3rd column: ground truth. 4th column: generated (dashed) and ground truth (solid) profiles along the vertical line shown in the 1st column. 5th column: mean reconstruction error, $ {\varDelta _u}\left( {{x_1}} \right) $, for each image (red line) and the average error, $ \left\langle {{\varDelta _u}\left( {{x_1}} \right)} \right\rangle $, over the test images (black curve). Note that ${x_1}$ and ${x_2}$ denotes the horizontal and vertical directions, respectively

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出版历程
  • 收稿日期:  2021-09-10
  • 录用日期:  2021-10-07
  • 网络出版日期:  2021-10-08
  • 刊出日期:  2021-10-26

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