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基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究

郭玉杰 吴晗浪 李薇 李迎港 吴剑晨

郭玉杰, 吴晗浪, 李薇, 李迎港, 吴剑晨. 基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究[J]. 工程力学, 2022, 39(8): 31-48. doi: 10.6052/j.issn.1000-4750.2021.04.0285
引用本文: 郭玉杰, 吴晗浪, 李薇, 李迎港, 吴剑晨. 基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究[J]. 工程力学, 2022, 39(8): 31-48. doi: 10.6052/j.issn.1000-4750.2021.04.0285
GUO Yu-jie, WU Han-lang, LI Wei, LI Ying-gang, WU Jian-chen. MODEL ORDER REDUCTION FOR NONLINEAR DYNAMIC ANALYSIS OF PARAMETERIZED CURVED BEAM STRUCTURES BASED ON ISOGEOMETRIC ANALYSIS[J]. Engineering Mechanics, 2022, 39(8): 31-48. doi: 10.6052/j.issn.1000-4750.2021.04.0285
Citation: GUO Yu-jie, WU Han-lang, LI Wei, LI Ying-gang, WU Jian-chen. MODEL ORDER REDUCTION FOR NONLINEAR DYNAMIC ANALYSIS OF PARAMETERIZED CURVED BEAM STRUCTURES BASED ON ISOGEOMETRIC ANALYSIS[J]. Engineering Mechanics, 2022, 39(8): 31-48. doi: 10.6052/j.issn.1000-4750.2021.04.0285

基于等几何分析的参数化曲梁结构非线性动力学降阶模型研究

doi: 10.6052/j.issn.1000-4750.2021.04.0285
基金项目: 国家自然科学基金面上项目(11972187)
详细信息
    作者简介:

    吴晗浪(1993−),男,贵州人,硕士生,主要从事等几何分析研究(E-mail: 473807649@qq.com)

    李 薇(1998−),女,河南人,硕士生,主要从事等几何结构动力学研究(E-mail: cyona@nuaa.edu.cn)

    李迎港(1997−),男,安徽人,硕士生,主要从事结构疲劳寿命研究(E-mail: liyinggang@nuaa.edu.cn)

    吴剑晨(1997−),男,安徽人,硕士生,主要从事航天器传热研究(E-mail: jianchenwu@nuaa.edu.cn)

    通讯作者:

    郭玉杰(1986−),男,江苏人,副教授,博士,主要从事结构计算力学研究(E-mail: yujieguo@nuaa.edu.cn)

  • 中图分类号: O342;TB122

MODEL ORDER REDUCTION FOR NONLINEAR DYNAMIC ANALYSIS OF PARAMETERIZED CURVED BEAM STRUCTURES BASED ON ISOGEOMETRIC ANALYSIS

  • 摘要: 模型降阶方法通过构造全阶模型的低阶近似模型有效提升了求解效率,同时也保留了原阶模型的主要信息从而保证了较高的计算精度。对于结构非线性以及参数化的模型降阶问题,常需要重复计算刚度矩阵等非线性以及参数依赖项,求解效率较低。此外,当参数化模型的几何形状改变时,往往需要重复进行CAD与有限元(FEA)模型的转换,这对于复杂结构较为耗时。等几何分析采用描述几何形状的非均匀有理B样条(NURBS)插值物理场,实现了CAD与FEA模型的统一,消除了两者之间繁琐的模型转换过程,其具有几何精确、高阶连续等优点,并且几何形状在细化过程中保持不变,非常适合于薄壁类结构的分析以及参数化表达。该研究结合等几何分析、特征正交分解(POD)以及离散经验插值方法(DEIM)研究参数化的平面曲梁结构的非线性动力学模型降阶问题。数值结果表明,基于等几何分析的POD-DEIM降阶模型能够显著提升平面曲梁结构的非线性动力学计算效率,并且该模型对于参数化以及变载荷等情形显示出了良好的适应性。
  • 图  1  NURBS曲线及NURBS基函数

    Figure  1.  NURBS curve and NURBS basis functions

    图  2  初始构型与当前构型

    Figure  2.  Reference and current configurations

    图  3  HHT-α时间积分算法流程图

    Figure  3.  HHT-α time integration scheme for nonlinear dynamic analysis

    图  4  两端固支半圆弧曲梁模型

    Figure  4.  Half-circle arc with clamped boundary conditions at both ends

    图  5  两端固支半圆弧曲梁位移收敛曲线(几何非线性)

    Figure  5.  Convergence studies of the half-circle arc beam(geometrically nonlinear)

    图  6  两端固支半圆弧曲梁奇异值

    Figure  6.  Singular values of displacement snapshots and stiffness matrix snapshots for polynomial degrees 2, 3, 4 for clamped half-circle arc

    图  7  两端固支半圆弧曲梁载荷-位移曲线

    Figure  7.  Load-displacement response of half-circle arc beam

    图  8  两端固支半圆弧曲梁中点轨迹对比图

    Figure  8.  Comparison of the paths traced by the mid-point of the half-circle arc beam obtained using the FOM and ROM

    图  9  两端固支半圆弧曲梁降阶与全阶模型位移差值图

    Figure  9.  Difference of mid-point displacement between FOM and ROM for half-circle arc beam

    图  10  两端固支半圆弧曲梁降阶、全阶模型计算时间对比

    Figure  10.  Comparison of computational time for different reduced order models of half-circle arc beam

    图  11  一端固支阿基米德螺旋梁

    Figure  11.  Archimedes spiral shaped beam with one end clamped

    图  12  一端固支阿基米德螺旋曲梁奇异值

    Figure  12.  Singular values of displacement snapshots and stiffness matrix snapshots for clamped Archimedes spiral beam

    图  13  一端固支阿基米德螺旋曲梁载荷位移曲线

    Figure  13.  Load-displacement curve of clamped Archimedes spiral arc beam

    图  14  一端固支阿基米德螺旋曲梁端点的轨迹对比图

    Figure  14.  Comparison of the paths traced by the end-point of the Archimedes spiral beam obtained using the FOM and ROM

    图  15  阿基米德螺旋曲梁端点降阶与全阶模型位移差值图

    Figure  15.  Difference of end-point displacement between FOM and ROM for the Archimedes spiral beam

    图  16  两端简支曲梁模型

    Figure  16.  Arc beam with simply supported boundary conditions at both ends

    图  17  两端简支曲梁奇异值

    Figure  17.  Singular values of displacement snapshots and stiffness matrix snapshots for simply supported arc beam

    图  18  两端简支曲梁载荷-位移曲线

    Figure  18.  Load-displacement curve of simply supported arc beam

    图  19  两端简支曲梁中点的轨迹对比图

    Figure  19.  Comparison of the paths traced by the mid-point of the arc beam obtained using the FOM and ROM

    图  20  两端简支曲梁降阶与全阶模型位移差值图

    Figure  20.  Difference of mid-point displacement between FOM and ROM for the arc beam

    图  21  两端简支曲梁频率响应对比图

    Figure  21.  Comparison of frequency response obtained using FOM and ROM

    图  22  两端简支曲梁载荷位移曲线($F = - 800\sin (36t)$)

    Figure  22.  Load-displacement curve of simply supported arc beam ($F = - 800\sin (36t)$)

    图  23  两端简支曲梁中点的轨迹对比图($F = - 800\sin (36t)$)

    Figure  23.  Comparison of the paths traced by the mid-point of the arc beam obtained using the FOM and ROM ($F = - 800\sin (36t)$)

    图  24  两端简支曲梁降阶与全阶模型位移差值图($F = - 800\sin (36t)$)

    Figure  24.  Difference of mid-point displacement between FOM and ROM for the arc beam ($F = - 800\sin (36t)$)

    图  25  两端简支曲梁频率响应对比图($F = - 800\sin (36t)$)

    Figure  25.  Comparison of frequency response obtained using FOM and ROM ($F = - 800\sin (36t)$)

    图  26  两端简支曲梁的参数化模型

    Figure  26.  Parametric model of simply supported curved beams

    图  27  控制点P2抽样分布图

    Figure  27.  Sampling positions of the control point P2

    图  28  参数化曲梁奇异值

    Figure  28.  Singular values of displacement snapshots and stiffness matrix snapshots for parameterized arc beam

    图  29  不同抽样方法所对应的位移载荷响应

    Figure  29.  Load-displacement response of half-circle arc beam

    图  30  参数化曲梁$\xi = 0.5$处的轨迹对比图

    Figure  30.  Comparison of the paths traced at the position of $\xi = 0.5$ for the parameterized arc beam obtained using the FOM and ROM

    图  31  参数化曲梁降阶与全阶模型位移差值图$(\xi = 0.5)$

    Figure  31.  Difference of the displacement between FOM and ROM for the parameterized arc beam $(\xi = 0.5)$

    图  32  参数化曲梁频率响应对比图

    Figure  32.  Comparison of frequency response obtained using FOM and ROM for the parameterized arc beam

    图  33  不同抽样方法所对应的位移载荷响应$(\xi = 0.5)$ ($F = - 800\sin (48t)$)

    Figure  33.  Comparisons of load-displacement histories of the parameterized arc beam at $(\xi = 0.5)$ ($F = - 800\sin (48t)$)

    图  34  不同抽样方法所对应的位移载荷响应$(\xi = 0.5)$ ($F = - 800\sin (48t)$, 时间步长:0.005 s)

    Figure  34.  Comparisons of load-displacement histories of the parameterized arc beam at $(\xi = 0.5)$($F = - 800\sin (48t)$, time step: 0.005 s)

    图  35  参数化曲梁$(\xi = 0.5)$处的轨迹对比图($F = - 800\sin (48t)$, 时间步长:0.005 s)

    Figure  35.  Comparison of the paths traced at the position of $(\xi = 0.5)$ for the parameterized arc beam obtained using the FOM and ROM ($F = - 800\sin (48t)$, time step: 0.005 s)

    图  36  参数化曲梁降阶与全阶模型位移差值图$(\xi = 0.5)$($F = - 800\sin (48t)$, 时间步长:0.005 s)

    Figure  36.  Difference of the displacement between FOM and ROM for the parameterized arc beam $(\xi = 0.5)$ ($F = - 800\sin (48t)$, time step: 0.005 s)

    图  37  参数化曲梁频率响应对比图($F = - 800\sin (48t)$, 时间步长:0.005 s)

    Figure  37.  Comparison of frequency response obtained using FOM and ROM for the parameterized arc beam ($F = - 800\sin (48t)$, time step: 0.005 s)

    表  1  两端固支半圆弧曲梁降阶模型计算时间对比

    Table  1.   Comparisons of computational time for reduced order models of half-circle arc beam

    计算模型全阶
    模型
    POD降阶模型POD-MDEIM
    降阶模型
    k = 2k = 3k = 5k = 15k = 20
    m = 15
    计算时间/s302.9208.9212.2234.3242.436.9
    时间比11.41.41.31.28.2
    下载: 导出CSV

    表  2  两端简支曲梁刚度矩阵MDEIM插值位置

    Table  2.   MDEIM Positions for stiffness matrix of simply supported arc beam

    位置$ {\phi }_{i} $行(i)列(j)单元编号e位置$ {\phi }_{i} $行(i)列(j)单元编号e
    2231317057646419,20,21,22
    2253317615676920,21,22,23
    268925256,7,8,97953727233,34,35,36
    369734349,10,11,128961818124,25,26,27
    369936349,10,11,1210081919128,29,30,31
    4371424011,12,13,1410753979730,31,32,33
    5599495114,15,16,171109110210031,32,33,34
    6607586017,18,19,20
    下载: 导出CSV

    表  3  两端简支曲梁刚度矩阵MDEIM插值位置

    Table  3.   MDEIM Positions for stiffness matrix of simply supported arc beam

    位置$ {\phi }_{i} $ij单元编号e位置$ {\phi }_{i} $ij单元编号e
    2433318953737522,23,24,25
    375232328,9,10,1111 738989830,31,32,33
    4237373610,11,1212 82710710733,34,35,36
    4841414111,12,13,1413 79511511536,37
    5082424312,13,1413 79711711536,37
    5814544915,16,1714 03711711736,37
    7135556017,18,1914 15811811837
    8108686820,21,22,23
    下载: 导出CSV

    表  4  参数化曲梁降阶模型计算时间对比

    Table  4.   Comparisons of computational time for reduced order models of parameterized arc beam

    控制点P2坐标计算时间/s速度提升率
    全阶
    模型
    POD-MDEIM
    降阶模型
    抽样1抽样2
    抽样1抽样2
    P2 = (7.5,9.5) m
    F = −800 N
    237.041.143.55.85.4
    P2 = (5.7,7.2) m
    F = −800 N
    242.139.037.66.26.4
    P2 = (6,5.5) m
    F = −800sin(48t) N
    402.078.171.65.15.6
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-04-16
  • 修回日期:  2021-07-05
  • 网络出版日期:  2021-08-06
  • 刊出日期:  2022-08-01

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