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矩阵带宽缩减技术在隐式间断有限元中的应用

李亮 吴颂平

李亮, 吴颂平. 矩阵带宽缩减技术在隐式间断有限元中的应用[J]. 北京航空航天大学学报, 2020, 46(3): 532-540. doi: 10.13700/j.bh.1001-5965.2019.0281
引用本文: 李亮, 吴颂平. 矩阵带宽缩减技术在隐式间断有限元中的应用[J]. 北京航空航天大学学报, 2020, 46(3): 532-540. doi: 10.13700/j.bh.1001-5965.2019.0281
LI Liang, WU Songping. Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin[J]. Journal of Beijing University of Aeronautics and Astronautics, 2020, 46(3): 532-540. doi: 10.13700/j.bh.1001-5965.2019.0281(in Chinese)
Citation: LI Liang, WU Songping. Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin[J]. Journal of Beijing University of Aeronautics and Astronautics, 2020, 46(3): 532-540. doi: 10.13700/j.bh.1001-5965.2019.0281(in Chinese)

矩阵带宽缩减技术在隐式间断有限元中的应用

doi: 10.13700/j.bh.1001-5965.2019.0281
基金项目: 

国家自然科学基金 91530325

详细信息
    作者简介:

    李亮, 男, 博士研究生。主要研究方向:间断有限元格式在计算流体力学中的应用

    吴颂平, 男, 博士, 教授, 博士生导师。主要研究方向:计算流体力学

    通讯作者:

    吴颂平, E-mail: wusping825@163.com

  • 中图分类号: O242.1;O35

Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin

Funds: 

National Natural Science Foundation of China 91530325

More Information
    Corresponding author: WU Songping, E-mail: wusping825@163.com
  • 摘要:

    为了数值求解二维Euler方程,以间断有限元方法作为空间离散、向后差分公式(BDF)作为时间离散。针对采用牛顿法求解源于隐式时间积分的非线性方程组,构造了相应的Jacobi矩阵,其具有阶数高、稀疏性强、数值非对称的特点。在每个时间步内,选择带预处理的广义极小残量(GMRES)方法求解线性方程组,预处理矩阵由不完全LU分解(ILU)方法构造。将矩阵带宽缩减技术应用于上述求解过程,无需额外的存储空间,就缩小了预处理矩阵与系数矩阵的差距,从而加快了GMRES方法的收敛、增大了可用的时间步长。通过求解典型的空气动力学问题,检验了该应用的有效性。

     

  • 图 1  稀疏矩阵和结点网络(带宽未经缩减)

    Figure 1.  Sparse matrix and corresponding node network(without bandwidth reduction)

    图 2  稀疏矩阵和结点网络(经RCM缩减带宽)

    Figure 2.  Sparse matrix and corresponding node network(with RCM bandwidth reduction)

    图 3  圆柱附近的网格

    Figure 3.  Grids around a cylinder

    图 4  带宽未经缩减的Jacobi矩阵

    Figure 4.  Jacobian matrix without bandwidth reduction

    图 5  带宽经过缩减的Jacobi矩阵

    Figure 5.  Jacobian matrix with bandwidth reduction

    图 6  圆柱周围的马赫数等值线(Ma=0.38, α=0°, ΔMa=0.05)

    Figure 6.  Mach number contours around a cylinder (Ma=0.38, α=0°, ΔMa=0.05)

    图 7  NACA0012翼型附近的网格

    Figure 7.  Grids around NACA0012 airfoil

    图 8  NACA0012翼型表面的压力系数(Ma=0.63, α=2°)

    Figure 8.  Pressure coefficient on surface of NACA0012 airfoil (Ma=0.63, α=2°)

    图 9  NACA0012翼型周围的马赫数等值线(Ma=0.63, α=2°, ΔMa=0.03)

    Figure 9.  Mach number contours around NACA0012 airfoil (Ma=0.63, α=2°, ΔMa=0.03)

    图 10  RAE2822翼型附近的网格

    Figure 10.  Grids around RAE2822 airfoil

    图 11  RAE2822翼型表面的压力系数(Ma=0.4, α=2.79°)

    Figure 11.  Pressure coefficient on surface of RAE2822 airfoil (Ma=0.4, α=2.79°)

    图 12  RAE2822翼型周围的马赫数等值线(Ma=0.4, α=2.79°, ΔMa=0.015)

    Figure 12.  Mach number contours around RAE2822 airfoil (Ma=0.4, α=2.79°, ΔMa=0.015)

    表  1  GMRES的迭代步数(圆柱绕流)

    Table  1.   Iteration steps of GMRES(flow around a cylinder)

    CFL 带宽未经缩减 带宽经过缩减
    5 6 4
    10 11 5
    20 15 6
    下载: 导出CSV

    表  2  预处理矩阵与系数矩阵之差的Frobenius范数(圆柱绕流)

    Table  2.   Frobenius norm for difference between preconditioner and coefficient matrix (flow around a cylinder)

    CFL 带宽未经缩减 带宽经过缩减
    5 40 34
    10 205 85
    20 40 157 203
    下载: 导出CSV

    表  3  GMRES的迭代步数(NACA0012翼型绕流)

    Table  3.   Iteration steps of GMRES(flow around NACA0012 airfoil)

    CFL 带宽未经缩减 带宽经过缩减
    5 5 4
    10 14 5
    20 未收敛 6
    下载: 导出CSV

    表  4  预处理矩阵与系数矩阵之差的Frobenius范数(NACA0012翼型绕流)

    Table  4.   Frobenius norm for difference between preconditioner and coefficient matrix(flow around NACA0012)

    CFL 带宽未经缩减 带宽经过缩减
    5 42 33
    10 280 86
    20 334 511 205
    下载: 导出CSV

    表  5  GMRES的迭代步数(RAE2822翼型绕流)

    Table  5.   Iteration steps of GMRES (flow around RAE2822 airfoil)

    CFL 带宽未经缩减 带宽经过缩减
    10 7 5
    20 10 5
    40 33 6
    下载: 导出CSV

    表  6  预处理矩阵与系数矩阵之差的Frobenius范数(RAE2822翼型绕流)

    Table  6.   Frobenius norm for difference between preconditioner and coefficient matrix (flow around RAE2822 airfoil)

    CFL 带宽未经缩减 带宽经过缩减
    10 70 63
    20 231 155
    40 349 645 369
    下载: 导出CSV
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  • 收稿日期:  2019-06-05
  • 网络出版日期:  2020-03-20

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