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基于拉格朗日插值的无网格直接配点法和稳定配点法

胡明皓 王莉华

胡明皓, 王莉华. 基于拉格朗日插值的无网格直接配点法和稳定配点法. 力学学报, 2023, 55(5): 1-11 doi: 10.6052/0459-1879-23-001
引用本文: 胡明皓, 王莉华. 基于拉格朗日插值的无网格直接配点法和稳定配点法. 力学学报, 2023, 55(5): 1-11 doi: 10.6052/0459-1879-23-001
Hu Minghao, Wang Lihua. Direct collocation method and stabilized collocation method based on Lagrange interpolation function. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(5): 1-11 doi: 10.6052/0459-1879-23-001
Citation: Hu Minghao, Wang Lihua. Direct collocation method and stabilized collocation method based on Lagrange interpolation function. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(5): 1-11 doi: 10.6052/0459-1879-23-001

基于拉格朗日插值的无网格直接配点法和稳定配点法

doi: 10.6052/0459-1879-23-001
基金项目: 国家自然科学基金资助项目(11972261, 12272270)
详细信息
    通讯作者:

    王莉华, 教授, 主要研究方向为计算力学. E-mail: lhwang@tongji.edu.cn

  • 中图分类号: O241.82

DIRECT COLLOCATION METHOD AND STABILIZED COLLOCATION METHOD BASED ON LAGRANGE INTERPOLATION FUNCTION

  • 摘要: 由于无网格法中大多数近似函数均为有理式, 不具有Kronecker delta性质, 因此难以精确地施加本质边界条件. 边界误差较大容易导致整个求解域求解结果精度低, 甚至引起数值不稳定现象. 文章在无网格直接配点法和稳定配点法中引入拉格朗日插值函数作为形函数, 构建了拉格朗日插值配点法(LICM)和拉格朗日插值稳定配点法(SLICM). 由于拉格朗日插值具有Kronecker delta性质, 可以像有限元法一样简单而精确地施加本质边界条件, 提高这两种方法的数值求解精度. 稳定配点法基于子域对强形式方程进行积分, 可以满足高阶积分约束, 即可以保证形函数在积分形式下也满足高阶一致性条件, 实现精确积分. 同时, 进行子域积分还可以减少离散矩阵的条件数, 从而提高算法的稳定性. 进一步提高拉格朗日插值稳定配点法的精度和稳定性. 通过数值算例验证这两种方法的精度、收敛性和稳定性, 结果表明基于拉格朗日插值的配点法的精度优于基于重构核近似的配点法, 拉格朗日插值稳定配点法的精度和稳定性均优于拉格朗日插值配点法.

     

  • 图  1  一维2节点和3节点拉格朗日插值形函数的节点分布图

    Figure  1.  Node distributions of the 1D 2-node and 3-node Lagrange interpolation shape functions

    图  2  二维4节点、6节点和9节点拉格朗日插值形函数的节点分布图

    Figure  2.  Node distributions of the 2D 4-node, 6-node and 9-node Lagrange interpolation shape functions

    图  3  p = 2时一维拉格朗日插值形函数

    Figure  3.  1D Lagrange interpolation shape function when p = 2

    图  5  p = 4时一维拉格朗日插值形函数

    Figure  5.  1D Lagrange interpolation shape function when p = 4

    图  4  p = 3时一维拉格朗日插值形函数

    Figure  4.  1D Lagrange interpolation shape function when p = 3

    图  6  稳定配点法中点的布置图

    Figure  6.  Points allocation in SCM

    图  7  一维直杆示意图

    Figure  7.  Description of the 1D rod problem

    图  8  一维直杆问题源点离散图

    Figure  8.  Discrete points of source points for the 1D rod problem

    图  9  一维直杆问题计算结果及误差

    Figure  9.  Numerical solutions for the 1D rod problem

    图  10  一维直杆问题收敛性分析

    Figure  10.  Convergence comparisons for the 1D rod problem

    图  11  一维直杆问题刚度矩阵条件数

    Figure  11.  Condition number comparisons for the 1D rod problem

    图  12  二维泊松问题源点离散图

    Figure  12.  Discrete points of source points for the 2D Poisson problem

    图  13  二维泊松问题域内位移解

    Figure  13.  Numerical solutions for the 2D Poisson problem

    图  14  二维泊松问题边界位移解

    Figure  14.  Numerical solutions of boundary for the 2D Poisson problem

    图  15  二维泊松问题收敛性分析

    Figure  15.  Convergence comparisons for the 2D Poisson problem

    图  16  二维泊松问题刚度矩阵条件数

    Figure  16.  Condition number comparisons for the 2D Poisson problem

    图  17  三维亥姆霍兹问题源点离散图

    Figure  17.  Discrete points of source points for the 3D Helmholtz problem

    图  18  三维亥姆霍兹问题域内位移解

    Figure  18.  Numerical solutions of the inner domain for the 3D Helmholtz problem

    图  19  三维亥姆霍兹问题边界位移解

    Figure  19.  Numerical solutions of boundary for the 3D Helmholtz problem

    19  三维亥姆霍兹问题边界位移解 (续)

    19.  Numerical solutions of boundary for the 3D Helmholtz problem (continued)

    图  20  三维亥姆霍兹问题收敛性分析

    Figure  20.  Convergence comparisons for the 3D Helmholtz problem

    图  21  三维亥姆霍兹问题刚度矩阵条件数

    Figure  21.  Condition number comparisons for the 3D Helmholtz problem

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出版历程
  • 收稿日期:  2023-01-01
  • 录用日期:  2023-03-29
  • 网络出版日期:  2023-03-30

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