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透射边界条件在波动谱元模拟中的实现:一维波动

邢浩洁 李鸿晶

邢浩洁, 李鸿晶. 透射边界条件在波动谱元模拟中的实现:一维波动[J]. 力学学报, 2017, 49(2): 367-379. doi: 10.6052/0459-1879-16-282
引用本文: 邢浩洁, 李鸿晶. 透射边界条件在波动谱元模拟中的实现:一维波动[J]. 力学学报, 2017, 49(2): 367-379. doi: 10.6052/0459-1879-16-282
Xing Haojie, Li Hongjing. IMPLEMENTATION OF MULTI-TRANSMITTING BOUNDARY CONDITION FOR WAVE MOTION SIMULATION BY SPECTRAL ELEMENT METHOD: ONE DIMENSION CASE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(2): 367-379. doi: 10.6052/0459-1879-16-282
Citation: Xing Haojie, Li Hongjing. IMPLEMENTATION OF MULTI-TRANSMITTING BOUNDARY CONDITION FOR WAVE MOTION SIMULATION BY SPECTRAL ELEMENT METHOD: ONE DIMENSION CASE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(2): 367-379. doi: 10.6052/0459-1879-16-282

透射边界条件在波动谱元模拟中的实现:一维波动

doi: 10.6052/0459-1879-16-282
基金项目: 

国家自然科学基金资助项目 51278245

详细信息
    通讯作者:

    2) 李鸿晶, 教授, 主要研究方向:地震工程学.E-mail:hjing@njtech.edu.cn

  • 中图分类号: P315

IMPLEMENTATION OF MULTI-TRANSMITTING BOUNDARY CONDITION FOR WAVE MOTION SIMULATION BY SPECTRAL ELEMENT METHOD: ONE DIMENSION CASE

  • 摘要: 多次透射公式(multi-transmitting formula,MTF)是一种具有普适性的局部人工边界条件,但其在近场波动数值模拟中一般与有限元法结合.由于波动谱元模拟的数值格式与有限元格式有极大的不同,传统的MTF在谱元离散格式中无法直接实现.为了使物理概念清楚、精度可控的多次透射人工边界条件能够适应波动谱元模拟的需求,首先指出多次透射边界与谱元离散格式结合的基本问题,并分析了空间内插和时间内插两种方案的可行性.然后从空间内插角度出发,提出基于拉格朗日多项式插值模式的MTF谱元格式,并采用一种简单内插方法实现高阶MTF.最后通过一维波动数值试验检验这些MTF谱元格式的精度,并讨论其数值稳定性.结果表明:对于一、二阶MTF,几种格式的精度相当;对于三、四阶MTF,基于谱单元位移模式插值的格式精度最高.相反,随着插值多项式阶次的升高,不同MTF格式的稳定临界值逐步降低,但是所有格式均在人工波速大大超过物理波速时才可能发生失稳.

     

  • 图  1  有限单元与谱单元对波场的空间离散

    Figure  1.  Spatial discretization of wave field using finite element or spectral element

    图  2  有限元、谱元离散网格中的MTF

    Figure  2.  MTF in the discrete grid of finite element and spectral element

    图  3  谱元离散网格的MTF时间内插方案

    Figure  3.  Temporal interpolation scheme of MTF in the discrete grid of spectral element

    图  4  基于三点抛物线内插的MTF谱元格式

    Figure  4.  MTF scheme based on parabolic interpolation applied in spectral element method

    图  5  基于M次多项式插值的MTF谱元格式

    Figure  5.  MTF scheme based on Mth-order polynomial interpolation applied in spectral element method

    图  6  半无限均匀弹性直杆的谱元离散模型

    Figure  6.  Discretized spectral element model for a semi-infinite straight uniform elastic rod

    图  7  人工边界节点位移时程 ( $c_{\rm a} = c)$

    Figure  7.  Displacement history of the artificial boundary node ( $c_{\rm a} = c)$

    图  8  人工边界节点位移时程 ( $c_{\rm a} = 2c)$

    Figure  8.  Displacement history of the artificial boundary node ( $c_{\rm a} = 2c)$

    图  9  人工边界节点位移时程 ( $c_{\rm a} = 0.5c)$

    Figure  9.  Displacement history of the artificial boundary node ( $c_{\rm a} = 0.5c)$

    图  10  MTF有限元格式的时域稳定条件

    Figure  10.  Time-domain stability condition of the finite element scheme of MTF

    图  11  MTF谱元格式的时域稳定条件

    Figure  11.  Time-domain stability condition of the spectral element scheme of MTF

    表  1  LSEM在参考单元上的节点坐标

    Table  1.   Node coordinates of LSEM in reference element

    表  2  不同MTF谱元格式的稳定临界值 (一维波动)

    Table  2.   Stability thresholds of several MTF spectral element schemes (1-D wave motion)

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  • 收稿日期:  2016-10-11
  • 网络出版日期:  2016-12-27
  • 刊出日期:  2017-03-18

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