IMPLEMENTATION OF MULTI-TRANSMITTING BOUNDARY CONDITION FOR WAVE MOTION SIMULATION BY SPECTRAL ELEMENT METHOD: ONE DIMENSION CASE
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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.Abstract: Multi-transmitting formula (MTF) is considered to be a universal local artificial boundary condition, which is generally employed in finite element simulation of near-field wave motion. Due to the great difference between spectral element method (SEM) and finite element method (FEM), the traditional numerical scheme of MTF cannot be simply adopted in SEM without any change. In order to make use of the advantages of MTF, i.e., clear physical mechanism and controllable accuracy, basic problems involved in the combination of MTF and SEM are discussed in this paper, then the feasibility of spatial or temporal interpolation schemes are investigated, respectively. From the view of spatial interpolation scheme, a set of numerical formulas of MTF based on Lagrange polynomial are proposed, where the higherorder MTF is implemented via a simple iteration process. The accuracy and stability of the above MTF schemes are examined by a standard 1-D spectral element model of wave motion. The numerical results show that all schemes have comparable accuracy for 1st-and 2nd-order MTF, and the MTF scheme based on spectral element displacement mode is superior to others for 3rd-or 4th-order MTF. On the contrary, the stability threshold descends with the growth of interpolation polynomials' order of different MTF schemes, but instabilities only occur under the unusual condition that artificial wave speed is far beyond the physical wave speed.
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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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