透射边界条件在波动谱元模拟中的实现:一维波动

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.

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

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 higher-order 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.