多层波导中矢量波动的时域人工边界条件

本文提出了一种近似的时域人工边界条件(artificial boundary condition, ABC)用来模拟含有瑞利阻尼的线弹性多层波导平面内的矢量波动,该ABC是时域稳定的, 且能与有限元法无缝耦合. 建立ABC的思路是,首先将多层波导的矢量波动方程简化为$x$方向和$y$方向解耦的两个标量波动方程;其次基于比例边界有限元法得到无限域$x$方向和$y$方向模态空间下半离散的频域动力刚度,再用矩阵连分式近似表示$x$方向和$y$方向的频域动力刚度;最后通过辅助变量技术将连分式时域化,从而分别得到人工边界上$x$方向和$y$方向的时域ABC.方法中影响计算精度和计算效率的参数有无限域的模态数$n$、连分式阶数$J$和人工边界远离兴趣域的距离$L$. 数值算例表明,仅需将被载荷激起的无限域的模态数$n$参与计算, 一般可以取$J$=3,$L$的取值基本与地下结构尺寸无关, 它与土层的总层高$H$成正比关系,关系系数与土层的材料参数有关. 

A TIME-DOMAIN ARTIFICIAL BOUNDARY CONDITION FOR VECTOR WAVE IN MULTILAYERED WAVEGUIDE

A time-domain artificial boundary condition (ABC) is proposed to simulate the in-plane vector wave in a linear elastic multilayered waveguide with Rayleigh damping. The ABC is stable and can be seamlessly coupled with the finite element method. First, the vector wave equations of the multilayered waveguide are simplified to two scalar wave equations, which are decoupled in both $x$ and $y$ directions. Then, based on the scaled boundary finite element method, semi-discrete frequency-domain dynamic stiffness in the modal space is obtained. The dynamic stiffness can be approximately expressed as matrix continued fraction. Finally, the continued fraction is converted to the time-domain ABC by introducing the auxiliary variable technique. In this method, the parameters affecting the calculation accuracy and efficiency include the mode number $n$, the order $J$ of continued fraction, and the distance $L$ from the artificial boundary to the region of interest. Numerical examples show that only the mode numbers of the infinite domain excited by the load have to be used. $J$=3 can be taken generally. The value of $L$ is independent of the size of the underground structure. But it is proportional to the total height $H$ of the soil layer, and the relation coefficient is related to the material parameters of the soil layer.