A numerical method for elastic wave scattering in multi-layered periodic media based on the scattering matrix and BEM
A numerical method for elastic wave scattering in multi-layered periodic media based on the scattering matrix and BEM
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DOI：
10.1299/mej.20-00364
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发表时间：
2021
影响因子：
0.5
通讯作者：
K. Matsushima;H. Isakari;Toru Takahashi;Toshiro Matsumoto
中科院分区：
文献类型：
--
作者：
K. Matsushima;H. Isakari;Toru Takahashi;Toshiro Matsumoto
本文提出了一种二维多层周期结构弹性波散射的数值分析方法。该方法基于边界元法（BEM）和表示周期结构附近入射波和出射波之间关系的散射矩阵。首先，我们用弹性问题解的平面波展开来定义散射矩阵。然后，我们给出了将边界积分方程组的解转化为散射矩阵元素的积分公式。本文提出的基于边界元的散射矩阵法可以将多层结构中的散射问题简化为基于每层散射矩阵的纯代数问题，与基于整个边界的网格划分和求解相应边界积分方程的方法相比，节省了计算资源。该方法基于相关矩阵的特征分解，以数值稳定的方式实现。此外，通过对散射矩阵的渐近分析，得到了半无限声子晶体的一些散射特性和声子带结构。数值算例表明，该方法不仅能准确地求解散射问题，还能计算出相关的特征模态，包括布洛赫模态和导波。
This paper presents a numerical method for analyzing elastic wave scattering by multi-layered periodic structures in two dimensions. The proposed method is based on the boundary element method (BEM) and scattering matrix representing a relationship between incoming and outgoing waves in the vicinity of a periodic structure. First, we define the scattering matrix using a plane-wave expansion of the solution of the elastic problem. Then, we show the integral formulae that convert the solution of a system of boundary integral equations into elements of the scattering matrix. The proposed scattering matrix method with the BEM enables us to reduce the scattering problem in a multi-layered structure to a purely algebraic one in terms of the scattering matrix of each layer, resulting into less requirement in computational resources than in the methods which are based on the meshing of an entire boundary and solution of the corresponding boundary integral equations. Such a procedure called layer-doubling method is implemented in a numerically stable manner based on the eigendecomposition of a relevant matrix. Moreover, some scattering properties and phononic band structures of semi-infinite phononic crystals are yielded through an asymptotic analysis on the scattering matrix. Some numerical examples are presented to demonstrate that the proposed method can solve accurately scattering problems and also can calculate some related eigenmodes including the Bloch modes and guided waves within stacked structures.