High-frequency homogenization for periodic media
High-frequency homogenization for periodic media
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DOI：
10.1098/rspa.2009.0612
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发表时间：
2010-08-08
影响因子：
3.5
通讯作者：
Pichugin, A. V.
中科院分区：
文献类型：
--
作者：
Craster, R. V.;Kaplunov, J.;Pichugin, A. V.
基于双尺度方法，发展了一个渐近过程，用于研究双周期非均匀介质中的波传播，该介质的特征长度尺度远小于微观结构的特征长度尺度。在周期性介质中，存在驻波的频率，其周期性为细胞的周期或两倍周期，在微尺度上出现。这些频率不属于经典均匀化理论所涵盖的低频有效性范围，这促使我们在对这些驻波进行扰动时使用术语“高频均匀化”。由此产生的长波方程推导出只明确依赖于宏观尺度，与微观尺度表示的积分量。这些方程精确地再现了布里渊区边缘附近的布洛赫模谱的行为，从而为“细胞共振”附近的周期性介质提供了一种明确的方法。这种模型方程的相似性，以高频率的长波长渐近，均匀的声波和弹性波导，有效的厚度共振附近的强调。几个说明性的例子被认为是开发的技术的有效性。
An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term 'high-frequency homogenization' when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of 'cell resonances'. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.