Boundary layers in the homogenization of a spectral problem in fluid-solid structures
Boundary layers in the homogenization of a spectral problem in fluid-solid structures
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DOI：
10.1137/s0036141096304328
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发表时间：
1998-03-01
影响因子：
2
通讯作者：
Conca, C
中科院分区：
文献类型：
--
作者：
Allaire, G;Conca, C
本文致力于对描述流固耦合周期结构振动的数学模型谱进行渐近分析。在之前的作品中 [Arch.理性机械。 Anal., 135 (1996), pp. 197-257]我们通过布洛赫波均匀化方法证明，在周期变为零的极限下，谱由三部分组成：宏观或均匀化谱、微观或布洛赫谱和第三个分量，即所谓的边界层谱。虽然前两个部分完全被描述为某个极限问题的谱，但后者仅仅被定义为与集中在边界上的特征向量序列相对应的极限特征值集合。本文的目的是借助一系列极限问题来明确表征该边界层谱，揭示周期性微观结构与域边界之间的密切联系。因此，我们获得了“完整性”结果，即特征值序列的所有可能渐近行为的精确描述，至少对于一类特殊的多边形域而言是如此。
This paper is devoted to the asymptotic analysis of the spectrum of a mathematical model that describes the vibrations of a coupled fluid-solid periodic structure. In a previous work [Arch. Rational Mech. Anal., 135 (1996), pp. 197-257] we proved by means of a Bloch wave homogenization method that, in the limit as the period goes to zero, the spectrum is made of three parts: the macroscopic or homogenized spectrum, the microscopic or Bloch spectrum, and a third component, the so-called boundary layer spectrum. While the two first parts were completely described as the spectrum of some limit problem, the latter was merely defined as the set of limit eigenvalues corresponding to sequences of eigenvectors concentrating on the boundary. It is the purpose of this paper to characterize explicitly this boundary layer spectrum with the help of a family of limit problems revealing the intimate connection between the periodic microstructure and the boundary of the domain. We therefore obtain a "completeness" result, i.e., a precise description of all possible asymptotic behaviors of sequences of eigenvalues, at least for a special class of polygonal domains.