Abstract While embedding acoustic black holes (ABHs) in plates has proved to be a lightweight and useful solution for noise and vibration reduction, their potential for wave manipulation is only in its beginnings. Recent studies seem to indicate that amazing properties can be attained with them. A plate with ABH indentations can be viewed as a single-phase metamaterial that can alter wave propagation direction thanks to the ABH power-law thickness profile. Some typical unconventional dispersion properties of multi-phase metamaterials, like collimation or negative refraction, can be recovered in a simple way. This work focuses on the collimation of flexural waves in plates by means of ABH arrangements. The first goal of the paper is to propose a predictive semi-analytical method to describe such phenomenon in an efficient manner. The method is also expected to allow one to perform quick parametric analyses for different ABH configurations. To that purpose, the Gaussian expansion method (GEM) is extended to deal with ABH phononic crystals on infinite plates. The Lagrangian for a single cell is built and a new basis of Gaussian functions is constructed satisfying the periodic boundary conditions of the problem. The Rayleigh-Ritz method is then adopted to compute the displacement field of annular and circular ABH arrays. The accuracy of the reconstructed GEM approach is validated by comparing dispersion curves and modal shapes with those obtained from finite element simulations. Equi-frequency contours are then used to determine the frequency bands prone to collimation. The focus is placed on annular ABH configurations which allow one to achieve collimation at lower frequencies than the circular ones. The second main contribution of this work therefore consists of a thorough analysis to characterize the influence of geometric parameters on the performance of ABH arrays. Moreover, new designs of annular ABH arrangements on plates are proposed for wave conduction, including curvature, and energy focusing in the long wavelength limit.

中文翻译：

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重建高斯基础以表征具有环形声学黑洞周期阵列的板中的弯曲波准直

摘要 虽然在板中嵌入声学黑洞 (ABH) 已被证明是一种轻量级且有用的降噪和减振解决方案，但它们用于波操纵的潜力才刚刚开始。最近的研究似乎表明它们可以实现惊人的特性。具有 ABH 压痕的板可以被视为单相超材料，由于 ABH 幂律厚度分布，它可以改变波的传播方向。多相超材料的一些典型的非常规色散特性，如准直或负折射，可以通过简单的方式恢复。这项工作的重点是通过 ABH 排列对板中弯曲波的准直。本文的第一个目标是提出一种预测性半分析方法，以有效的方式描述这种现象。该方法还有望让人们对不同的 ABH 配置进行快速参数分析。为此，高斯展开法 (GEM) 被扩展到处理无限板上的 ABH 声子晶体。构建了单个单元的拉格朗日函数，并构建了满足问题周期性边界条件的新的高斯函数基。然后采用 Rayleigh-Ritz 方法计算环形和圆形 ABH 阵列的位移场。通过将频散曲线和模态形状与从有限元模拟中获得的曲线和模态形状进行比较，验证了重建 GEM 方法的准确性。然后使用等频轮廓来确定易于准直的频带。重点放在环形 ABH 配置上，该配置允许在比圆形配置更低的频率下实现准直。因此，这项工作的第二个主要贡献包括彻底分析以表征几何参数对 ABH 阵列性能的影响。此外，还提出了在板上的环形 ABH 布置的新设计，用于波传导，包括曲率和长波长范围内的能量聚焦。