This paper presents a boundary element method (BEM) for computing the energy transmittance of a singly-periodic grating in 2D for a wide frequency band, which is of engineering interest in various fields with possible applications to acoustic metamaterial design. The proposed method is based on the Pad\'e approximants of the response. The high-order frequency derivatives of the sound pressure necessary to evaluate the approximants are evaluated by a novel fast BEM accelerated by the fast-multipole and hierarchical matrix methods combined with the automatic differentiation. The target frequency band is divided adaptively, and the Pad\'e approximation is used in each subband so as to accurately estimate the transmittance for a wide frequency range. Through some numerical examples, we confirm that the proposed method can efficiently and accurately give the transmittance even when some anomalies and stopband exist in the target band.

翻译：本文提出了一种边界元法，用于计算二维单周期光栅在宽频带内的能量透射率。该方法在工程领域具有广泛的应用价值，尤其适用于声学超材料设计中的潜在场景。所提方法基于响应的Padé逼近近似，通过新型快速边界元法计算评估近似所需的高阶声压频率导数，该快速边界元法结合了快速多极子算法与分层矩阵方法，并辅以自动微分技术。目标频带通过自适应方式划分，并在每个子带内使用Padé逼近以准确估计宽频范围内的透射率。数值算例表明，即使目标频带内存在异常现象和阻带，所提方法仍能高效、精确地给出透射率。