An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.

中文翻译：

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基于精确三角谱元法的复杂几何声学问题数值模拟

开发了一种精确的三角谱元方法 (TSEM) 来模拟复杂计算域中的声学问题。引入Fekete点和Koornwinder-Dubiner多项式，在本方法中使用三角形单元来代替传统谱元法(SEM)中的四边形单元。提高了离散化复杂几何的效率，同时保持了 SEM 的高精度。求解了从线性化欧拉方程 (LEE) 导出的二阶控制方程的弱形式，并实现了完美匹配层 (PML) 边界条件。三个具有解析解的基准问题被用来证明指数收敛速度，方便地实现实体壁边界条件，并分别在本方法的复杂几何结构中进行离散化。还介绍了亥姆霍兹谐振器 (HR) 的应用，以证明在实际工程中使用本方法的可能性。HR 的数值共振频率与理论结果非常吻合。