Many problems in fluid mechanics and acoustics can be modelled by Helmholtz scattering off poro-elastic plates. We develop a boundary spectral method, based on collocation of local Mathieu function expansions, for Helmholtz scattering off multiple variable poro-elastic plates in two dimensions. Such boundary conditions, namely the varying physical parameters and coupled thin-plate equation, present a considerable challenge to current methods. The new method is fast, accurate and flexible, with the ability to compute expansions in thousands (and even tens of thousands) of Mathieu functions, thus making it a favourable method for the considered geometries. Comparisons are made with elastic boundary element methods, where the new method is found to be faster and more accurate. Our solution representation directly provides a sine series approximation of the far-field directivity and can be evaluated near or on the scatterers, meaning that the near field can be computed stably and efficiently. The new method also allows us to examine the effects of varying stiffness along a plate, which is poorly studied due to limitations of other available techniques. We show that a power-law decrease to zero in stiffness parameters gives rise to unexpected scattering and aeroacoustic effects similar to an acoustic black hole metamaterial.

中文翻译：

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多可变多孔弹性板散射的 Mathieu 函数边界谱方法，适用于超材料和声学

流体力学和声学中的许多问题都可以通过 Helmholtz 对多孔弹性板的散射进行建模。我们开发了一种边界谱方法，基于局部 Mathieu 函数展开的搭配，用于亥姆霍兹在二维中从多个可变多孔弹性板散射。这种边界条件，即变化的物理参数和耦合的薄板方程，对当前的方法提出了相当大的挑战。新方法快速、准确且灵活，能够计算数千（甚至数万）个 Mathieu 函数的展开，从而使其成为所考虑几何的有利方法。与弹性边界元方法进行了比较，发现新方法更快、更准确。我们的解决方案表示直接提供了远场方向性的正弦级数近似值，并且可以在散射体附近或散射体上进行评估，这意味着可以稳定有效地计算近场。新方法还允许我们检查沿板的不同刚度的影响，由于其他可用技术的局限性，这方面的研究很少。我们表明，刚度参数的幂律减小到零会产生类似于声学黑洞超材料的意外散射和气动声学效应。由于其他可用技术的限制，这方面的研究很少。我们表明，刚度参数的幂律减小到零会产生类似于声学黑洞超材料的意外散射和气动声学效应。由于其他可用技术的限制，这方面的研究很少。我们表明，刚度参数的幂律减小到零会产生类似于声学黑洞超材料的意外散射和气动声学效应。