This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with |$\tanh $| functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

中文翻译：

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用于穿孔筛网的快速且光谱准确的数值方法（适用于Robin边界条件）

本文考虑使用顺应性边界条件来提供共线板有限阵列的均质模型，从而对多孔筛网或光栅进行建模。多孔筛网正式具有Dirichlet和Neumann边界条件的混合，而均质化模型具有Robin边界条件。穿孔的屏幕形成了散射理论中的典型模型，其应用范围从电磁学到航空声学。在这种情况下，对掺入较大结构中的穿孔介质的兴趣促使人们质疑均质化边界条件的适当性，尤其是当均质化模型更改了在屏幕极端边缘考虑的结行为时。为了促进有效的研究，我们考虑了三种解决Helmholtz方程的数值方法：一套共线刚性板的精确问题的统一变换和维纳-霍夫迭代方法（问题的几何困难意味着这种方法以指数形式收敛是至关重要的）和一种新颖的Mathieu函数搭配方法来考虑沿单个板的​​长度施加可变的柔度。我们详细介绍了每种方法的相对性能和实际考虑因素。通过将使用均化边界条件获得的解与共线板的问题进行比较，我们验证了即使对板的数量有限，先前的理论研究中给出的恒定顺应性也适合获得良好的解估计，即使对于中等数量的板也是如此。渐进政权。| $ \ tanh $ | 函数减小了远场近似中的误差（如果我们足够渐近，则渐近状态）。我们还发现，甚至在渐近状态下，板数和波数也有显着影响。最后两点表明对最终效果进行建模以实现高度准确的结果的重要性。