Higher order numerical methods for acoustic scattering problems with locally perturbed periodic structures
Final Report Abstract
Time-harmonic wave scattering problems in (locally perturbed) bi-periodic structures are important topics in application areas such as optics or photonic crystals. From the mathematical point of view, the problem is modeled by the Helmholtz equation in a half space above a bounded bi-periodic surface (assume it is periodic in x1 and x2 directions) in a three-dimensional space, with a boundary condition on the bi-periodic surface and a proper radiation condition in the x3 direction which guarantees the well-posedness of the problem. The numerical simulation for such a problem is particularly interesting but also very challenging due to the unbounded domain. To solve the problem by the finite element method, we need to approximate the scattering problem by one defined in a bounded domain. With a proper truncation technique, such as the transparent boundary condition (TBC) or the perfectly matched layers (PML), we first reduce the unbounded domain into a bi-periodic waveguide which is bounded in the x3 direction. Then applying the Floquet-Bloch transform, the new problem is rewritten into an equivalent family of quasi-periodic problems, which are suitable to be solved by the finite element method. The TBC results in a complex singular integral in a square due to the Rayleigh anomalies. To approximate the integral numerically, we developed a high-order tailor-made quadrature method. The PML, on the other hand, is no longer exact but gets rid of singularities from the TBC. But since it is an approximation, we also work on the error estimations to guarantee the convergence of this method. Finally, the method is also extended to locally perturbed periodic surfaces based on a domain transformation. An efficient numerical method is also developed based on the Schur complement. We have also presented numerous numerical examples to illustrate our theoretical results.
Publications
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Numerical methods for scattering problems in periodic waveguides. Numerische Mathematik, 148(4), 959-996.
Zhang, Ruming
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Spectrum Decomposition of Translation Operators in Periodic Waveguide. SIAM Journal on Applied Mathematics, 81(1), 233-257.
Zhang, Ruming
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Exponential Convergence of Perfectly Matched Layers for Scattering Problems with Periodic Surfaces. SIAM Journal on Numerical Analysis, 60(2), 804-823.
Zhang, Ruming
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Does PML exponentially absorb outgoing waves scattering from a periodic surface?
W. Lu, K. Shen & R. Zhang
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Higher-Order Convergence of Perfectly Matched Layers in Three-Dimensional Biperiodic Surface Scattering Problems. SIAM Journal on Numerical Analysis, 61(6), 2917-2939.
Zhang, Ruming
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A high‐order numerical method for solving non‐periodic scattering problems in three‐dimensional bi‐periodic structures. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 104(9).
Arens, Tilo; Shafieeabyaneh, Nasim & Zhang, Ruming
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The PML-method for a scattering problem for a local perturbation of an open periodic waveguide. Numerische Mathematik, 157(2), 717-748.
Kirsch, Andreas & Zhang, Ruming
