Junior Research Group: Waves in Periodic Structures
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This project is devoted to the investigation of time-harmonic acoustic scattering problems with (locally perturbed) periodic inhomogeneous layers above impenetrable plates in three dimensional spaces. The scattering problems are modeled by Helmholtz equations in unbounded domains, both the theoretical analysis and the numerical solution of which are very challenging.
The main tool involved in this project is the Floquet-Bloch transform, which has been proven to be very powerful for scattering problems with periodic structures in two dimensional spaces. The first objective is to analyze continuity and regularity of the Bloch transformed field with respect to the quasi-periodicity parameter, where the Dirichlet-to-Neumann map plays an important role. The second goal is to propose a high order numerical method for scattering problems with periodic layers, based on the regularity results established for the quasi-periodic Bloch transformed problems. In contrast to the 2D case, the singularities of the Bloch transformed fields are no longer localized in a finite number of points, but cover a union of singular circles. Thus a straightforward extension of the high order numerical method for the 2D case may not be appropriate for the 3D case, and new ideas will be required. The third goal is to develop an efficient numerical method for locally perturbed periodic layers. Either a coupled finite element method or a discretization of the Lippmann-Schwinger equation will be applied.
Related publications and preprints
- R. Zhang, Numerical method for scattering problems in periodic waveguides. Numer. Math., 148(4), 959-996, 2021. https://doi.org/10.1007/s00211-021-01229-0
- R. Zhang, Numerical Methods for Plane Waves Scattered by Locally Perturbed Periodic Surfaces. Accepted by Comput. Math. Appl., 2021. https://arXiv.org/abs/1801.01761
- R. Zhang, Spectrum decomposition of translation operators in periodic waveguide, SIAM J. Appl. Math. 81(1), 233-257, 2021. https://doi.org/10.1137/19M1290942
- R. Zhang, High order methods to simulate scattering problems in locally perturbed periodic waveguides. Submitted to SISC. https://arxiv.org/submit/3753573
- R. Zhang, Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces. Submitted to SINUM. https://arxiv.org/abs/2107.02032