Associated Project 7 • Wave propagation in 3D bi-periodic structures (6/2021 - 4/2023)

Bi-periodic structure
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 modelled by Helmholtz equations in unbounded domains, both the theoretical analysis and the numerical solution of which are very challenging.
The main tool involved nullin 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.
Bi-periodic structure
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.

Publications

  1. and . The PML-method for a scattering problem for a local perturbation of an open periodic waveguide. Numer. Math., 157(2):717–748, April . URL https://doi.org/10.1007/s00211-025-01456-9. [bibtex]

  2. , , and . A high-order numerical method for solving non-periodic scattering problems in three-dimensional bi-periodic structures. ZAMM Z. Angew. Math. Mech., 104(9):e202300650, September . URL https://doi.org/10.1002/zamm.202300650. [preprint] [files] [bibtex]

  3. . On the scattering of a plane wave by a perturbed open periodic waveguide. Math. Methods Appl. Sci., 46(9):10698–10718, June . URL https://doi.org/10.1002/mma.9147. [preprint] [bibtex]

  4. . High order complex contour discretization methods to simulate scattering problems in locally perturbed periodic waveguides. SIAM J. Sci. Comput., 44(5):B1257–B1281, October . URL https://doi.org/10.1137/21M1421532. [preprint] [bibtex]

  5. . Exponential convergence of perfectly matched layers for scattering problems with periodic surfaces. SIAM J. Num. Anal., 60(2):804–823, April . URL https://doi.org/10.1137/21M1439043. [preprint] [bibtex]

  6. . A scattering problem for a local pertubation of an open periodic waveguide. Math. Methods Appl. Sci., 45(10):5737–5773, February . URL https://doi.org/10.1002/mma.8137. [preprint] [bibtex]

  7. . Numerical methods for scattering problems in periodic waveguides. Numer. Math., 148(4):959–996, August . URL https://doi.org/10.1007/s00211-021-01229-0. [preprint] [bibtex]

  8. . Spectrum decomposition of translation operators in periodic waveguide. SIAM J. Appl. Math., 81(1):233–257, February . URL https://doi.org/10.1137/19M1290942. [bibtex]

  9. and . Near-field imaging of locally perturbed periodic surfaces. Inverse Problems, 35(11):114003, October . URL https://doi.org/10.1088/1361-6420/ab2e8f. [preprint] [bibtex]

Former staff
Name Title Function
Shafieeabyaneh, Nasim M.Sc. Doctoral researcher
Zhang, Ruming Dr. Junior research group leader