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Dr. Ruming Zhang

Welcome to my personal homepage! You can find my latest research and teaching activities below.

Current List of Courses
Semester Titel Typ
Winter Semester 2020/21 Lecture
Summer Semester 2019 Lecture
Winter Semester 2018/19 Lecture

Lecture in Winter Semester 2020/21


Wave propagation in periodic structures

This lecture introduces theoretical analysis and numerical methods to simulate wave propagation in periodic structures. This is an interesting topic in both mathematics and other areas, such as nano-technology. In this lecture, we will study the well-posedness of this kind of problems and discuss the properties of the solutions. We will also introduce numerical methods to simulation the physical processes. The content of this lecture is listed below.

Content:'

  • Theoretical and numerical analysis of quasi-periodic scattering problems (integral equation method, variational method)
  • Floquet-Bloch Transform
  • Wave propagating in open periodic waveguides
  • Wave propagating in closed periodic waveguides (Floquet theory, eigenvalue problems)
  • Numerical simulation of wave propagating in periodic waveguides (integral equation method, finite element method)

Preliminary course: functional analysis, partial differential equation, numerical analysis
Due to the current situation of the corona virus, the teaching will be carried out online.


Project

DFG research grant (No. 433126998)

Higher order numerical methods for acoustic scattering problems with locally perturbed periodic structures
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 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. https://gepris.dfg.de/gepris/projekt/433126998?language=en


Research interest

  • Direct scattering problems in unbounded structures.
    • Variational approach of the scattering problems in perturbed periodic structures, via the Floquet-Bloch transform.
    • Limiting absorption principle in scattering problems in periodic waveguides.
    • Integral equation methods for scattering problems from periodic or rough surfaces.
    • Finite element methods for scattering problems from perturbed periodic surfaces or waveguides.
  • Inverse problems in perturbed periodic structures
    • Optimization and regularization methods for inverse scattering problems in periodic structures.
    • Fast imaging methods for inverse scattering problems: linear sampling method, factorization method, etc.
  • Fast algorithms for evaluations of quasi-periodic Green's functions.
  • Numerical method for evaluations of eigenvalues in a linear system.

Publications

  1. 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
  2. X. Liu, R. Zhang, Near-field imaging of locally perturbed periodic surfaces, Inverse Problems 35(11), Special Issue in Memory of Professor Armin Lechleiter, 114003, 2019. https://doi.org/10.1088/1361-6420/ab2e8f
  3. R. Zhang, High order numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 40(4), A2286-A2314, 2018. https://doi.org/10.1137/17M1144945
  4. B. Zhang, R. Zhang, An FFT-based algorithm for efficient computation of quasiperiodic Green's functions for the Helmholtz and Maxwell's equations, SIAM J. Sci. Comput., 40(3), B915-B941, 2018. https://doi.org/10.1137/18M1165621
  5. A. Lechleiter, R. Zhang, The reconstruction of a local perturbation in periodic structures, Inverse Problems, 34, 035006, 2018. https://doi.org/10.1088/1361-6420/aaa7b1
  6. R. Zhang, J. Sun, The reconstruction of obstacles in a waveguide using finite elements, J. Comput. Math., 36(1), 29-46, 2018. http://doi.org/10.4208/jcm.1610-m2016-0559
  7. R. Zhang, B. Zhang, A new integral equation formulation for scattering by penetrable diffraction gratings, J. Comput. Math., 36(1), 110-127, 2018. http://doi.org/10.4208/jcm.1612-m2016-0501
  8. R. Zhang, J. Sun, C. Zheng, Reconstruction of a penetrable obstacle in periodic waveguides, Comput. Math. Appl., 74(11), 2739 - 2751, 2017. https://doi.org/10.1016/j.camwa.2017.08.028
  9. A. Lechleiter, R. Zhang, A Floquet-Bloch transform based numerical method for scattering from locally perturbed periodic surfaces, SIAM J. Sci. Comput., 39(5), B819-B839, 2017. https://doi.org/10.1137/16M1104111
  10. A. Lechleiter, R. Zhang, Non-periodic acoustic and electromagnetic scattering from periodic structures in 3D, Comput. Math. Appl., 74(11), 2723 - 2738, 2017. https://doi.org/10.1016/j.camwa.2017.08.042
  11. M. Li, R. Zhang, Near-field imaging of sound-soft obstacles in periodic waveguides, Inverse Problems and Imaging, 11(6), 1091-1105, 2017. http://doi.org/10.3934/ipi.2017050
  12. A. Lechleiter, R. Zhang, A convergent numerical scheme for scattering of aperiodic waves from periodic surfaces based on the Floquet-Bloch transform, SIAM J. Numer. Anal., 55(2), 713-736, 2017. https://doi.org/10.1137/16M1067524
  13. R. Huang, A. Struthers, J. Sun, R. Zhang, Recursive integral method for transmission eigenvalues, J. Comput. Phys., 327, 830-840, 2016. https://doi.org/10.1016/j.jcp.2016.10.001
  14. G. Sun, R. Zhang, A sampling method for the reconstruction of a periodic interface in a layered medium, Inverse Problems, 32, 075005, 2016. https://doi.org/10.1088/0266-5611/32/7/075005
  15. J. Yang, B. Zhang, R. Zhang, Near-field imaging of periodic interfaces in multilayered media, Inverse Problems, 32, 035010, 2016. https://doi.org/10.1088/0266-5611/32/3/035010
  16. J. Li, G. Sun, R. Zhang, The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71(7), 1491-1502, 2016. https://doi.org/10.1016/j.camwa.2016.02.031
  17. R. Zhang, J. Sun, Efficient finite element method for grating profile reconstruction, J. Comput. Phys., 302(1), 405-419, 2015. https://doi.org/10.1016/j.jcp.2015.09.016
  18. R. Zhang, B. Zhang, Near-field imaging of periodic inhomogeneous media, Inverse Problems, 30, 045004, 2014. https://doi.org/10.1088/0266-5611/30/4/045004
  19. J. Yang, B. Zhang, R. Zhang, Reconstruction of penetrable grating profiles, Inverse Problems and Imaging, 7(4), 1393-1407, 2013. https://doi.org/10.3934/ipi.2013.7.1393
  20. J. Yang, B. Zhang, R. Zhang, A sampling method for the inverse transmission problem for periodic media, Inverse Problems, 28, 035004, 2012. https://doi.org/10.1088/0266-5611/28/3/035004

Preprints

  1. R. Zhang, Numerical method for scattering problems in periodic waveguides. Submitted to Numer. Math., 1st round revision. https://arxiv.org/abs/1906.12283
  2. R. Zhang, Numerical methods for scattering problems from multi-layers with different periodicities. Submitted to Numer. Methods Partial Differential Eq. https://arXiv.org/abs/1806.05063
  3. R. Zhang, Numerical methods for quasi-periodic incident fields scattered by locally perturbed periodic surfaces. Preprint. https://arXiv.org/abs/1801.01761
  4. R. Zhang, A Bloch transform based numerical method for the rough surface scattering problems. Preprint. https://arXiv.org/abs/1805.11484
  5. R. Zhang, The study of the Bloch transform of the total fields diffracted by locally perturbed periodic surfaces. Preprint. https://arxiv.org/abs/1708.07560