WIAS Preprint No. 2002, (2014)

Finite element method to fluid-solid interaction problems with unbounded periodic interfaces


  • Hu, Guanghui
  • Rathsfeld, Andreas
  • Yin, Tao

2010 Mathematics Subject Classification

  • 78A45 35Q74 74F10 35B27


  • fluid-solid interaction, periodic structure, variational approach, Helmholtz equation, Lamé system, convergence analysis, Rayleigh expansion


Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

Appeared in

  • Numer. Methods Partial Differential Equations, 32 (2016) pp. 5--35.

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