Publications

Preprints

• P. Amenoagbadji, S. Fliss, P. Joly, Time-harmonic wave propagation in junctions of two periodic half-spaces, accepted for publication in Pure and Applied Analysis, 2025.
  ABSTRACT

  We are interested in the Helmholtz equation in a junction of two periodic half-spaces. When the overall medium is periodic in the direction of the interface, Fliss and Joly (2019) proposed a method which consists in applying a partial Floquet-Bloch transform along the interface, to obtain a family of waveguide problems parameterized by the Floquet variable. In this paper, we consider two model configurations where the medium is no longer periodic in the direction of the interface. Inspired by the works of Gérard-Varet and Masmoudi (2011, 2012), and Blanc, Le Bris, and Lions (2015), we use the fact that the overall medium has a so-called quasiperiodic structure, in the sense that it is the restriction of a higher dimensional periodic medium. Accordingly, the Helmholtz equation is lifted onto a higher dimensional problem with coefficients that are periodic along the interface. This periodicity property allows us to adapt the tools previously developed for periodic media. However, the augmented PDE is elliptically degenerate (in the sense of the principal part of its differential operator) and thus more delicate to analyse.

  ARXIV HAL

Articles in peer-reviewed journals

• P. Amenoagbadji, S. Fliss, P. Joly, Wave propagation in one-dimensional quasiperiodic media, Communications in Optimization Theory (2023) 17, "Special Issue on numerical analysis and control dedicated to the memory of Professor Roland Glowinski", 2023.
  ABSTRACT

  This work is devoted to the resolution of the Helmholtz equation \( -(\mu\, u')' - \rho\, \omega^2\, u = f \) in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.

  JOURNAL ARXIV HAL

Thesis manuscript

• P. Amenoagbadji, Wave propagation in quasiperiodic media, thesis defended on 12/13/2023, before the jury composed of D. Gérard-Varet (President), E. Bonnetier and A. Levitt (Reviewers), B. Delourme, S. Guenneau, and C. Le Bris (Examiners), S. Fliss et P. Joly (Supervisors).
  HAL thèses

Oral communications