Guillaume Legendre

Perfectly Matched Layers for the Convected Helmholtz Equation

In this paper, we propose and analyze perfectly matched absorbing layers for a problem of time-harmonic acoustic waves propagating in a duct in the presence of a uniform flow. The absorbing layers are designed for the pressure field, satisfying the convected scalar Helmholtz equation. A difficulty, compared to the Helmholtz equation, comes from the presence of so-called inverse upstream modes which become unstable, instead of evanescent, with the classical Berengers perfectly matched layers (PMLs). We investigate here a PML model, recently introduced for time-dependent problems, which makes all outgoing waves evanescent. We then analyze the error due to the truncation of the domain and prove that the convergence is exponential with respect to the size of the layers for both the classical and the new PML models. Numerical validations are finally presented.

Perfectly Matched Layers for Time-Harmonic Acoustics in the Presence of a Uniform Flow

This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian--Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included.

WELL-POSEDNESS OF THE DRUDE–BORN–FEDOROV MODEL FOR CHIRAL MEDIA

We consider a chiral medium in a bounded domain, enclosed in a perfectly conducting material. We solve the transient Maxwell equations in this domain, when the medium is modeled by the Drude-Born-Fedorov constitutive equations. The input data is located on the boundary, in the form of given surface current and surface charge densities. It is proved that, except for a countable set of chirality admittance values, the problem is mathematically well-posed. This result holds for domains with non-smooth boundaries.

Regularization of the Time-Harmonic Galbrun’s Equations

This work concerns the mathematical analysis and the finite element approximation of the time harmonic linearized Galbrun’s equations, which modelize the acoustic propagation in presence of a mean flow [3, 7]. This problem has not been satisfactorily solved until now, particularly in the frequency domain, although it could be a main contribution to the industrial objective of noise reduction, in aeronautics for instance.