Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation

Philippe Marchner (1, 2, 3, 4) , Hadrien Beriot (2) , Xavier Antoine (4, 1) , Christophe Geuzaine (3)

1 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization
2 Siemens Industry Software S.r.l.
3 Institut Montefiore - Département d'Electricité, Electronique et Informatique (Liège)
4 IECL - Institut Élie Cartan de Lorraine

Perfectly Matched Layers (PMLs) appear as a popular alternative to non-reflecting boundary conditions for wave-type problems. The core idea is to extend the computational domain by a fictitious layer with specific absorption properties such that the wave amplitude decays significantly and does not produce back reflections. In the context of convected acoustics, it is well-known that PMLs are exposed to stability issues in the frequency and time domain. It is caused by a mismatch between the phase velocity on which the PML acts, and the group velocity which carries the energy of the wave. The objective of this study is to take advantage of the Lorentz transformation in order to design stable perfectly matched layers for generally shaped convex domains in a uniform mean flow of arbitrary orientation. We aim at presenting a pedagogical approach to tackle the stability issue. The robustness of the approach is also demonstrated through several two-dimensional high-order finite element simulations of increasing complexity.

Mots clés: Perfectly matched layers; stability; Lorentz transformation; convected Helmholtz equation; cross flow; high-order FEM

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Informatique [cs]; Modélisation et simulation; Acoustique [physics.class-ph]

Philippe Marchner, Hadrien Beriot, Xavier Antoine, Christophe Geuzaine. Stable Perfectly Matched Layers with Lorentz transformation for the convected Helmholtz equation. Journal of Computational Physics, 2021, ⟨10.1016/j.jcp.2021.110180⟩. ⟨hal-02556182v2⟩