Wave Propagation : Mathematical Study and Simulation

Identity card

POEMS (french acronym whose translated meaning is Wave Propagation : Mathematical Study and Simulation) is a Mixed Research Unit (UMR 7231), since january 2005, whose 3 supervisions are CNRS, ENSTA-Paris and INRIA. This unit is at the same time an INRIA Project (part of INRIA Saclay Center), a tema of the Applied Mathematical Unit of ENSTA-Paris, inside FMJH (Jacques Hadamard Mathematical Foundation) and a CNRS Unit, dependent on INSIS department (Engineering Science and Systems Institute). The unit is located at ENSTA-Paris, in campus of <a href='https://www.ip-paris.fr'>Institut Polytechnique de Paris</a>, Palaiseau, France.

POEMS director is Anne-Sophie Bonnet-Ben Dhia.

In POEMS, there are about fifteen researchers or faculty members, 2 engineers and about fifteen students (PhD or post-docs).

Objectives

POEMS scientific activities are devoted to the development of mathematical and numerical studies about wave propagation, in all domains of Physics and Mechanics, namely electro-magnetic waves, acoustic waves, elastic waves or gravitational waves.

POEMS work concerns the modelling of complex problems, the mathematical analysis of these models (generally governed by Partial Differential Equations), the development of approximation methods and of computation softwares.

Research activities

Our main research themes are as follows :

• Obtaining asymptotically effective models for wave propagation. The aim is to take into account the presence of thin layers, small heterogeneities or microstructures of the medium, in an approximate model that is less costly to solve numerically than the initial model obtained by techniques such as connected asymptotic expansion or double-scale homogenization.
• Applications of spectral theory to the analysis of wave propagation. We are particularly interested in the study of waveguides, whether closed or open, uniform or periodic. Depending on the case, the aim is to prove the existence of gaps in the essential spectrum, and the existence or non-existence of eigenvalues in the gaps or dips in the essential spectrum.
• The development of specific methods for the treatment of wave propagation or diffraction in unbounded domains. We develop methods for transparent or absorbing conditions (Dirichlet-to-Neumann, Perfectly Matched Layers) as well as integral methods.
• Design and implementation of efficient numerical methods for wave propagation. In particular, we develop domain decomposition approaches and Fast Multipole Method (FMM) and H-Matrix methods for integral equations.
• Inverse Diffraction Problems: The goal is to detect and/or identify objects based on the measurement of the far field diffracted by these objects when illuminated by an incident wave. In particular, we develop sampling methods based on the concepts of linear sampling or topological gradient.

The main areas of application are as follows:

• The study of electromagnetic wave propagation in non-classical media such as metamaterials or plasmas. For example, we are interested in the design of stable PML layers in dispersive media.
• Simulation of elastic wave propagation in the subsurface. The aim is to study site effects (amplification of seismic motion by topography, geological structure, etc.) or soil-structure interactions.
• Sound propagation in a flow. This subject, which is part of the field of aeroacoustics, is involved in efforts to reduce aircraft noise.
• Non-destructive testing of structures using ultrasonic waves. We are particularly interested in NDT of guide structures such as pipes or plates.

At last, we develop through a partnership with IRMAR, the XLiFE++ library. This is a versatile finite element library, containing specific tools for solving problems on unbounded domains, such as Dirichlet-to-Neumann conditions or coupling with BEM.

We also develop COFFEE, a 3D solver for linear elastodynamics based on fast BEMs (full implementation in Fortran 90). The 3-D elastodynamic equations are solved with the boundary element method accelerated by the multi-level fast multipole method or H-matrix based solvers. The fundamental solutions for the infinite or half-space are used. A boundary element-boundary element coupling strategy is also implemented so multi-region problems (strata inside a valley for example) can be solved.