## Section: New Results

### Waveguides, resonances, and scattering theory

#### Modelling of non-homogeneous lossy coaxial cable for time domain simulation.

Participants : Sébastien Impériale, Patrick Joly.

In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. This question has been motivated by our collaboration with CEA-LIST about the numerical modeling of non-destructive testing (for the detection of cracks in metallic bodies for instance) by ultra-sounds and more precisely the modeling of piezo-electric transducers (see section 6.1.6 ). The complete description of such devices often requires an accurate modeling of the supply process, which includes the propagation of the electric current along coaxial cables. This question appears as an independent sub-problem.

The main characteristic of such coaxial cables is that their transverse directions are very small with respect to their length as well as the wavelength. As a consequence, one would like to use a simplified 1D model as an effective (or homogenized) model for electromagnetic propagation. In this work, we construct and justify rigorously such a model by way of an asymptotic analysis of time harmonic 3D Maxwell' s equations in such a structure. The effective model appears as a generalized wave equation with additional time convolution terms that take into account electric and magnetic losses. By this way, we justify and extend some models proposed in the electrical engineering literature, in particular the well-known telegraphist's equation. The properties of our limit model in time domain has been analyzed and a stable discretization process has been proposed. Numerical simulation in academic simulations exhibit some exotic phenomena of "dispersive dissipation".

Our further investigations of the subject will be developed in the framework of the new ANR project SODDA in collaboration with the team Sysiphe (M. Sorine).

#### Study of lineic defect in periodic media

Participant : Sonia Fliss.

This works deals with one dimensional infinite perturbation - namely line defects - in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a selfadjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann (DtN) approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. In opposition to existing methods, this method is exact but there is a price to be paid : the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. An article presenting the method and its properties is being written, the numerical study is in progress in collaboration with Kersten Schmidt and Dirk Klindworth from the Technische Universität Berlin.

Our further investigations of the subject will be developped in the framework of a new DGA project.

#### A new approach for the numerical computation of non linear modes of vibrating systems

Participants : Anne-Sophie Bonnet-Ben Dhia, Jean-François Mercier.

A collaboration with Cyril Touzé and François Blanc (Unité de Mécanique, ENSTA). The simulation of vibrations of large amplitude of thin plates or shells requires the expensive solution of a non-linear finite element model. The main objective of the proposed study is to develop a reliable numerical method which reduces drastically the number of degrees of freedom. The main idea is the use of the so-called non-linear modes to project the dynamics on invariant subspaces, in order to generate accurate reduced-order models. Cyril Touzé from the Unité de Mécanique of ENSTA has derived an asymptotic method of calculation of the non-linear modes for both conservative and damped systems. But the asymptotically computed solution remains accurate only for moderate amplitudes. This motivates the present study which consists in developing a numerical method for the computation of the non-linear modes, without any asymptotic assumption. This is the object of a collaboration with Cyril Touzé, and new results have been obtained during the post-doc of François Blanc in the Unité de Mécanique of ENSTA. The partial differential equations defining the invariant manifold of the non-linear mode are seen as a vectorial transport problem : the variables are the amplitude and the phase (a, $\varphi $) where the phase $\varphi $ plays the role of the time. In the case of conservative systems, a finite difference scheme is used and an iterative algorithm is written, to take into account the $2\pi $-periodicity in $\varphi $ which is seen as a constraint. An adjoint state approach has been introduced to evaluate the gradient of the coast function. The method has been validated in a simple example with two degrees of freedom. Good agreement with an alternative method, the continuation of periodic solutions method, has been found. Currently the method is extended to the case of damped systems. The main difficulty is that, due to a change of variables, the $2\pi $-periodicity does not hold anymore and new constraints more complicated to implement must be considered. Numerical implementation is still under progress.