Section: New Results

Waveguides, resonances, and scattering theory

Localized modes in periodic waveguides

Participants : Anne-Sophie Bonnet-Ben Dhia, Bérangère Delourme, Sonia Fliss, Sergei Nazarov, Elizaveta Vasilevskaia.

The general objective is the study of localized modes in locally perturbed periodic media. We investigate the existence theory of such modes as well as their numerical computations. We can distinguish two types of problems.

Numerical computation of guided modes in periodic media with line defects. We are interested in the propagation of guided modes that propagate in the direction of the line defect (which is parallel to one of the periodicity directions of the unperturbed medium) and decrease exponentially in the transverse directions. We aim at computing these modes and their dispersion relation. Last year, we developed a method based on the use of the DtN approach introduced in the PhD thesis of S. Fliss and the resolution of "operator pencil" eigenvalue problems. This year, in collaboration with Kersten Schmidt, we have made a numerical comparison of this new method with the more standard supercell method.

Existence of localized modes in closed periodic waveguides. We consider a propagation medium which is infinite and periodic in one space dimension and bounded in the transverse ones. We investigate the question of the influence of a local defect on the existence of localized modes. Once again this reduces to a selfadjoint eigenvalue problem in an unbounded domain.

The first problem that we studied is in the framework of the PostDoc of Bérangère Delourme. We have considered general locally perturbed periodic media for which we focus on determining sufficient conditions on the periodic media or the local defect so that it exists at least one eigenvalue below the essential spectrum of the underlying perfectly periodic operator. These sufficient conditions are based on Min-Max theory and an appropriate choice of test functions. We were able to validate these existence conditions thanks to the numerical method based on the use of DtN operators. For situations where the periodic "reference medium" is closed to a simple "limit medium" fo which all calculations can be made by hand, we show that these conditions could be really simple and explicit using perturbation theory and asymptotic expansions of the eigenvalues. We are investigating now the extension of this approach to sufficient conditions for existence of guided modes inside the essential spectrum.

The second case, that is investigated in the framework of the PhD thesis of E. Valisevskaia, is the case where the propagation medium is a thin structure (the thinness being characterized by the parameter $\epsilon$) whose limit is a periodic graph. This is for instance the case of a symmetric ladder as illustrated by figure . If Neumann boundary conditions are considered, it is well known (see in particular the works by Exner, Kuchment) the the limit model when $\epsilon$tends to is the Helmholtz equation on the graph (1D Helmholtz equations on each branch competed by continuity and Kirchoff transimission conditions at each node) . For this limit problem, the underlying operator does not present any spectral gaps but can be written, due to the symmetry of the problem, as the sum of two operators, each of which having an infinity of spectral gaps. This allows us to look for eigenvalues in these spectral gaps, induced by symmetric and localized perturbations of the limit graph model. This can de done for instance by modifying (symetrically) the Kirchoff conditions on two symmetric nodes of the graph. In the limit process mentionned above, this would correspond to modifying the width of the rung that joins these two points in the original problem. First existence results have been obtained in this direction. In a further step, one can expect, by asymptotic analysis, to get corresponding existence results for the original problem, at least for $\epsilon$ small enough.

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.

Harmonic wave propagation in locally perturbed periodic waveguide

Participants : Sonia Fliss, Patrick Joly.

We work on the expression and the asymptotic behaviour of the Green function for time harmonic wave equation in two-dimensional periodic waveguide. This enables us to define a radiation condition and show well-posedness of the Helmholtz equation set in a periodic waveguide. The redaction of an article is ongoing.

This analysis is one of the main tool to solve inverse problems in locally perturbed periodic waveguide (see section 6.6.1 ) when the data are far field measurements of scattering problems.

One challenging perspective of this work is to extend these results to periodic problems in free space.

Finite element approximation of modes of elastic waveguides immersed in an infinite fluid

Participants : Anne-Sophie Bonnet-Ben Dhia, Cédric Doucet, Christophe hazard.

This work is developped in collaboration with Vahan Baronian (CEA). We are developping numerical tools to simulate ultrasonic non-destructive testing in elastic waveguides. This particular topic aims at finding an efficient way of coupling semi-analytical finite element methods and perfectly matched layers (PMLs) to compute modes of elastic waveguides embedded in an infinite fluid.

During our numerical investigations, we noticed that the semi-analytical mixed finite element formulation proposed in the PhD thesis of V. Baronian may lead to the computation of spurious modes. We overcame this problem in the following way: instead of approximating components of stress tensors by means of first-order finite elements of class ${𝒞}^0$, we decided to use zeroth-order discontinuous ones. This simple modification seems not only to stabilize the discretization step, but also to approximate modes more accurately in comparison with the classical semi-analytical finite element formulation. Last but not least, we observed a meaningful improvement of the approximation of the continuous spectrum of stretched operators related to PMLs. Besides, previous results (in the PhD thesis of B. Goursaud) about the best way of designing PMLs to simulate wave propagation in open acoustic waveguides have been confirmed by our numerical experiments on immersed elastic structures.

Further investigations need to be carried out to explain these phenomena. Especially, a theoretical analysis still remains to be done.