Section: New Results

Numerical methods for time domain wave propagation

Coupling Retarded Potentials and Discontinuous Galerkin Methods for time dependent wave propagation problems

Participant : Patrick Joly.

This topic is developed in collaboration with J. Rodriguez (Santiago de Compostela) in the framework of the contract ADNUMO with AIRBUS. The general objective was to use time-domain integral equations - or retarded potentials - as a tool for contructing transparent boundary conditions for wave problems in unbounded media, by coupling them to an inerior volumic method, namely the Discontinuous Galerkin (DG) method.

Since last year, our new goal is to extend the method proposed in a previous work for DG with central fluxes to the case of upwind fluxes, while preserving most of the good properties of the original method from both theoretical (stability via energy dissipation - instead of energy conservation) and practical points of view. We have designed a method that achieves this goal at the only prize of a small deterioration of the CFL condition. The method has been successfully implemented and the numerical results clearly emphasize the superiority of upwing fluxes for taking into account the convection terms in the linearized Euler equations in aeroacoustics, the privileged application.

At the same time, we have used similar ideas for treating physical boundary conditions involving differential (in time) impedance operators.

Solving the Homogeneous Isotropic Linear Elastodynamics Equations Using Potentials and Finite Elements.

Participants : Aliénor Burel, Marc Duruflé, Patrick Joly.

This topic is the subject of the first part oh th PhD thesis of A. Burel. Its aim is to use the classical theoretical decomposition of the elastodynamic displacement into two potentials referring to the pressure wave and the shear wave, and use it in a numerical context. Last year, a method has been proposed for solving the Dirichlet problem (clamped boundary), successfully analyzed and implemented. For free boundary conditions, we have proposed an original method considereing these boundary conditions as a perturbation of the Dirichlet conditions. The natural adaptation of the variational formulation used in the case of the Dirichlet problems presents nice theoretical properties and leads to satisfactory numerical results for the time harmonic problem. However, the implementation for the time dependent problem reveals severe instability phenomena that seem to be already present in the semi-discrete (in space) problem. In order to understand the cause of these instability (and possibly remedy them) we are currently performing the Kreiss analysis of the half-space problems in the case where $Q_1$ finite elements are used on the same uniform square grid for both P-waves and S-waves potentials.

Time domain analysis of Maxwell's equations in Lorentz materials

Participants : Maxence Cassier, Lucas Chesnel, Christophe Hazard, Patrick Joly, Valentin Vinoles.

This is the time-domain counterpart of the research done at Poems about frequency domain analysis of metamaterials (see also the section 6.2.7 ) in the framework of the ANR Project Metamath. One fundamental question is the link between the two problems via the limiting amplitude principle, in particular in the cases where the time harmonic problem fails to be well posed problem in the standard framework. This occurs at certain frequencies (see section ) when one considers a transmission problem between a Lorentz material and a standard one.

We are investigating this question from both theoretical and numerical points of view. This is also the object of a collaboration with B. Gralak from the Institut Fresnel in Marseille.

Modeling and numerical simulation of a piano.

Participants : Juliette Chabassier, Marc Duruflé, Sébastien Imperiale, Patrick Joly.

The defense of the PhD thesis of Juliette Chabassier, in March, has marked one of the most spectacular achievements in Poems for the past years, concerning the "complete" physical and mathematical modeling of a grand piano and its computer simulation. This is the result of a quite interdisciplinary work in collaboration with Antoine Chaigne (UME, ENSTA). We refer the reader to the three previous activity reports of Poems for a more detailed description of the scientific developments that have led to the implementation of a parallel code for the simulation of the piano. Using this code, M. Duruflé and J. Chabassier have realized a bank of synthetic sounds that can be used for playing scoreboards (using MIDI files for instance). For more details, and also other additional information about the work, we refer the reader to the Web page : http://modelisation.piano.free.fr .

Although already quite satiafactory, the results obtained by the present version of the code show that there is still room for the improvement of our piano model. One of the ideas consists in improving the quality of the model for the hammers and that is why J. Chabassier and M. Duruflé have proposed an enriched model involving the virations of the hammer's shank. We expect to achieve further progress in this direction through our participation to the ITN (Initial Training Network) European project BATWOMAN (Basic Acoustics Training and Workprogram on Methodologies for Acoustics Network) that has been submitted lst November. This projects regroups 11 partners from 7 different contries and gathers academic people with industrials of the donain, including Steinway.

As a theoretical complement to the numerical developments, we have led a systematic theoretical study of the numerical method used in our code for computing string's vibrations. Our concern was to develop a new implicit time discretization, which is associated with finite element methods in space, in order to reduce numerical dispersion while allowing the use of a large time step. We proposed a new $\theta$-scheme based on different $\theta$-approximations for the flexural and shear terms of the equations, which allows to reduce numerical dispersion while relaxing the stability condition. In particular, we gave some insights of innovative proofs of stability by energy techniques that provide uniform estimates with respect to the CFL number. Theoretical results have been illustrated with numerical experiments corresponding to the simulation of a realistic piano string.

Numerical methods in electromagnetism

Participant : Patrick Ciarlet.

Collaborations with Eric Chung, Tang Fei Yu and Jun Zou (Chinese University of Hong Kong, China), Philippe Ciarlet (City University of Hong Kong, China) Haijun Wu (Nanjing University, China), Stefan Sauter and Corina Simian (Universität Zürich).

The numerical approximation of electromagnetic fields is still a very active branch of research. Below, three lines of work are briefly reported.

Edge finite elements are widely used in 2D/3D electromagnetics, however they approximate very weakly the divergence of the fields. In a recent work with H. Wu & J. Zou, we proposed a method that allows one to approximate the divergence accurately in $H^{-s}$-norms ($1/2<s<1$).

Discontinuous Galerkin finite elements are also very popular, as they allow one to design fast (and accurate) methods to solve PDEs. Jointly with E. Chung and T. F. Yu, we designed a numerical method to solve the 2D/3D time-dependent Maxwell equations, using a high order staggered DG method in the spirit of those introduced by E. Chung and B. Engquist. The method has been analyzed on Cartesian meshes and its generalization to unstructured meshes is under way.

A few years ago, we proposed with Philippe Ciarlet a method to solve some problems in linear elasticity intrinsically. With S. Sauter, C. Simian and Philippe Ciarlet, we studied a similar approach that can be applied to 2D electrostatics. It consists in solving the problem in the electric field directly, using exact or local curl-free approximation of the field. Within this framework, we have been able to derive a general method that allows one to derive intrinsic conforming and non-conforming finite element spaces to compute the electrostatic potential. Generalization to 3D electrostatics and linear elasticity is under way.