## Section: New Results

### Modeling

#### Implementation of a non-reflecting boundary condition on ellipsoidal boundary

Participants : Hélène Barucq, Anne-Gaëlle Saint-Guirons, Sébastien Tordeux.

The modeling of wave propagation problems using finite element methods usually requires the truncation of the computational domain around the scatterer of interest. Absorbing boundary condition are classically considered in order to avoid spurious reflections. This year we have implemented and tested an exact condition based on a non local Dirichlet to Neumann operator in the context of the Helmholtz equation posed on an elongated domain.

#### Explicit computation of the electrostatic energy for an elliptical charged disc

Participants : Sophie Laurens, Sébastien Tordeux.

In [32] , We have described a method to obtain an explicit expression for the electro- static energy of a charged elliptical infinitely thin disc. The charge distribution is assumed to be polynomial. Such explicit values for this energy are fundamen- tal for assessing the accuracy of boundary element method codes. The main tools used are an extension of Copson’s method and a diagonalization, given by Leppington and Levine, of the single-layer potential operator associated with the electrostatic potential created by a distribution of charges on an elliptical disc.

#### A new modified equation approach for solving the wave equation

Participants : Cyril Agut, Hélène Barucq, Henri Calandra, Julien Diaz, Florent Ventimiglia.

The new method involving $p$-harmonic operator described in section 3.2 has been presented in [17] . We have proved the convergence of the scheme and its stability under a CFL condition. Numerical results in one, two and three-dimensional configurations show that this CFL condition is slightly greater than the CFL condition of the second-order Leap-Frog scheme.

In the framework of the PhD thesis of Florent Ventimiglia, we are now considering the extension of this technique to the first order formulation of the acoustic and elastodynamic equations. A numerical analysis of performance in 1D indicates that, for a given accuracy, this method requires less storage than the High-Order ADER Schemes for and similar computational costs. We are now implementing this algorithm in 3D in order to confirm this analysis and to assess its performance in an RTM framework on realistic configurations.

#### Stability Analysis of an Interior Penalty Discontinuous Galerkin Method for the Wave equation

Participants : Cyril Agut, Hélène Barucq, Julien Diaz.

The Interior Penalty Discontinuous Galerkin Method [72] , [69] , [83] we use in the IPDGFEM code requires the introduction of a penalty parameter. Except for regular quadrilateral or cubic meshes, the optimal value of this parameter is not explicitely known. Moreover, the condition number of the resulting stiffness matrix is an increasing function of this parameter, but the precise behaviour has not been explicited neither. We have carried out a theoretical and numerical study of the pnealization parameter and of the CFL condition for quadrilateral and cubic meshes, this results have been presented in a paper accepted in M2AN [16]

#### Higher Order Absorbing Boundary Conditions for the Wave Equation

Participants : Hélène Barucq, Juliette Chabassier, Julien Diaz.

The numerical simulation of wave propagation is generally performed by truncating the propagation medium and the team works on new ABCs, trying to improve the performance of existing conditions. Following the analysis performed in [23] , we have considered the issue of constructing high-order ABCs for the Helmholtz equation. Now, to derive conditions of order greater than two is really technical. In addition, when the coefficients representing the geological properties of the medium are not regular, the method of construction of ABCs is not completely justified. That is why we turned to the construction of conditions that take into account all the characteristics of the diffraction phenomenon and not only waves that propagate like in the case of standrad ABCs. This is what we call enriched ABCs. A research report is being written, an article should be submitted in 2013. During 2012, a publication for the acoustic wave equation has been accepted in M3AS [23] and a second one has been submitted.

#### Multiperforated plates in linear acoustics

Participants : Abderrahmane Bendali, M'Barek Fares, Sophie Laurens, Estelle Piot, Sébastien Tordeux.

Acoustic engineers use approximate heuristic models to deal with multiperforated plates in liners and in combustion chambers of turbo-engines. These models were suffering from a lack of mathematical justifications and were consequently difficult to improve. Performing an asymptotic analysis (the small parameter is the radius of the perforations), we have justified these models and proposed some improvement. Our theoretical results have been compared to numerical simulations performed at CERFACS (M'Barek Fares) and to acoustical experiments realized at ONERA (Estelle Piot). Two papers have been published in 2012 [27] , [30] .

#### Performance Assessment of IPDG for the solution of an elasto-acoustic scattering problem

Participants : Hélène Barucq, Rabia Djellouli, Élodie Estecahandy.

We present a solution methodology for the direct elasto-acoustic scattering problem that falls in the category of Discontinuous Galerkin methods. The method distinguishes itself from the existing methods by combining high-order Discontinuous Galerkin approximations, local stabilizations for the coupled problem and the use of curved element edges on the boundaries. We present some numerical results that illustrate the salient features and highlight the performance of the proposed solution methodology on the resonance phenomenon existing in the elastic scatterer for simple geometries such as circles. Moreover, the designed method ensures a convergence order with a gain of two order of magnitude compared to polygonal boundaries, and a potential to address both mid- and high-frequency regimes. These results have been presented to ECCOMAS 2012 [44] and to two workshops [42] [43] .

#### Operator Based Upscaling for Discontinuous Galerkin Methods

Participants : Hélène Barucq, Théophile Chaumont, Julien Diaz, Christian Gout, Victor Péron.

Scientists and engineers generally tackle problems that include multiscale effects and that are thus difficult to solve numerically. The main difficulty is to capture both the fine and the coarse scales to get an accurate numerical solution. Indeed, the computations are generally performed by using numerical schemes based on grids. But the stability and thus the accuracy of the numerical method depends on the size of the grid which must be refined drastically in the case of very fine scales. That implies huge computational costs and in particular the limitations of the memory capacity are often reached. It is thus necessary to use numerical methods that are able to capture the fine scale effects with computations on coarse meshes. Operator-based upscaling is one of them and we present in [22] a first attempt to adapt that technique to a Discontinuous Galerkin Method (DGM). We consider the Laplace problem as a benchmark and we compare the performance of the resulting numerical scheme with the classical one using Lagrange finite elements. The comparison involves both an accuracy analysis and a complexity calculus. This work shows that there is an interest of combining DGM with upscaling.

#### Asymptotic Modeling for Elasto-Acoustics

Participants : Julien Diaz, Victor Péron.

We present in [65] equivalent conditions and asymptotic models for the diffraction problem of elasto-acoustic waves in a solid medium surrounded by a thin layer of fluid medium. This problem is well suited for the notion of equivalent conditions : since the thickness of the layer is small with respect to the wavelength, the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the third order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads us to solve only elastic equations. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

Questions regarding the implementation of the conditions have been addressed carefully. Indeed, the boundary conditions have been integrated without changing the structure of the code Hou10ni.

This work has been presented in four international conferences and Workshops : Aquitaine-Euskadi Workshop on Applied Mathematics; First Russian-French Conference on Mathematical Geophysics, Mathematical Modeling in Continuum Mechanics and Inverse Problems; Workshop HPC-GA; Twelfth International Conference Zaragoza-Pau on Mathematics.

A paper with numerical results for the elasto-acoustic problem with a thin layer and a variable thickness is in preparation.

#### Asymptotic modeling in electromagnetism

Participants : François Buret, Monique Dauge, Patrick Dular, Laurent Krähenbühl, Victor Péron, Ronan Perrussel, Clair Poignard, Damien Voyer.

The following results rely on a problematic developed in section
3.2 , item **Asymptotic modeling**.

In the paper [28] , eddy current problems are addressed in a bidimensional setting where the conducting medium is non-magnetic and has a corner singularity. For any fixed skin depth we show that the flux density is bounded near the corner, unlike the perfect conducting case. Then as the skin depth goes to zero, the first two terms of a multiscale expansion of the magnetic potential are introduced to tackle the magneto-harmonic problem. The heuristics of the method are given and numerical computations illustrate the obtained accuracy.

In a forthcoming paper, we describe the magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator. This work has been presented in the international conference WCCM 2012.

#### Asymptotic models for penalization methods in porous media

Participants : Gilles Carbou, Victor Péron.

We investigate a Stokes-Brinkman problem with Beavers and Joseph transmission conditions, adapted to a penalization method in porous media. We exhibit a WKB expansion for the solution of the fluid-porous interface problem. The main interest is to derive equivalent models for the penalization method. We explicit the first terms of the WKB expansion for the flow and the pressure in the subdomains. Each asymptotics of the flow writes as a sum of a tangential boundary layer term plus a standard term in the porous region. From the benefits or these boundary layers, we infer a collection of elementary transmission problems satisfied by the standard parts of the asymptotics for the flow and the pressure. As a consequence of the penalization of the Laplacian operator which applies to the flow in the porous media, a degenerate operator of order zero applies to the elementary velocities appears in the porous region. The main difficulty concern the proof of elliptic regularity up to the interface for the solution of each elementary problem, since exotic conditions for the flow and the pressure appears along the interface. Our strategy consists to adapt a proof of elliptic regularity for the solution of a Darcy problem set in homogeneous media and developed by Boyer-Fabrie.

#### Asymptotic modeling in electromagnetism

Participants : Marc Duruflé, Victor Péron, Clair Poignard.

We investigate asymptotic models for 3D transmission problems in electromagnetism with homogeneous thin layers (uniform thickness). We exhibit Generalized Impedance Boundary Conditions of order 1 when the thin layer is symmetric and non-symmetric with respect to its mean surface. We present also a limit model for a resistive thin layer, and an equivalent model of order 1 for large contrast in conductivities through the thin layer. We write all these models in a general form. Questions regarding the implementation of the conditions have been addressed carefully. Numerical results with the high-order finite element library Montjoie illustrate the accuracy of the asymptotic models. A paper is in preparation.

#### Absorbing Boundary Conditions for Tilted Transverse Isotropic Elastic Media

Participants : Hélène Barucq, Lionel Boillot, Henri Calandra, Julien Diaz.

The simulation of wave propagation in geophysical media is often performed in domains which are huge compared to the wavelenghts of the problem. It is then necessary to reduce the computational domain to a box. When considering acoustic or elastic isotropic media, this can be done by applying an Absorbing Boundary Condition (ABC) or by adding a Perfectly Matched Layer (PML). However, a realistic representation of the Earth subsurface must include anisotropy and, in particular, the so-called Tilted Transverse Isotropy. Perfectly Matched Layers are known to be unstable for this kind of media and, to the best of our knowledge, no ABC have been proposed yet. We have thus proposed a low-order ABC for TTI media.

This ABC has been constructed for elliptic TTI media, where the slowness curve of the P-Wave is a rotated ellipse. Then, an appropriate change of variable can be applied in order to transform this ellipse into a circle. The main idea consists in imposing the isotropic ABC in the new system of coordinates and to apply the inverse change of variable in order to obtain the elliptic TTI ABC. We have compared numerically the reflections generated by this new ABC in TTI domain to the ones generated by the classical first order ABC in isotropic domains. The results show that the new ABC performs as well as the classical first order one. Moreover, this ABC seems to be also well-suited to non elliptic TTI media. These results have been presented at the Congrès Français d'Acoustique [35] , at two workshops [39] , [45] .

#### Efficient solution methodology based on a local wave tracking strategy for high-frequency Helmholtz problems.

Participants : Mohamed Amara, Sharang Chaudhry, Julien Diaz, Rabia Djellouli, Steven Fiedler.

We have designed a new and efficient solution methodology for solving high-frequency Helmholtz problems. The proposed method is a least-squares based technique that employs variable bases of plane waves at the element level of the domain partition. A local wave tracking strategy is adopted for the selection of the basis at the regional/element level. More specifically, for each element of the mesh partition, a basis of plane waves is chosen so that one of the plane waves in the basis is oriented in the direction of the propagation of the field inside the considered element. The determination of the direction of the field inside the mesh partition is formulated as a minimization problem. Since the problem is nonlinear, we apply Newton's method to determine the minimum. The computation of Jacobians and Hessians that arise in the iterations of the Newton's method is based on the exact characterization of the Fréchet derivatives of the field with respect to the propagation directions. Such a characterization is crucial for the stability, fast convergence, and computational efficiency of the Newton algorithm. These results are part of the Master thesis of Sharang Chaudhry (student à CSUN) and have been presented to the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS, Vienna, 2012).