Section: New Results
Accelerated numerical solvers for largescale wave problems
Fast solution procedures are of critical importance for industrial applications such as nondestructive testing, electromagnetic compatibility testing and seismic risk assessment. In these examples, the wavelength is very small in comparison to the characteristic length of the problems, which leads to extremely expensive numerical procedures if standard methods are used. To address the fast numerical solution of largescale waves problems, we work at the same time on numerical methods, algorithmic issues and implementation strategies to speed up solvers.
Nonoverlapping Domain Decomposition Method (DDM) using nonlocal transmission operators for wave propagation problems.
Participants : Patrick Joly, Emile Parolin.
The research in this direction was mainly concerned by the extension to the electromagnetic setting of the linear convergence theory of nonoverlapping DDM that relies on nonlocal transmission operators. The principal task was to propose, analyse and implement some candidate nonlocal operators satisfying the assumptions of the theory. There were two main propositions:

Integral operator for the electromagnetic setting: the operator is available in closed form and its structure lead naturally to a localizable form via truncation of the kernel to limit the effective computational cost while retaining its good properties. The construction of such an operator turned out to be somewhat difficult due to the particular functional setting of Maxwell's equations.

DtN based nonlocal operator: the operator is computed by solving auxiliary coercive problems in the vicinity of the transmission interface. The computational cost remains moderate as the implementation no longer involve dense matrix blocks from the integral operators but rather lead to augmented sparse linear systems. Initially developed for the electromagnetic setting, the approach is appealing as it provided a unified formalism that can be applied both to Helmholtz and Maxwell equations and proved to be efficient in numerical experiments.
Another important research direction is created by the technical and theoretical difficulty posed by junction points, which are points where three or more subdomains abut. Xavier Claeys recently proposed a method to deal with this specific issue, based on the multitrace formalism, which led to a joint collaboration on the subject. The main idea is to perform a global exchange operation, on the whole skeleton, rather than a local pointtopoint exchange. The preliminary numerical results recently obtained are promising.
An efficient domain decomposition method with crosspoint treatment for Helmholtz problems
Participant : Axel Modave.
This is a collaboration with X. Antoine (IECL, Nancy), A. Royer (ULiège) and C. Geuzaine (ULiège). The parallel finiteelement solution of largescale timeharmonic scattering problems is addressed with a nonoverlapping domain decomposition method (DDM). It is well known that the efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on highorder absorbing boundary conditions (HABCs) are well suited for configurations without cross points (where more than two subdomains meet). In this work, we extend this approach to efficiently deal with cross points. Twodimensional finiteelement results are presented.
Modelling the fluidstructure coupling caused by a farfield underwater explosion using a convolution quadrature based fast boundary element method.
Participants : Marc Bonnet, Stéphanie Chaillat, Damien MavaleixMarchessoux.
This study is done in collaboration with Bruno Leblé (Naval Group). It aims at developing computational strategies for modelling the impact of a farfield underwater explosion shock wave on a structure, in deep water. An iterative fluidstructure coupling is developed to solve the problem. Two complementary methods are used: the Finite Element Method (FEM), that offers a wide range of tools to compute the structure response; and the Boundary Element Method (BEM), more suitable to deal with large surrounding fluid domains. We concentrate on developing (i) a fast transient BEM procedure and (ii) a transient FEMBEM coupling algorithm. The fast transient BEM is based on a fast multipoleaccelerated Laplacedomain BEM (implemented in the inhouse code COFFEE), extended to the time domain by the Convolution Quadrature Method (CQM). In particular, using empirical approximations for the solution of integral problems involving large (complex) frequencies has been found to yield satisfactorily accurate solutions while saving significant amounts of computational work. The transient BEMFEM coupling (under progress) will be based on a blockSOR iterative approach, for which a preliminary investigation shows the existence of relaxation parameters that ensure convergence.
Asymptotic based methods for very high frequency problems.
Participant : Eric Lunéville.
This research is developed in collaboration with Marc Lenoir and Daniel Bouche (CEA).
It has recently been realized that the combination of integral and asymptotic methods was a remarkable and necessary tool to solve scattering problems, in the case where the frequency is high and the geometry must be finely taken into account.
In order to implement the highfrequency approximations that we are developing as part of these hybrid HF/BF methods, we have introduced new geometric tools into the XLiFE++ library, in particular splines and BSplines approximations as well as parameterizations to access quantities such as curvature, curvilinear abscissa, etc. We have also started to interface the OpenCascad library to the XLiFE++ library, which will eventually allow us to manage more complex geometric situations (cylinder and sphere intersection for example). In parallel, we have completed the implementation of 2D HF approximations in the shadowlight transition zone based on the Fock function. Diffraction by a 2D corner is in progress.