Section: New Results

Integral equations

Fast solution of the BEM system in 3-D frequency-domain elastodynamics

Participants : Stéphanie Chaillat, Patrick Ciarlet, Luca Desiderio.

The main advantage of the Boundary Element Method (BEM) is that only the domain boundaries are discretized leading to a drastic reduction of the total number of degrees of freedom. In traditional BE implementation the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix. Using the $\mathscr{H}$-matrix arithmetic and low-rank approximations (performed with Adaptive Cross Approximation), we derive a fast direct solver for the BEM system in 3-D frequency-domain elastodynamics. We assess the numerical efficiency and accuracy on the basis of numerical results obtained for problems having known solutions. In particular, we study the efficiency of low-rank approximations when the frequency is increased. The efficiency of the method is also illustrated to study seismic wave propagation in 3-D domains. This is done in partnership with SHELL company in the framework of the PhD of Luca Desiderio.

OSRC preconditioner for 3D elastodynamics

Participant : Stéphanie Chaillat.

This work is done in collaboration with Marion Darbas from University of Picardie and Frédérique Le Louer from Technological University of Compiègne. The fast multipole accelerated boundary element method (FM-BEM) is a possible approach to deal with scattering problems of time-harmonic elastic waves by a three-dimensional rigid obstacle. In 3D elastodynamics, the FM-BEM has been shown to be efficient with solution times of order $O(NlogN)$ per iteration (where $N$ is the number of BE degrees of freedom). However, the number of iterations in GMRES can significantly hinder the overall efficiency of the FM-BEM. To reduce the number of iterations, we propose a clever integral representation of the scattered field which naturally incorporates a regularizing operator. When considering Dirichlet boundary value problems, the regularizing operator is a high-frequency approximation to the Dirichlet-to-Neumann operator, and is constructed in the framework of the On-Surface Radiation Condition (OSRC) method. This OSRC-like preconditioner is successfully applied to Dirichlet exterior problems in 3D elastodynamics.

Boundary Integral Formulations for Modeling Eddy Current Testing

Participants : Marc Bonnet, Audrey Vigneron.

This work was a part of the PhD thesis of Audrey Vigneron, and has been done in collaboration with Edouard Demaldent from CEA-List. It concerns the simulation of eddy current non-destructive testing, which aims to assess the presence of defects (cut, corrosion ...) in a conductive, and possibly magnetic, medium. We propose a simple block-SOR solution method for the PMCHWT-type Maxwell integral formulation, that is well suited for the low-frequency, high-conductivity limit typical of eddy current testing methods. We also derive an asymptotic expansion of the Maxwell integral formulation in powers of some relevant (small) non-dimensional number and show its relation to Hiptmair's eddy current integral formulation. Both aspects are validated on 3D numerical experiments.