Combining Analytic Preconditioner and Fast Multipole Method for the 3-D Helmholtz Equation

International audienceThe paper presents a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering the Combined Field Integral Equation (CFIE). We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. We first investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. We then apply the resolution algorithm to various and significant test-cases using a GMRES solver. The OSRC-preconditioning technique is combined to a Fast Multipole Method in order to deal with high-frequency 3-D cases. This variety of tests validates the effectiveness of the method and fully justifies the interest of such a combination

Eddy current interaction between a probe coil and a conducting plate

International audienceConsider a coil above a conducting plate. The interaction between the probe-coil and the plate is modeled by a quasi-static approximation of Maxwell's equations: the eddy current model. The associated electromagnetic transmission boundary-value problem can be solved by the integral equations method. However, the discretization of integral operators gives dense, complex and ill-conditioned linear systems. We present here a method to compute the reaction field and the coil impedance variation by solving only surface partial differential equations

Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles

An introduction to operator preconditioning for the fast iterative integral equation solution of time-harmonic scattering problems

International audienceThe aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations

Integral Equations and Iterative Schemes for Acoustic Scattering Problems

International audienceThe aim of this chapter is to provide an introduction to the iterative Krylov solution of integral equations for time-harmonic acoustic scattering. From the point of view of computational methods, considering large frequencies in acoustics is challenging since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. Most specifically, we develop the problematic and explain some partial solutions through analytical preconditioning for high frequency scattering and the introduction of new combined field integral equations

Numerical Modeling and High Speed Parallel Computing: New Perspectives for Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring

This paper deals with microwave tomography for brain stroke imaging using state-of-the-art numerical modeling and massively parallel computing. Microwave tomographic imaging requires the solution of an inverse problem based on a minimization algorithm (e.g. gradient based) with successive solutions of a direct problem such as the accurate modeling of a whole-microwave measurement system. Moreover, a sufficiently high number of unknowns is required to accurately represent the solution. As the system will be used for detecting the brain stroke (ischemic or hemorrhagic) as well as for monitoring during the treatment, running times for the reconstructions should be reasonable. The method used is based on high-order finite elements, parallel preconditioners from the Domain Decomposition method and Domain Specific Language with open source FreeFEM++ solver

Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

International audienceIt has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies