Bertrand Thierry

A quasi-optimal domain decomposition algorithm for the time-harmonic Maxwell's equations

This paper presents a new non-overlapping domain decomposition method for the time harmonic Maxwells equations, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Magnetic-to-Electric operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented.

Wide Frequency Band Numerical Approaches for Multiple Scattering Problems by Disks

Efficient, robust and accurate algorithms are proposed for solving the multiple scattering problem by M circular obstacles for the whole spectrum of frequency. The representation of the solution is based on an integral equation formulation next solved by using Fourier basis. Numerical examples are provided to show that the approaches are efficient.

Far Field Modeling of Electromagnetic Time Reversal and Application to Selective Focusing on Small Scatterers

A time harmonic far field model for closed electromagnetic time reversal mirrors is proposed. Then, a limit model corresponding to small perfectly conducting scatterers is derived. This asymptotic model is used to prove the selective focusing properties of the time reversal operator. In particular, a mathematical justification of the decomposition of the time reversal operator (DORT) method is given for axially symmetric scatterers.

μ-diff: An open-source Matlab toolbox for computing multiple scattering problems by disks

The aim of this paper is to describe a Matlab toolbox, called µ-diff, for modeling and numerically solving two-dimensional complex multiple scattering by a large collection of circular cylinders. The approximation methods in µ-diff are based on the Fourier series expansions of the four basic integral operators arising in scattering theory. Based on these expressions, an efficient spectrally accurate finite-dimensional solution of multiple scattering problems can be simply obtained for complex media even when many scatterers are considered as well as large frequencies. The solution of the global linear system to solve can use either direct solvers or preconditioned iterative Krylov subspace solvers for block Toeplitz matrices. Based on this approach, this paper explains how the code is built and organized. Some complete numerical examples of applications (direct and inverse scattering) are provided to show that µ-diff is a flexible, efficient and robust toolbox for solving some complex multiple scattering problems.

GetDDM: An open framework for testing optimized Schwarz methods for time-harmonic wave problems☆

We present an open finite element framework, called GetDDM, for testing optimized Schwarz domain decomposition techniques for time-harmonic wave problems. After a review of Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples for Helmholtz and Maxwell’s equations, is freely available online for further testing.

An open source domain decomposition solver for time-harmonic electromagnetic wave problems

We present a flexible finite element solver for testing optimized Schwarz domain decomposition techniques for the time-harmonic Maxwell equations. After a review of non-overlapping Schwarz domain decomposition methods and associated transmission conditions, we discuss the implementation, based on the open source software GetDP and Gmsh. The solver, along with ready-to-use examples, is available online for further testing.

A remark on the single scattering preconditioner applied to boundary integral equations

This article deals with boundary integral equation preconditioning for the multiple scattering problem. The focus is put on the single scattering preconditioner, corresponding to the diagonal part of the integral operator, for which two results are proved. Indeed, after applying this geometric preconditioner, it appears that, firstly, every direct integral equations become identical to each other, and secondly, that the indirect integral equation of Brakhage-Werner becomes equal to the direct integral equations, up to a change of basis. These properties imply in particular that the convergence rate of a Krylov subspaces solver will be exactly the same for every preconditioned integral equations. To illustrate this, some numerical simulations are provided at the end of the paper.

Spectral and condition number estimates of the acoustic single-layer operator for low-frequency multiple scattering in dense media

The aim of this paper is to derive spectral and condition number estimates of the single-layer operator for low-frequency multiple scattering problems. This work extends to dense media the analysis initiated in Antoine and Thierry (in press) [19]. Estimates are obtained first in the case of circular cylinders by Fourier analysis and are next formally adapted to disks, ellipses and rectangles in the framework of boundary element methods. Numerical simulations validating the approach are also given.

µ-diff: A matlab toolbox for multiple scattering problems by disks

The aim of this paper is to describe a Matlab toolbox, called µ-diff, which is designed for spectrally solving two-dimensional complex multiple scattering by a large collection of circular cylinders. The formulation is based on the Fourier series expansions of the four basic integral operators arising in scattering problems. Based on these expressions, a spectrally accurate finite-dimensional solution of multiple scattering problems can be simply obtained for complex media even when many scatterers are considered as well as large frequencies. The efficient solution of the final linear system to solve makes use of preconditioned Krylov subspace solvers for block Toeplitz matrices.

Numerical study of a spectral approach to solve integral equation formulation of the high‐frequency acoustic multiple scattering problem by disks.

This aim of this talk is to propose a new numerical method for solving the multiple scattering problem of an acoustic wave by circular cylinders. The situation where the wavelength is small compared to the characteristic size of the cylinders is analyzed in detail here. It is known that this framework implies specific numerical difficulties since it leads to the numerical solution of a large size linear system. The proposed approach is based on computing the scattered field as the solution of an integral equation. Writing this integral equation in a Fourier basis and using a suitable truncation of these Fourier series and a preconditioned Krylov iterative solver (GMRES), an efficient and robust numerical procedure is built. Computational simulations will be provided for solving high frequency multiple scattering problems by random configurations of circular obstacles.