**6. References**

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170 Trends in Electromagnetism – From Fundamentals to Applications

is essential to preserve the numerical stability of the solver, as can be observed comparing

Density of *S* = 2% threshold Density of *L* GMRES(30) GMRES(50) 1.0e-3 0.30 29 29

Density of *S* = 4% *MILU* Density of *L* GMRES(30) GMRES(50) 1.0e-3 0.39 26 26

Density of *S* = 6% *MILU* Density of *L* GMRES(30) GMRES(50) 1.0e-3 0.46 24 24

Table 6. Number of iterations of GMRES using a multilevel inverse-based ILU factorization

We have discussed some fast iterative solution techniques for solving surface boundary integral equations. High-frequency simulations of large structures are extremely demanding for scalable solvers and large computing resources. We have reviewed recent advances for the class of Krylov subspace methods, sparse approximate inverses, incomplete *LU* factorizations. Other approach have been applied in this area of research. Multigrid methods are provably optimal algorithms for solving various classes of partial differential equations. Attempts to apply these techniques to dense linear systems have obtained mixed success. Early experiments on boundary element equations are reported with geometric versions on simple model problems, typically the hypersingular and single-layer potential integral operators arising from the Laplace equation (Bramble et al. (1994); Petersdorff & Stephan (1992); Rjasanow (1987)). Multigrids require a hierarchy of nested meshes to setup the principal components of the algorithm, *i.e.* a coarsening strategy to decrease the number of unknowns, grid transfer operators to move from a grid to another one, coarse grid operators and smoothing procedure, see e.g. Hackbusch (1985). Thus they are difficult to implement. On the other hand, algebraic multigrid algorithms use only single grid information extracted from either the graph or the entries of the coefficient matrix and are nearly as effective as geometric algorithms in reducing the number of iterations, see e.g Braess (1995); Brandt (1999); Ruge & Stüben (1987); Vanek et al. (1996). Langer et al. propose to apply an auxiliary sparse matrix reflecting the local topology of the mesh on the fine grid to setup all the components of the multigrid algorithm in a purely algebraic setting (Langer et al. (2003)). This *gray-box* approach is fairly robust on model problems and maintains the algorithmic and memory complexity of the M-V product operation (Langer & Pusch (2005)), thus it is well suited to be combined with

as preconditioner. The model problem is the same as in Table 5.

MLFMA. See also Carpentieri et al. (2007) for another multigrid-type solver.

Preconditioners based on wavelet techniques are also receiving interest. The wavelet compression of integral operators with smooth kernels yields nearly sparse matrices with at most <sup>O</sup>(*<sup>n</sup>* log*<sup>a</sup> <sup>n</sup>*) nonzero entries, where *<sup>a</sup>* is a small constant that depends on the operator and the wavelet used, see e.g. earlier work by Beylkin et al. (1991); Dahmen et al. (1993); Harbrecht & Schneider (2004); Hawkins et al. (2007); Lage & Schwab (1999). The compressed

results in Table 5 and Table 6.

**5. Concluding remarks**


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**0**

**8**

*France*

**Time Reversal for Electromagnetism:**

<sup>2</sup>*CNRS, UMR 6602, LASMEA, F-63177, Aubière*

Ibrahim El Baba1,2, Sébastien Lalléchère1,2 and Pierre Bonnet1,2

**Applications in Electromagnetic Compatibility**

<sup>1</sup>*Clermont University, Blaise Pascal University, BP 10448, F-63000, Clermont-Ferrand*

ElectroMagnetic Compatibility (EMC) is the branch of electromagnetism that studies generation, propagation and reception of involuntary electromagnetic energy in reference to the undesirable effect (electromagnetic interference) that this energy can induce. Since 1996, date of the directive 89/336/CEE (Directive 89/336/CEE, 1989) compulsory implementation concerning the electromagnetic compatibility (called CE) in Europe, and for much longer in

Most electrical and electronic equipment may be considered as sources of interference because it generates electromagnetic perturbations that pollute the environment and may disrupt the operation of other equipment (victims). The EMC is the ability of a device, equipment or system to operate satisfactorily within its electromagnetic environment and without producing itself an intolerable electromagnetic disturbance to anything in this environment. EMC hence controls the electromagnetic environment of the electronic equipment. To this end, EMC tackles several issues. Firstly, are the emission problems related to the generation of unwanted electromagnetic energy from a source and the measures that should be taken to reduce the generation of such disturbances and to prevent the escape of any remaining energy to the external environment. To verify that the perturbation level does not exceed a threshold value defined by standards, we measure the electric and/or magnetic fields radiated at a certain distance in the case of electromagnetic emissions, the voltage and/or current in the case of conducted disturbances. Secondly the susceptibility problems refer to the proper functioning of electrical equipment in presence of unplanned electromagnetic field. In the tests, we inject perturbation (conducting/radiating mode) on a device and check its good operation. Thirdly, for interference/noise disturbances, the EMC solutions are mainly obtained by addressing both the emissions and the vulnerability problems. This means minimizing the interference source levels and hardening the potential victims (shielding for

For measurements, EMC provides as test facilities different tools, the most popular are: the Anechoic Chamber (AC) (Emerson, 1973) and the Mode Stirred Reverberation Chamber (MSRC) (Corona et al., 2002; Hill, 1998). The AC is a cavity whose aim is to simulate the free space. Its walls are covered with ferrite tiles and/or polyurethane pyramids loaded with carbon absorbing electromagnetic waves and preventing their reflection. The second tool has grown in popularity over the past twenty years due to its ability to provide a

United States, EMC has been playing an increasingly important role.

**1. Introduction**

example).

