**Numerical Methods in Electromagnetism**

40 Will-be-set-by-IN-TECH

152 Trends in Electromagnetism – From Fundamentals to Applications

The financial support of the JINR Dubna - IFIN-HH Magurele-Bucharest project no.

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**4. Acknowledgment**

**5. References**

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1984).

**1. Introduction**

to use approximate numerical techniques.

An accurate numerical solution of Electromagnetic scattering problems is critically demanded in the simulation of many real-life applications, such as in the design of industrial processes and in the study of wave propagation phenomena. Electromagnetic (EM) scattering problems address the physical issue of computing the diffraction pattern of the EM radiation that is propagated by a complex body, illuminated by an incident wave. An explicit solution is possible only for simple targets, e.g. for spherical bodies; complicated geometries impose

**Fast Preconditioned Krylov Methods for** 

*University of Groningen, Institute of Mathematics and Computing Science,* 

**Boundary Integral Equations** 

**in Electromagnetic Scattering** 

**7**

Bruno Carpentieri

*Groningen The Netherlands* 

Until the emergence of high-performance computing in the early eighties, the analysis of scattering problems was afforded by using approximate high frequency techniques such as the shooting and bouncing ray method (SBR) (Lee et al. (1988)). Ray-based asymptotic methods like SBR and the uniform theory of diffraction are based on the idea that EM scattering becomes a localized phenomenon as the size of the scatterer increases with respect to the wavelength. In the last decades, due to impressive advances in computer technology and the introduction of innovative algorithms with limited computational and memory requirement, a more rigorous numerical solution has become possible for many practical applications.

Finite-difference (FD) (Kunz & Luebbers (1993); Taflove (1995)), finite-element (FE) (Silvester & Ferrari (1990); Volakis et al. (1998)) and finite-volume (FV) methods (Bonnet et al. (1998); Botha (2006)) can be used to discretize the Maxwell's equations into a finite volume surrounding the scatterer, giving rise to sparse systems of linear equations. Upon inversion of the system, a solution is computed for all excitations. More recently, alternative approaches based on integral equations are becoming increasingly popular for solving high-frequency EM scattering problems. They reformulate the Maxwell's equations in the frequency domain and solve for the electric and the magnetic currents induced on the surface of the object. Thus integral methods require only a simple description of the surface of the target by means of triangular facets (see an example of discretization in Figure 1). This means that a 3D problem is reduced to solving a 2D surface problem, simplifying considerably the mesh generation especially in the case of moving objects. No artificial boundaries need to be imposed and boundary conditions are automatically satisfied in the case of perfectly conducting objects. Another interest to use surface discretizations is that they noticeably reduce the effect of grid dispersion errors. Grid dispersion errors occur when a wave has a different phase velocity on
