## . Electromagnetic Scattering Problems

The most challenging electromagnetic scattering problems are found in stealth related applications. A common quantity of interest is the monostatic radar cross section (RCS) expressing the power flow density carried away by the scattered field from the scatterer relative to the power flow density carried by an illuminating incident field of given direction and wave length λ. The most challenging problems include non-smooth geometries, such as the fan blades of a jet engine and local very thin dissipative material coatings. Consider that a typical channel length is O(100 λ) and a typical cross section diameter is O(10 λ). Predicting the RCS in several hundred monostatic directions for a real jet engine air intake or exhaust outlet poses a huge computational challenge. Using a minimum of ten standard linear hexahedrons per wavelength, we face meshes with 107 elements for wave resolution, the resolution of the geometry and material discontinuities not taken into account. The hp-adaptive finite element method is undoubtedly a competitive candidate. For channels with simple geometry and perfect conducting walls, waveguide-based modematching methods have proven to be very cost efficient. We should remember that surprisingly accurate RCS predictions have been reported using these simple generalized scattering matrix type of models. The illuminating field is projected on a set of inward traveling waves and an expression for the outgoing waves in the same cross section is constructed. Kirchhoff’s classical aperture integration formula is then used to obtain the far-field from which the RCS is determined. However, this method falls short if dissipative materials are present and the channel has a varying cross section and a complicated termination. Note also that the mode-matching method essentially is a bounded domain method. The exterior problem is decoupled and handled by Kirchhoff’s method. The approach presented in this chapter is far more sophisticated and general.