Method of Boundary Integral Equations in the Problem of Diffraction of a Monochromatic Electromagnetic Wave by a System
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GRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
Method of Boundary Integral Equations in the Problem of Diffraction of a Monochromatic Electromagnetic Wave by a System of Perfectly Conducting and Piecewise Homogeneous Dielectric Objects E. V. Zakharov1∗ and A. V. Setukha1,2∗∗ 1
2
Lomonosov Moscow State University, Moscow, 119991 Russia Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333 Russia e-mail: ∗ [email protected], ∗∗ [email protected] Received March 15, 2020; revised March 15, 2020; accepted May 14, 2020
Abstract—We consider the three-dimensional problem of diffraction of a monochromatic electromagnetic wave by a system of objects of various physical nature, including dielectric bodies (domains), perfectly conducting bodies, and perfectly conducting screens. The perfectly conducting bodies may be placed in an exterior medium or immersed into the dielectric domains. In addition, the perfectly conducting screens may reside at the interface of the dielectric domains, being part of each of those. We give a boundary value problem for the Maxwell equations that describes the electromagnetic field under consideration. Integral representations are derived for the electromagnetic field in terms of surface integrals, and the boundary value problem is reduced to a system of boundary integral equations containing weakly and strongly singular surface integrals. The integral equations are written on the dielectric and perfectly conducting parts of the interface between the dielectric domains, on the surfaces of the perfectly conducting bodies, and on the perfectly conducting screens. DOI: 10.1134/S0012266120090062
INTRODUCTION The boundary integral equation method is widely used in problems of diffraction of monochromatic electromagnetic waves both when theoretically exploring the solvability of these problems and when constructing efficient numerical methods of their solution. In the latter case, an important advantage of this method is that the computational mesh is constructed only on surfaces that are the exterior boundaries of bodies and the interfaces between parts of bodies with different dielectric properties. In this case, the electrodynamic equations outside the bodies and conditions at infinity are satisfied automatically. The classical problem of diffraction of a monochromatic wave by a perfectly conducting body bounded by a closed surface is well known to be reducible to a Fredholm integral equation of the second kind, which can be used both to prove the unique solvability of the problem and to construct numerical methods for its solution [1, 2]. At the same time, the problems of diffraction of electromagnetic waves by thin perfectly conducting screens cannot be reduced to equations of the kind, because equations to which these problems are reducible must ensure that the boundary condition is satisfied on both sides of the surface. In the mid-twentieth century, Maue [1, pp. 115–116 of the Russian translation] reduced such problems to a boundary integral equation of the first ki
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