İki boyutlu saçılma problemi için sayısal çözüm yöntemleri

Abstract

Bu tezde düzlemsel dalga ile aydınlatılmış bir dielektrik cisimden: saçılan alanın sayısal olarak nasıl hesaplanabileceği konu edilmiştir. îlk önce saçılan alanı gelen alan cinsinden ifade eden bir integral denklem oluşturulmaktadır. Bu integral denklem üzerinde dielektrik süreksizliğin sebep olacağı etki, indüksiyon akımlarının distribüsyon anlamında yorumlanması ile berraklaştırılmaktadır. Tezde integral denklem lineer hale getirilerek moment metodu ile çözülmüştür. Ayrıca Bölüm 3 'de moment metodu ile Finite element metodunu birleştiren hybrid metodla çözüm yöntemi incelenip moment metodu ile kıyaslanmıştır. Sayısal sonuçlar bölümünde, dielektrik silindir için analitik çözümle kıyaslama suretiyle sayısal sonuçların doğruluğu gösterilip çeşitli saçıcılar için hesaplanan saçılan alan değerleri verilmiştir.

The methods of Numerical Solution for 2-D Scattering Problem The development of many biological applications in electromagnetics, such as microwave imaging and hyper thermia treatment, requires fast and accurate computati on of the fields in inhomogeneous lossy dielectrics. Analytical methods exit only for such simple geometries as cylindrical and spherical ones and are mainly used to check the accuracy of numerical solution. Fig. 1. shows an inhomogeneous lossy dielectric cy linder with arbitrarily shaped cross section Dv illumina ted by a TM polarized incident field E^. The total elec tric field is denoted by Ej. An exp(-jWt) time dependen ce is implied. *?» Eg=Eg.Uz Fig.l. Geometry of the 2-D TM Cylinder. It is well known that the scattered field is gene rated by currents induced inside the object that radiate into the homogeneous enterior region. Let us define the function k(r) at point r, proportional to induced cur rent t (r), as K(r) = ET(r).[e(r)-l] - t (r). DWe, (1) Where e (r) is the relative permittivity of the inner region respectively, ET(r) is given by Em(r) = E.(r) + (k2-grad.div)J7 G(r,r ' ) K(r ' )ds ' (2) Dv Where e(r,r*) is the free space (the exterior medium) Green's function. - vi - By applying the differential operator grad div on the Green function. We obtain the following integral ecruation E"(r) = ET(r)-k2 // G(r,r ' )K(r« )ds' 1 Dv -PV // grad,div'G(r,r').K(r')ds,+ \.K. (r) (3) Dv z Where PV represents the principal value and grad.' and div' refer to derivatives with respect to the primed coordinates. In order to preserve the convolutional form of this integral equation during numerical computation. We apply the method of moments with a pulse basis and point-matching. To do so, the object divided into N ele- menter square cells with width A. It follows that the electric field and the dielectric properties are taken as constant in each cell and that equality is enforced at the cell centers. The resulting lineer system is then E.(rn) = [Ae(rn) + l]ET(rR) 2 N -k. I Ae(ri).ET(ri). // G(r.r'Jds' ' i=l sA. N -E Ae(r,).E"(r1). // grad".div'. G(r",r' )ds ' (4) i=l ixi sA^ n where xn xi r : ( ) observation point r. : ( ) source Yn Yi point and sA^^ is the area of the elementary cell -* vii - centered at point r.. It has been suggested that better results could be obtained with an integral equation in which the char ge contribution appears explicitly. Its expression below is deduced form (2) : E. (r) - E",(r)-k2 // GdTjr'KMr'îds* DV // grad'GCrjr'Jdiv'Mr'Jds* (5) DV Here, the first integral is associated with the polarization current and the second with the charge. If we assume that the cross section Dv can be divided into M homogeneous subregions Vm, the divergence of the func tion K (r) in the second integral is nonzero only at the interfaces between subregions. This integral then redu ced to a line integral along such interfaces and (5) is given by 2 N Ei(r) = ET(r)-lc E Aem. // ET (r. ). G(r,r » ) ds ' m=l sm m M - E Aem- / ET(r') -n'.grad1 GCr^'JdA1 (6) m=l c m Where em is the relative permittivity of region Vm. The quantity n' is the outward -directed unit normal vector. sm is the surface area of region Vm, and Cm is the boundary of region Vm. By dividing each homogeneous subregion into elementary cells and using the same method of discreti zation as above, we obtain the linear system. 2 N E. (r ) = E-(rn)-kz E Ae(r,).E(r.) // G(r,r')ds' i=l sA. i N -E AelrJ.EdJ / n'grad' G(r,r ' )dV (7) i-1 cA± n - vxii - Where cA. is the boundary of the elementary cell cente red at point r^. In this t, we introduce a new integral formula tion by using a generalized function formalism. The derivatives of the discontinous function K(r) must be estimated carefully. The generalized function formalism allows us to take into account such discontinuities and to obtain an integral formulation where the jump $ in the function l/e(r) appears at any boundary c of discon tinuity. Thus div K (r) is written as div K(r) - {div K(r)> + ^-\-^c <8) Where { } means that the derivatives are taken in the function sense and