Iteration Method in the Theory of Nonhomogeneous Dielectrics

Abstract

The problem of finding the relationship between an unknown field E and the field E,, which is the solution of a similar boundary value problem for an auxiliary medium,is solved for anarbitrary (in the sense of the distribution of the nonhomogeneities and their dielectric properties) nonhomogeneous medium of volume V. The Poisson equation for the nonhomogeneous medium corresponds to the integral equation for the field E . Methods of functional analysis ere used to show that the solution of the equation can be represented in the form of the Neumann series, which in general satisfies the conditions of uniform convergence. Certain limitations on the dielectric permittivities of the auxiliary medium are derived as sufficient conditions for the convergence of the series. ,&Wi IlpOH3BOJIbHOfi (B CMbICJIe PaCnpeHeneHMR HeOnHOpOHHOCTefi M UX HM3neKTpMseCHMX CBOmCTB) HeOHHOpOAHOfi CpeHbl o6.be~a v PeUIeHa 3maW 0 HaXOmAeHMIl CBRBH MeXAy HeM3BeCTHbIM AOJIeM E M lTOJIeM E,, HBJIRIOILWMCR PeUIeHEleM MAeHTUsHOfi KpaeBOfi 3aHaqM EJIH BCnOMOraTeJIbHOfi CpeAbI. YpaBHeHMIO nyaCCOHa AJIH HeOnHOpOAHOfi CpeAbI nOCTaBJIeHO B COOTBeTCTBMe MHTeI' paJIbHOe YpaBHeHMe AJIR IIOJIR E . rIpM IIOMOLUEl anna-paTa (PyHKuHoHamHoro a~a n m a n o~a 3 a~0 , s-ro peweHne 3~0 r o ypaBHemrr npencTa-BMMO B (PopMe pRna HefiMaHa, AJIH KoToporo B 061uew cnysae BH~OJIHRIOTCR YCJIOBMR no;lysemx orpamsemfi, HaKnanbIsaeMMe Ha guanempmecme npomuaeMocm BCAO-paBHOMepHOfi CXOHMMOCTM. B Ka4eCTBe AOCTaTOsHbIX yCJIOBIlfi CXOAMMOCTM 3 T O r 0 PRna MOraTeJIbJlHOfi CpeAbI.

Electromagnetic Wave Scattering in Inhomogeneous Media with Spatial Dispersion

“…8) Using the coordinate dependence of the field ( E ) in the form (1.6), let us go from the operator equation (2.8) in r-space to the tensor equation [lo] ; * 73 --( E i .…”

“…Similar conditions for a stationary problem were obtained in [8]. I n addition, the choice of the parameter e, is determined by considerations of convenience when solving the dispersion equation (1.8).…”

A full-field image conversion method for the inverse conductivity problem with internal measurements

“…. , d, can be assumed to be small, in a sense discussed later on, then the featured integral operator can be inverted using a Neumann series [24,28]. Therefore, the unknown conductivity map is approximated by the following series expansion:…”

“…While our overarching goal is the treatment of fullfield data pertaining to solid mechanics, the proposed method is discussed here in the context of the scalar conductivity model which has the same mathematical structure as the elasticity model yet avoids resorting to tensorial notations. Note that a Fourier-based forward solution method corresponding to this model is discussed in [24,25]. In [26, ch.…”

Superfunctions and the algebra of subspace collections and their association with rational functions of several complex variables

“…, P n where P i consists of those square integrable fields which are nonzero only within component i. This formulation, in terms of a Z(n) subspace collection, evolved out of earlier Hilbert space formulations of the problem (Fokin 1982;Kohler and Papanicolaou 1982;Papanicolaou and Varadhan 1982;Golden and Papanicolaou 1983;Kantor and Bergman 1984;Dell'Antonio, Figari, and Orlandi 1986) and can easily be extended to the elastic, thermoelastic, piezoelectric, and poroelastic equations of multiphase and polycrystalline materials (see, for example, Chapter 12 in Milton (2002)). The formulation has proved to be particularly important in the theory of exact relations of composite materials (Grabovsky 1998;Sage 1998: Grabovsky andMilton 1998;Grabovsky, Milton, and Sage 2000;Grabovsky 2004) (see also Chapter 17 in Milton (2002)) where one seeks microstructure independent relations satisfied by effective tensors.…”