Reduced surface integral equations

Laplacian potential fields in stratified media are usually analyzed using an integral equation for an unknown function over the union of all the interfaces between regions with different homogeneous materials. In this paper, the field problem is solved using a reduced integral equation involving a single unknown function over only the boundary of the source region. The new integral equation is derived by introducing surface operators to express the potential and its normal derivative on each interface in terms of a single unknown function over the same interface. These operators and the corresponding single functions are obtained recursively, from one interface to the next. Thus, a substantial decrease in the amount of necessary numerical computation and computer memory is achieved especially for systems containing identical layered bodies where the reduction operators are only constructed for one of the bodies. The purpose of this paper is to derive reduced integral equations by directly applying the interface conditions and to show their high computational efficiency for systems of layered bodies. PACS Nos.: 02.30.Rz, 02.70.Pt, 41.20.Cv Resume: On analyse generalement Jes champs de potentiel de Latt'a..c~ dans Jes milieux stratifies a !'aide d'une equation integrale pour une fonction inconnue sur !'union de toutes !es interfaces entre Jes regions avec differents materiaux homogenes. Nous solutionnons ici le probJeme de champ en UtiJisant UJl~ esuation integraJe reduite impJiquant une seuJe fonction inconnue sur la surface de 1r.~atifce. La nouvelle equation integrale est obtenue ,, en introduisant des operateurs de surface pour ecrire le potentiel et sa derivee normale sur chaque interface en terme d'une seule fonction inconnue sur la meme interface. Ces operateurs et la fonction inconnue sont obtenus de fa\:on recursive, d'une surface a l'autre. Nous reduisons ainsi de fa\:on significative Jes besoins en calcul numerique et en memoire, specialement pour Jes systemes qui contiennent des couches identiques oil Jes operateurs de reduction sont construits pour une seule couche. Notre but iciest d'obtenir des equations integrales reduites en appliquant directement !es conditions lirnites aux interfaces et de demontrer la haute efficacite de la methode pour !es systemes stratifies. [Traduit par la Redaction] Received 'Ntl'f\lmf,>!•' 200f. Accepted 17 January 2007. Published on the NRC Research Press Web site at http://cjp.nrc.ca/ on 19 February 2007. I.R. Ciric. Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail: [email protected]). Can. J. Phys. 84: 1049-1061 (2006) doi: 10.1139/P07-008 © 2006 NRC Canada 1050 Can. J. Phys. Vol. 84, 2006