Modified Combined Field Integral Equations for Electromagnetic Scattering

Steinbach, Olaf; Windisch, Markus

doi:10.1137/070698063

Cited by 18 publications

(14 citation statements)

References 27 publications

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“…This problem can be overcome by using regularized CFIE that rely on compact operators which map between Dirichlet and Neumann traces. This was first employed for theoretical investigations [28] and, more recently, used for the design of stable Galerkin boundary element methods [5,3,22,23,35,2]. Our approach is inspired by [3].…”

Section: Single-trace Combined Field Integral Equationmentioning

confidence: 99%

Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Claeys

Hiptmair

2014

Integr. Equ. Oper. Theory

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We study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions. Following rA. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer. Math., 100 (2005), pp. 1-19s we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of rX. Claeys and R. Hiptmair, Multi-trace boundary integral formulation for acoustic scattering by composite structures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 1163-1201s to situations with Dirichlet boundary conditions.

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Section: Single-trace Combined Field Integral Equationmentioning

confidence: 99%

Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Claeys

Hiptmair

2014

Integr. Equ. Oper. Theory

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“…The main difficulty stems from the lack of compactness of the magnetic field integral operators (sometimes referred to as electromagnetic double layer operators) in the case of Lipschitz boundaries. Alternative boundary integral equations [41,57] use regularizers that act precisely on the magnetic field integral operators and lead to formulations whose operators are compact perturbations of coercive operators. The latter property ensures the well posedness of the aforementioned regularized boundary integral equations in Lipschitz domains.…”

Section: Introductionmentioning

confidence: 99%

Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

Boubendir

Turc

2014

Computers & Mathematics with Applications

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We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations for the solution of three dimensional frequency domain electromagnetic scattering problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined magnetic and electric dipole layer potentials that involves a composition of the latter type of boundary layers with regularizing operators. The regularizing operators are of two types: (1) modified versions of electric field integral operators with complex wavenumbers, and (2) principal symbols of those operators in the sense of pseudodifferential operators. We show that the boundary integral operators that enter these CFIER formulations are Fredholm of the second kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering problems. We present a spectral analysis of CFIER operators with regularizing operators that have purely imaginary wavenumbers for spherical geometries-we refer to these operators as Calderón-Ikawa CFIER. Under certain assumptions on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing operators, we show that the ensuing Calderón-Ikawa CFIER operators are coercive for spherical geometries. These properties allow us to derive wavenumber explicit bounds on the condition numbers of Calderón-Ikawa CFIER operators. When regularizing operators with complex wavenumbers with non-zero real parts are used-we refer to these operators as Calderón-Complex CFIER, we show numerical evidence that those complex wavenumbers can be selected in a manner that leads to CFIER formulations whose condition numbers can be bounded independently of frequency for spherical geometries. In addition, the Calderón-Complex CFIER operators possess excellent spectral properties in the high-frequency regime for both convex and non-convex scatterers. We provide numerical evidence that our solvers based on fast, high-order Nyström discretization of these equations converge in very small numbers of GMRES iterations, and the iteration counts are virtually independent of frequency for several smooth scatterers with slowly varying curvatures.propagation of waves across large numbers of volumetric elements [9]. An important computational alternative to finite-difference and finite-element approaches is found in boundary integral methods. Numerical methods based on integral formulations of scattering problems enjoy a number of attractive properties as they formulate the problems on lower-dimensional, bounded computational domains and capture intrinsically the outgoing character of scattered waves. Thus, on account of the dimensional reduction and associated small discretizations (significantly smaller than the discretizations required by volumetric finite-element or finite-difference approximations), in conjunction with available fast solvers [12, 15-17, 19, 20, 26, 27, 50, 51, 55], numerical algor...

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“…A similar approach has been proposed in [26], but only for the acoustic case. A related technique, the boundary element tearing and interconnecting (BETI) method (a boundary element counterpart of the FETI method) has been developed by Steinbach et al for strongly elliptic problems [28,30,36]. Its extension to Maxwell's equations was pursued by Windisch in his Ph.D. thesis [39], Chapter 8, but effective preconditioning for this formulation remains open.…”

Section: Introductionmentioning

confidence: 99%

Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

Claeys

Hiptmair

2012

ESAIM: M2AN

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Abstract.Since matrix compression has paved the way for discretizing the boundary integral equation formulations of electromagnetics scattering on very fine meshes, preconditioners for the resulting linear systems have become key to efficient simulations. Operator preconditioning based on Calderón identities has proved to be a powerful device for devising preconditioners. However, this is not possible for the usual first-kind boundary formulations for electromagnetic scattering at general penetrable composite obstacles. We propose a new first-kind boundary integral equation formulation following the reasoning employed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] for acoustic scattering. We call it multi-trace formulation, because its unknowns are two pairs of traces on interfaces in the interior of the scatterer. We give a comprehensive analysis culminating in a proof of coercivity, and uniqueness and existence of solution. We establish a Calderón identity for the multi-trace formulation, which forms the foundation for operator preconditioning in the case of conforming Galerkin boundary element discretization.Mathematics Subject Classification. 78A45, 35A35, 35Q60, 78M15.

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Modified Combined Field Integral Equations for Electromagnetic Scattering

Cited by 18 publications

References 27 publications

Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects

Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems

Electromagnetic scattering at composite objects : a novel multi-trace boundary integral formulation

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