Electric Field Integral Equation Operator Research Articles

A hierarchical preconditioner for the electric field integral equation on unstructured meshes based on primal and dual Haar bases

A new hierarchical basis preconditioner for the electric field integral equation (EFIE) operator is introduced. In contrast to existing hierarchical basis preconditioners, it works on arbitrary meshes and preconditions both the vector and the scalar potential within the EFIE operator. This is obtained by taking into account that the vector and the scalar potential discretized with loop-star basis functions are related to the hypersingular and the single layer operator (i.e., the well known integral operators from acoustics). For the single layer operator discretized with piecewise constant functions, a hierarchical preconditioner can easily be constructed. Thus the strategy we propose in this work for preconditioning the EFIE is the transformation of the scalar and the vector potential into operators equivalent to the single layer operator and to its inverse. More specifically, when the scalar potential is discretized with star functions as source and testing functions, the resulting matrix is a single layer operator discretized with piecewise constant functions and multiplied left and right with two additional graph Laplacian matrices. By inverting these graph Laplacian matrices, the discretized single layer operator is obtained, which can be preconditioned with the hierarchical basis. Dually, when the vector potential is discretized with loop functions, the resulting matrix can be interpreted as a hypersingular operator discretized with piecewise linear functions. By leveraging on a scalar Calderón identity, we can interpret this operator as spectrally equivalent to the inverse single layer operator. Then we use a linear-in-complexity, closed-form inverse of the dual hierarchical basis to precondition the hypersingular operator. The numerical results show the effectiveness of the proposed preconditioner and the practical impact of theoretical developments in real case scenarios.

Fast Analysis of Electromagnetic Scattering From Conducting Objects Buried Under a Lossy Ground

Electromagnetic scattering from conducting objects is investigated for the applications of underground detection. The air–soil composite is modeled by a classical half space where the dyadic Green's function can be defined and formulated. The electric field integral equation (EFIE) is employed to guarantee the accuracy and robustness of the analysis for arbitrarily shaped scatterers. The internal resonance of EFIE is first studied under typical working conditions (frequencies, soil water contents, etc.). It is shown that this spurious resonance is much alleviated due to loss of the soil. Next, the unbounded and ill-posed spectrum of the EFIE operator is remedied by a novel localized half-space Calderon preconditioner by further exploring the lossy nature of the background. Such preconditioning is extremely useful for objects containing multiscale features. Finally, the half-space multilevel fast multipole algorithm based on real-image approximation is adopted to accelerate the computation for electrically large scatterers. Several examples in the applications of subsurface sensing are demonstrated to validate the efficiency and accuracy of this method.

Sparse tensor edge elements

We consider the tensorized operator for the Maxwell cavity source problem in frequency domain. Such formulations occur when computing statistical moments of the fields under a stochastic volume excitation. We establish a discrete inf-sup condition for its Ritz-Galerkin discretization on sparse tensor product edge element spaces built on nested sequences of meshes. Our main tool is a generalization of the edge element Fortin projector to a tensor product setting. The techniques extend to the surface boundary edge element discretization of tensorized electric field integral equation operators.

Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole Algorithms

Calderon preconditioners have recently been demonstrated to be very successful in stabilizing the electric field integral equation (EFIE) for perfect electric conductors at lower frequencies. Previous authors have shown that, by using a dense matrix preconditioner based on the Calderon identities, the low frequency instability is removed while still maintaining the inherent accuracy of the EFIE. It was also demonstrated that the spectral properties of the Caldero-n preconditioner are conserved during discretization if the EFIE operator is discretized with Rao-Wilton-Glisson expansion functions and the preconditioner with Buffa-Christiansen expansion functions. In this article we will show how the Calderon multiplicative preconditioner (CMP) can be combined with fast multipole methods to accelerate the numerical solution, leading to an overall complexity of O(N logN) for the entire iterative solution. At low frequencies, where the CMP is most useful, the traditional multilevel fast multipole algorithm (MLFMA) is unstable and we apply the nondirectional stable plane wave MLFMA (NSPWMLFMA) that resolves the low frequency breakdown of the MLFMA. The combined algorithm will be called the CMP-NSPWMLFMA. Applying the CMP-NSPWMLFMA at open surfaces or very low frequencies leads to certain problems, which will be discussed in this article.

Nullspaces of MFIE and CalderÓn Preconditioned EFIE Operators Applied to Toroidal Surfaces

Magnetic field integral equation (MFIE) and Calderon preconditioned electric field integral equation (EFIE) operators applied to toroidal surfaces have nontrivial nullspaces in the static limit. The nature of these nullspaces is elucidated and a technique for generating a basis for them presented. In addition, the effects of these nullspaces on the numerical solution of both frequency and time-domain MFIE and CalderOacuten preconditioned EFIEs are investigated. The theoretical analysis is accompanied by corroborating numerical examples that show how these operators' nullspaces affect real-world problems.

A Numerical Implementation of a Modified Form of the Electric Field Integral Equation

The details of a Galerkin discretization scheme for a modified form of the electric field integral equation are outlined for smooth, three-dimensional, perfectly conducting scatterers. Limitations of the divergence conforming finite-element bases in preserving the self-stabilizing properties of the electric field integral equation operator are indicated. A numerically efficient alternative is outlined which relies on an operator-based Helmholtz decomposition. The condition number of the resulting matrix equation is demonstrated to be frequency independent for scattering from a perfectly conducting sphere at various frequencies.

Electromagnetic Computation Using Parametric Geometry

ABSTRACT Electromagnetic formulations are presented that fully utilize the curvilinear geometry information available in parametric geometry codes. These parametric geometry codes (surface patch codes) are in wide use and the formulations presented can be used with any of these geometry codes Independent of the differences between versions. The ultimate goal is a three dimensional moment: method surface formulation enabling computation for bodies composed of metallic and nonmetallic portions. This goal requires that both the magnetic field integral equation operator and the electric field integral equation operator be encoded. To facilitate the presentation of the parametric geometry formulation of these operators, the simpler parametric geometry formulation for the physical optics integral is first presented. Additional background material based on two dimensional analysis is presented to address issues of faceting versus curvilinear and basis function choice.

On the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane

It is demonstrated by specific examples that eigenvalues of the electric field integral equation (EFIE) operator for finite extent objects in lossless media can have branch points in the complex frequency s -plane. Specifically, it is shown from the analysis of the spheroidal wave equation in the s -plane that branch points are present in the eigenvalues of the integral operator associated with the scalar scattering problem from a perfectly conducting spheroid. Similarly, it is concluded from the analysis of the Mathieu's differential equation that branch points, besides that associated with the infinite extent of the object, exist in the s -plane behavior of the eigenvalues of the EFIE operator for a perfectly conducting infinite elliptic cylinder. A proof is given that branch point singularities cannot occur in the eigenvalues of EFIE operators for structures in which geometrical symmetry completely determines the eigenfunctions (and hence they are frequency independent). It is conjectured that branch points may always be present when sufficient object symmetry is lacking. This conjecture is supported by the fact that branch points appear when a sphere is deformed into a spheroid or when a circular cylinder is deformed into an elliptic one. An analogous phenomenon has been observed in circuit and transmission line problems. For example, it is shown that when the taper parameter of an exponential transmission line goes to zero (the line becomes uniform and thus a symmetrical structure), the branch points in the eigenvalues of the impedance matrix move away to infinity.

LARGE FREQUENCY ASYMPTOTIC PROPERTIES OF RESOLVENT KERNELS

ABSTRACT ABSTRACT A conjecture on the large complex frequency asymptotic behaviour of the resolvent kernel of the electric field integral equation operator is presented. The conjecture is based on a detailed examination of the corresponding large frequency behaviour of a matrix approximant to the operator. From this analysis it is concluded that the resolvent decays exponentially on a sequence of concentric circular contours of increasing radius threading between poles in the left half plane. The decay rate is proportional to the distance between observation and source points.