Kristof Cools

Time Domain CalderÓn Identities and Their Application to the Integral Equation Analysis of Scattering by PEC Objects Part II: Stability

Novel time domain integral equations for analyzing scattering from perfect electrically conducting objects are presented. They are free from DC and resonant instabilities plaguing standard electric field integral equation. The new equations are obtained using operator manipulations originating from the Calderon identities. Theoretical motivations leading to the construction of the new equations are explored and numerical results confirming their theoretically predicted behavior are presented.

Accurate and Conforming Mixed Discretization of the MFIE

In this letter, a novel discretization scheme for the magnetic field integral equation is presented. The new scheme is designated “mixed” because it uses Rao-Wilton-Glisson functions to expand the current density and Buffa-Christiansen functions to test the magnetic field radiated by the candidate solution. The convergent nature of the proposed mixed MFIE is theoretically proven, and numerical results showing that the proposed method yields more accurate results than the classical one are presented.

A CalderÓn Multiplicative Preconditioner for the Combined Field Integral Equation

A Calderon multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Calderon-preconditioned CFIEs, a localization procedure is employed to ensure that the equation is resonance-free. The iterative solution of the linear system of equations obtained via the CMP-based discretization of the CFIE converges rapidly regardless of the discretization density and the frequency of excitation.

On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries

All known integral equation techniques for simulating scattering and radiation from arbitrarily shaped, perfect electrically conducting objects suffer from one or more of the following shortcomings: (i) they give rise to ill-conditioned systems when the frequency is low (ii) and/or when the discretization density is high, (iii) their applicability is limited to the quasi-static regime, (iv) they require a search for global topological loops, (v) they suffer from numerical cancellations in the solution when the frequency is very low. This work presents an equation that does not suffer from any of the above drawbacks when applied to smooth and closed objects. The new formulation is obtained starting from a Helmholtz decomposition of two discretizations of the electric field integral operator obtained by using RWGs and dual bases respectively. The new decomposition does not leverage Loop and Star/Tree basis functions, but projectors that derive from them. Following the decomposition, the two discretizations are combined in a Calderon-like fashion resulting in a new overall equation that is shown to exhibit self-regularizing properties without suffering from the limitations of existing formulations. Numerical results show the usefulness of the proposed method both for closed and open structures.

Time Domain CalderÓn Identities and Their Application to the Integral Equation Analysis of Scattering by PEC Objects Part I: Preconditioning

Time domain electric field integral equations often are used to analyze transient scattering from perfect electrically conducting objects. When discretized using marching-on-in-time recipes they give rise to linear systems of equations that can be solved for the induced currents for all time steps. Unfortunately, when the scatterer is approximated by increasingly dense meshes, the condition number of these systems grows rapidly, slowing down the convergence of iterative solvers. Here, time domain Calderon identities are derived and subsequently used to construct a Calderon-preconditioned time domain electric field integral equation that can be discretized even with dense meshes using Buffa-Christiansen basis functions. Numerical results that demonstrate the effectiveness and accuracy of the proposed method are presented.

Improving the MFIE's accuracy by using a mixed discretization

The scattering of time-harmonic electromagnetic waves by perfect electrical conductors (PECs) can be modelled by several boundary integral equations, the magnetic and electric field integral equations (MFIE and EFIE) being the most prominent ones[1]. These equations can be discretized by expanding current distributions in terms of Rao-Wilton-Glisson (RWG) functions defined on a triangular mesh approximating the scatterers surface and by testing the equations using the same RWG functions [2].

A Calderón Multiplicative Preconditioner for the PMCHWT Integral Equation

Electromagnetic scattering by penetrable bodies often is modelled by the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) integral equation. Unfortunately the spectrum of the operator involved in this equation is bounded neither from above or below. This implies that the equation suffers from dense discretization breakdown; that is, the condition numbers of the matrix resulting upon discretizing the equation rise with the mesh density. The electric field integral equation, often used to model scattering by perfect electrically conducting bodies, is susceptible to a similar breakdown phenomenon. Recently, this breakdown was cured by leveraging the Calderón identities. In this paper, a Calderón preconditioned PMCHWT integral equation is introduced. By constructing a Calderón identity for the PMCHWT operator, it is shown that the new equation does not suffer from dense discretization breakdown. A consistent discretization scheme involving both Rao-Wilton-Glisson and Buffa-Christiansen functions is introduced. This scheme amounts to the application of a multiplicative matrix preconditioner to the classical PMCHWT system, and therefore is compatible with existing boundary element codes and acceleration schemes. The efficiency and accuracy of the algorithm are corroborated by numerical examples.

A Space-Time Mixed Galerkin Marching-on-in-Time Scheme for the Time-Domain Combined Field Integral Equation

The time domain combined field integral equation (TD-CFIE), which is constructed from a weighted sum of the time domain electric and magnetic field integral equations (TD-EFIE and TD-MFIE) for analyzing transient scattering from closed perfect electrically conducting bodies, is free from spurious resonances. The standard marching-on-in-time technique for discretizing the TD-CFIE uses Galerkin and collocation schemes in space and time, respectively. Unfortunately, the standard scheme is theoretically not well understood: stability and convergence have been proven for only one class of space-time Galerkin discretizations. Moreover, existing discretization schemes are nonconforming, i.e., the TD-MFIE contribution is tested with divergence conforming functions instead of curl conforming functions. We therefore introduce a novel space-time mixed Galerkin discretization for the TD-CFIE. A family of temporal basis and testing functions with arbitrary order is introduced. It is explained how the corresponding interactions can be computed efficiently by existing collocation-in-time codes. The spatial mixed discretization is made fully conforming and consistent by leveraging both Rao-Wilton-Glisson and Buffa-Christiansen basis functions and by applying the appropriate bi-orthogonalization procedures. The combination of both techniques is essential when high accuracy over a broad frequency band is required.

High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative Preconditioning of the EFIE

A new high-order Calderón multiplicative preconditioner (HO-CMP) for the electric field integral equation (EFIE) is presented. In contrast to previous CMPs, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of high-order quasi curl-conforming basis functions. Like its predecessors, the HO-CMP can be seamlessly integrated into existing EFIE codes. Numerical results demonstrate that the linear systems of equations obtained using the proposed HO-CMP converge rapidly, regardless of the mesh density and of the order of the current expansion.

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.