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

et al. 2014

Abstract-The standard and mixed discretizations for the Magnetic Field Integral Equation (MFIE) and the Müller Integral Equation (MUIE) are investigated in the context of lowfrequency (LF) scattering problems involving simply connected scatterers. It is proved that, at low frequencies, the frequency scaling of the nonsolenoidal part of the solution current can be incorrect for the standard discretization. In addition, it is proved that the frequency scaling obtained with the mixed discretization is correct. The reason for this problem in the standard discretization scheme is the absence of exact solenoidal currents in the rotated RWG finite element space. The adoption of the mixed discretization scheme eliminates this problem and leads to a well-conditioned system of linear equations that remains accurate at low frequencies. Numerical results confirm these theoretical predictions and also show that, when the frequency is lowered, a finer and finer mesh is required to keep the accuracy constant with the standard discretization.

“…This expression leads to the scalings in equation (31). For the case of the 4 by 4 block matrix arising in the MUIE, it is advantageous to proceed in two steps.…”

Section: Discussionmentioning

confidence: 99%

“…For a proof, we refer to [31]. Integral operator (1) has a very well-known behavior at low frequencies that is used in the loop star decomposition for the EFIE [32].…”

Section: Lf Behavior Of Integral Operatorsmentioning

confidence: 99%

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Low-Frequency Scaling of the Standard and Mixed Magnetic Field and Müller Integral Equations

Bogaert

et al. 2014

“…Furthermore, it supports sourceless constant or linear in time divergence free solutions (DC instabilities) [16], [17]. For simply connected geometries, DC instabilities can be eliminated by switching to the Calderón preconditioned TD-EFIE [18] and applying the so-called dot-trick [16]. However, for multiply connected geometries, the dot-trick EFIE still supports static solutions [19] and is therefore susceptible to DC instability.…”

Section: Introductionmentioning

confidence: 99%

A DC Stable and Large-Time Step Well-Balanced TD-EFIE Based on Quasi-Helmholtz Projectors

Beghein

2015

Abstract-The marching-on-in-time solution of the time domain electric field integral equation (TD-EFIE) has traditionally suffered from a number of issues, including the emergence of spurious static currents (DC instability) and ill-conditioning at large time steps (low frequencies). In this contribution, a spacetime Galerkin discretization of the TD-EFIE is proposed, which separates the loop and star components of both the equation and the unknown. Judiciously integrating or differentiating these components with respect to time leads to an equation which is free from DC instability. By choosing the correct temporal basis and testing functions for each of the components, a stable marchingon-in-time system is obtained. Furthermore, the scaling of these basis and testing functions ensure that the system remains wellconditioned for large time steps. The loop-star decomposition is performed using quasi-Helmholtz projectors in order to avoid the explicit transformation to the unstable bases of loops and stars (or trees), and to avoid the search for global loops, which is a computationally expensive operation.Index Terms-time domain, electric field integral equation, DC instability, low frequency breakdown.

“…It should also be noted that bothΛl andHh are RWG coefficients of solenoidal functions. Given the decomposition in (15), a dual decomposition exists for the BC functions ( [31], [32], [20], [21]): given an arbitrary BC coefficient vectorÁ, the following decomposition holds…”

mentioning

confidence: 99%

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

Bogaert

et al. 2013

111

137

Abstract-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,