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# The figures shows max (T ) and min (T ) as a function of the maximum spectral index 1/ for a sphere of unit radius.

Source publication

Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily, however, it is possible to leverage effective preconditioning and regularization st...

## Context in source publication

**Context 1**

... discretized operator inherits the spectral properties of the analytic EFIE operator, though the effect of the basis function should be removed by considering the spectrum of G −1 , T . Figure 9 shows the spectrum of the EFIE operator discretized on a sphere, confirming that indeed the condition number scales as 1/ 2 . ...

## Citations

... High-order basis functions also provide a faster convergence to the physical solution when refining the mesh [7], [8]. Nevertheless, despite the use of a more accurate framework, the EFIE suffers from ill-conditioning and loss of significant digits at lowfrequency [9]. ...

... It consists in a change of basis that allows to reorganize the system into solenoidal and non-solenoidal contributions and to rescale them appropriately to cure the problematic behavior of the EFIE at low-frequency. However, the construction of the solenoidal basis functions can be burdensome because it requires the identification of the global cycles; a task that might be challenging when modeling complex geometries [9]. ...

... More recently, the method of the quasi-Helmholtz projectors has been developed for the RWG case [9]. The method generates orthogonal projectors over the solenoidal and nonsolenoidal subspaces from the computation of the Star matrix, without having to explicitly identify the cycles. ...

The accuracy of the electric field integral equation (EFIE) can be substantially improved using high-order discretizations. However, this equation suffers from ill-conditioning and deleterious numerical effects in the low-frequency regime, often jeopardizing its solution. This can be fixed using quasi-Helmholtz decompositions, in which the source and testing elements are separated into their solenoidal and non-solenoidal contributions, then rescaled in order to avoid both the low-frequency conditioning breakdown and the loss of numerical accuracy. However, standard quasi-Helmholtz decompositions require handling discretized differential operators that often worsen the mesh-refinement ill-conditioning and require the finding of the topological cycles of the geometry, which can be expensive when modeling complex scatterers, especially in high-order. This paper solves these drawbacks by presenting the first extension of the quasi-Helmholtz projectors to high-order discretizations and their application to the stabilization of the EFIE when discretized with high-order basis functions. Our strategy will not require the identification of the cycles and will provide constant condition numbers for decreasing frequencies. Theoretical considerations will be accompanied by numerical results showing the effectiveness of our method in complex scenarios.

... Singular values of G −1 SG −1 N and singular values of G −1 S α G −1 N where S α is computed using (8) and (9), ordered by the singular vectors of the Laplace-Beltrami operator.the effectiveness of the filtering procedure in the Calderón precontitioned TE-EFIE[6] ...

Recent contributions showed the benefits of operator filtering for both preconditioning and fast solution strategies. While previous contributions leveraged laplacian-based filters, in this work we introduce and study a different approach leveraging the truncation of appropriately chosen spectral representations of operators' kernels. In this contribution, the technique is applied to the operators of the 2D TE- and TM-electric field integral equations (EFIE). We explore two different spectral representations for the 2D Green's function that lead to two distinct types of filtering of the EFIE operators. Numerical results corroborate the effectiveness of the newly proposed approaches, also in the Calder\'on preconditioned EFIE

... Tackling all the issues of both ies simultaneously (for scattering scenarios involving dielectric and pec bodies) is actively researched [Adrian et al. 2016;Guzman et al. 2017;Adrian 2018;Chhim et al. 2018;G.-Y. Zhu et al. 2019;Merlini 2019;Chhim et al. 2020;Merlini et al. 2020;Hofmann et al. 2021;Adrian et al. 2021]. ...

Boundary integral equations for the analysis of scattering and radiation scenarios with reduced computational effort, increased reliability, and increased accuracy are discussed. The contributions consist of two parts: first, the simulation of scenarios involving perfectly conducting objects and second, the post-processing of measured radiated and scattered fields for near-field far-field transformations, involving settings with complex measurement data, echo suppression, and phase retrieval.

... The electric field integral equation (EFIE) provides the solution of electromagnetic scattering and radiation problems for perfectly electrically conducting (PEC) objects. At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. ...

... At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. These decompositions require to search for global loops, a challenge even when junctions are not present. ...

... The operators T A,κ and T ϕ,κ are the rotated vector and scalar potentials on the surface [1] and n is the oriented unit normal over Γ. On edges that form a junction between n triangles, we define n − 1 independent RWGs on n − 1 couples of triangles as is often done [2]. ...

p>The electric field integral equation (EFIE) is known to suffer from ill-conditioning and numerical instabilities at low frequencies (low-frequency breakdown). A common approach to solve this problem is to rely on the loop and star decomposition of the unknowns. Unfortunatelly, building the loops is challenging in many applications, especially in the presence of junctions. In this work, we investigate the effectiveness of quasi-Helmholtz projector approaches in problems containing junctions for curing the low-frequency breakdown without detecting the global loops. Our study suggests that the performance of the algorithms required to obtain the projectors in the presence of junctions is maintained while keeping constant the number of sheets per junction. Finally, with a sequence of numerical tests, this work shows the practical impact of the technique and its applicability to real case scenarios.</p

... The electric field integral equation (EFIE) provides the solution of electromagnetic scattering and radiation problems for perfectly electrically conducting (PEC) objects. At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. ...

... At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. These decompositions require to search for global loops, a challenge even when junctions are not present. ...

... The operators T A,κ and T ϕ,κ are the rotated vector and scalar potentials on the surface [1] and n is the oriented unit normal over Γ. On edges that form a junction between n triangles, we define n − 1 independent RWGs on n − 1 couples of triangles as is often done [2]. ...

p>The electric field integral equation (EFIE) is known to suffer from ill-conditioning and numerical instabilities at low frequencies (low-frequency breakdown). A common approach to solve this problem is to rely on the loop and star decomposition of the unknowns. Unfortunatelly, building the loops is challenging in many applications, especially in the presence of junctions. In this work, we investigate the effectiveness of quasi-Helmholtz projector approaches in problems containing junctions for curing the low-frequency breakdown without detecting the global loops. Our study suggests that the performance of the algorithms required to obtain the projectors in the presence of junctions is maintained while keeping constant the number of sheets per junction. Finally, with a sequence of numerical tests, this work shows the practical impact of the technique and its applicability to real case scenarios.</p

... The electric field integral equation (EFIE) provides the solution of electromagnetic scattering and radiation problems for perfectly electrically conducting (PEC) objects. At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. ...

... At lowfrequency, the unbalanced coefficients scaling of the EFIE lead to ill-conditioned linear systems and to numerical instabilities [1]. This is the so-called low-frequency breakdown and it is standardly solved using quasi-Helmholtz approaches such as loop and star decomposition [1] where the unknowns are separated into their solenoidal and non-solenoidal parts. These decompositions require to search for global loops, a challenge even when junctions are not present. ...

... The operators T A,κ and T ϕ,κ are the rotated vector and scalar potentials on the surface [1] and n is the oriented unit normal over Γ. On edges that form a junction between n triangles, we define n − 1 independent RWGs on n − 1 couples of triangles as is often done [2]. ...

p>The electric field integral equation (EFIE) is known to suffer from ill-conditioning and numerical instabilities at low frequencies (low-frequency breakdown). A common approach to solve this problem is to rely on the loop and star decomposition of the unknowns. Unfortunatelly, building the loops is challenging in many applications, especially in the presence of junctions. In this work, we investigate the effectiveness of quasi-Helmholtz projector approaches in problems containing junctions for curing the low-frequency breakdown without detecting the global loops. Our study suggests that the performance of the algorithms required to obtain the projectors in the presence of junctions is maintained while keeping constant the number of sheets per junction. Finally, with a sequence of numerical tests, this work shows the practical impact of the technique and its applicability to real case scenarios.</p

... and the magnetic field integral operator (MFIO) [23] ...

... First we will show that quasi-Helmholtz projectors can successfully regularize the Steklov-Poincaré operators in both discretizations presented here. This is proven in (19) and (20) where we exploited standard cancellation properties of projectors on solenoidal spaces [23] (i.e., P ΛH T h = T h P ΛH = P ΣH T h = T h P ΣH = 0) from which T h = P Σ T h P Σ and T h = P Λ T h P Λ . In addition in (20) we used the result ∥P Σ −G T /2 + K −1 P Λ ∥ = O k 2 which follows from ∥P Σ −G T /2 + K P Λ ∥ = O k 2 (proven in Section IV.B.1 of [23]) after following a similar procedure as the one in Appendix B of [23]; in (19) ...

... This is proven in (19) and (20) where we exploited standard cancellation properties of projectors on solenoidal spaces [23] (i.e., P ΛH T h = T h P ΛH = P ΣH T h = T h P ΣH = 0) from which T h = P Σ T h P Σ and T h = P Λ T h P Λ . In addition in (20) we used the result ∥P Σ −G T /2 + K −1 P Λ ∥ = O k 2 which follows from ∥P Σ −G T /2 + K P Λ ∥ = O k 2 (proven in Section IV.B.1 of [23]) after following a similar procedure as the one in Appendix B of [23]; in (19) ...

The inverse source problem in electromagnetics has proved quite relevant for a large class of applications. When it is coupled with the equivalence theorem, the sources are often evaluated as electric and/or magnetic current distributions on an appropriately chosen equivalent surface. In this context, in antenna diagnostics in particular, Love solutions, i.e., solutions which radiate zero-fields inside the equivalent surface, are often sought at the cost of an increase of the dimension of the linear system to be solved. In this work, instead, we present a reduced-in-size single current formulation of the inverse source problem that obtains one of the Love currents via a stable discretization of the Steklov-Poincaré boundary operator leveraging dual functions. The new approach is enriched by theoretical treatments and by a further low-frequency stabilization of the Steklov-Poincaré operator based on the quasi-Helmholtz projectors that is the first of its kind in this field. The effectiveness and practical relevance of the new schemes are demonstrated via both theoretical and numerical results.

... Among the wellestablished formulations, the electric field integral equation (EFIE) plays a crucial role, both in itself and within combined field formulations [6]. The EFIE, lamentably, becomes illconditioned when the frequency is low or the discretization density high [7]. These phenomena-respectively known as the low-frequency and h-refinement breakdowns-cause the solution of the EFIE to become increasingly challenging to obtain, as the number of iterations of the solution process grows unbounded, which jeopardizes the possibility of achieving an overall linear complexity. ...

... In their standard incarnations they do, however, require the use of a dual discretization and global loop handling, because global loops reside in the static null-space of the Calderón operator. The introduction of implicit quasi-Helmholtz decompositions via the so called quasi-Helmholtz projectors [27], when combined with Calderón approaches, led to the design of several wellconditioned formulations, free from static nullspaces (see [7], [27]- [29] and references therein) and, in some incarnations, free from the need of performing a barycentric refinement [30]. Quasi-Helmholtz projectors have shown to be an effective and efficiently computable tool for performing quasi-Helmholtz decompositions, but, by themselves, they can only tackle the low-frequency breakdown and must be combined with Calderón-like strategies that involve multiple operators, to obtain h-refinement spectral preconditioning effects. ...

... Both Buffa-Christiansen [38] and Chen-Wilton [39] elements can be used for this dual discretization. For the sake of brevity, we will omit the explicit definitions of the dual elements that will be denoted by {g n } n in the following; the reader can refer to [7] and references therein for a more detailed treatment. We will also need the definition of the standard and dual Gram matrices whose entries are [G] mn = f m , f n and [G] mn = g m , g n . ...

Quasi-Helmholtz decompositions are fundamental tools in integral equation modeling of electromagnetic problems because of their ability of rescaling solenoidal and non-solenoidal components of solutions, operator matrices, and radiated fields. These tools are however incapable, per se, of modifying the refinement-dependent spectral behavior of the different operators and often need to be combined with other preconditioning strategies. This paper introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: the filtered Loop-Star functions and the quasi-Helmholtz Laplacian filters. Because they are capable of manipulating large parts of the operators' spectra, new families of preconditioners and fast solvers can be derived from these new tools. A first application to the case of the frequency and h-refinement preconditioning of the electric field integral equation is presented together with numerical results showing the practical effectiveness of the newly proposed decompositions.

... To assess the influence of a current component on the far field (FF), we utilize the stabilized evaluation [25] ...

... Choosing it similar to the one suggested in [25] as ...

... Similarly, for the quasi-Helmholtz projectors, we introduce [25] = ...

p>The accurate solution of quasi-Helmholtz decomposed electric field integral equations (EFIEs) in the presence of arbitrary excitations is addressed: Depending on the specific excitation, the quasi-Helmholtz components of the induced current density do not have the same asymptotic scaling in frequency, and thus, the current components are solved for with, in general, different relative accuracies. In order to ensure the same asymptotic scaling, we propose a frequency normalization scheme of quasi-Helmholtz decomposed EFIEs which adapts itself to the excitation and which is valid irrespective of the specific excitation and irrespective of the underlying topology of the structure. Specifically, neither an ad-hoc adaption nor a-priori information about the excitation is needed as the scaling factors are derived based on the norms of the right-hand side (RHS) components and the frequency. Numerical results corroborate the presented theory and show the effectiveness of our approach.</p

... To assess the influence of a current component on the far field (FF), we utilize the stabilized evaluation [25] ...

... Choosing it similar to the one suggested in [25] as ...

... Similarly, for the quasi-Helmholtz projectors, we introduce [25] = ...

p>The accurate solution of quasi-Helmholtz decomposed electric field integral equations (EFIEs) in the presence of arbitrary excitations is addressed: Depending on the specific excitation, the quasi-Helmholtz components of the induced current density do not have the same asymptotic scaling in frequency, and thus, the current components are solved for with, in general, different relative accuracies. In order to ensure the same asymptotic scaling, we propose a frequency normalization scheme of quasi-Helmholtz decomposed EFIEs which adapts itself to the excitation and which is valid irrespective of the specific excitation and irrespective of the underlying topology of the structure. Specifically, neither an ad-hoc adaption nor a-priori information about the excitation is needed as the scaling factors are derived based on the norms of the right-hand side (RHS) components and the frequency. Numerical results corroborate the presented theory and show the effectiveness of our approach.</p