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# The vector field of an RWG function. The vector denotes the directed edge, + and − denote the domains of the cells, + and − denote vertices on the edge , and + and − are the vertices opposite to the edge .

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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...

## Contexts in source publication

**Context 1**

... the convention depicted in Figure 1. Following a Petrov-Galerkin approach, we obtain the system of equations ...

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... is the th cell of the mesh, following the conventions depicted in Figure 1. With the definition of these matrices, the quasi-Helmholtz decomposition in (33) can be equivalently written as ...

**Context 3**

... projectors ensure that the overall system matrix is wellconditioned also on multiply-connected geometries. Figure 10 shows that the condition number of the preconditioned system matrix in (128) is asymptotically bounded. A similar result holds even on the more challenging NASA almond benchmark [114] (Figure 11). ...

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... 10 shows that the condition number of the preconditioned system matrix in (128) is asymptotically bounded. A similar result holds even on the more challenging NASA almond benchmark [114] (Figure 11). ...

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... is the Gram matrix of piecewise linear dual basis functions as defined in [45] and patch basis functions with Condition number EFIE LS-EFIE P-EFIE P-CMP-EFIE RF-CMP-EFIE MFIE P-CMP-MFIE CMP-EFIE Fig. 10. Condition number of the system matrices as a function of 1/, which is proportional to the maximum spectral index, for a cube of side 1 m and a frequency of 10 7 Hz. The labels "CMP-EFIE", "CFIE", "RF-CMP-EFIE", "P-CMP-MFIE", and "P-CMP-CFIE" refer to the standard Calderón EFIE (126), the conforming CFIE, the refinement-free Calderón ...

**Context 6**

... allows the use of CG, which in exact arithmetic guaranties convergence, and has a lower computational cost with respect to CGS or GMRES [53]. Figure 13 shows the number of iterations for the refinementfree preconditioned compared with (128), a standard EFIE and a loop-tree preconditioned EFIE. It displays that the number of iterations become bounded independently from , this confirming the dense-discretization stability of the formulation. ...

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... a suitably chosen and in high-frequency scenarios, the scaling of the projectors must be set to a unitary value [96]. Figure 12 shows that the CFIE formulation is free from interior resonances and Figure 13 the number of iterations as a function of the spectral index 1/. For the presented results, we have weighted the EFIE and MFIE part equally. ...

**Context 8**

... a suitably chosen and in high-frequency scenarios, the scaling of the projectors must be set to a unitary value [96]. Figure 12 shows that the CFIE formulation is free from interior resonances and Figure 13 the number of iterations as a function of the spectral index 1/. For the presented results, we have weighted the EFIE and MFIE part equally. ...

**Context 9**

... this combined formulation has the advantage of not exhibiting a static nullspace. This is because, unlike the standard the Calderón EFIE (126), the Calderón EFIE stabilized with projectors (128) has no static nullspace, which can additively cure that of the MFIE and results in equation (142) that is free from spurious resonances, as illustrated in Figure 14. While both Calderón-Yukawa CFIEs (141) and (142) are free from the dense-discretization breakdown (i.e., when all parameters are kept constant expect from , the condition number can be asymptotically bounded for → 0), the condition number cannot be bounded independently from . ...

**Context 10**

... was seen in the previous sections, the spectrum of the electromagnetic boundary integral operators can be separated into parts: one for the solenoidal functions and the other for the non-solenoidal functions. The O ( 2/3 ) growth of the condition number is due to a maximum singular value in the solenoidal part of the spectrum that grows as O ( 1/3 ) and a minimum singular value in the non-solenoidal part of the spectrum that decreases as O ( −1/3 ). This effect is related to the high-frequency breakdown in the scalar Helmholtz equation in acoustics, where both the Dirichlet problem solved with a standard CFIE and the Neumann problem solved with a Calderón-Yukawa CFIE have condition numbers that grow as O ( 1/3 ) on a sphere [163], [164]. ...

**Context 11**

... speaking, a mesh is ill-shaped if it has narrow triangles (see Figure 15 for a mesh with narrow triangles; for a more nuanced discussion on measures for the quality of meshes, see [169]). Ill-shaped meshes lead to higher condition numbers of the discretized operator compared with a discretization based on a well-shaped mesh, that is, where the angles of each triangle have roughly the same size. ...

**Context 12**

... a consequence, the condition number grows as O ( 2 ). The singular values of the EFIE operator on a sphere have been represented on Figure 16. A way to solve this problem is by scaling the solenoidal part by −1 and by scaling the non-solenoidal part by . ...

**Context 13**

... a consequence, the condition number grows as O ( −2 ). The spectrum of the EFIE have been represented on Figure 17 for different frequencies to illustrate this growth of the condition number as the discretization density increases. This problem can be solved using Calderón preconditioning: as → +∞ (or equivalently as → 0), the singular values of (T ) 2 are ...

**Context 14**

... MFIE suffers from the same problem. The resonant frequencies can be read on Figure 16 with the first one appearing at ≈ 2.74. Also, it is clear from Figure 16 that the higher is, the more common the resonant frequencies are. ...

**Context 15**

... resonant frequencies can be read on Figure 16 with the first one appearing at ≈ 2.74. Also, it is clear from Figure 16 that the higher is, the more common the resonant frequencies are. The result of the resonances on the spectrum is clear on Figure 17 where there is a finite number of singular values that are unbounded from below at a fixed frequency. ...

**Context 16**

... it is clear from Figure 16 that the higher is, the more common the resonant frequencies are. The result of the resonances on the spectrum is clear on Figure 17 where there is a finite number of singular values that are unbounded from below at a fixed frequency. ...

**Context 17**

... result is a growth of the condition number faster than a constant times 2/3 where numerical evidences on the spherical harmonics show that this limit is in practice attained. The singular values of the Calderón-Yukawa CFIE have been represented on Figure 18 as well as the asymptotic bounds that are derived in the following sections. ...

## Citations

... 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

... 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

... Hence, we can recognize different branches of eigenvalues of (Z − K Z ): on one side, those associated to N , ( + +1 ) λ , , diverging towards minus infinity with the order of the corresponding eigenfunction, and, on the other side, those associated to S , ( −1 + −1 +1 ) λ , , converging toward 0 for → ∞. To obtain a statement on how the condition number grows in ℎ , we note that the maximum degree supported by a mesh resolution ℎ behaves asymptotically as = O ( ℎ −1 ) [3]. We then find ...

We present a Calder\'on preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calder\'on schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios.

... One strategy to obtain an accurate discretization of the RHS leverages a Taylor series expansion to set the static contribution to zero when tested with solenoidal functions [8], [11], [12]. Notably, this approach is independent of the topology of the underlying geometry. ...

... Analogously, with e nsol = P Σ e ex , we get O (| ex |) = O (∥e nsol ∥). Hence, condition (12) can be expressed equivalently as ...

... The results for a plane-wave excitation are shown in Fig. 11 (a). The reference solution is determined by stabilizing the RHS with a Taylor series expansion following [12]. The comparison with the stabilization via scalar potentials shows good agreement over the whole frequency range, clearly demonstrating the correctness of the proposed scalar potential formulation for multiply-connected geometries. ...

p>In order to accurately compute scattered and radiated fields in the presence of arbitrary excitations, a low-frequency stable discretization of the right-hand side (RHS) of a quasi-Helmholtz preconditioned electric field integral equation (EFIE) on multiply-connected geometries is introduced, which avoids an ad-hoc extraction of the static contribution of the RHS when tested with solenoidal functions. To obtain an excitation agnostic approach, our ansatz generalizes a technique to multiply-connected geometries where the testing of the RHS with loop functions is replaced by a testing of the normal component of the magnetic field with a scalar potential. To this end, we leverage orientable global loop functions that are formed by a chain of Rao-Wilton-Glisson (RWG) functions around the holes and handles of the geometry, for which we introduce cap surfaces that allow to uniquely define a scalar potential. We show that this approach works with open and closed, orientable and non-orientable geometries. The numerical results demonstrate the effectiveness of this approach.</p

... One strategy to obtain an accurate discretization of the RHS leverages a Taylor series expansion to set the static contribution to zero when tested with solenoidal functions [8], [11], [12]. Notably, this approach is independent of the topology of the underlying geometry. ...

... Analogously, with e nsol = P Σ e ex , we get O (| ex |) = O (∥e nsol ∥). Hence, condition (12) can be expressed equivalently as ...

... The results for a plane-wave excitation are shown in Fig. 11 (a). The reference solution is determined by stabilizing the RHS with a Taylor series expansion following [12]. The comparison with the stabilization via scalar potentials shows good agreement over the whole frequency range, clearly demonstrating the correctness of the proposed scalar potential formulation for multiply-connected geometries. ...

p>In order to accurately compute scattered and radiated fields in the presence of arbitrary excitations, a low-frequency stable discretization of the right-hand side (RHS) of a quasi-Helmholtz preconditioned electric field integral equation (EFIE) on multiply-connected geometries is introduced, which avoids an ad-hoc extraction of the static contribution of the RHS when tested with solenoidal functions. To obtain an excitation agnostic approach, our ansatz generalizes a technique to multiply-connected geometries where the testing of the RHS with loop functions is replaced by a testing of the normal component of the magnetic field with a scalar potential. To this end, we leverage orientable global loop functions that are formed by a chain of Rao-Wilton-Glisson (RWG) functions around the holes and handles of the geometry, for which we introduce cap surfaces that allow to uniquely define a scalar potential. We show that this approach works with open and closed, orientable and non-orientable geometries. The numerical results demonstrate the effectiveness of this approach.</p

... A popular choice as a forward solution equation is the Electric Field Integral Equation (EFIE). Although this formulation is widespread, its numerical solution comes with its own set of challenges: the linear systems that stem from its discretization are dense and illconditioned, with a condition number that grows with the inverses of the frequency and the average mesh edge length [1]. ...

This paper focuses on fast direct solvers for integral equations in the low-to-moderate-frequency regime obtained by leveraging preconditioned first kind or second kind operators regularized with Laplacian filters. The spectral errors arising from boundary element discretizations are properly handled by filtering that, in addition, allows for the use of low-rank representations for the compact perturbations of all operators involved. Numerical results show the effectiveness of the approaches and their effectiveness in the direct solution of integral equations.