We propose a high-order-accurate Nyström collocation discretization of the combined field integral equation (CFIE) on smooth surfaces that explicitly enforces surface current density continuity across patch boundaries via sparse mapping matrices. Consequently, the incorporation of line integrals is avoided, which often spoils the convergence of iterative solvers. Our choice of the collocation scheme resembles a mixed conforming discretization, as known from Galerkin discretizations employing primal and dual basis functions; however, no additional computational overhead is incurred, as is common with dual basis functions. Numerical results show that the breakdown at resonance frequencies is overcome, and the high accuracy obtained away from those frequencies is restored.

One of the most effective means to precondition the electric field integral equation (EFIE) discretized with Rao-Wilton-Glisson (RWG) functions is the multiplicative Calderón preconditioner employing Buffa-Christiansen (BC) functions as a basis dual to the RWG basis. It results in a formulation that is free from the dense-discretization and the low-frequency breakdown. To generalize the multiplicative Calderón preconditioner from the low-order BC and RWG basis to higher orders, we utilize B-spline-based basis functions and establish the first explicit high-order dual basis. It can be regarded as a generalization of the BC functions to arbitrary polynomial degrees and constitutes a fundamental building block for other approaches that rely on a dual basis. Numerical results for the obtained preconditioner demonstrate a low and constant number of generalized minimum residual (GMRES) iterations independent of the number of unknonws and the polynomial degree for canonical and realistic perfectly electrically conducting (PEC) scatterers; a key to enable the full potential of higher-order bases.

We present a multilevel B-spline-based fast integral method for the solution of the electric field integral equation (EFIE), combining fast Fourier transformation (FFT)-compatible kernel interpolation with robust high-order interpolation. Existing FFT-accelerated global Lagrange-based approaches rely on equidistant interpolation points and can, therefore, suffer from Runge-type instabilities at high interpolation orders, limiting robust high-accuracy compression. In contrast, B-splines on equidistant knot vectors overcome these instabilities and enable robust high-order interpolation for accurate matrix compression. Replacing Lagrange interpolation by B-spline interpolation is, however, non-trivial: B-spline coefficients do not coincide with function values at the interpolation points, and the associated sampling matrices can become ill-conditioned. To address these challenges, we introduce a knot-removal stabilization strategy, combined with exact interlevel transfers based on knot insertion, yielding accurate, well-conditioned multilevel interpolation. Moreover, we propose a factorization strategy that preserves the null space of the scalar potential operator up to machine precision and is compatible with low-frequency preconditioning techniques. Numerical results for both canonical and realistic geometries demonstrate robust high-order interpolation without the breakdown observed for Lagrange-based approaches and confirm 𝒪(N) complexity.

Solving radiation and scattering problems, particularly those involving perfectly electrically conducting objects, by leveraging the electric field integral equation (EFIE) in conjunction with the boundary element method, is a widely used approach. However, a significant drawback of this method is the resulting densely populated system matrix, which restricts its application to problems with a small number of unknowns due to computational and storage constraints. To address this limitation, matrix-free acceleration techniques such as the propagating plane-wave based multilevel fast multipole method (MLFMM) (W. Chew, E. Michielssen, J. M. Song, and J. M. Jin, Fast and Efficient Algorithms in Computational Electromagnetics. Artech House, Inc., Jul. 2001), the adaptive cross approximation (ACA) (K. Zhao, M. Vouvakis, and J.-F. Lee, “The Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems,” IEEE Transactions on Electromagnetic Compatibility, vol. 47, no. 4, Nov. 2005), or the interpolatory $\mathcal{H}^{2}$-method (W. Hackbusch and S. Börm, “ $\mathcal{H}^{2}$-matrix approximation of integral operators by interpolation,” Applied Numerical Mathematics, vol. 43, no. 1-2, Oct. 2002; D. T. Schobert and T. F. Eibert, “Low-Frequency Surface Integral Equation Solution by Multilevel Green's Function Interpolation With Fast Fourier Transform Acceleration,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 3, Mar. 2012) have been developed. These techniques reduce the storage and computational complexity of a matrix-vector product from quadratic to (quasi-)linear. As a consequence, they enable the efficient solution of larger and more complex problems when combined with an iterative solver.

The electric field integral equation (EFIE) is a widely employed method for solving electromagnetic scattering and radiation problems. However, it is susceptible to two significant challenges: the low-frequency breakdown and the dense-discretization breakdown, which lead to poor or no convergence of iterative solvers.

We introduce a fast Fourier transform (FFT) accelerated interpolatory $\mathcal{H}^{2}$-method for solving the electric field integral equation (EFIE) at low frequencies, which overcomes drawbacks of existing interpolatory approaches. By interpolating the kernel of the integral equation using a set of equidistant Bsplines instead of Lagrange polynomials, the Runge phenomenon and related numerical instabilities can be avoided. Leveraging the three-level Toeplitz structure of the kernel, computational complexity and storage requirements can be minimized employing FFTs. To maintain an octree-based multilevel algorithm, we partition basis functions spanning multiple adjacent clusters and precorrect near-intraction computations. The efficacy of the proposed algorithm is demonstrated through numerical results, highlighting its potential for efficient and accurate EFIE solutions at low frequencies.

The multilevel fast multipole method (MLFMM) is widely employed as an acceleration scheme for solving timeharmonic scattering problems in conjunction with the electric field integral equation (EFIE). However, the efficiency of the MLFMM can be compromised when densely discretizing the geometry: there is a lower limit on the smallest box-size of the MLFMM octree, which spoils the computational efficiency. To overcome this limitation, we propose a hybrid scheme that combines the MLFMM with an interpolatory $\mathcal{H}^{2}$-method. The $\mathcal{H}^{2}$-method utilizes B-splines to interpolate the relevant Green's function and leverages a fast Fourier transform (FFT) acceleration to further reduce the computational complexity. We demonstrate the effectiveness of this hybrid scheme and its practical applicability through numerical results.

To reconstruct equivalent electric surface currents from irregularly distributed observations of the fields radiated by a device under test (DUT), we investigate a low-frequency stabilized formulation based on a quasi-Helmholtz decomposition: Rao-Wilton-Glisson (RWG) basis functions are employed for the equivalent surface currents, dipoles as probe antennas, and the relation between probes and sources is established via integral operators. Since, in general, a rectangular linear system of equations (LSE) is obtained, we analyze both the normal-residual system of equations (NRE) and the normal-error system of equations (NEE) for an iterative solution with a generalized minimum residual (GMRES) solver. Specifically, our analysis shows that a self-adaptive normalization scheme is needed in both cases, where the scheme itself can remain the same.

A wideband stable surface electric and magnetic current reconstruction method based on electric and magnetic field measurements for the characterization of a device under test (DUT) by extending a high-frequency formulation to the static limit is presented. In general, surface current reconstructions are suffering from low-frequency instabilities leading to a catastrophic loss of accuracy. We overcome these instabilities by leveraging quasi-Helmholtz (qH) decompositions and deriving a suitable self-adaptive, blackbox-like normalization technique taking into account the actual radiation characteristics obtained by the measurement data. The results from experimental measurements of a printed circuit board (PCB) show the low-frequency stability of the formulation.

A discretization of the electric field integral equation (EFIE) with div-conforming hierarchical B-splines is presented. In contrast to the standard B-spline discretizations of the EFIE, our scheme allows local refinements of the basis functions. This results in a more efficient scheme, in particular, when the geometry is described by curvilinear surface patches that vary significantly in their area, a scenario that is typically encountered in multi-scale scenarios. Numerical results show the effectiveness of the presented approach.

In order to low-frequency stabilize the isogeometrically discretized electric field integral equation (EFIE) based on B-splines, we derive quasi-Helmholtz projectors to form a preconditioner. To this end, we show how the projectors can be obtained efficiently for arbitrary polynomial orders of the basis functions. The approach is valid for single- and multi-patch descriptions of the geometry, which can be open or closed as well as simply or multiply connected without the need to search for global loops. Numerical results demonstrate the derived preconditioner's effectiveness in obtaining accurate scattered and radiated fields.

The hitherto first conforming higher-order discretization of the magnetic field integral equation (MFIE) is presented. Non-conforming discretizations of the MFIE lead to a loss of accuracy and make it impossible to obtain low-frequency stable formulations. So far, however, only a lowest-order conforming discretization of the MFIE has been presented by leveraging Buffa-Christiansen (BC) functions, which are dual to the lowest-order Rao-Wilton-Glisson (RWG) functions, limiting the speed of convergence. We obtain a conforming high-order discretization by utilizing B-spline-based basis functions and establishing a set of dual basis functions, which can be regarded as a generalization of the (low-order) BC functions known from triangular meshes. Numerical results demonstrate the effectiveness of the proposed discretization scheme.

In order to low-frequency stabilize the electric field integral equation (EFIE) when discretized with divergence conforming B-spline-based basis and testing functions in an isogeometric approach, we propose a corresponding quasi-Helmholtz preconditioner. To this end, we derive i) a loop-star decomposition for the B-spline basis in the form of sparse mapping matrices applicable to arbitrary polynomial orders of the basis as well as to open and closed geometries described by single-patch or multipatch parametric surfaces (as an example, nonuniform rational B-splines (NURBS) surfaces are considered). Based on the loop-star analysis, we show ii) that quasi-Helmholtz projectors can be defined efficiently. This renders the proposed low-frequency stabilization directly applicable to multiply-connected geometries without the need to search for global loops and results in better-conditioned system matrices compared with directly using the loop-star basis. Numerical results demonstrate the effectiveness of the proposed approach.

The electric field integral equation (EFIE) is widely employed to determine the field that is scattered from perfectly electrically conducting (PEC) structures. However, it is known to suffer from a low-frequency breakdown. In order to overcome this breakdown for a B-spline based (isogeometric) discretization of arbitrary polynomial order of the EFIE employing the method of moments, we propose a loop-star decomposition of the discretized surface current density resulting in a preconditioner involving solely sparse matrices. The proposed decomposition is applicable to open and closed simply-connected surfaces described by a single or by multiple patches. To verify the correctness of the proposed method, numerical examples are provided.

We present a multiplicative Calderon preconditioner for the electric field integral equation (EFIE) when discretized with B-spline-based basis functions, that is, the resulting formulation is free from the dense-discretization breakdown. We obtain the preconditioner by establishing a set of suitable dual basis functions, which can be explicitly expressed as a superposition of a refined discretization, as is known from the (low-order) Buffa-Christiansen (BC) functions. In contrast to the BC functions, our approach applies to arbitrary polynomial degrees of the basis functions for single- and multi-patch (curvilinear) descriptions of the geometry, which can be open or closed. Numerical results demonstrate the optimal nature of the derived preconditioner.

Scattering problems in electromagnetics have a multitude of applications, including, but not limited to, remote sensing and radar systems, electromagnetic compatibility, and antenna design. A natural way to address these problems in free space are integral equations such as the electric field integral equation (EFIE). The EFIE can be approximated by a linear system of equations using the boundary element method (BEM). Typically, in the first step, a mesh is created on the surface of the geometry. Currents on the mesh are approximated by a superposition of basis functions. This results in a geometric discretization error that can be prevented by utilizing exact parametric descriptions of the geometry surface. One commonly employed technique involves the use of non-uniform rational B-splines (NURBS), where the surface is described by a union of patches. Basis functions are subsequently established using the same parametric representation by mapping them from a parametric space onto the patches [1]. This aligns well with common industrial design schemes that focus on computer-aided design (CAD) tools and integrates the design and simulation process. Existence and uniqueness of the solution of the BEM for the EFIE using a set of divergence-conforming basis functions have been established in [2].

The low-frequency stability of inverse surface source field transformations based on the electric field integral operator (EFIO) together with the boundary element method (BEM) is investigated. In order to overcome the observed breakdown, we study a stabilization scheme based on quasi-Helmholtz projectors. This scheme incorporates a self-adaptive normalization and a stabilized right-hand side (RHS) evaluation which are found of critical importance to determine all fields accurately as shown by the simulation of different inverse surface source reconstructions.

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.

Low-frequency preconditioned boundary integral equations based on quasi-Helmholtz decompositions are widely used to obtain the radiated or scattered fields by a finite structure over a wide frequency range. Specifically, for perfectly electrically conducting (PEC) structures, the conformingly discretized magnetic field integral equation (MFIE) plays a crucial role. However, a careful analysis and, potentially, ad-hoc adaptions are necessary for each excitation to ensure that all fields are obtained accurately. To avoid such cumbersome analyses, we propose an excitation agnostic and self-adaptive frequency normalization scheme. To this end, the appropriate scaling factors are derived based on the norms of the right-hand side (RHS) components without requiring any ad-hoc adaptions or any a-priori knowledge about the excitation. Numerical results demonstrate the effectiveness of this approach to obtain accurate fields.

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 approach 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 function. 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 suitable scalar function. We show that this approach works with open and closed, orientable and non-orientable geometries. The numerical results demonstrate the effectiveness of this approach.