The Method of Auxiliary Sources (MAS) in Computational Electromagnetics: A Comprehensive Review of Advancements over the Past Two Decades

Abstract

This paper presents a comprehensive review of research conducted on the Method of Auxiliary Sources (MAS) over a period of the last 22 years, i.e., since the last up-to-date survey was published. MAS is a very attractive numerical technique due to its simple algorithmic structure and the generally low computational cost it requires in terms of memory and CPU time; this is why it has been applied to a vast variety of cases, as concluded by the long citations list included. After a short introduction summarizing the fundamental concepts of the method, references since 2002 are categorized, briefly described, and commented on. This work is intended to assist every researcher who is involved in MAS computations, providing an exhaustive, to the best of the authors’ knowledge, list of related publications.

1. Introduction

The Method of Auxiliary Sources (MAS) is a well-established numerical technique, applicable to the solution of elliptic Boundary Value Problems (BVPs) typically encountered in various fields of Physics and Engineering, such as electromagnetics. Formally, it is closely related to surface integral equation formulations, such as the Method of Moments (MoM), although technically it is not based on the same mathematical concepts. It was originally conceived by a number of scholars in parts of the former Soviet Union (nowadays the Republic of Georgia) [1,2,3] and was further developed at the National Technical University of Athens (NTUA), Greece, via a fruitful international cooperation starting in the 1990s [4]. Similar or even almost identical methods have been independently proposed by different research groups in other countries under various names, such as the Multifilament Current Model Method (MFCM) [5] or the Discrete Singularity Method [6], mainly for addressing electromagnetic (EM) scattering problems. Another closely related method (practically almost identical to MAS) is the Source Model Technique (SMT), which is further analyzed below. Moreover, MAS may be considered as a special case of the Generalized Multipole Technique (GMT) or Multiple Multipole Technique (MMT) [7]. All these methods share the same fundamental concept: the EM BVP is formulated in terms of discrete fictitious currents (the “auxiliary sources”—ASs), placed away from physical boundaries, and not in terms of continuous equivalent currents located on the actual interfaces, which is the core assumption of MoM, for instance. Since the ASs are pointwise, there is no need for elaborate surface integrations with singular kernels, which are necessary in MoM.

The concept of expanding the solution of a BVP in terms of fundamental solutions of the governing partial differential equation lies at the core of many widely used numerical methods, which efficiently model wave phenomena arising in diverse application areas. Specifically, the basic idea behind the methods of fictitious/auxiliary sources, fundamental solutions, and the aforementioned SMT concerns the representation of the approximate solution of the BVP as a finite superposition of the fields of fictitious/auxiliary sources located outside the problem’s domain, with amplitudes determined by the boundary conditions on the actual physical boundary. More information on this concept and its outgrowth are discussed below.

The most important advantages of MAS over MoM include algorithmic simplicity and lower computational cost, making it attractive for the numerical solution of problems related to EM scattering, antenna modeling, waveguide structures, etc. In the early 2000s, the salient features of MAS had already been appreciated, and a multitude of applications were described in the literature. A comprehensive review of published research papers in EM, until 2002, was given in [8], the metrics of which can be found in [9]. An analogous review, focused on the applications of MAS in various physical phenomena (not restricted to EM propagation) was given in [10]. In [8], the basic mathematical formulation was described first, and a comparison to boundary element methods (such as MoM) was performed, illustrating all their relative advantages and disadvantages. Furthermore, several aspects of MAS applications were highlighted, supported by all relevant references, including the following:

(a)

AS placement and MAS optimization;

(b)

MAS computational cost;

(c)

MAS/MoM hybridization;

(d)

Development of holographic imaging;

(e)

MAS optimization via complex image theory techniques;

(f)

Scattering by edges;

(g)

Dielectrically coated metal scatterers modeled by the Standard Impedance Boundary Condition (SIBC);

(h)

Radar Cross Section (RCS) evaluation of jet engine inlets;

(i)

Development of modified versions of MAS.

Since 2002, when [8] was published, MAS has been further refined, in terms of theoretical issues related to its convergence rate and efficiency, and its applications have been extended to several complex configurations. The purpose of this paper is to present an exhaustive survey of the literature related to MAS published since 2002. For earlier references we refer the reader to [8]. The present paper is organized as follows: Section 2 addresses theoretical issues, such as the optimal location of the ASs and the convergence of field values vs. current values. Section 3 describes the generalizations of the results to non-circular cross sections, namely generalized cylindrical geometries, irregular shapes, and infinite/open boundaries, as well as applications in inverse scattering and extension to 3D including Bodies of Revolution (BORs). Section 4 deals with applications to typical EM problems (e.g., antenna modeling, waveguides, and resonators), as well as more sophisticated EM issues, such as multilayers, metamaterials, chiral and anisotropic materials, detection of Unexploded Ordnance (UXO), etc. Section 5 presents MAS variants and hybrids, such as time-domain versions and Fast Multipole Method hybrids, as well as indicative applications to acoustics. Finally, Section 6 summarizes the manuscript and proposes future work.

2. Theoretical Aspects of MAS with Respect to Convergence and Efficiency

Suppose we are interested in the solution of an EM problem, such as scattering, radiation, or waveguide propagation. The geometry may involve physical objects, typically (but not necessarily restricted to) Perfectly Electric Conductors (PECs) or dielectrics. The basic concept of MAS is the representation of the unknown EM fields as a weighted superposition of elementary fields radiated by fictitious ASs. These fields are proportional to the Green’s function (or the fundamental Helmholtz solution) corresponding to the problem considered, and they are always located outside the domain of field computation, and not on its boundaries, which is the standard case in boundary element techniques, such as MoM. To determine the unknown weights of the ASs, the appropriate boundary condition is enforced on a set of collocation points (CPs) on the boundary, leading to a linear system, the solution of which yields the numerical values of the weights.

To illustrate the MAS formulation geometrically, Figure 1 depicts the simplest possible case, i.e., a two-dimensional (2D), transverse to the z axis, PEC scatterer illuminated by a time-harmonic, incident electric field ${\mathit{E}}^i$. Let time convention be $\mathrm{e}\mathrm{x}\mathrm{p}\left(j\omega t\right)$, which is suppressed as usual. Let the PEC boundary surface be S. Region I (outside S) is occupied by a homogenous dielectric (typically vacuum or air) with permittivity $\epsilon$, whereas Region II (inside S) is originally filled with the PEC material. To calculate the scattered fields ${\mathit{E}}^s$ and ${\mathit{H}}^s$ in Region I, PEC is removed and its effect is assumed to be equivalent to a set of ASs ($N$ in total) placed on an auxiliary surface S’, inside Region II, which is usually similar to the boundary surface S. According to the aforementioned concept, in the TM incidence case, each AS, located at ${\mathit{r}}_n^{\prime }$, radiates an elementary, z-oriented electric field with intensity equal to

$$E_n\left(\mathit{r}\right)=-j\left(w_n/4\right)H_0^{\left(2\right)}\left(k\left|\mathit{r}-{\mathit{r}}_n^{\prime }\right|\right)$$

(1)

where $H_0^{\left(2\right)}$ denotes the Hankel function of zero order and the second kind, $k$ is the wavenumber pertaining to Region I, $\mathit{r}$ is the observation location, and $w_n$ is the corresponding unknown weight to be determined. To this end, the total electric field is enforced to satisfy the PEC boundary condition at the CPs located at ${\mathit{r}}_m$ on S; therefore, a linear system is constructed, the solution of which yields the values of $w_n$, $n=1,\dots ,N$, namely

$$\left[\mathit{Z}\right]\left\{\mathit{w}\right\}=-\left\{{\mathit{E}}^i\right\}$$

(2)

where $\left\{\mathit{w}\right\}$ is a column vector (of length $N$) containing the weights $w_n$ and $\left\{{\mathit{E}}^i\right\}$ is a column vector containing the incident electric field values evaluated at the CPs, whereas the elements of the interaction matrix $\left[\mathit{Z}\right]$ are given by

$$Z_{m,n}\equiv -\left(j/4\right)H_0^{\left(2\right)}\left(k\left|{\mathit{r}}_{\mathit{m}}-{\mathit{r}}_n^{\prime }\right|\right)$$

(3)

If the number of CPs is also equal to $N$, then the system is square and solution is sought through standard techniques essentially equivalent to the inversion of matrix $\left[\mathit{Z}\right]$; if it is greater than $N$, then the system is over-determined, and the solution is found in the least square sense via the pseudo-inverse of $\left[\mathit{Z}\right]$.

Starting with the fundamental formulation above, the following modifications take place, according to the geometry configuration of each particular problem: for TE incidence on a 2D PEC problem, the ASs radiate z-oriented magnetic fields with intensity equal to

$$H_n\left(\mathit{r}\right)=-j\left(w_n/4\right)H_0^{\left(2\right)}\left(k\left|\mathit{r}-{\mathit{r}}_n^{\prime }\right|\right)$$

(4)

For a PEC scatterer covered by a thin dielectric layer, the Standard Impedance Boundary Condition (SIBC) is a useful approximation to avoid excessive mathematical complications [8]; the AS placement remains the same, but, according to SIBC, the continuity of a linear combination of the electric and magnetic field intensities must be enforced at the CPs. For an exclusively dielectric scatterer, additional ASs must be defined on an auxiliary surface S″ located in Region I to represent the fields within Region II (filled with the dielectric), and boundary conditions at CPs must be enforced separately for both the electric and magnetic fields. In the 3D counterparts of the aforementioned cases, the auxiliary surface S′ is defined in an analogous way, and the ASs are formed as pairs of dipoles, which are perpendicular to each other and simultaneously tangential to S′, radiating elementary fields proportional to the 3D Green’s function, i.e., equal to

$${E_n\left(\mathit{r}\right)=w}_n\mathrm{e}\mathrm{x}\mathrm{p}\left(-jk\left|\mathit{r}-{\mathit{r}}_n^{\prime }\right|\right)/\left(4\pi \left|\mathit{r}-{\mathit{r}}_n^{\prime }\right|\right)$$

(5)

The unknown weights are determined again via the enforcement of the appropriate boundary condition and solution of the resulting linear system of equations.

Since the ASs never coincide with the CPs, the argument of the Green’s function involved in the elements of matrix $\left[\mathit{Z}\right]$ never vanishes, meaning that the Green’s function singularity does not cause computational problems, a situation which is typical in MoM, thus facilitating straightforward matrix filling and almost trivial computer code implementation. However, the price to pay is that convergence is not always guaranteed. Early publications [8] prove that, although the set of functions related to the elementary fields of the ASs is complete, the corresponding Dirichlet problem has a solution if the auxiliary surface encloses all the singularities of the analytical continuation of the scattered field. A detailed explanation of the situation was presented in [11,12], along with a method to find the exact location of the singularities inside the boundary S, based on a representation of S in the complex plane. The method was applied to an ellipse and a multi-foil and was further extended in [13] to piecewise smooth boundaries essentially composed of different shapes connected together to form a closed S. Although this singularity tracing technique can even be implemented in a numerical sense for arbitrary 2D scatterers, is not yet known to be applicable in 3D problems.

Apart from convergence difficulties related to the singularities of the scattered field, MAS is sometimes known to demonstrate numerical instability, even if the theoretical, aforementioned AS location criteria are met. A rigorous explanation of this phenomenon was originally presented in [14], where the fundamental problem of TM scattering from a PEC circular cylinder shown in Figure 2 was thoroughly investigated. Specifically, the MAS linear system was inverted analytically by taking advantage of its circulant matrix properties and the concomitant possibility to calculate all its eigenvalues and the condition number exactly. The main conclusions, illustrated in Figure 3 below (taken from [14]) are as follows:

(a)

When $ka$ approaches any zero of an integer-order Bessel function and $N$ is adequately large, the error in the satisfaction of the boundary condition increases (physically corresponding to a resonance behavior). These resonances are depicted as isolated spikes in Figure 3a; they are inherent to MAS and are independent of the arithmetic precision of the computer environment, affecting even the predicted analytical error.

(b)

The condition number, as well as the numerical error of the system, increases with no upper bound for very large numbers of unknowns $N$, as well as for small radius $a$, resulting in unstable numerical solutions in both cases, i.e., exceedingly large computational errors, although the predicted analytical error remains low.

Based on the results produced in [14], a long series of papers investigated modifications of the problem at hand. In [15], the TE case for the same structure was analyzed and similar conclusions were drawn, revealing that the resonances were again associated with the roots of the Bessel function, and not to its derivative, contrary to what is heuristically expected. In [16], the results were extended to a dielectric circular cylinder, and although the mathematical analysis was much more complicated, involving matrix singular values instead of eigenvalues, the conclusions were surprisingly pretty much the same, in addition to the fact that the external auxiliary surface required (lying in Region I) could be placed almost anywhere without restriction. In [17], a similar investigation was carried out for a PEC cylinder covered with a thin dielectric layer, amenable to the SIBC, where it was demonstrated that, in addition to the errors occurring in the PEC case, numerical instabilities are expected for particular values of the coating permittivity.

Although exploitation of the circulant matrix properties related to circular cylinders is not straightforwardly applicable to 3D problems, so-called 2.5D geometries can indeed be treated; in [18], oblique incidence was combined with the scatterer described in [14], demanding the use of vector ASs to satisfy all boundary conditions. The mathematical analysis revealed, for the first time, resonances related to the zeros of the derivatives of integer-order Bessel functions, in addition to the Bessel functions themselves. Moreover, the condition number, as well as the two kinds of error (analytical and computational) expectedly depended on the incidence angle. A direct extension of [18] to dielectric cylinders was presented in [19], where similar conclusions were drawn, including the freedom to choose the outer auxiliary surface at any distance from the origin.

Feasibility to analytically invert circulant matrices was a key feature in [14,15,16,17,18,19], a fact that was rigorously investigated in [20] for the Laplace equation (e.g., electrostatic problems for a circular disk). Like in time-harmonic problems, Ref. [20] presented a complete error, convergence, and stability analysis. Moreover, circulant matrices’ invertibility was crucial in the rigorous substantiation of a largely unexpected but highly interesting and exceptionally useful finding: under certain conditions, even if MAS currents diverge as $N\to \infty$ because of AS misplacement, radiated fields outside the scatterer may converge. This phenomenon was originally described in [21], where a PEC cylinder, similar to [14], was illuminated by a filamentary source (instead of a plane wave) and convergence was investigated for various radii of the auxiliary surface. Circulant matrices were inverted via a Discrete Fourier Transform (DFT); the standard, discrete MAS problem was compared to a continuous MAS problem amenable to a Fredholm integral equation of the first kind, and the final conclusion was that even if the auxiliary surface does not enclose the singularities of the scattered field, MAS fields in Region I of Figure 1 still converge. MAS currents indeed diverge, as the theory predicts and is demonstrated by erratic oscillations; however, this fact is of lesser importance, since currents are only an intermediate result; fields are the actual sought-after quantities. The current oscillations observed in [21] were asymptotically quantified in [22], where MAS behavior for the PEC cylinder was compared to that of an infinite ground plane. It was concluded that oscillations in the real part of the current due to divergence were inherent to the method when singularities are not enclosed and have to be separated from possible numerical instabilities due to ill-conditioning. For plane-wave incidence, oscillations obviously do not occur, since singularities migrate to the center of the cylinder and are thus inevitably enclosed by the auxiliary surface. Additionally, the behavior of the near field calculated close to the auxiliary surface was studied in [23], where oscillations are found to be also unavoidable when very close to the ASs. An extension of [21,22] to dielectric cylinders was presented in [24], whereas [25,26] discussed the similarities and dissimilarities between MAS and the Extended Integral Equation-EIE (also known as the Null Field Method—NFM); the latter does not demonstrate oscillatory behavior, as opposed to the combination of MAS and EIE [27]. The conclusions drawn in [21,22,23,24] were generalized in [28] so that they also apply to the dual situation, i.e., a cavity interior problem; the extension to off-set ASs and Magnetostatic problems was given in [29], whereas the entire convergence analysis was summed up in [30].

In addition to [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], where important findings about the circular cylinder were shown, independent researchers have proposed various methods to optimally locate the ASs for special cases. In [31], large scatterers were investigated, and the auxiliary surface was proposed to coincide with the caustic surface of Geometrical Optics. An analogous proposal, based on Physical Optics and Love’s equivalent principle, was found in [32]. The case of a large PEC ellipse was handled in [33], where ASs were placed on confocal ellipses, while their number $N$ was compared to the Nyquist sampling number $N_0=2L/\lambda$, where $L$ is the length of the scattering ellipse and $\lambda$ is the wavelength. It was found that the error in the boundary condition decreases for smaller $N/N_0$ and smaller auxiliary surfaces, although the latter should not shrink too much, for reasons already explained. An entirely different perspective was invoked in [34], where the location of the auxiliary surface was optimized on the basis of the condition number of the over-determined liner system. A singular value decomposition (SVD) was performed, troublesome singular values were discarded, and the optimal auxiliary surface configuration (radius and $N$) for a circular cylinder was based on the maximization of the number of remaining singular values. Another approach was adopted in [35], where the locations of the ASs were determined simultaneously with their coefficients, leading to a nonlinear least-squares problem. Moreover, the Level-Set Technique, originating from Fluid Dynamics, was utilized in [36] to optimize MAS. The Level-Set approaches are further discussed hereinafter from other perspectives.

Apart from the important issue of the location of the ASs, another crucial topic is its computational cost and complexity. The simplicity in the construction of the linear system matrix promises a low number of numerical operations, but the question is how it compares to established boundary element methods, such as MoM. The first attempt to quantify complexity was [37], where the focus was to minimize the computational cost of various numerical techniques including MAS. Furthermore, Ref. [38] performed a detailed analysis of the computational cost of MAS and its modified version, called MMAS [39], and compared to MoM. The main conclusion was that in terms of complexity (when $N\to \infty$), all schemes are of the same order, namely $O\left(N^3\right)$. However, in terms of actual computational cost, for finite $N$ and the same discretization density, both MAS and MMAS are superior to MoM. Even when MoM requires a lower number of unknowns for the same accuracy, MAS and MMAS are still less consuming than MoM under a certain threshold of $N$. It should be stressed that the aforesaid comparisons and observations pertaining to complexity and computational cost were limited to the solution of the matrix equations ensuing from each method (i.e., the derivation of the currents). However, the computational cost of the routines aiming at the computation of the EM fields outside/inside scatterers is much more revealing of the superiority of MAS against MoM, since the MAS fields are trivial to compute (mere superposition of the ASs fields), whereas MoM fields require current integration over the scatterer, usually performed numerically. This topic is most interesting by itself and will be examined in a future study.

3. Extension to Generic Geometries

3.1. Gradual Modifications of the Standalone, Circular Cylinder

The theoretical results extracted in [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36] facilitated the application of MAS to more generic cylindrical geometries after suitable adaptation. In [40], the analytical eigenfunction solution for an impedance cylinder illuminated by a plane wave was manipulated in such a way that, after truncation, it was shown to be equivalent to the MAS solution, and MAS unknown weights were expressed in terms of eigenfunction coefficients. The analysis was extended to illumination by an infinite line source in [41]. The location of the ASs was determined on the basis of a Genetic Algorithm (GA) in [42], while dielectric cylinders of an arbitrary cross section were analogously investigated in [43], where the GA was trained to minimize the maximum of the boundary-condition errors. The first generation of the GA sequence mandatorily contains a chromosome associated with a typical MAS placement procedure. Scattering from arrays of PEC cylinders was the objective in [44], where coupling was restricted to neighboring cylinders only to reduce the computational cost, while an extension of [44] to dielectric arrays was presented in [45]. A similar approach was invoked in the case of a “jacket” structure, i.e., a dielectric cylinder containing eccentric, dielectric inclusions of permittivity different to the hosting cylinder [46].

Unfortunately, the circulant matrix behavior is a property of the circular cylinder configuration only, and therefore the MAS system cannot be inverted analytically on these grounds when the cross section ceases to be a circle. However, it is reasonable to expect that, when this cross section is not too different from a circle (i.e., if it is considered to be some sort of perturbed circle), the solution properties should not deviate too much from the circular case. Indeed, this was exactly the main observation of [47], where PEC and dielectric circular-like shapes were investigated. Specifically, the computational error and the condition number were estimated for an ellipse, a “daisy”, and a superellipse, and their plots were strongly reminiscent of the circular case presented in [14,15,16,17,18,19]. The conclusions of [47] were found to be applicable to analogous impedance structures, including an additional Cassinian ellipse in [48], where the Radar Cross Section (RCS) of the scatterers was also computed. To alleviate the difficulties caused by high condition numbers observed in [47,48], two different preconditioning schemes were proposed in [49]. According to the first technique, the MAS matrix for the perturbed geometry is Taylor-expanded and the three first terms are found to be still circulant, meaning that they can be diagonalized just like in the circular case. For small perturbations, higher-order terms are neglected and system multiplication from the left by the inverse of the unperturbed MAS matrix yields a modified linear system, whose matrix is basically a perturbation of the unit matrix and therefore prone to numerically stable inversion. According to the second technique, preconditioning is achieved by multiplying the perturbed MAS system from the left by the inverse of the eigenvector matrix.

3.2. Irregular Shapes

Additionally, concerning EM scattering by 2D non-circular shapes, general aspects of the MAS application and implementation for the case of TM-polarized waves were investigated in [50]. Specific cases were considered, for which the singularities of the analytic continuation of the scattered field into the interior of the scatterer can be analytically determined, and it was shown that the amplitudes (currents) of the sources diverge and oscillate when the auxiliary curve does not enclose these singularities, but, still, the electric fields computed by means of the divergent and oscillatory currents are convergent and correct. A representative example of the application of MAS for a scattering problem involving a non-circular boundary is illustrated in Figure 4 corresponding to a rounded-triangular PEC scatterer [50]. The two panels of Figure 4 show two choices of auxiliary curves, the first enclosing the singularities of the analytical continuation of the scattered field in the scatterer’s interior, and the second does not enclose them. As demonstrated in [50], auxiliary currents resulting from the configuration of Figure 4a are smooth, while auxiliary currents resulting from the configuration of Figure 4b diverge and oscillate between large negative and positive values. Importantly, the electric fields generated by both configurations are convergent and correct; the correctness of the fields was verified by using independent numerical solvers. The analysis and conclusions of [50] were extended in [51] for the case of TE-polarized waves, where also a large-$N$ asymptotic formula for the divergent MAS currents was established and applications to Electromagnetic Compatibility (EMC) were pointed out, including the modeling of shielding properties for inhomogeneous wires, composed of various metals. These EMC applications referred to a PEC scattering configuration coated by a suitable dielectric layer being able to control the scattered field in the external region.

Additionally, plane-wave scattering by 2D models of a raindrop was solved using MAS in [52], while MAS was employed in [53] for the analysis of TM plane-wave scattering from infinite, conducting, or dielectric cylinders, including curved wedges with the latter defined as intersections of circular arcs. Like in earlier publications related to straight edges, ASs were allowed to approach the tip closely, being simultaneously densely packed. The optimization of the parameters of MAS for scattering by an infinite arbitrarily shaped cylinder was performed in [54] by means of a so-called Level-Set Technique. The scattering problem of E-polarized waves by PEC 2D objects of arbitrary piecewise smooth shapes was investigated by means of MAS in [55].

Deviating to further irregular shapes, the SMT (practically almost identical to MAS, as already mentioned) for determining the modes guided along infinite chains of metallic nanowires, when they are embedded in layered media, was presented in [56]. An SMT-based computational tool for the analysis of EM scattering by surface grooves and slits was presented in [57]. The development of MAS to the problem of plane-wave scattering by dielectrics with a periodically modulated shape was analyzed in [58], where applications to frequency-selective surfaces were also examined. In [59], an SMT computational technique was proposed for coping with an array of penetrable cylinders of smooth arbitrary cross sections and partially buried in a penetrable half-space substrate.

3.3. Problems with Infinite/Finite Open Boundaries

The majority of the works using MAS focus on closed and, most importantly, finite geometries. Notable exceptions include [60,61,62,63,64,65,66,67,68,69,70], whereas many more have confronted infinite boundaries but not in depth. In [63], it was numerically demonstrated that fields calculated via MAS exhibit systematic far-field inaccuracies with respect to the analytically calculated field for the case of a filamentary current close to an imperfectly conducting plane. Additional numerical experiments have revealed that the same behavior occurs for a variety of sources, including vertical and horizontal elementary dipoles, as well as wire antennas. Notes and remarks on the application of MAS to such problems can be found in [62]. A theoretical basis to the aforementioned observations can be found in [70]. It was found that this issue is a result of failure to locate the singularities and utilize them as positions for ASs. This manifestation of the non-optimal character of MAS solutions that have not localized the singularities challenges the boundary-condition error as the sole mean to evaluate the quality of MAS solutions. In other words, relatively small or controllable boundary-condition errors do not always and necessarily guarantee the convergence and accuracy of MAS solutions.

With regard to problems with finite open boundaries (e.g., a PEC strip/plate excited by a plane wave), these can be treated in the same way as the ones with closed boundaries with the aid of properly selected air–air interfaces, which are introduced so as to yield closed equivalents [64]. As an exemplary case, one can start with the problem of a plane wave impinging onto a curved PEC strip of infinite length, the open boundary of which can be conveniently closed with the aid of an air–air interface, so as to form an “equivalent” circular cylinder. The structure obtained can be treated via standard MAS modeling, with the only critical modification pertaining to the necessity for satisfying the mixed boundary conditions thus ensuing, namely, the PEC boundary conditions on the strip and the continuity conditions on the trivial air–air interface. Apart from this fact, the formulas for obtaining the unknown EM fields in each region (that is, inside and outside the equivalent cylinder) are identical to the ones describing the fields in the respective regions of a penetrable cylinder. This study has been carried out by the authors and will be the subject of a future paper.

3.4. Inverse Problems

As is perhaps less well-known, MAS has been applied to inverse problems, mainly inverse scattering ones. Efficient MAS algorithms were developed in [4], where, by exploiting scattered field singularities, visualization methods for large scatterers were proposed. In [31], a numerical MAS scheme was presented, which reconstructs a field up to its singularities and, thus, leads into the optimized solutions of the pertinent inverse scattering problems. Additionally, some other early references can be found in [8]. Since then, a small number of works have appeared in the literature, covering grating defects’ detection [71] and inverse scattering by wires on dielectric substrates [72], as well as the determination of body shape and complex permittivity of 2D scatterers [73].

3.5. 3D Problems

The methods presented in Section 2 took advantage of several 2D geometric symmetries and prompted attempts to extend their use to 3D (when possible) cases and in particular, to Bodies of Revolution (BOR), also known as axisymmetric geometries. Electrostatic problems governed by the Laplace equation in such situations were solved in [74], where the properties of block-circulant matrices were exploited. Furthermore, the scattering counterpart for a PEC BOR coated by a thin dielectric layer amenable to SIBC was analyzed in [75], where the auxiliary surface was determined according to [11,12]. A similar approach was invoked for a PEC BOR coated by a chiral medium in [76]. An entirely different AS concept was applied to a dielectric BOR in [77], where fictitious current full rings replaced the standard discrete ASs. Various additional 3D applications were cited in [8]; however, the issue of optimal ASs location for arbitrary 3D geometries remains unresolved. Even in canonical geometries, such as a sphere, the semi-analytical inversion of the MAS matrix has not been achieved yet, in contrast to the circular cylinder, since the matrix fails to be circulant. This is a very challenging and is a key research topic of great significance with respect to MAS efficiency in realistic 3D configurations.

4. Advanced EM Applications

4.1. Antenna Modeling

During the first few decades of their systematic evolvement, MAS and the related techniques were mainly applied to scattering problems [8]. Indeed, early MAS formulations were written down within contexts pertaining to EM scattering only, whereas a very small number of published works mentioned radiation problems. However, during the past two decades, an increasing number of published studies have focused on antenna modeling and on radiation problems tackled via MAS.

The aforementioned works cover thin-wire antennas and arrays of wire elements [78,79,80,81,82,83,84] and elementary radiating elements in close vicinity to infinite ground planes [60,61,62,63,70], as well as other radiating structures [85,86,87,88,89], such as patches, dielectric-resonator radiators, and horns.

The application of MAS to such problems is somewhat less typical in the sense that thin structures and infinite interfaces impose additional challenges that are often treated in an ad hoc manner. For example, an unconventional MAS applied to the classic radiation problem of a vertical electric dipole above an infinite lossy ground is the one presented in [60] and shown in Figure 5 (showing two sets of ASs for the description of the EM fields in the regions above and below the imperfect ground plane); instead of using tangential (to the interface between the two regions of the problem) elementary electric dipoles as ASs, the circular arrays of elementary electric dipoles perpendicular to the interface were used, so as to inherently exploit the cylindrical symmetry of the problem. The density of the dipoles in the arrays and the distance between the arrays were heuristically obtained, so as to reach convincing numerical accuracy and stability. More on classic antenna theory problems involving elementary radiators close to infinite boundaries has been presented and discussed in [61,62,63,70]. The impact of finite ground planes on the radiation from horn antennas was investigated by MAS in [90].

4.2. Waveguides and Resonators

MAS has also proved to be a fast and effective numerical procedure for the analysis of waveguides and resonators. In [91], the method of fundamental solutions (MFSs) in conjunction with the SVD, was applied for the computation of cutoff wavenumbers of elliptical waveguides. MAS with external excitation was proposed in [92] for analyzing arbitrarily shaped hollow waveguides. The method is based on the physical response of a waveguide to excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. A MAS-based technique was presented in [93] for analyzing the modes characteristics of a half-mode substrate-integrated waveguide. In [94] and [95], the MFCM mode analysis was proposed for 2D metallic waveguides having arbitrary contours and that are filled with isotropic or anisotropic materials. In [96], it was shown that MAS with an external or an internal excitation source leads to the accurate computation of the cutoff wavenumbers (eigenvalues) of cylindrical waveguides (simply or multiply connected waveguides with circular, elliptical, and rounded-triangular cross sections). In particular, when an internal source is used, the eigenvalues are accurately computed without employing a regularization procedure. Moreover, in [97], it was demonstrated that the eigenvalues of the waveguide can be determined, in a simpler manner, by the means of the auxiliary currents’ vector norm (ACVN). In [98], it was shown that MAS can be effective in computing the propagation constants (eigenvalues) of a cylindrical dielectric waveguide with the core of the arbitrary cross section, provided that two excitation sources (an electric and a magnetic current filament lying within the core) are employed to excite hybrid modes of the dielectric waveguide; the modified method was, thus, named MAS with Two Excitation Sources (MAS-TES). The theoretical framework, as well as numerical implementations of MAS with external or internal excitations sources, with applications in electromagnetic scattering, mode analysis, and shielding problems, were presented in [99]. Furthermore, a method of dipole representation (MDPR) was proposed in [100] for determining the cutoff wavenumbers of arbitrarily shaped waveguides. MDPR is a method similar to MAS with an excitation source, which is based on the solution of the 2D Laplace BVP with rational functions (dipole representations) as the basis functions. The ACVN-based MAS with an excitation source was extended in [101] and employed for the computation of the cutoff wavenumbers of arbitrarily shaped, PEC waveguides, having anisotropic permittivity and permeability in-fills.

4.3. Multilayered Media

MAS has also been applied for the solution of EM BVPs comprising planarly and cylindrically multilayered media. In [102], MAS was developed by employing elementary electric/magnetic currents placed on auxiliary surfaces to compute the fields resulting from the primary ionospheric current and secondary currents induced in the layered earth. An ASs scheme was presented in [103] for the calculation of EM fields inside a cavity filled with a multilayered dielectric. ASs and collocation points were introduced on the sidewalls of the cavity where the boundary conditions need to be imposed. Each considered AS was a source of the omnidirectional radiation of the waveguide’s eigenmode. EM scattering by a two-layered cylinder and a cylinder buried in a two-layered earth medium (a problem of practical use in subsurface imaging) was analyzed in [104] by a quasi-point matching method with fictitious sources (QPM-FS). An extended MAS (EMAS) was presented in [105] for scattering problems by multilayered structures (dielectric shells, multilayered dielectric cylinder of different materials, and conducting cylinders coated with a dielectric or a metamaterial). A MAS-based method, which offers an alternative to Sommerfeld integrals methods, and is used for the calculation of a point source radiation in a multilayered background, was presented in [106]. This method lies on the decomposition of reflected and transmitted fields on a basis of secondary sources with amplitudes determined by applying boundary conditions at each multilayer interface, yielding an over-determined system of equations. ASs’ numerical schemes for the analysis of stratified media and the design of resonant cavities antennas were elaborated upon in [107].

4.4. Anisotropic and Chiral Scatterers, Metamaterials and Graphene

MAS and its closely related variants have also been applied to solve scattering and radiation problems by complex media and metamaterials. The fundamental concepts of MAS are identical to the isotropic case; the difference lies in the Green’s function, which is obviously more complicated to derive. Photonic crystals made of bi-isotropic or chiral materials were treated in [108], whereas left-handed media were the analysis topic in [109]. Three-dimensional scattering from uniaxial objects with a smooth boundary using a multiple infinitesimal dipole method (MIDM) was investigated in [110]. The implementation of the MFCM to analyze the scattering from homogeneous anisotropic cylinders was developed in [111], where monostatic and bistatic scattering widths of cylinders with and without sharp edges were investigated. In [112], an MFCM incorporating the generalized sheet transition condition (GSTC) was presented to simulate 2.5D cylindrical metasurfaces with arbitrary cross sections. Concerning chiral media, MFS schemes were developed in [113] for EM scattering by a penetrable chiral obstacle in an achiral environment, in [114] for scattering by a perfectly conducting body embedded in a chiral medium, and in [115] for direct and inverse scattering problems corresponding to a penetrable chiral body within a chiral environment. Furthermore, the application of MAS in metamaterial structures was studied in [116], where the effect of a conducting rod of arbitrary shape on a metamaterial slab with a low refractive index was analyzed. An efficient method for the accurate computation of the response of photonic crystal filters was developed in [117] by combining the Model-Based Parameter Estimation with a Multiple Multipole Program and the MAS.

Recently, modified and enhanced versions of MAS have also been implemented for the investigation of structures covered by graphene. Specifically, in [118], ideal monolayer graphene was evaluated as a shielding material for cylindrical configurations in the RF/microwave region by using MAS for the numerical modeling of the involved structures and replacing the graphene layer by an Impedance Matrix Boundary Condition (IMBC). Additionally, in [119], the shielding effectiveness of anisotropic graphene was examined in the RF/microwave region. The graphene layer was replaced by a surface current boundary condition (SCBC) and MAS was employed to analyze planar and cylindrical shielding configurations.

4.5. Detection of Burried Objects and UXO

It is also worth pointing out that MAS-based techniques have been exploited effectively for detecting buried objects and classifying Unexploded Ordnance (UXO) correctly. In [120], MAS was developed for the solution of wide-band EM Induction (EMI) problems involving highly conducting and possibly permeable metallic objects modeling buried UXO. In [121], a hybrid algorithm, with standard MAS and MAS with Thin-Skin Approximation (TSA), was introduced and applied for, solving the EMI forward problem, while, in [122], interaction phenomena were investigated between highly conducting and permeable metallic UXO objects in the EMI frequency range (i.e., from 10 s of Hz up to 100 s of kHz). Furthermore, in [123], MAS numerical schemes were developed to study the frequency dependence of the EMI response by different classes of UXO scatterers. An enhanced variant of MAS was established in [124] for a fast and accurate representation of responses for 3D heterogeneous objects placed in an arbitrary time-varying field. Finally, a MAS-inspired orthonormalized volume magnetic source model was invoked in [125] for UXO discrimination.

5. MAS Variants and Hybridizations

Throughout the past two decades, MAS and the several closely related methods discussed herein and in [8] have notably evolved into a broad spectrum of variants and hybrids possessing certain advantages over traditional MAS. The recent variants of MAS-related methods include notable advancements, such as modified and improved-accuracy versions [11,12,39,52,61,68,126,127] (including the MMAS discussed above), spurious-free implementations [128], regularized variants [129], and iterative implementations exploiting randomly distributed sources [130], as well as combined-source solutions (solutions based on both electric- and magnetic-type sources) [131].

Another important feature of the multitude of MAS variants is the easiness of their hybridization with other techniques of computational electromagnetics, but also with techniques from other areas of science. The ensuing hybrids expand the range of applicability of MAS and permit its adaptation to more demanding cases/problems. With regard to the hybridizations with MoM, these mostly aim at incorporating thin structures like wires [132,133], coping with sharp corners [134], and enhancing the accuracy and convergence rate of the solutions for problems involving closely positioned illuminating sources [135]. A similar hybrid technique for the analysis of thin-wire antennas and arrays is that of [136,137], which utilizes ASs whose currents are obtained by applying reaction matching instead of point matching. Another promising hybrid is that with the finite-element method (FEM), which has been applied to second-order scalar elliptic BVPs in [138]. Similarly, the hybridization of MAS with the Mode-Matching Technique (MMT) seems auspicious for solving problems involving waveguide openings, apertures, and horns [89]. More extensive knowledge and experience is now available for the hybridization of MAS with the Fast Multipole Method (FMM) [139,140], and the similar scattering-matrix method [141]. In essence, the fundamental concept of FMM can be used to effectively group the interactions between ASs and CPs and, subsequently, to approximate them by simpler asymptotic formulas, so as to achieve computational efficiency, without notably compromising numerical accuracy, especially for large-scale problems that involve electrically large scatterers or multitudes of electromagnetically coupled objects. Many numerical experiments have shown the efficacy of the said MAS-FMM hybrid and have revealed huge savings in the associated computational cost [139,140,141,142,143]. By definition, the MAS-FMM hybrid is more efficient than standard MAS only for electrically large geometries. Also, accuracy may be depleted for small geometries, due to the inadequate accuracy of the FMM approximation [143]. Furthermore, its main challenge may be slow convergence, due to possible ill-conditioning of the linear system. The remedy to this situation is suitable preconditioning, greatly enhancing the efficiency of the scheme [142].

Apart from the many variants and hybrids discussed above, there exists an increasing number of works applying MAS-related methods in conjunction with evolutionary techniques like Particle Swarm Optimization (PSO) [144], GAs [42,43,104,145], and the Level-Set Technique [36,54,146], mainly in an effort towards systematic and automated procedures for selecting optimal/suboptimal attributes/parameters regarding the placement of ASs and CPs.

Although MAS-related methods are generally considered to be frequency-domain ones, a time-domain variant was introduced in [147], where the MOT (Marching-On-in-Time) procedure was applied to a scattering problem. The adjusted ASs invoked were triangular time pulses with unknown coefficients, whose values were still determined through the solution of a linear system. Another time-domain approach was followed in [148], in which MAS was repeatedly applied to approximate the response of a scatterer to a Gaussian pulse spanning the frequency range from 0 to 6 GHz. The diffraction of a periodic pulse radiated by a horn antenna and impinging on a circular dielectric object was investigated in the time domain by MAS in [149]. Time-domain applications are extremely limited so far and cover 2D circular cylinders only; results for arbitrary geometries and 3D configurations are still unknown, and constitute an open field of future research. Again, the key issue to be resolved remains the optimal location of the ASs.

To emphasize the versatility of the MAS family, their applicability to non-EM Physics problems involving wave propagation is worth mentioning. In 3D acoustics, the application of MAS was analytically and numerically investigated in [150], where a soft sphere excited by an external point source and a soft ellipsoid excited by a plane wave were analyzed. The direct acoustic scattering problem of a point source field by a penetrable spheroidal scatterer hosting an impenetrable spheroidal body was considered in [151], and it was solved numerically by combining a Vekua transformation technique (providing analytical solutions of the associated Helmholtz equation) with the MAS employed to represent the contribution of the inhomogeneity. Finally, a MAS variant involving internal auxiliary source–sinks (MIASS) for 2D interior Dirichlet problems was presented in [152].

6. Discussion, Conclusions and Future Work

Judging from the very long list of papers cited in this review, along with the vast variety of different applications described herein, the salient features of MAS as a computational tool cannot be overemphasized. Its advantages over standard, perhaps more mature, integral-equation techniques, such as MoM and its fast variants like the Multi-Level Fast Multipole Algorithm (MLFMA), have been exploited by several research groups worldwide to produce accurate and dependable numerical data at a very low computational cost (including algorithmic simplicity). Nevertheless, further work is necessary before MAS is established as a completely robust method. The optimal location of the ASs in arbitrary geometries is still an open issue that has not yet been resolved for all kinds of geometrical configurations. Moreover, 3D applications are still extremely few compared to 2D, while combinations with techniques, such as tensor decomposition, domain decomposition, and other numerical linear algebra-based methods would be very interesting to investigate. In particular, domain decomposition would greatly improve computational savings, since the AS density would vary within the geometry, being significantly reduced at locations close to smooth sections of the scatterer. Therefore, there is still plenty of room left for both young and veteran researchers to extend MAS towards all sorts of practical EM problems encountered in EMC, Communications, Remote Sensing, Biomedical Imaging, etc.