THE METHOD OF AUXILIARY SOURCES (MAS)
IN COMPUTATIONAL ELECTROMAGNETICS
Dimitra I. Kaklamani
Department of Electrical and Computer Engineering National Technical University of Athens
9 Iroon Polytechniou Str., GR-15780, Zografos, Athens, Greece Tel: +301 7722277, Fax: +301 7723557, E-mail: [email protected]
Key words: Method of Auxiliary Sources, Computational Electromagnetics, Boundary Value Problems.
Abstract. The Method of Auxiliary Sources (MAS) is a numerical technique, alternative to the traditional surface integral equation formulation, for solving elliptic boundary value problems, appearing in electromagnetic scattering analysis, antenna modelling, waveguiding structures etc. This paper provides an overview of MAS, as applied to computational electromagnetics. The fundamentals of MAS are presented and its advantages over the Method of Moments are highlighted, while special emphasis is given to advanced topics and recent developments of MAS.
1 INTRODUCTION
The Method of Auxiliary Sources (MAS) is an advanced and highly promising numerical technique, for solving elliptic boundary value problems. It constitutes a feasible alternative to the traditional surface integral equation formulation and possesses significant advantages, concerning numerical stability, computational accuracy and easy implementation that make it attractive for numerical modelling problems appearing in electromagnetic (EM) scattering analysis, antenna modelling, waveguiding structures etc. MAS has been introduced, named and developed for years by Georgian researchers, at the Republic of Georgia (former Soviet Union)1-27, while, recently, a significant co-operation with Greek researchers has been also developed in the same area28-43. It is important to emphasise that, the same method has been independently developed by other research groups all over the world under different names, such as "The Current Model Method"44-52 or "The Discrete Singularity Method"53-55, mainly for treating EM scattering problems. The common idea in all these methods is that, the EM boundary value problem is not formulated in terms of continuous equivalent surface currents flowing on the same surfaces, where the corresponding boundary conditions are enforced, but in terms of discrete fictitious currents, the "auxiliary sources" (ASs), situated at some distance away from the physical boundaries.
Indeed, in typical integral equation techniques56, by applying the equivalence principle, the EM field inside a homogeneous, isotropic and linear region/domain of the structure under investigation can be expressed in terms of a known impressed field (the excitation) and unknown equivalent electric and magnetic continuous currents distributed over its boundary surface, which are employed in order to model the field discontinuity across the boundary surface. Then, by expressing the corresponding boundary conditions in terms of the impressed field and the equivalent continuous currents, different types of surface integral equations are obtained, which, in the general case, are numerically solved via the Moment Method (MoM)56-58.
Unlike MoM, MAS does not account for current discontinuities on the boundaries, but directly constructs the unknown EM fields in each domain with the assistance of fictitious, equivalent point sources, the ASs, displaced with respect to the boundaries. These ASs are chosen so that their fields are elementary analytical solutions to the boundary value problem. The actual EM fields in each domain are then expressed as weighted superpositions of these analytical solutions and the unknown expansion coefficients are determined through utilisation (usually point-matching) of the relevant boundary conditions.
It should be noted that, the concept of field approximation by means of a linear combination of analytically known field functions is not unique to the MAS approach alone. The thin wire approximation, where the current is modelled as a filament flowing along the axis of the wire, as well as the classical Mie solution for a sphere, with the unknowns being discrete multipole sources at the centre of the sphere, have conceptual similarities to MAS. However, they are both restricted to specific geometries. The innovation that enables application to general geometries is the use of multiple origins. The idea has been successfully used for years in Electrostatics, known as "The Charge Simulation Method"59, according to
which fictitious discrete line charges are distributed at multiple origins outside the region, where the electrostatic field is to be computed. The potentials of the fictitious charges are particular solutions of the Laplace and Poisson equations and their magnitudes are determined by satisfying the boundary conditions at discrete points on the boundary. Accordingly, in Electrodynamics, various numerical methods, which are based on the "Extended Boundary Condition Method" (EBCM)60 and are often known as "Generalized Multipole Techniques" (GMT’s) or "Multipole Multipole Techniques" (MMT’s), simulate EM fields by means of cylindrical and spherical wave multipole functions, up to some specified order, centred at multiple origins, for treating two-dimensional (2D) and three-dimensional (3D) problems respectively61-68. Although independently evolved, MAS could, in a sense, be considered as a special case of the GMT, in which only poles of zero order are activated, forming a set of fictitious, but otherwise physically interpretable and analytically simpler sources.
The present paper is organised as follows: In section 2, the fundamentals of MAS are presented, by giving two generalised EM scattering examples and the advantages of MAS over the MoM are highlighted. Advanced topics on MAS are discussed in section 3, while recent developments/modifications of MAS in diverse scattering, inverse scattering and radiation problems are presented in section 4. The paper is completed by the concluding section 5.
2 THE FUNDAMENTALS OF THE METHOD OF AUXILIARY SOURCES
According to the most widely used up to today version of MAS, which will be referred to as "standard" MAS, the radiating ASs are chosen to be either current filaments for 2D problems, generating fields proportional to a Hankel function (2D Green’s function), or pairs of elementary dipoles for 3D problems, generating fields proportional to the 3D Green’s function. The members of each pair are perpendicular to each other and, simultaneously, tangential to the auxiliary surface, to account for the magnetic field discontinuity across the auxiliary surface. In a standard MAS formulation, the ASs are homogeneously distributed on auxiliary surfaces, conformal to the physical boundaries. The EM fields in each domain are expressed as weighted superpositions of the EM fields generated by all the ASs, with unknown expansion coefficients, which are determined by point-matching the relevant boundary conditions at discrete collocation points (CPs) on the physical boundaries. The distribution of the CPs is, again, homogeneous and their number is, usually, equal to the number of ASs, although, sometimes, it is considered preferable to overdetermine the linear system of equations and solve it "as well as possible" (e.g. in the sense of "least squares")66.
For a better understanding of the fundamentals of standard MAS, two generalised problems of EM scattering of an external known electric field Einc by a perfect electric conductor (PEC) and a homogeneous isotropic dielectric scatterer are considered. The corresponding geometries are shown in Figs. 1 and 2, respectively.
physical surface S PEC E inc μ0, ε infinite homogeneous space μ0, ε mathematical surface S E s + E inc H s + H inc ASs E inc auxiliary surface S΄ (a) (b)
Figure1. (a) PEC scatterer of smooth surface S illuminated by a known external field Εinc inside an infinite
homogeneous and linear space with dielectric permittivity ε and magnetic permeability µ0. (b) Equivalent MAS
model: the PEC scatterer does not exist, the ASs radiate inside an infinite homogeneous and linear space with dielectric permittivity ε and magnetic permeability µ0 and they are located on an auxiliary surface S′ enclosed by
the (fictitious) physical surface S, while the CPs, on which the boundary condition is satisfied are located on the (fictitious) S.
In the first problem, the PEC, with a smooth external surface S, is located inside an infinite homogeneous isotropic and linear space, with dielectric permittivity ε and magnetic permeability µ0 (Fig. 1(a)). The ASs radiate, in the absence of the PEC, inside an infinite homogeneous isotropic and linear space, with dielectric permittivity ε and magnetic permeability µ0 and they are located on an auxiliary surface S′, enclosed by the (fictitious) physical surface S (Fig. 1(b)). Then, the unknown scattered field E is described ass
∑
∑
= ⋅ = n n n n s n s E G a E (1)where E denotes the electric field of the n-th AS, with sn
n
G being known analytic solutions of the corresponding wave equation† and a being unknown coefficients to be determined. Byn imposing the satisfaction of the boundary condition (zeroth total electric field tangential to S) at discrete CPs on the (fictitious) physical surface S, a system of linear equations is derived in terms of a , whose solution gives the unknown coefficients and, consequently, the unknownn scattered field E . Existence and uniqueness issues of the MAS solution have been explicitlys addressed in47.
† Equation (1), as well as the following equations (2) and (3), are given in the most general format of 3D
problem, where the unknown coefficients are vectors and the known analytic solutions of the wave equation (Green’s functions) are dyadics.
physical surface S region II μ0, ε E inc region I μ0, ε0 infinite homogeneous space μ0, ε0 mathematical surface S E s + E inc H s + H inc
first ASs set E inc auxiliary surface S΄ (a) (b) infinite homogeneous space μ0, ε mathematical surface S E II, H II
second ASs set auxiliary
surface S˝
(c)
Figure2. (a) Homogeneous isotropic and linear dielectric scatterer with dielectric permittivity ε, magnetic permeability μ0 and smooth surface S illuminated by a known external field Εinc in the free space. (b) Equivalent
MAS model for describing the scattered field in region I: the dielectric scatterer does not exist, the ASs radiate in the free space and they are located on an auxiliary surface S′ enclosed by the (fictitious) physical surface S, while
the CPs, on which the boundary conditions are satisfied are located on the (fictitious) S. (c) Equivalent MAS model for describing the field in region II: the dielectric scatterer does not exist, the ASs radiate inside an infinite homogeneous and linear space with dielectric permittivity ε and magnetic permeability µ0 and they are located on
an auxiliary surface S˝ enclosing the (fictitious) physical surface S, while the CPs, on which the boundary conditions are satisfied are located on the (fictitious) S.
In the second problem, a homogeneous isotropic and linear dielectric scatterer with dielectric permittivity ε, magnetic permeability µ0 and smooth external surface S, is located in the free space (Fig. 2(a)). Now, two sets of ASs are needed for the MAS formulation: one set of ASs, radiating in the free space in the absence of the dielectric scatterer and located on an auxiliary surface S′, enclosed by the (fictitious) physical surface S (Fig. 2(b)) and a second set of ASs, radiating again in the absence of the dielectric scatterer, but inside an infinite space filled by the material of the dielectric scatterer and located on an auxiliary surface S′′, enclosing the (fictitious) physical surface S (Fig. 2(c)). Then, the unknown scattered field Es
in region I is described as a sum of the fields of the first set of ASs
∑
∑
= ⋅ = n I n I n n s n s E G a E (2)where E denotes the electric field of the n-th AS of the first set, with sn a being unknownIn coefficients to be determined and G being known analytic solutions of the wave equationIn for E , while the unknown field s E in region II is described as a sum of the fields of theII second set of ASs
∑
∑
= ⋅ = n II n II n n II n II E G a E (3)where E denotes the electric field of the n-th AS of the second set, with IIn a being unknownIIn coefficients to be determined and G being known analytic solutions of the wave equationIIn for E . By imposing the satisfaction of the boundary conditions for the tangential to S totalII EM field at discrete CPs on the (fictitious) physical surface S, a coupled system of linear equations is derived in terms of a and In a , whose solution gives the unknown coefficientsIIn and, consequently, the unknown fields E and s E .II
The salient feature of MAS, that renders the technique very efficient and accurate, is the non-vanishing distance between source and observation points (i.e. between ASs and CPs). This displacement with respect to the boundaries practically eliminates the Green’s function singularity problem of a typical MoM kernel, forming a set of smooth functions on the boundaries. Moreover, the technique is conceptually very simple to be implemented: by choosing a finite number of AS and matching the boundary conditions at a discrete set of points, a matrix equation instead of an integral equation is automatically derived, without any MoM transformation needed. Furthermore, since each solution in the set is analytically known, there is no need to integrate currents in order to determine fields at any stage of the solution, i.e. filling the kernel, checking the results, computing near- and far-fields etc.
3 ADVANCED TOPICS ON THE METHOD OF AUXILIARY SOURCES
The choice of the type of ASs is very crucial in implementing MAS. In principle, any set of fictitious sources generating fields, which are analytical solutions of the Maxwell’s equations, are acceptable. As already presented in the previous section, in standard MAS, for 2D problems, the radiating AS are chosen to be current filaments of infinite length, generating fields proportional to a Hankel function (2D Green’s function), as
zˆ )) ( k ( H a ) ( En ρ = n (o2) ρ−ρn (4)
where an e+jωt time harmonic dependence is assumed, En(ρ) is the electric field of the n-th AS at the observation point ρ, a is the unknown weight for the n-th AS, n H(o2)(⋅) is the Hankel function of zeroth order and second kind, k is the wavenumber of the medium in which the ASs radiate, ρn is the position vector of the n-th AS and zˆ is the unit vector along the transverse axis of the 2D problem. For 3D problems, in standard MAS, the radiating ASs are usually chosen to be pairs of elementary dipoles, generating fields proportional to the 3D Green’s function. The members of each pair are perpendicular to each other and, simultaneously, tangential to the auxiliary surface, to account for the magnetic field discontinuity across the auxiliary surface. Nevertheless, other types of ASs can be used, depending on the specific problem (type of excitation, geometry etc.).
Other important aspects in implementing MAS are the location and number of the ASs, as well as the number and distribution of the CPs on the boundaries. These issues are, in principal, dependent on the type of excitation, the geometry and the electrical dimensions of the structure and affect significantly the efficiency of MAS. Many researchers have used empirical rules in determining essential parameters, such as the location of the ASs‡. Although the formulation allows for any number/location/distribution, in a standard MAS implementation, the radiating ASs (current filaments or Hertzian dipoles) are usually homogeneously distributed on fictitious auxiliary surfaces, usually conformal to the actual surfaces of the structure. However, non-uniform distributions, as well as non-conformal auxiliary surfaces have been proven to be useful, e.g. for treating edges in scattering problems or close to the excitation region in radiation problems. Moreover, the distance between the auxiliary surface and the actual surface of the structure -especially in conjunction with the distance between two adjacent ASs- affects significantly the efficiency of the MAS solution. It has been proved that the convergence rate and the accuracy of a MAS code is dependent on the relative location of the ASs with respect to the singularities of the actual field simulated by these sources29-31,49,69. If there are no singularities of the field lying between the surface on which the AS are placed and the actual surface of the structure, then a converging MAS solution is guaranteed. If there are singularities lying between the auxiliary surface and the actual surface of the structure, then the MAS solution would not be necessarily convergent. It should be noted though49 that, even in the latter case, where mathematically admissible solution does not exist, a standard MAS formulation can still be followed and results sufficiently accurate for some engineering purposes might be derived before encountering numerical instabilities. The optimum solution, in the sense of the minimum number of used ASs, can be derived, if the ASs are located exactly at the singularities of the actual field simulated by these ASs29-31. This remark illustrates that MAS, although a numerical technique, possesses to a certain degree some physical insight to the examined EM problem. This is, otherwise, an exclusive advantage of the high-frequency asymptotic techniques, versus traditional numerical techniques. Moreover, it is, obviously, a remark of major
importance in treating large scale EM problems, where standard MAS requires a huge number of ASs. In most practical problems, though, the exact location of the field singularities is not known a priori and, moreover, this location is often frequency dependent. In EM scattering problems, for example, the singularities of the scattered field depend on the operation frequency, the geometry of the scatterer and the smoothness of the incident external field on the scatterer surface. Therefore, a further theoretical investigation of the above mentioned issue is of major importance, with a possible use of image theory and analytical optimisation techniques, which have already been successfully applied in the past to canonical geometries, based on analytical continuation concepts49 and caustic surfaces theory29. Discussion on specific examples illustrating these ideas will be carried out during the Conference.
As far as the collocation points are concerned, their number is usually taken to be equal to the number of AS, although in some cases it is preferable to over-determine the derived matrix equation and solve it to provide a least-squares fit to the boundary condition66. In any case, what is checked is the error on the boundary condition by increasing the number of AS and/or collocation points, because between the matching points the mismatching may be quite big. Specific convergence examples will be presented and discussed at the Conference.
4 RECENT DEVELOPMENTS OF THE METHOD OF AUXILIARY SOURCES
Over the last three decades the capabilities of standard MAS have been extensively studied and its accuracy has been significantly improved, mainly for treating EM scattering problems. Recent developments of MAS that will be presented at the Conference include the following topics:
− Hybridisation of MAS with MoM35-39 in solving 3D EM scattering structures consisted of isolated conducting and dielectric parts. The electric fields inside the dielectric parts are represented by pulse piecewise functions, while ASs are employed to represent the currents on the conducting parts.
− Development of a holographic imaging technique31 for solving inverse scattering problems. Standard MAS is modified, using "absorbing" ASs.
− Scattering by edges34. Standard MAS is modified, using non-uniform ASs’ distribution and auxiliary surfaces non-conformal to the physical boundaries in the vicinity of the edges.
− Treatment of dielectrically coated 2D conducting scatterers modelled by a Standard Impedance Boundary Condition (SIBC)34, with special emphasis on a complexity analysis of MAS compared to MoM. Although SIBC is not expected to be valid in the vicinity of an edge, the developed MAS/SIBC model remains sufficiently accurate for a wide class of scattering structures including wedges as well.
− Radar Cross Section (RCS) evaluation of jet engine inlets, using MAS in conjunction with modal field representation and exploitation of the inlet cylindrical periodicity33.
− Development of a modified version of MAS, suitable for treating thin structures and open domains40-43. The essence of the proposed modification is that, instead of the EM fields generated by the fictitious current ASs, it is the currents themselves and the charges on the ASs, which are used as unknowns. The technique is employed to analyse microstrip
antenna elements of planar and conformal array systems, while derived results are further optimised, by using non-uniform grids and different types of basis functions.
5 CONCLUSIONS
This paper gives an overview of MAS -a numerical technique, alternative to the traditional surface integral equation formulation, for solving elliptic boundary value problems- as applied to computational electromagnetics. The fundamentals of MAS have been presented and its advantages over the Method of Moments have been highlighted, while special emphasis has been given to advanced topics and recent developments of MAS, which prove its efficiency and its capability of becoming a power computational tool in treating a wide range of EM scattering and EM radiation problems.
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