Classical Delta Function Dyadic Approach

Dyadic representations enable solutions of electromagnetic problems to be more compactly written. Common representations of the dyadic Green’s function used in EM problems in- clude: spatial representation in terms of derivatives of the common scalar Green’s function and the eigenfunction representation in terms of vector wave functions of the geometry. Recall that the Green’s function of a wave equation is a solution of the wave equation for a point source. When the solution due to a point source is known, the solution to the general source can be obtained by the principle of linear superposition [69, p.24].

The prior section performed a direct integration of the vector wave equation using the second vector Green’s theorem. This section considers a method based on the use of Dyadic analysis, known to give more compact representations for complex scattering problems.

Recall the vector wave equation as

∇ × ∇ × E(r) − k2 E(r) = iωµJe(r) (3.2.22)

The Dyadic Green’s function satisfies

∇ × ∇ × G(r, r0) − k2 G(r, r0) = Iδ(r − r0) (3.2.23)

In similar fashion to earlier, we use the Second Vector-Dyadic Green’s theorem, written Z V £¡ ∇ × ∇ × P¢· Q − P ·¡∇ × ∇ × Q¢¤dV = Z S ˆ n ·£P ס∇ × Q¢+¡∇ × P¢× Q¤dS (3.2.24)

Substituting (3.2.22), (3.2.23) in (3.2.24) and applying the singularity exclusion approach, E(r) = lim δ→0 Z V −Vδ ¡

∇2E(r0) + k2E(r0)¢· IG(r, r0) dV + I S £ ˆ n(r) × E(r0) · (∇ × G(r, r0)¤dS0 + I S ©

(ˆn(r0) × ∇ × E(r0)) · IG(r, r0) +£n(rˆ 0) · E(r0)∇ · IG(r, r0)

− ˆn(r0) · IG(r, r0)∇ · E(r0)¤ªdS0 (3.2.25)

Recognizing that for any r ∈ V ,

∇2E(r0) + k2E(r0) = iωµ ·

I +∇∇

k2

¸

· Je(r) (3.2.26)

and the conventional electric field Green’s dyadic, Gee is the solution of

Gee(r, r0) = _4π1 · I +∇∇ k2 ¸ G(r, r0), for r06= r (3.2.27)

where G(r, r0_{) is the usual free space scalar Green’s function. We can reduce the volume}

integral term to the key electric field equation of interest [70],

E(r) = iωµ0

Z

V

J(r0) · Gee(r, r0) dV0 (3.2.28)

known to be valid for r outside of V and is valid inside V as well. If we let the surface S extend to infinity and apply the radiation condition, the surface integrals in (3.2.25) disappear, leaving the volume integral of the form in equation (3.2.28).

In deriving the equation for the electric field in terms of the electric dyadic Green’s function, two methods of solution exist, one integral formula approach using a classical delta-function approach, and one without. Historically, there has been much confusion in that different solution approaches yielded differing results, where uniqueness requires the same result [71, 72, 70, 73, 74]. The fundamental difference involved the specification of the principal volume, and the associated manipulation of integral and differential operators in the field derivations. At the source point, this interchange becomes invalid [74].

For a differential operator L, its inverse is an integral operator, which can be assumed to have a kernel g(r; r0). The integral operator is defined as [14, pp. 45-51]

L−1u(r) =

Z

By interchanging the order of integration and differentiation

u(r) = L−1Lu(r) =

Z

Lg(r, ζ)dζ (3.2.30)

which implies that

Lg(r, r0) = δ(r, r0) (3.2.31) In a rigorous sense, the operator interchange above could not be performed, however, the dirac- delta function is a symbolic function, and has special properties that enable such treatment. These properties were investigated by Schwarz in the Theory of Distributions, presented by Dirac [15]. In the strictest sense, however, special accommodations needs to be made if dyadic functions or operators are involved. This is the equivalent of a special treatment for the handling of the exclusion volume. Van Bladel [70] concluded that this dyadic Green’s function is insufficient at the source point to determine the correct value of E(r), requiring the addition of a dyadic representing a source contribution. Another name for depolarizing dyadic L is “normalized dyadic solid angle”.

Yaghjian [74] reformulated equation (3.2.28) to take this source dyadic into account,

E(r) = iωµ0lim δ → 0

Z V −Vδ Gee· J dV0+L · J_iω² 0 (3.2.32) where L = 1 4π Z Sδ ˆ n e_R0 R2 dS0 (3.2.33)

Note that ˆn is the unit normal out of the principal volume and e_R0 is the unit vector from r0

to r. As Yaghjian points out [74, p. 252], the dyadic L, which depends on the geometry of the the principal volume compensates exactly to produce the unique E(r), regardless of the geometry of said volume.

3.3

Conclusions

In section 3.1 we introduced the Green’s function method with potential theory as the basis; Poisson’s equation is solved, giving a particular and complimentary solution that accommo- date Dirichlet and Neumann boundary conditions on the surface of a region of interest. A progressive introduction of Green’s theorem is given, with the initial focus on the scalar static problem. We build on these foundations in section 3.1.3 with the solution of the scalar wave problem, defined as an inhomogeneous scalar wave integral equation; the mathematics of Green’s functions and Green’s theorem are evolved. The surfaces and regions constructed are consistent with those in all radiation and scattering problems reviewed in this document.

The discontinuity, where source equals observation point, is handled by use of a singularity exclusion volume and limit calculations. For the scalar wave equation, it was shown that the vector problem was reduced to a simplified scalar form, due to separation of variables. Section 3.2 reviews derivations of frequency-domain (time harmonic) integral equations in vector and dyadic form specific to the general radiation and scattering problem for a perfectly conducting body. A direct integration method, considered equivalent to that of Stratton and Chu was presented. The second vector Green’s theorem is applied, with the typical singularity exclusion surface used. With the application of boundary conditions for perfect electric conductors, the Stratton-Chu representation is reduced to the well-known magnetic field integral equation (MFIE) and electric field integral equation (EFIE) form. This form is used extensively in chapters 4 and 5 for Spectral Expansion methods.

A review of integral equation theory applied to the Electromagnetics field would not be complete without dyadic analysis and the electric Green’s dyadic. Two approaches were presented, the classical delta function dyadic approach, and Yaghjian’s “combined integral formula” approach. Some of the contention surrounding these two methods was discussed; Yaghjian’s formulation, which introduces the source dyadic has stood the test of time. The root cause of the problem in the delta function approach was shown to be in the interchange of operators and the handling of the principal volume. The dependency on the shape of the source region, and a treatment of the dyadic approach is given in section 3.2.2.

Operator Analysis of EFIE and

MFIE

This chapter draws together a diverse range of topics in integral theory and functional anal- ysis with a cohesive treatment in preparation for the application of spectral expansion and resolvent methods. In chapter 3, a progressive development of integral equation theory for the scalar, vector and dyadic inhomogeneous equations was presented. In this chapter, we define an operator theory framework for EFIE and MFIE equations, study operator properties, the space on which the operators act, and its topological and algebraic structure.

The focus of the latter part of this thesis is on the EFIE and MFIE, with emphasis on numerical application of appropriate spectral expansions. This form is later discretized for numerical evaluation. The fundamental theories and formulations that require investigation are the properties of the (integral) operators considered, the spaces in/on which they are applied, and the expansion of points in the function space in terms of its basis elements. This chapter presents directed analyses particular to the external radiation and scattering problem from complex structures.

The mathematics of function spaces used throughout the document are introduced in sec- tion 4.1, starting with Hilbert space. A primer on algebraic and topological structure applied to function spaces is presented, including definitions of continuity, completeness and con- vergence. For the reader familiar with functional analysis, this section can be skipped. To facilitate handling of derivatives and integral equations, we review Lebesgue integrability, generalized distributions and Sobolev Spaces. More attention is given to Sobolev space in the region and on the boundary of a region, with a motivation for its use tied to power considerations in electromagnetic problems.

In section 4.3, we define accurate (mathematically justifiable) representations of integral equa- tions for scattering from a complex body within a function space with the appropriate struc- ture, including domains, norms and inner products. Application of the Stratton-Chu integral equations to the complex structures in this document, as appropriate for generic complex structures is included. Operator forms of the EFIE and MFIE are defined, followed by an analysis of the operator characteristics.

The mathematical constructs of bases are discussed in section 4.2, beginning with Schauder bases on Banach spaces to orthonormal bases and root vectors for Hilbert spaces. Derivation of operator properties used for later spectral analysis are reviewed in section 4.4. Compactness and nonselfadjointness are the 2 key characteristics that will be utilized in spectral expansion methods in chapter 5. The characteristics of the integral operator L, and in particular, the case of nonselfadjoint, compact and complete operators will be defined. Manipulations of nonselfadjoint operators to Hilbert-Schmidt forms as used in certain cases will be evolved. The theoretical analysis of nonselfadjoint operators and the applicability of orthonormality, Riesz basis with brackets and techniques for handling root vectors and Jordan Chains are also discussed in section 4.4.

Hadamard’s properties for well-posed solutions to partial differential equations are given in section 4.5, with a pointed analysis of cause and effect. Known deficiencies with respect to ill-posedness and nonuniqueness of the solutions of the integral equations are discussed. In section 4.6.2, we review regularization methods as potential stabilization mechanisms for these ill-posed problems, discussing analytical techniques as well as discretization approaches, suitable for a SuperNEC integral equation solution.

4.1

Function Space for our Problem

This section provides some of the basic mathematical constructs of the spaces used throughout this document. The mathematical literature dealing with real, complex and functional analy- sis is voluminous and varies in level of complexity. The electromagnetics literature uses these tools, often without clear definitions and typically not explaining why these mathematical tools are used. Lengthy and abstract definitions can be provided for Hilbert spaces, Banach spaces, inner-product, normed, linear, metric, function, sequence and topological spaces. We summarize the most useful constructs here.