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ELECTROMAGNETIC MODELING OF COMPOSITE METALLIC AND DIELECTRIC STRUCTURES USING HIGHER ORDER BASIS FUNCTIONS

1.0 SUMMARY

The mathematical equations and the unique properties of the higher order basis functions involved in the simulation software HOBBIES (Higher Order Basis Based Integral Equation Solver) are presented in this chapter. The theory and the solution methodology of using Surface Integral Equations (SIEs) for analyzing the radiation and scattering from composite metallic and dielectric structures in the frequency domain is introduced first, followed by the unique properties of the higher order basis functions. The higher order basis functions involved in this case are nothing but simple polynomials of varying degrees. These basis functions are now used to describe the unknowns over electrically large subdomain patches, in contrast to the conventional piecewise linear basis functions, popularly known as the Rao–Wilton–Glisson (RWG) basis functions over sub wavelength triangular patches. Use of these higher order basis—namely use multiple basis functions over the same electrically large patch—in contrast to use of a single basis over an electrically small patch can lead to a significant reduction in the total number of unknowns in addition to several other interesting properties as described in this chapter. Typically, for a higher order basis, only 10–20 unknowns per wavelength squared of surface area are needed, leading to a reduction of an order of the magnitude of the size of the impedance matrix that needs to be solved. Hence, problems using the subsectional basis that require a supercomputer to solve can easily be solved on a laptop computer. Also, electrically large problems can easily be handled using modest computer resources, whereas the same problems cannot be solved on large computers using the subsectional basis because the matrix sizes will be extremely large! For example, if one wishes to analyze a metallic cube with each dimension of four times the wavelength, using 10 subsections per wavelength in a piecewise sub sectional basis will lead to a total of approximately 57,600 unknowns, whereas with the higher order basis, it will use approximately 2,700 unknowns and the total solution time on a laptop PC will be less than a minute! Many other interesting and salient features of the higher order basis are also discussed in this chapter.

1.1 INTEGRAL EQUATIONS FOR DIELECTRIC STRUCTURES

The main objective of a frequency-domain electromagnetic field analysis of a structure is to evaluate the distribution of the electric and the magnetic fields inside and/or outside the structure as a function of frequency, when some impressed sources or incident fields are given. This objective can be achieved in several different ways. One is based on the solution of the equivalent currents on the boundary surfaces of the structure using the SIE, which is the approach that will be used and is discussed in this chapter.

Various forms of the SIE formulations have been derived to solve for the surface currents using a numerical procedure. One of the popular forms of this surface integral equation is called the Poggio–Miller–Chang–Harrington–Wu (PMCHW) [1-3] formulation and is widely used for the analysis of three-dimensional composite metallic and dielectric structures. The derivation of the PMCHW formulation is presented in this section for completeness. We first start with the derivation of the SIE for a pure dielectric structure, and then it is extended to the analysis of composite metallic and dielectric structures. As shown in Figure 1.1, a homogeneous dielectric body with a permittivity ε⁽ⁱ⁾ and permeability μ⁽ⁱ⁾ is placed in an infinite homogeneous medium with material parameters ε^(e) and μ^(e). The structure of interest is now illuminated by an incident plane wave denoted by (E_inc, H_inc), where E_inc and H_inc are the electric and magnetic fields incident on the structure from the external region, respectively.

Figure 1.1. Electromagnetic scattering from a dielectric body.

The equivalent principle is now applied in two consecutive steps to obtain the equivalent fields for the exterior and interior regions, respectively.

Using the equivalent theorem, we first establish the model that is specific for the computation of the external scattered fields, given the incident fields. Assume that the interior fields in the dielectric body are equal to zero; then the material of the internal region can be replaced by the same material as that of the exterior region. To preserve the continuity of the fields across the dielectric boundary, equivalent electric and magnetic currents, J^(e) and M^(e), should be placed on the surface S of the body, as illustrated in Figure 1.2 (a). From the boundary conditions for the electric and magnetic fields, we have

where S₊ implies the fields are evaluated on the exterior side of the surface S. is the unit normal vector directed outward from the interior to the exterior region. E^(e) and H^(e) denote the scattered electric and magnetic fields in the exterior region, respectively.

The next step is to apply the equivalent principle again to obtain the surface currents that will generate only the fields interior to the structure. The exterior fields are set to be zero as our interest is only in the interior fields, and the exterior medium is now replaced by the same material as in the dielectric structure. Similarly, equivalent electric and magnetic currents, J⁽ⁱ⁾ and M⁽ⁱ⁾, are now placed on the surface of the structure to compensate for the field discontinuity across the boundary, as shown in Figure 1.2 (b). In this case, the boundary conditions are

where S_ implies that the fields are evaluated at the interior side of the surface S. E⁽ⁱ⁾ and H⁽ⁱ⁾ denote the scattered electric and the magnetic fields in the interior region, respectively.

The equivalent model in Figure 1.2 (a) preserves the exterior fields of the original problem, while that in Figure 1.2 (b) refers to the interior fields only. By generating a model with the exterior part from Figure 1.2 (a) and the interior part from Figure 1.2 (b), a new model for the original problem is constructed and it is shown in Figure 1.2 (c). Since in the original problem, the tangential components of the electric and the magnetic fields are continuous, the following boundary conditions must hold:

Equations (1.5) and (1.6) are generally called the “PMCHW formulation”. Substituting Eqs. (1.1)–(1.4) into Eqs. (1.5) and (1.6) yields the expressions for the currents as:

As a result, there are only two unknowns in Eqs. (1.5) and (1.6) that are required to be solved, i.e., J and M. Once we obtain the above currents, the scattered fields can be easily evaluated.

Figure 1.2. An equivalent model for the analysis of scattering from a dielectric body: (a) the equivalent model for the exterior region, (b) the equivalent model for the interior region, (c) the equivalent model for the entire region.

1.2 A GENERAL FORMULATION FOR THE ANALYSIS OF COMPOSITE METALLIC AND DIELECTRIC STRUCTURES

The presence of inhomogeneous dielectric structures can always be categorized by a combination of various piecewise homogeneous dielectric bodies [1-3]. Therefore, any composite metallic and dielectric structure can be represented as an electromagnetic system consisting of a finite number of finite-sized, linear, homogeneous, and isotropic regions, situated in an unbounded linear, homogeneous, and isotropic environment, as shown in Figure 1.3 (a). Most often, this environment is assumed to be vacuum, but it can also be a different medium.

Some regions in Figure 1.3 can be perfect electric conductors (PECs). In the interior of any such region, the electromagnetic field is always zero. These regions can be referred to as zero-field regions and are collectively denoted by the region 0 as shown in Figure 1.3. Metallic wires and plates can be considered as special cases of the zero-field regions. Plates forming an open surface can be regarded as a degenerate case of a zero-field region, where one dimension (thickness) of the region is zero.

In all the remaining regions, electromagnetic fields exist. These regions will collectively be referred to as non-zero-field regions. The total number of such regions is denoted by n. The medium representing region i is characterized by the complex permittivity ε⁽ⁱ⁾ and permeability μ⁽ⁱ⁾, i = 1,…, n, which include all the relevant losses. In any of these regions, impressed electric and magnetic fields, and , i = 1,…, n, may exist whose angular frequency is ω.

Consider an arbitrary region with a nonzero impressed electromagnetic field, e.g., region i as shown in Figure 1.3 (a). According to the surface equivalence theorem, the effect of all the sources outside the region i can be replaced by equivalent currents placed at the boundary surface of region i; in which case, the fields outside region i become zero, as shown in Figure 1.3 (b). The region outside region i is denoted as region 0 = i (region 0 with respect to region i, in the equivalent problem for region i). Since the field outside region i is zero, it can be homogenized with respect to region i; i.e., it can be filled by the same material as region i. Thus, a multiple-region problem (consisting of n regions) can be decomposed into n single-region problems.

The densities of equivalent currents at the boundary surface between regions i and j are

where is the unit normal vector directed from region j to region i, and E⁽ⁱ⁾ and H⁽ⁱ⁾ are the electric and the magnetic fields at the boundary surface, just inside region i.

Figure 1.3. Decomposition of a multiple-region problem into single-region problems: (a) the original problem, (b) the equivalent problem for region i.

If the equivalent currents for region j, J_sji and M_sji, are considered, they are evaluated according to Eq. (1.9), but with the indices i and j interchanged, i.e.,

Since there are no currents across regions i and j, the fields H⁽ⁱ⁾ and H^(j), as well the fields E⁽ⁱ⁾ and E^(j), satisfy the following boundary conditions:

After expressing the field vectors in terms of the equivalent currents, and using Eqs. (1.9) and (1.10), along with , one observes that the equivalent currents are related by

Thus, n single-region problems are mutually coupled through the conditions of Eqs. (1.11) and (1.12). It has been observed that satisfaction of these conditions guarantees the uniqueness of the solution for the sources and the fields alike. In that case, the main objective of the frequency-domain analysis is to evaluate the distribution of the equivalent electric and the magnetic currents at the boundary surfaces of the single-region problems, which satisfy the conditions expressed in Eqs. (1.11) and (1.12).

The field integral equations can be derived from the boundary conditions given by Eq. (1.11). Starting with Eq. (1.11b), the total electric field in region i can be expressed as

where E⁽ⁱ⁾(J_sik, M_sik) represents the scattered field inside region i, which is produced by the currents placed on the boundary surface between regions i and k, and the corresponding incident field is . The scattered fields, just inside region i, due to the currents placed on the boundary surface of regions i and k, are given by

where L⁽ⁱ⁾ and K⁽ⁱ⁾ are the operators defined by

In Eq. (1.15a and b), X_sik can be either the electric or the magnetic current, and

where g⁽ⁱ⁾(r,r′) is the Green's function for the homogenous medium i; r_ik, which is denoted as r′ in the Green's function, is the vector position of the source point and r is the vector position of the field point. The divergence operator ∇_sik acts on r_ik (on surface S_ik) and the gradient operator ∇ acts on r. After substituting Eqs. (1.13) and (1.14a) that hold for regions i and j into Eq. (1.11b), the integral equation can be expressed in the following form:

In a similar manner, another integral equation using the boundary condition of Eq. (1.11a) can be obtained, which is dual to Eq. (1.16a) and is given by

These two sets of equations represent a general form of the PMCHW formulation. For the case when one of the two regions sharing a common boundary surface is a PEC, the magnetic currents are equal to zero at the boundary surface and, therefore, the first of the equations above degenerates into the electric field integral equation (EFIE).

Note that the EFIE provides a solution not only for closed metallic bodies but also for open metallic surfaces and metallic wires. In particular, for the case dealing with wires, the EFIE is based on the extended boundary conditions and in addition invokes the thin-wire approximation. So for the analysis of electromagnetic radiation and scattering from arbitrary structures composed of wires and plates, a set of PMCHW and EFIE equations for the unknown electric and magnetic currents can be obtained. This set of equations is solved using the MoM methodology. To obtain an efficient method for the analysis of radiation and scattering from any composite structures, special care should be taken when choosing the basis functions, which needs to be characterized in two distinct different steps: geometric modeling and approximation of the relevant currents.

1.3 GEOMETRIC MODELING

Curves and curved surfaces provide a convenient mathematical means of describing a geometric model. Instead of using drawings, metal strips, or clay models, designers can use these mathematical expressions to represent the surfaces used on airplane wings, automobile bodies, machine parts, or other smooth curves and surfaces.

Parametric curves and surfaces are widely used in the traditional way to model a geometry structure. At this moment, the most popular technique for target description uses the NURBS (Non Uniform Rational B-Spline) to represent curves and surfaces. This technique is currently used in the aeronautic, automobile, ship, and other industries because it provides a great advantage in describing a complex objects' geometrical representation and manipulation. The NURBS scheme is able to manipulate both free-form surfaces and primitive quadric surfaces (cylinders, spheres, cones, etc.) with a small number of patches and, therefore, uses a small amount of information. For instance, a primitive quadric surface, like a cylinder, can be modeled perfectly as only one NURBS surface; a complex body, such as a complete aircraft, can be described with nearly with all its details, and with an accuracy of 1 mm, using only a few hundred NURBS patches. Today, many of the available computer-aided geometric design (CAGD) tools provide descriptions of the objects of interest in terms of NURBS curves and surfaces [4, 5].

In this section, we briefly review parametric curves and surfaces, followed by an introduction of NURBS curves and surfaces.

1.3.1 Parametric Curves

A parametric curve in space has the following form:

where , , and are three real-valued functions. Thus, f(u) maps a real value u defined in the closed interval [0, 1] to a point in space as shown in Figure 1.4. The domain of these real functions and the vector-valued function does not have to be [0, 1]. Thus, for each u in [0, 1], there corresponds a point [x(u), y(u), z(u)] in space. Note that if a function is removed from the definition of , has two coordinate components and becomes a curve in the coordinate plane.

As an example of a parametric curve, consider the circular helix defined as follows:

Figure 1.5 plots the curve in [0, 10 × π]. The initial point is (10, 0, 0), and the endpoint is (10, 0, 2 × 10 × π). Note that this curve lies on the surface of a cylinder of radius a, and its axis is oriented along the z-axis.

1.3.2 Parametric Surfaces

A parametric surface differs from a parametric curve only in that it has two parametric directions (u and v) instead of one (Figure 1.6). Parametric surfaces are defined by a set of three functions, one for each coordinate, as follows:

where parameters u and v are in certain domains. For our purpose, we shall assume both u and v are in the range between 0 and 1.

Figure 1.4. Mapping a parametric curve to an object space.

Figure 1.5. Circular helix (can be used to model a helical antenna).

Thus, (u, v) is a point in the square defined by (0, 0), (1, 0), (0, 1) and (1,1) in the uv-coordinate plane. Figure 1.6 illustrates this concept. A parametric surface patch can be considered as a union of (an infinite number of) curves. There are many ways to form these unions of curves, but the simplest one is the so-called isoparametric curves. Given a parametric surface f(u, v), if u is fixed to a value, say 0.1, and v varies, this generates a curve on the surface whose u coordinate is a constant. This is the isoparametric curve in the v direction with u = 0.1. Similarly, by fixing v to a value and letting u vary, we obtain an isoparametric curve whose v direction is a constant. Therefore, u be fixed at 0, 0.1, 0.2, …, 0.9, and 1, we shall have 11 isoparametric curves f(0, v), f(0.1, v), f(0.2,v), …, f(0.9,v), and f(1, v). These curves sweep out the surface if we let u change from 0 to 1 continuously. Similarly, the isoparametric curves generated by varying v also cover the surface. Figure 1.7 shows a few isoparametric curves in both directions.

Figure 1.6. Mapping a parametric surface to the object space.

Figure 1.7. Isoparametric curves on a parametric surface.

The most widely used parametric surfaces consist of cone, cylinder, sphere, ellipsoid, plane, and so on.

The surfaces of revolution provide another important class of surfaces that can be easily parameterized. For example, if the graph z = f(x), a ≤ x ≤ b is rotated about the z-axis, then the resulting surface has a parameterization

As an example for the surface of revolution, we use a parametric surface to model the reflector of a parabolic antenna surface, as illustrated in Figure 1.8.

Figure 1.8. The parametric surface showing the reflector of a parabolic antenna with a feeding horn.

1.3.3 NURBS Curves

In this section, the NURBS concepts are introduced in terms of curves, which are lines in three-dimensional (3D) space, such as a helix. Once one can define a NURBS curve, generating a NURBS surface is an extension of the previous concept.

A NURBS curve or surface is parametric—that is, the equations that describe it depend on certain variables (or parameters) that are not explicitly part of the geometry. A NURBS curve is described in terms of a parameter, u. For example, a nonuniform rational B-spline curve is defined by

where N_i,p are the B-spline basis functions, i corresponds to the ith control point, p is the order of the basis functions, P_i are the control points, and w_i are the weights associated with P_i . The B-spline basis functions are defined as

where t_i are the knots forming a knot vector T = {t₀, t₁ …..t_m}.

The responsibility of a user is to define the components that make up the NURBS curve parametric functions. Instead of explicitly specifying the equations, one can specify the following three things:

• Control points (or, Control vertex)
• Knot sequence
• Order—implicitly defined by the number of control points and the number of knots

Control points are points in object space that affect the shape of the curve. More precisely, increasing the value of w_i will move the curve toward the control point P_i. In fact, all affected points on the curve will also be shifted in the direction of P_i. When w_i approaches infinity, the curve will pass through the control point P_i. On the other hand, decreasing the value of w_i will move the curve away from the control point P_i. The control points are used to derive the shape of the curve, and they create the hull and usually reside off the curve. A NURBS curve example is shown in Figure 1.9.

Figure 1.9. A NURBS curve (the hull connects the control points together).

The knot sequence defines how the control points affect the curve. The knot sequence is simply a list of nondecreasing numbers. These numbers determine whether the curve passes through and interpolates between some of the control points (an interpolating curve) or passes near the control points (an approximating curve).

The order of a curve determines the form of the parametric equations. It affects how smooth the curve is going to be defined. The order of the curve also affects how the curve behaves when a control point is moved.

1.3.4 NURBS Surfaces

A NURBS surface is a rational piecewise polynomial parametric surface. The mathematic representation of a NURBS surface is defined as:

where P_i,j are the control points, i corresponds to the ith control point along the u direction, and j corresponds to the jth control point along the v direction. W_{i, j} are the weights associated with the control points. k and l are the degree of the surface. And the basis functions N_{i, k}(u) and N_j,l(v) are defined by Eq. (1.22).

NURBS modeling has many advantages for modeling any physical structure. For example, it may result in:

• More accurate characterization of the geometry of the structure of interest
• Use of fewer numbers of patches to model the structure as shown in Figure 1.10

1.4 MOM MODELING OF THE STRUCTURES

1.4.1 From the Physical Geometry of the Structure to a CEM Model

The curves and surfaces introduced in Section 1.3 has to be meshed so as to limit the electrical size of the patches on which the basis functions will be introduced in a MoM context, as shown in Figures 1.11 and 1.12. A flexible geometric modeling in a CEM context can be achieved by using truncated cones to model wire-like structures and bilinear patches to characterize other arbitrarily shaped surfaces. Therefore, no matter what type of structures one encounters in a particular model to discretize geometry in the MoM context, one simply needs three building blocks. They are, respectively, wires, bilinear surfaces, and wire-surface junction models for wires connected to surfaces. The flow graph from starting with a geometry model and to a discretized structure to be analyzed by the MoM is illustrated in Figure 1.13.

A conducting wire will then be composed of piecewise truncated cones with a finite radius at each end that may be different.

Figure 1.10. Aircraft modeled with NURBS surfaces.

Figure 1.11. A NURBS curve is meshed into piecewise linear segments.

Figure 1.12. A sphere with four NURBS surfaces is meshed into quadrilateral patches.

Figure 1.13. Transfer from a pure geometry model to a MoM model.

A bilinear patch is the simplest non-flat (curved) surface because it is completely defined through four corner points. Let P_0,0, P_1,0, P_0,1, and P_1,1 be four points in 3D space. We want to construct a NURBS representation of the surface obtained by bilinearly interpolating between the four-line segment, P_0,0P_1,0, P_0,1P_1,1, P_0,0P_0,1 and P_1,0P_1,1. Clearly, the desired (non-rational) surface is given by

Equation (1.24) represents a linear interpolation between the opposite boundary lines in each of the two directions. It is discussed here because its four boundary curves are straight lines and because the coordinates of any point on this surface can be derived through linear interpolations. Since this patch is completely defined by its four corner points, it cannot have a very complex shape. Nevertheless, it may be highly curved. If the four corners are coplanar, the bilinear patch defined by them is flat.

A wire-surface junction model can be derived from the description of a wire and a bilinear surface.

Detailed information about the generation of wires, bilinear surfaces, and so on will be illustrated next.

1.4.2 Right-Truncated Cones to Model Wire Structures

Note that any structure composed of conducting wires can be modeled with thin PEC wires that meet the requirements of a thin-wire approximation. In a thin wire approximation, no circumferential variation of the current is assumed to exist on the wire structure. Second, a valid wire segment should have a length-radius ratio of at least 10. A right-truncated cone is determined by the position vectors and the radii of its starting points and the endpoints, characterized by r₁, a₁ and, r₂, a₂, respectively. This is shown in Figure 1.14, where r₁ represents the position vector of the beginning of the cone, s is a local coordinate along the cone reference generatrix, and s₁ and s₂ are the s-coordinates of the beginning and the end of the cone generatrix.

Figure 1.14. A right-truncated cone to model a wire segment.

The reason for adopting the s-coordinate system rather than the z-coordinate system is that it will be assumed that the surface current density vector has only the s-component and is not expressed in terms of z if the s- and z-axes are normal to each other. In order to define the parametric equations for the surface of a cone, a local cylindrical coordinate system where the z-coordinate axis coincides with the cone axis will be adopted. In that case, the parametric equation for the surface of the cone surface can be written as

where s is the local coordinate along a cone generatrix, p is the circumferential angle measured from the x-axis, and i_ρ(p) is the radial unit vector perpendicular to the cone axis.

Under some specific circumstances, the truncated cone degenerates into a right cylinder (a₁ = a₂), an ordinary cone (a₂ = 0), a flat disk (a₂ = 0, r₁ = r₂), or a frill (r₁ = r₂), if the appropriate parameters are chosen. The right-truncated cone and its degenerate forms can also be used for the modeling of cylindrical wires with a stepped variation of the radius, as shown in Figure 1.15(a), as well as for flat and conical wire ends, as shown in Figure 1.15(b) and (c).

Figure 1.15 Geometrical modeling of wires along with their tips: (a) wire that has a curvilinear axis and a variable radius, (b) cylindrical wires with flat ends, (c) flat, frill-like, or conical changes of the wire radius with conical wire ends.

1.4.3 Bilinear Surfaces to Model Arbitrarily Shaped Surfaces

Metallic and dielectric surfaces will be modeled by bilinear surfaces. A bilinear surface is, in general, a nonplanar quadrilateral, which is defined uniquely by its four arbitrarily spaced vertices, as shown in Figure 1.16. Hence, it can be used for efficient modeling of both flat and curved surfaces. The parametric equation of such an isoparametric element can be written as

where r₁₁, r₁₂, r₂₁, and r₂₂ are the position vectors of the four vertices, and p and s are the local coordinates. After some elementary transformations, this equation can be rewritten as

where

Figure 1.16. An example of a bilinear surface.

Depending on the values of the vectors r_p, r_s, and r_ps, a bilinear surface can take different degenerate forms: a flat quadrilateral (r_p, r_s, and r_ps are coplanar), rhomboid (r_ps = 0), or rectangle (r_ps = 0 and ).

Although a bilinear surface for the general case is curved, all the p- and s-coordinate lines are straight lines. This is precisely why such surfaces are termed “bilinear”. Note that, owing to this property, a bilinear surface cannot be concave or convex, but only inflected or planar.

The maximum allowable length for the edges of the bilinear surfaces is at most two wavelengths. The current distribution over such large patches can still be successfully approximated by entire domain basis function expansions consisting of polynomials. If the length of an edge for any patch is longer than two wavelengths, that patch is subdivided automatically without the intervention of the user into a set of sub patches in such a manner that only edges longer than two wavelengths are subdivided into a minimum number of edges shorter than two wavelengths.

1.4.4 Wire-to-Surface Junction

Consider the junction of wires (modeled by right-truncated cones as discussed in Section 1.4.2) and surfaces (modeled by bilinear surfaces as given in Section 1.4.3), as shown in Figure 1.17.

Figure 1.17 Characterization of a wire-to-surface junction.

One end of each wire and one edge of each surface are collocated in an electrically small domain, called the junction domain. All these ends are assumed to be interconnected by enforcing the continuity of the current in the junction region. At a junction, the following holds:

• The total current flowing out of the junction is zero
• Sum total of all the partial currents flowing through the ends of the wires and the edges of the surfaces satisfies the Kirchhoff's current law

It is obvious that this junction model can be applied not only to wire-to-surface junctions but also to nontrivial wire-to-wire junctions and surface-to-surface junctions, if the ends of the wires and the edges of the surfaces are collocated in an electrically small domain. Hence, such a model of the junction is referred to as a localized junction model.

All wire-to-surface junctions are grouped into two classes:

• A wire-to-surface junction
• A combined wire-to-surface junction

A wire-to-surface junction contains the end of one wire end and one electrically short surface edge collocated in an electrically small domain. Any other wire-to-surface junction is represented as a combination of simple wire-to-surface junctions. Hence, they are referred to as combined wire-to-surface junctions.

All combined wire-to-surface junctions are grouped into three subclasses (as shown in Figure 1.18).

Figure 1.18. Combined wire-to-surface junctions: (a) the junction of a wire and a corner of a surface, (b) the junction of a wire and the edge of a surface (that is not electrically short), (c) the junction of a wire in the middle of a surface.

1.5 DESCRIPTION OF HIGHER ORDER BASIS FUNCTIONS

An efficient functional approximation for the unknown currents can be achieved by using entire-domain expansions consisting of linear combinations of polynomials that automatically satisfy the equation of continuity. This guarantees the continuity of the current at arbitrary multiple metallic and/or dielectric junctions.

1.5.1 Current Expansion along a Thin PEC Wire

Consider an arbitrarily shaped, perfectly conducting wire structure, where one assumes that the surface current J_s(s) is a function of the coordinate s only; i.e.,

where ∂r(p,s)/∂s represents the unitary vector along the generatrix of the truncated cone and is the corresponding unit vector. The total current intensity per unit length along the cone can be defined as

Thus, the corresponding charge per unit length can be expressed as

from which the continuity of the charge and the current for a truncated cone can be obtained as

Current distributions along the wires are approximated by polynomials that automatically satisfy the equation of continuity at the ends of the wires and junctions. The approximation for the current takes the form of a polynomial, and can be written as

where a_i are the unknown coefficients and N_s is the order of the approximation. Coefficients a₀ and a₁ can be expressed in terms of the other unknown coefficients a_i(i = 2,…, N_s) so that it satisfies the right boundary condition for the current at the ends of the wires; i.e., I₁ = I(−1) and I₂ = I(1). By using these boundary conditions, the expansion for the current in Eq. (1.32) can be rewritten as

where the node basis function, N(s), and the segment basis functions, S_i(s) (i = 2,…,N_S), are expressed as

Note that all the basis functions in this expansion are equal to zero at the beginning and at the end of a wire, except the (node) basis function corresponding to the unknown values of the current I₁(I₂), which is equal to one at the beginning (end) of the wire. The continuity equation at the beginning (end) of a free wire end, which is not connected to any other structure, is satisfied by omitting the first (second) term in the expansion shown above. The continuity equation at a junction of the ith and the jth wires (the end of the ith wire coincides with the beginning of the jth wire) can be automatically satisfied if independent basis functions corresponding to I_2i, and I_1j are replaced by a unique basis function in the form of a doublet as

Doublets are basis functions defined along two interconnecting wires. Similarly, basis functions corresponding to the unknowns a_i (i = 2,.., N_s) defined on a single segment are termed singletons. Note that such triangle doublets (usually used in subdomain piecewise linear approximation) represent a special case of an entire-domain approximation, obtained for N_s = 1. For a general case, the entire-domain approximation of currents along a complex thin-wire structure can be represented as a combination of overlapping doublets and singletons.

As an example, Figure 1.19 illustrates the singletons and the doublets along a generalized wire structure connected to other wires. In Figure 1.19, s(1) and s(−1) are the nodes of each wire, where different subscripts indicate that the nodes belong to different wires. For example, s₃(−1) is the node located at s = −1 of the third wire. The doublets drawn in this figure are given by Eq. (1.35), and the singletons along the wires are characterized by Eq. (1.34b).

Figure 1.19. Sketch of the piecewise linear and quadratic along a wire.

The equation of continuity at the junction of two or more wires is satisfied by grouping the corresponding node basis functions into doublets. These doublets actually represent the usual triangle (rooftop) basis functions. One doublet automatically satisfies the continuity equation at a junction of two wires, and two doublets automatically satisfy the continuity equation at a junction of three wires, and so on.

It is important to note that in this case, the continuity of the charge is guaranteed except at the feed points where the applied voltage should be related to the discontinuity of the charge. In addition, since the charges are continuous, it provides more stable results for the near-field.

1.5.2 Current Expansion over a Bilinear Surface

The surface current over a bilinear surface is decomposed into its p- and s-components. However, the p-component current can be treated as the s-component current defined over the same bilinear surface with interchanged p- and s-coordinates. Thus, for the general case, a distribution of the surface currents can be represented by a sum of s-components defined over bilinear surfaces that overlap or are interconnected. The initial approximations for the s-components of the electric and the magnetic currents over a bilinear surface are expressed as

where N_p and N_s are the degrees of approximations along the coordinates; a_ij and b_ij are the unknown coefficients; and F_ij are the basis functions. They are mathematically characterized by

where α_p and α_s are the unitary vectors, and are the corresponding unit vectors, and r(p,s) is given by Eq. (1.27). The relevant expressions in Eq. (1.37a) can be rearranged to satisfy automatically the continuity equation, assuming that there are no line charges present at the ends of the elements and at junctions. Let us assume the following:

In that case,

and the expansion for the electric current can be written in the form

Next, introduce an alternative set of unknowns through c_i1 and c_i2(i = 0,…, N_p), which are defined by

The coefficients for any i, a_i0, and a_i1 can be expressed in terms of the other unknown coefficients, a_ij(j = 2, 3, …, N_s), c_i1 and c_i2. In this case, the expansion for the electric currents is given by Eq. (1.38) and can be written as

where the edge basis functions, E_i(p,s) (i = 0,…,N_p), and the patch basis functions, P_ij(p,s)(i = 0,…,N_p; j = 2,…, N_s), are expressed as

where N(s) and S(s) are defined in Eq. (1.34). When k = 1, the first edge located along s = −1 of a bilinear patch is connected to an edge of another bilinear patch, and when k = 2, the second edge located along s = 1 of a bilinear patch is connected to an edge of another bilinear patch.

The basis functions E_i2 and P_ij are equal to zero along the first edge defined by s = −1, and the basis functions E_i1 and P_ij are equal to zero at the second edge defined by s = 1. This means that the equation of continuity at the first (second) edge is automatically satisfied by omitting the basis functions E_i2(E_i1) and that the basis functions E_i1(E_i2) take part in satisfying the continuity equation along the interconnected first (second) edge.

If a surface patch is not connected to the other patches, then the current distribution over that patch is approximated only by the basis functions P_ij. In what follows, such a current expansion will be termed as a patch expansion and the basis functions P_ij will be termed the patch basis functions. It is important to note that the basis functions E_i1 and E_i2 are used to satisfy the equation of continuity along the interconnected edges, and therefore, they will be referred to as the edge basis functions. All edge basis functions that are used for the satisfaction of the equation of continuity at some junction will be grouped with the junction expansion functions. This implies that in the case of complex structures, the current distribution is approximated by patch and junction expansion functions.

To display the shape of the patch basis function and the edge basis function, the first several terms of Eq. (1.41) are shown in Figure 1.20. In Figure 1.20, i and j are the order of the basis function, and k is either 1 for the first edge or 2 for the second edge of a bilinear patch as mentioned before.

Figure 1.20(a)–(d) depicts the different patch basis functions. Recall that in this section, only the s-component current is considered. It is shown that the patch basis functions tend to zero at the edge, where s = −1 or s = 1. The purpose of the negative sign in front of the term pⁱS_j(s) is used for convenience. Figure 1.20(e)–(h) depicts the different edge basis functions. Edge basis functions are specified along different edges. If the second edge located along s = 1 of a patch as shown in Figure 1.20(e) is interconnected to the first edge located along s = −1 of another patch as shown in Figure 1.20(f), then the two edge basis functions can be grouped into one doublet. The doublet has to be of the same order of p. Similarly, if the second edge of a patch, as shown in Figure 1.20 (g), is interconnected to the first edge of another patch, as shown in Figure 1.20 (h), then these two edge basis functions can be grouped into a doublet. Again, note that the doublet has to be of the same order of p for both cases. In this way, the current at the junctions of the edges between different patches can be expressed in terms of the edge basis functions.

Note that the edge basis functions E_i1 and E_i2 have the same form with respect to the appropriate edge. (If the s coordinate is replaced by its negative value, then the basis function E_i1 is transformed into the basis function E_i2, and vice versa.) Therefore, any junction can be considered to be composed of isoparametric elements interconnected along the side s = −1.

Figure 1.20. Several types of patch and edge basis functions: (a)–(d) are patch basis functions, (e)–(h) are edge basis functions.

In that case, the corresponding junction expansion consists of the edge basis functions of the form E_i1 multiplied by the unknown coefficients of the form c_i1 . To simplify the notation, index “1” will be omitted from c_i1 and E_i1 . The normal component of the ith edge basis function going out of the junction can be expressed as

All interconnected isoparametric elements can be described by the same parametric equation along the junction. In that case, they also have the same unitary vectors along the junctions; thus, the expressions |α_p(p, −1)| are equal for all interconnected isoparametric elements. It is thus concluded that all edge basis functions defined over interconnected elements have the same normal component going out of the junction. Note that the equation of continuity along this edge must be satisfied independently of each order i (i.e., for different values of the order p).

Because of this property, it is possible to satisfy exactly the continuity equation at the junction by a proper combination of the edge basis functions without introducing fictitious line charges. Along the edges of free patches, the appropriate boundary condition is satisfied by omitting the corresponding edge basis functions (of all orders). The continuity equation for the ith order at the junction of two or more patches is satisfied by grouping the corresponding edge basis functions of the ith order into doublets of the ith order. These surface doublets represent generalized rooftop basis functions, which can be considered to be an extension of the doublets for the wires. One doublet of the ith order automatically satisfies the continuity equation for the ith order at a junction of two patches, two doublets of the ith order automatically satisfy the continuity equation for the ith order at a junction of three patches, and so on.

For the case of multiple dielectric junctions, the continuity equation must be satisfied for each region. In that case, one doublet of the ith order automatically satisfies the continuity equation for the ith order at a junction of two patches in each region. However, all these doublets must enter the final solution with the same weighting coefficients. Hence, all these doublets are grouped into one multiplet basis function of the ith order. For the case of composite metallic and dielectric junctions, the equation of continuity for the ith order is satisfied by the proper combination of doublets and multiplets.

1.6 TESTING PROCEDURE

The integral equations are solved by using Galerkin's method, which is a special implementation of the Method of Moments (MoM). Use of the higher order basis functions results in a very efficient code, requiring only 3–4 unknowns per wavelength along electrically long wires, and 10–20 unknowns per wavelength squared of surface area for large metallic surfaces, enabling solution of real-life problems on personal computers.

1.6.1 Testing Procedure for Thin PEC Wires

As mentioned in Section 1.2, for the case where one of the two regions sharing a common boundary surface is a PEC, the magnetic currents are equal to zero at that boundary surface and the first of the PMCHW equations degenerates into the EFIE. For the case of PEC wires, only the EFIE is needed.

The area element for PEC wires dS can be expressed as

Note that the local p-coordinate system has been chosen to be circular. It is convenient to choose the origin of the p-coordinates for any s in the following way. Consider a plane that contains the field point and a representative plane of symmetry of that circle. The intersection of this plane and the circle will always be chosen as the origin of the p-coordinates corresponding to the chosen s-coordinate. In this case, R(p) = R(−p) so that the integration over p can be performed from 0, rather than from − π, to π . According to Eq. (1.14), the fields finally become

where the magnetic vector potential A(r) and the electric scalar potential Φ(r) are given by

In Eq. (1.45), the Green's function is given by with . The distance between the source and the field point is R = |r − r′(p,s)| in the expression for the Green's function, and r′(p,s) can be obtained using Eq. (1.25a). The vector is given by Eq. (1.28b).

Often, for simplicity in the analysis, an exact expression of the Green's function is replaced by the reduced kernel. The difference is that in the exact kernel, there is an integration along the circumferential direction that is not there in the reduced kernel. This simplifies the numerical computations significantly with negligible loss in accuracy if the wire structure is considered to be composed of electrically thin wires. The distance between the source and the field point in this case is defined by

and since R_a is no longer a function of p, the expressions for the electric scalar potential and the magnetic vector potential reduce to the following form:

Thus, the expressions for the electric and magnetic fields are given by

where

If the usual boundary condition for the tangential components of the electric field is applied, i.e., (E + E_inc)_tan = 0, then the elements of the impedance matrix due to the singletons will be given by

where

and I_i(s) is the ith testing function, A_sj(s) is the s component of the magnetic vector potential, and Φ_sj(s) is the electric scalar potential along the generatrix of the truncated cone due to the jth basis function along the direction of the generatrix of the truncated cone.

If the extended boundary condition, (E_z + E_inc,z) = 0, is used, the same expression is valid, except that the s-coordinate should be substituted by the z-coordinate.

To evaluate the line integrals originating in the potentials and field vectors with the approximation of currents in the form of polynomials containing powers of functions, special care needs to be employed for the evaluation of the following integrals:

where is the Green's function.

The integral in Eq. (1.52) can be separated into real and imaginary parts. The imaginary parts of the integrand,

can be efficiently numerically integrated using the Gauss–Legendre quadrature formulas as these functions are well behaved.

For the real part of the integrand , and , there is an integrable singularity. To decrease the computational time and increase simultaneously the accuracy in the evaluation of the integrals, the quasi-singularity of the kernel should be softened prior to the numerical integration of these integrals. First, h(s) = sⁱ is expanded in a Taylor series about s = s₀, to yield

Then, let R₀ = βR_a, and expand cosR₀ in Re(I_pi) and cos(R₀) + R₀ sin(R₀) in the expression of Re(I_qi) by a MacLaurin series about R₀ = 0. After some rearrangements of the terms, the singularity in the integrand of Re(I_pi) and Re(I_qi) can be analytically evaluated.

1.6.2 Testing Procedure for Bilinear Surfaces

To determine the unknown current coefficients over PEC and composite dielectric structures, the first of the coupled integral equations, Eq. (1.16a), is tested with the basis functions of the electric current, and the second of the coupled integral equations, Eq. (1.16b), is tested by using the basis functions of the magnetic current. The resulting matrix elements represent the linear combinations of two types of impedance integrals, and , defined as

where L and K are linear operators given by Eq. (1.15). F_k is the kth basis function, and the surface over which it is defined is represented by S_k; F_l is the lth testing function, and the surface over which it is defined is denoted by S_l. Before applying the operators in the expression above, the notation of Eq. (1.15) is changed; the order number of the region is omitted, the index of the boundary surface ik is replaced by the index of the basis function k, and the position vector of the field point r is replaced by the position vector of the testing surface r_l. After applying the operators L and K and after some suitable transformation, the integrals shown above, which are encountered in the evaluation of the elements of the impedance matrix, are obtained in a symmetric form as follows:

Since the patch basis functions, doublets, and multiplets are linear combinations of the initial polynomial basis functions, given by Eq. (1.36), the elements of the impedance matrix due to the initial basis functions are evaluated according to Eqs. (1.56) and (1.57) with the basis functions F_k and F_l replaced by the initial basis functions F_ikjk and F_iljl. According to Eq. (1.37a), these initial basis functions are written in the form

where p_k and s_k (p_l and s_l) are the local p- and s-coordinates of the kth (lth) element and α_pk and α_sk (α_pl and α_sl) are the corresponding unitary vectors. Similarly, and are easily found, keeping in mind that

Finally, the surface elements dS_k and dS_l, which correspond to the initial basis functions, can be written in the form

The final expressions for the elements of the impedance matrix are given by

Since the basis and the testing functions are represented in the form of power series, the corresponding elements of the impedance matrix can be written as

where

In Eq. (1.67), the vectors r_ck, r_pk, r_sk, and r_psk (r_cl, r_pl, r_sl, r_psl) define the kth (lth) bilinear surface according to Eq. (1.27). Recall that these expressions are derived for the basis and testing functions directed along the s-coordinate. If the basis and/or testing functions directed along the p-coordinate are used, the same expressions are obtained for each type of element of the impedance matrix, except that the vectors r_pk and r_sk, and/or vectors r_pl and r_sl, and the subscripts i and j, are interchanged.

Note that evaluation of the elements of the impedance matrix is now reduced to evaluating only two classes of integrals given by Eqs. (1.64) and (1.66). Special care is now devoted to the efficient evaluation of these integrals.

The evaluation of these integrals cannot be performed analytically. Pure numerical evaluation suffices if the kth and the lth surfaces are not close to each other. If they are either close, or coincide, then quasi-singular and singular parts of these integrals are extracted and evaluated analytically, while the remainder is evaluated numerically. However, more attention needs to be paid to the coordinate transformation between the Cartesian system and the local bilinear system. In all cases, the Gauss–Legendre formulas are used for numerical evaluations. The main part of the central processing unit (CPU) time used for evaluation of these integrals is due to the evaluation of the Green's function at the sampling points of the integrand.

To minimize the computation time, for each pair of integration points, one belonging to the kth surface and another one to the lth surface, the Green's function is evaluated only once for all the integrals in Eqs. (1.56) and (1.66). Thus, a relatively short matrix filling time is realized. Since the order of the integration formula along one coordinate is directly proportional to the order of the current approximation along this coordinate, the fill time for one matrix element depends only slightly on the order of the approximation.

The electric and magnetic fields due to the s-component of the electric surface–current–density vector and the assumed current expansion over any bilinear surfaces take the form

where

Using these expressions, one can analyze the various interactions in and between the structures consisting of both truncated conical wires and bilinear surfaces.

1.7 MODELING OF THE EXCITATIONS

Scatterers are excited by plane waves. Assume that the wave is incident on the structure from the direction defined by the unit vector k_inc, as shown in Figure 1.21. Let the wave be elliptically polarized (let the amplitude of the electric field vector be complex, e.g., E_i0). Finally, let us adopt the coordinate origin as the phase reference point.

The expression for the (complex) electric field of the wave is then of the form

The unit vector n_i is completely determined by the azimuth angle ϕ and the elevation angle θ related to the direction of arrival of the incident wave. The complex electric field vector is completely determined by its ϕ- and θ-components.

Figure 1.21. A plane wave excitation.

Antennas are excited by voltage generators. Two types of generators are considered:

• Delta-function generator
• Coaxial line excitation

The delta-function generator is a point-like ideal voltage generator. This generator can be defined in several ways. The simplest way is to require that the potential difference between two infinitesimally close points of the generator terminals be equal to the desired electromagnetic field of the generator.

Theoretically speaking, this generator cannot be used for excitation of cylindrical wires. Let us consider the excitation region of a cylindrical dipole antenna driven by an ideal voltage generator, as shown in Figure 1.22 (a). When the distance between the ends of the dipole δ tends to zero, a delta-function generator is obtained. It is obvious that the capacitance and the corresponding capacitive currents between the two driven dipole feed points are infinitely large. Low-order approximation of the current along the dipole cannot follow this sharp rise in the current intensity in the immediate vicinity of the excitation if the dipole is not relatively fat. Such an imperfect modeling of the excitation can result in an accurate value for the input impedance even when using a delta-function generator. Finally, note that this generator can be used for excitation of conical wire segments. This means that relatively fat wires having conical bases, as shown in Figure 1.22 (b), can also be excited by a delta-function generator.

Coaxial line excitation is approximated by a frill of a TEM (transverse electromagnetic) magnetic current. Let us consider the simplest case of a coaxial line excitation. This is a vertical cylindrical monopole antenna above a ground plane fed by a coaxial line, the radii of which are a and b, as shown in Figure 1.23 (a). The wave generated in the line is practically a pure TEM wave up to a certain distance below the line opening. Near the opening, higher order modes exist in addition to the TEM mode. However, if β a < 0.1 (β is the free-space phase constant), these higher order modes can be neglected without sacrificing the accuracy of the value related to the admittance. By assume that this condition is met, the electric field vector in the annular opening of the line has only a radial component of the form

where V is the voltage at the opening of the coaxial line, and ρ is the distance of the point in the opening considered from the z-axis.

Figure 1.22. A delta-function generator: (a) cylindrical wires, (b) conical wires.

According to the equivalence theorem, the electromagnetic field above the ground plane is not changed if a perfectly conducting plane is extended to cover the opening of the coaxial line and an annular layer of the surface magnetic currents is placed where the opening was, immediately above the plane, as shown in Figure 1.23 (b). These magnetic currents have only the circular component, the intensity of which with respect to the reference direction (shown in the figure) is given by

The image theory is applied to obtain an equivalent system, as shown in Figure 1.23 (c). In this system, the magnetic current intensity of the frill is twice of that in Figure 1.23 (b). These magnetic currents are the source of the impressed electric field for the symmetrical dipole antenna.

The impressed electric field due to the TEM magnetic current frill shown in Figure 1.23 (c), at a point P (x, 0, z), has a x- and z-components, the intensity of which are

where g(r) is the Green's function in the free space, r is the distance between the source and the field point, and ϕ is the usual azimuth angle related to the spherical coordinate.

A coaxial line excitation can also be used if the monopole antenna is placed above the finite ground plate. In this case, this excitation is modeled in the same way as shown in Figure 1.23 (b), except that the infinite ground plane is replaced by a finite ground plate. Note that the impressed magnetic currents produce discontinuous distribution of the excitation field over the plate. The intensity of the electric field component tangential to the plate below the frill is equal to half the intensity of the surface magnetic currents and is zero elsewhere. As a result, the electric charge distribution and the first derivative of the surface electric current are also discontinuous over the plate.

Figure 1.23. Coaxial line excitation of antennas above a ground plane: (a) a monopole antenna above a ground plane, (b) electrical equivalent of a coaxial line excitation represented by a TEM magnetic current frill, (c) an equivalent dipole antenna system corresponding to the monopole antenna above a ground plane.

Such behavior of charges and currents can be taken into account if the wire-to-surface junction in the excitation domain is modeled by a wire and eight plates (four plates below the frill and four plates for the rest of the original plate), as shown in Figure 1.24. This means that a TEM magnetic frill at the wire-to-surface junction requires additional segmentation of the surface. (Remember that a wire-to-surface junction without a TEM frill is modeled by a wire and four plates, as shown in Figure 1.18.)

Figure 1.24. Coaxial line excitation of an antenna located above a finite plate.

This additional segmentation can be avoided if the theorem of transfer of excitation is applied. According to this theorem, admittance of the antenna is not changed if the frill excitation is transferred to the wire and the frill removed. (After the transfer, there is no excitation over the plate and the impressed electric field along the wire is doubled.) Since the excitation over the plate is not discontinuous anymore, the additional partitioning required by the frill becomes unnecessary.

1.8 EXAMPLES ILLUSTRATING THE REQUIREMENTS OF THE GEOMETRICAL MODELING

As a first example, consider the modeling of a thin conducting wire. One can consider a thin wire or more exactly a cylindrical wire that can be modeled by several surfaces as shown in Figure 1.25. The half-wave dipole is 1 m long, and its radius is 5 cm.

The radar cross section (RCS) of this structure is computed as a function of frequency. Two different models are used. The first one is a thin wire model as the length-to-radius ratio is 20. The second is a more accurate surface model representing the cylinder by several surfaces. It is seen that as the radius of the wire exceeds approximately 0.01λ, the thin wire approximation breaks down. The thin wire approximation assumes that the current flow is along the axis of the wire and that there is no circumferential variation of the current. Also, the length-to-radius ratio cannot become too small, say less than 3.

Figure 1.25. Monostatic RCS of a wire as a function of frequency: (a) a wire model, (b) a surface model for a conducting cylinder, (c) RCS of the models.

Next, consider the gain of a circular loop antenna of radius 1 m made by a wire of radius 1 mm that is computed by approximating the loop by an n-sided inscribed polygon, as a function of frequency. The antenna model is shown in Figure 1.26, and the antenna gain is shown in Figure 1.27. The antenna gain is computed along the x-axis in the plane of the loop. It is seen that to obtain an accurate result for the gain, the maximum dimension of any linear segment of the polygon should not exceed a half wavelength. If the length of a linear segment of the n-sided polygon inscribed within the circle is greater than 0.5λ, then the number of segments comprising the polygon should be increased by one, or equivalently, it should be n + 1-sided inscribed polygon. The maximum length of a linear segment is constrained as follows: when n = 4 for circles whose circumference is smaller than one wavelength; for each increase of the circumference for up to a half wavelength, the number of segments n should be increased by one.

Figure 1.26. A circular loop antenna of radius a = 1 m, made of a thin wire of radius r = 1 mm, and excited by a point generator.

Figure 1.27. Gain of the circular loop antenna versus frequency, obtained for various numbers of segments n of a polygon inscribed into the circle of an equivalent radius.

Now, we consider the monostatic RCS of a metallic sphere with a radius of a = 1.0 m as a function of frequency. The results are compared when the surface is divided by n = 2, 4, 6, 8 per quarter circumference, which results in m = 24, 86, 220, and 372 quadrilateral patches, respectively, as shown in Figure 1.28. It is observed from Figure 1.29 that the frequency at which the RCS becomes unstable is related to the value of n. When n = 2, 4, and 6, the RCS become unstable at 150 MHz, 300 MHz, 450 MHz, respectively, and it can be characterized by λ_unstable = 4a/n . It is seen that to obtain an accurate result for the gain, the maximum dimension of any linear segment of the polygon should not exceed half a wavelength. If the length of a linear segment of the n-sided polygon inscribed within the circle is greater than 0.5λ, then the number of segments comprising the polygon should be increased by one, or equivalently, it should be n + 1 -sided inscribed polygon.

Figure 1.28. A sphere whose surface is divided by (a) n = 2, (b) n = 4, (c) n = 6, (d) n = 8 per quarter of the circumference.

Figure 1.29. Monostatic RCS of a spherical scatterer (a = 1.0 m) versus frequency. The geometric model of the sphere consists of m = 24, 86, 220, and 372 quadrilateral patches.

1.9 EXAMPLES ILLUSTRATING THE SALIENT FEATURES OF THE HIGHER ORDER BASIS FUNCTIONS

It is well known that the electric field integral equation breaks down corresponding to an analysis frequency that is one of the internal resonances of a closed perfectly conducting structure. As a first example, consider the electromagnetic scattering from a conducting cube of side 2 m, as shown in Figure 1.30. This structure has an internal resonant frequency at around 106 MHz. The advantage of the higher order basis is illustrated in Figure 1.31. If the cube is discretized into 24 quadrilateral patches [Figure 1.30 (a)] the analysis displays a defect. However, as the number of quadrilaterals is increased from 24 to 54 [Figure 1.30 (b)] to 96 [Figure 1.30 (c)], the defect becomes extremely localized as shown in Figure 1.31. If the integral accuracy of the solution procedure is enhanced to the next level, it is shown in Figure 1.32, even using 6 patches, illustrates that this defect becomes extremely localized. And finally, if the empty conducting cube is filled with a lossy dielectric material of dimension 0.1λ [Figure 1.30 (d)], then again similar conclusions can be drawn as is clear from Figure 1.33.

Figure 1.30. A 2 m cube discretized into (a) 24 patches, (b) 54 patches, (c) 96 patches, (d) 24 patches and filled with a small lossy dielectric cube.

Figure 1.31. Monostatic RCS of a 2-m cube simulated at 100 points between 105.5 MHz and 106.5 MHz using single-precision, normal integral accuracy and normal current expansion using 24, 54, and 96 quadrilateral patches.

Figure 1.32. Monostatic RCS of a 2-m cube simulated at 100 points between 105.5 MHz and 106.5 MHz using a total of six quadrilateral patches. It is shown that as the integral accuracy is increased, the defect also becomes highly localized.

Figure 1.33. Monostatic RCS of a 2-m cube simulated at 100 points between 105.5 MHz and 106.5 MHz using 24 quadrilateral patches. It is shown that the defect becomes difficult to observe when a lossy dielectric cube of dimension 0.1λ is inserted into the structure.

The second example will show that the higher order basis functions can result in a significant reduction in the number of unknowns while analyzing a structure without any loss of accuracy. Consider the computation of the input impedance of a 50-Ω transmission line consisting of two plates 17 inches long, 1 inch wide and are separated by 0.16 inches. For simplicity we will assume that they are located in free space. The transmission line at one end is connected to a 50-Ω load using two thin wires, and the input impedance is computed at the other end of the line at the feed point of two thin wires connected to the transmission line. In Figure 1.34 (a), the input impedance of the transmission line is calculated by using the classical subdomain basis functions using 135 patches with 542 unknowns, and the real and the imaginary parts of the impedance are plotted in Figures 1.35 and 1.36, respectively. The number of patches implies the discretization of the central quadrilateral on each line. In Figures 1.34 (b), the patches in the middle of the transmission line are meshed by 1 inch by 1 inch quadrilaterals using 15 quadrilateral patches using 148 unknowns, and the results for the real and the imaginary parts for the input impedance are also plotted in Figures 1.35 and 1.36, respectively. The last example in Figure 1.34 (c) considers an extreme case using only one quadrilateral in the middle part of size 15 inch by 1 inch, and the unknown number is 84. The reason such cases can be dealt with in this methodology as for the higher order basis applied on a flat surface the quadrilateral can have dimensions of 2λ × 2λ. For plate sizes larger than 2λ, the program automatically subdivides the plates as it is not efficient to use higher order polynomials without going into numerical instability. Figures 1.35 and 1.36 plot the real and the imaginary parts of the input impedance of the transmission line. The interesting observation that can be made from the plots of Figures 1.35 and 1.36, that all three sets of results are acceptable for the computed input impedance of the line, illustrating that use of the entire domain basis functions over large-sized patches can still get acceptable results as quickly as the number of unknowns has been greatly reduced, in the use of the higher order basis.

Figure 1.34. A transmission line with different numbers of meshed patches: (a) 135 patches, (b) 15 patches, (c) 1 patch.

The third example deals with the electromagnetic scattering from six flat plates of different sizes. The plate is located in the Y-Z plane and is irradiated by an x-polarized incident wave. The six sizes of the plates are given in Table 1.1. The next line shows the order of the current approximation used to carry out an accurate analysis for the RCS. The third row lists the total number of unknowns used to analyze the structure. The final row provides the number of unknowns required per square wavelength to analyze the plates. The interesting thing about the use of the higher order basis for the analysis of flat plates is that the number of unknowns per square wavelength is reduced in the analysis as the size of the structure is increased until it reaches a size of 2λ, and then the structure is automatically subdivided. This is a salient feature of the higher order basis.

Finally, we consider the electromagnetic scattering from a conducting cube of size 4λ on the side. Using the triangular patch basis functions, one will need approximately 300 unknowns per surface area of a plate (using 10 unknowns per wavelength, a plate of size 1λ will require approximately 300 unknowns corresponding to the edges of the triangular patches to discretize the structure) of a square wavelength × (6 sides) × (2 components of the currents) × (4.0 λ − actual size of the cube)² = 57,600 unknowns. One will need a supercomputer to solve this problem using MoM only which may take approximately 1.5 day to solve. However, using a higher order basis, it will take 2700 unknowns and the solution for one scattering angle can be obtained in less than a minute on any laptop computer. This example illustrates the advantage of using a higher order basis over a subsectional basis.

Figure 1.35. Real part of Z₁₁ for the transmission lines of Figure 1.34.

Figure 1.36. Imaginary part of Z₁₁ for the transmission lines of Figure 1.34.

Table 1.1 Characteristics for the analysis of a flat plate using a higher order basis.

1.10 PERFORMANCE OF THE PMCHW FORMULATION USING HIGHER ORDER BASIS FUNCTIONS FOR DIFFERENT VALUES OF THE DIELECTRIC CONSTANTS

In this section, we study the performance of the PMCHW formulation given by (1.16) in Section 1.2, using a higher order basis and when using materials of different values of the dielectric constants. As an example, consider a dielectric cube of size 1 m and of permittivity ε_r = 1, when it is irradiated by a plane incident wave at different frequencies. Theoretically there should be no scattering from this structure. However, due to various numerical approximations carried out in the computations, there will always be some scattered fields. The residual fields are plotted in Figure 1.37 along with the RCS of a conducting cube of exactly the same size. It is seen that for all the cases, the RCS due to the residual fields are always less than 50 dB with respect to the conducting cube. This figure clearly illustrates when compared with Figure 3 of [6] that this current methodology is much more accurate (by at least 30 dB in most regions) over the formulation of [3]. Hence, all methodologies using higher order basis are not the same; some are more accurate than others.

In a similar vein, it would be interesting to see how the PMCHW methodology performs when using a higher order basis for analyzing high dielectric constant materials. Let us consider a cube of side 0.2λ oriented along the three coordinate axes and located in free space. It is irradiated by a x-polarized plane wave incident from the negative z-axis. The cube is made of a dielectric material whose relative permittivity is ε_r = 1 − j10^k, for k = 2,3,4. A second-order approximation is used for the currents located on the surface of the dielectric resulting in N = 96 for the total number of unknowns. Figure 1.38 plots the bistatic RCS in the XZ-plane as a function of the angle θ. It is interesting to note that as the value of k increases, and even for ε_r = 1 − j10000, the results for the dielectric cube are quite stable and accurate and approach the RCS of a conducting cube as expected.

Figure 1.37. Residual RCS of a dielectric cube of size 1 m as a function of frequency and compared with that from a conducting cube.

Figure 1.38. Bistatic RCS of a dielectric cube of different relative permittivities ε_r = 1 − j10^k, for k = 2,3,4 and compared with that from a conducting cube.

1.11 PERFORMANCE OF THE PMCHW FORMULATION AT VERY LOW FREQUENCIES USING HIGHER ORDER BASIS FUNCTIONS FOR DIELECTRIC BODIES

It is well known that a subsectional basis breaks down at low frequencies due to the discontinuity of the charge and for various other numerical approximations used to discretize and solve an electric field integral equation. That is why the loop-star basis function was introduced in [7, 8].

The higher order basis breaks down at a much lower frequency than the unmodified subsectional basis as we will illustrate. To study the stability of the higher order basis at very low frequencies for dielectric bodies of the PMCHW formulation, we compare the RCS of an object with its electric polarizability to illustrate that with the higher order basis one can easily analyze structures even when the free space wavelength is around 100–10000 times the size of the object.

As illustrated in [9] the electric polarizability α_n of a dielectric sphere is related to its monostatic RCS, σ_RCS, by where with the free space parameters μ₀, ε₀ and V is the volume of the body. In the three figures of Figure 1.39 we model a quarter of a dielectric sphere of 1 m radius and the relative permittivity is ε_r = 10. The xOz anti-symmetry plane and the xOy symmetry plane are used to compensate for a whole sphere and simplify the calculation. The incident wave is coming from ϕ = θ = 0° direction with E_ϕ = 1 V, and E_θ = 0, as shown in Figure 1.39. The results shown in Figure 1.40 are the normalized polarizability estimates which are calculated using the relation described above and the RCS results using the higher order basis. The estimates are also compared with the exact value for the normalized polarizability, which is α_n =2.25. Three different models for the sphere are used in Figure 1.39. For the denser grids, the surface is divided into 384 and 216, and in the coarse model, into 96 quadrilaterals. However, symmetry was also exploited in the calculations by using two symmetry planes. One could thus reduce the surface to be meshed, into one fourth. The lowest order basis functions were used, two per edge (one for the electric and one for the magnetic current). Each patch has four edges but each edge is shared by two patches. Hence, there are four basis functions per patch. This means that (since the symmetry gave us a reduction into one fourth), the number of unknowns in these three cases is 384, 216, and 96, exactly as the total number of patches. The relative error of the low frequency analysis is also given by Figure 1.40. As we can see, very good estimates for the static polarizability of an object can be found form the dynamic calculations with HOBBIES. The electrical parameters of the sphere, given with reasonable calculation times, an accuracy of three significant digits in α_n, can be achieved, as long as we are around the proper frequency. It is important that the analysis is done in sufficiently low frequencies so that Rayleigh scattering dominates. In Figure 1.40, we can clearly observe this high-frequency divergence starting at around 10 MHz. This makes sense, because then the wavelength is 30 m, and the wavelengths that correspond to higher frequencies than that corresponding to the value that is close to the diameter of the sphere (2 m). On the other hand, this methodology is not meant for static calculations, and therefore a low-frequency breakdown can be seen in the results, particularly in the kilohertz region, and the accuracy of the estimates starts to get bad quickly.

It is obvious that a dynamic solver fails in the statics limit. Against this trivial observation, one must appreciate the accuracy of the solution which remains good even when the size of the object is four orders of magnitude smaller than the wavelength, implying that the analysis is good until the radius of the dielectric sphere is approximately 0.0003λ₀.

Figure 1.39. The meshed sphere used in the calculation of the polarizability. The dielectric sphere has a relative permittivity of ε_r = 10. The radius of the sphere is 1 m. Note the use of the two symmetry planes. The number of quadrilaterals is 24, 54 and 96 in one quadrant in (a), (b) and (c), respectively.

Figure 1.40. The estimate of the polarizability of a dielectric sphere with ε_r = 10, calculated over a broad frequency range. The radius of the sphere is 1 m. The sphere was approximated with three different plate-models, corresponding to 96, 216 and 384 patches over the whole sphere. In the lower part of the figure is the relative error of the calculations.

1.12 EVALUATION OF ANTENNA AND SCATTERER CHARACTERISTICS

Once the unknown coefficients of current expansion are determined, the required electrical properties of the structure analyzed can be computed with relative ease. In this work, the following characteristics are evaluated:

• Network parameters
• Current distribution
• Near-field
• Far-field

1.12.1 Network Parameters

An antenna is a structure with at least one port, excited at its ports by at least one ideal voltage generator. With respect to the generators, the antenna behaves like a passive network. (For example, an antenna array is considered to be a multiport antenna, and the present technique properly takes into account all the coupling between the array elements.)

When an N-port antenna is analyzed in operation mode ANTENNA (one generator at a time), the analysis is performed N-times. In the kth analysis, a 1-V generator is connected to the kth port, while other ports are short-circuited. Then the complex port currents represent admittance parameters Y_ik, i = 1,..,N. Once the Y-parameters are determined, impedance and scattering parameters are easily calculated by using the following matrix equations:

where [50Ω] is a diagonal matrix whose all diagonal elements are equal to 50 Ω. It is important to point out that the classical S-parameters are not the suitable parameters to deal with antennas or when the characteristic impedance is complex as it may yield non-intuitive results [10]. In that case, it is better to use the scattering wave parameters, which provide results that are easily interpretable, particularly when a system is conjugately matched.

1.12.2 Current Distribution

Once the unknown coefficients of the current expansion are determined, current along any wire or over any surface can be evaluated over a specified grid of points with relative ease, by using expressions given in Sections 1.5.1 and 1.5.2.

1.12.3 Near-Field

Once the unknown coefficients for the current expansion are determined, the near-field in the vicinity of the wires and surfaces can be evaluated on a specified grid of points with relative ease, by using expressions given in Sections 1.5.

1.12.4 Far-Field

If the distance from the field point to the antenna is much greater than the maximal dimension of the antenna and much greater than the wavelength, the electric field vector is written in the form

where r, ϕ, and θ are spherical coordinates and β is the free-space phase coefficient. The vector e(ϕ, θ) will be referred to as the normalized electric field. Note that the (normalized) electric field vector has only ϕ- and θ-components. The corresponding magnetic field vector is expressed as

where Z₀ is the wave impedance of the vacuum and i_r is the unit vector in the radial direction. The corresponding Poynting vector is given by

From the above equations, we see that the electric field vector, the magnetic field vector, and the Poynting vector can be easily evaluated at any far-field point if the value of the normalized electric field is known.

Since the perfect conducting antennas in the vacuum have an efficiency equal to 1, the antenna directive gain and the antenna power gain with respect to an isotropic radiator (a hypothetical antenna that radiates uniformly into the whole space) are given by the same expression

where P_fed is the power fed to the antenna; that is,

where N is the number of antenna ports, V_i are complex voltages at these ports, I_i are complex currents at these ports, and the asterisk denotes the complex conjugate. Note that gain is expressed in unnamed units. Hence, it is also expressed in decibels as

The scatterer radar cross section (RCS) is evaluated as

where E(r,ϕ,θ) is a scattered far electric field vector given by Eq. (1.76) and E_i(ϕ_i, θ_i) the electric field vector of the incident plane wave. In the general case, when the directions of the illuminating wave and of the scattered wave do not coincide, the above expression defines the bistatic RCS. If these two directions coincide, the monostatic RCS is defined. Most often, RCS is divided by the wavelength squared (i.e., σ/λ² is evaluated). Note that such a normalized RCS is expressed in unnamed units. Hence, it can be also expressed in decibels in the same way as gain in Eq. (1.81).

1.13 CONCLUSION

In this chapter, the geometrical modeling of complex electromagnetic structures is implemented using a right-truncated cone and bilinear surface for wire and surfaces, respectively. The wire-surface junctions are dealt with quasi-static approximation. Higher order basis functions are defined over wires and bilinear surfaces. Detailed testing procedures to obtain the impendence matrix are discussed in detail. Two typical excitations, wave and generator, are introduced for scattering and radiation problems, respectively. At the end of this chapter, formulations for the antenna and scattering characteristics are discussed.

REFERENCES

[1] B. H. Jung, T. K. Sarkar, S. W. Ting, Y. Zhang, Z. Mei, Z. Ji, M. Yuan, A. De, M. Salazar-Palma, and S. M. Rao, Time and Frequency Domain Solutions of EM Problems Using Integral Equations and a Hybrid Methodology, IEEE-Wiley Press, Hoboken, NJ, 2010.

[2] Y. Zhang and T. K. Sarkar, Parallel Solution of Integral Equation Based EM Problems in the Frequency Domain. IEEE-Wiley Press, Hoboken, NJ, 2009.

[3] B. M. Kolundzija, J. S. Ognjanovic, and T. K. Sarkar, WIPL-D: Electromagnetic Modeling of Composite Metallic and Dielectric Structures, Software and User's Manual, Artech House, Boston, 2000.

[4] http://mathworld.wolfram.com/NURBSCurve.html.

[5] http://mathworld.wolfram.com/NURBSSurface.html.

[6] B. Kolundzija, J. Ognjanovic, and T. Sarkar, “On the Limits of WIPL-D Code,” 18^th Annual Review of Progress in Applied Electromagnetics Conference (ACES), Naval Postgraduate School, CA, Mar. 18-22, 2002, pp. 615-622.

[7] J. S. Lim, S. M. Rao, and D. R. Wilton, “A Novel technique to Calculate the Electromagnetic Scattering by Surfaces of Arbitrary Shape,” National URSI Conference, Ann Arbor, MI, June 1993.

[8] S. Uckun, T. K. Sarkar, S. M. Rao, and M. Salazar-Palma, “A Novel Technique for Analysis of Electromagnetic Scattering from Microstrip Antennas of Arbitrary Shape,” IEEE Trans, on Microwave Theory and Techniques, Vol. 45, No. 4, pp.485-491, Apr. 1997.

[9] A. Sihvola, T. K. Sarkar, and B. Kolundzija, “From Radar Cross Section to Electrostatics,” IEEE Antennas and Wireless Propagation Letters, Vol. 3, 2004, pp. 324 – 327.

[10] T. K. Sarkar and M. Salazar-Palma, “An Exposition on the Choice of the Proper S-Parameters in Characterizing Transmission Lines with a Complex Characteristic Impedance and a General Methodology to Compute Them,” IEEE Antennas and Propagation Magazine, Feb. 2013, (to be published).