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Integral Equation Methods

Since Harrington first proposed the method of moments (MoM) in 1967 [1], MoM has become one of the most important numerical methods using the integral equation approache in computational electromagnetics. The MoM of Harrington is a method which transforms a functional operator equation describing the physical problem into a matrix equation by first approximating the unknown functions by a set of expansion functions with a set of unknown coefficients and then performing a scalar (or symmetric) product on the operator equaiton with selected testing functions. Today, more than two and a half decades after Harrington's now well-known book [3] was published, the MoM has been enriched by many researchers and many new features have been added and many new application areas have been explored. Here, we only discuss MoM in scattering and radiation applications.

An integral equation can be derived for the induced current in the scatterers or for the equivalent current on the scatterers. When the unknowns are volume currents in the scatterer the associated integral equation is called the volume integral equation, whereas when the unknowns are surface currents or equivalent surface currents on the surface of the scatterer the associated integral equation is called the surface integral equation (or boundary integral equation). We can convert the integral equation into a matrix equation by discretizing the unknown currents. Since the unknown quantities to be solved for in an integral equation formulation are restricted to be on or within the scatterers, the number of unknowns in this formulation is generally smaller than the number of those found in the differential equation formulation. This is particularly true of the surface integral equation formulation, in which the number of unknowns can be much smaller than the number unknowns in differential equation formulations and volume integral equation formulations.

The MoM can be applied in both time domain and frequency domain. Principal differences in applying the method of moments in the two domains are primarily in the formulation and solution steps, and hence much of the software developed for one approach can be used for the other. In the time domain, the unknown equivalent current or field must be discretized in both time and space. Local equivalent induced currents depend only on the local excitation and, due to propagation delay effects, to the past history of the other equivalent currents on the object. Since the equivalent current can be locally updated without knowledge of the updated values of the remaining currents, the solution can proceed in a marching on-in-time fashion. The potentials used to compute time domain fields are retarded potentials, and hence one must store the history of the equivalent current distribution over the object for an interval of time equal to the longest transit time across the object.

Which approach to apply depends on the quantities of interest. It is easier to model dispersive material characteristics in the frequency domain, and relatively little additional expense is involved in obtaining the response due to multiple monochromatic excitations once the moment matrix is obtained and factored. However, nonlinear characteristics are more easily modeled in the time domain. In this thesis, only frequency domain formulations are considered.

One of the important issues in applying numerical methods is how to model the geometry of a scatterer of specified material. The geometry of an electromagnetic boundary value problem is defined by specifying the spatial dependence and electrical parameters of all materials. Let us review some common geometric models.

The wire-grid modeling approach has been remarkably successful in many problems, particularly those requiring the prediction of far-field quantities such as radiation patterns and radar cross sections [4]. In addition to resulting in a surface model that is easy to describe to the computer, the technique has the advantage that all numerically computed integrals are essentially one-dimensional.

Piecewise linear straight line segments are commonly used to approximate wires [5], the cross sections of two dimensional cylinders [6], and bodies of revolution [8][7]. Only the nodal coordinates and the connectivity between nodes are needed to completely specify the geometry.

The most notable example of a wire-grid modeling code is the widely used Numerical Electromagnetics Code (NEC) [9] developed at Lawrence Livermore Laboratory as an extremely versatile general-purpose user-oriented code. The code can treat complex wire configurations which model either surfaces or multi-wire antennas in the frequency domain. The Livermore group has also developed the Thin Wire Time Domain (TWTD) code, which has similar capabilities for solving transient problems directly in the time domain. Apparently because of computer limitations, this code has not been used extensively to model surfaces, however.

Despite its simplicity and generality, the wire-grid modeling approach is not well-suited for calculating near-field and surface quantities such as surface current and input impedance. Other difficulties encountered in wire-grid modeling include the occasional presence of fictitious loop currents in the solution, difficulties with internal resonances [10], and the problem of relating computed wire currents to equivalent surface currents. The accuracy of wire-grid modeling has also been questioned on theoretical grounds [11]. These difficulties have provided incentives for developing the surface patch approach as an alternative to the wire-grid technique.

Surfaces are often modeled using planar rectangular and triangular elements, producing piecewise linear models of the surface [15][14][13][12]. The elements are specified by enumerating their boundary edge vertices. The order of these vertices may be used to implicitly specify, via the right hand rule, the sense of the unit vector, normal to the face. On orientable surfaces, one should ensure that the normal vector is always on the same side of the surface; this is accomplished by requiring that the common edge between every pair of adjacent faces is traversed in opposite directions as one travels around the boundaries of the two faces in the sense of their orientation. In surface modeling, the position of a vertex relative to the others can be specified by listing all the vertices connected to it by means of boundary edges. The most popular method is that of Rao, Wilton, and Glisson [14], which uses flat triangles. Some most popular general purpose computer programs for electromagnetic scattering and radiation with 3-D arbitrarily-shaped surface are used RWG techniques [17][16][15].

The existence of artificial creases in such surface models leads to erroneous edge scattering. Because of this, and because a more efficient or accurate surface model is desirable for other reasons, a mechanism for putting current basis functions on curved surfaces is needed. The use of flat facets to model a nonplanar surface creates unnecessary man-made discretization errors in the solution. Such errors can be important, for example, in near-field calculations when the observation point is on or close to the object surface such that effects of surface roughness are more easily seen. Progress has recently been made in developing more precise geometric models which take the surface curvature into consideration [22][2][21][20][19][18].

Graglia introduced a finite element type of parametric element in the method of moment analysis [19][18]. He proposed parametric (curved) elements which are generated by distorting simple forms (such as triangles, rectangles, tetradrons, rectangular prisms, etc.) so as to obtain other elements of more flexible shape which better match the object to be approximated.

To demonstrate the practical usefulness of parametric elements, he developed a program, based on a point matching technique, to study scattering from penetrable cylinders of arbitrary shape. In a practical example, he showed the superiority of parametric elements in the method of moments solution of a volume integral equation. These elements permit a better geometrical description of the scatterer, reduce the number of unknowns, shorten computation times, and give results more accurate than those provided by the commonly used planar elements.

Sancer [20], reported that researchers at Northrop use a parametric element model in their MoM code to get accurate RCS predictions with a minimum number of surface patches. A pulse basis function on each parametric element and point matching are used for both MFIE and EFIE. The disadvantage of Sancer's approach is that the pulse basis functions had line charges at the patch boundaries.

At Hughes Research Laboratory, Wandzura [21] constructed basis functions for representing currents on curved surfaces using differential geometry. These basis functions are appropriate for method of moments solution of boundary integral equations. They maintain the essential properties of the basis functions of Rao-Wilton-Glisson (RWG) [14], while allowing higher order basis functions (more variables per patch). The use of these functions is expected to result in a large reduction in the computational resources required to solve a given problem for a fixed level of accuracy. Wandzura claimed that his basis functions can be reduced to those of RWG. But we have not see his implementation yet.

The parametric model used in this report is the one proposed and developed by Wilkes and Cha [2] at Syracuse Research Corporation. Wilkes and Cha's parametric surface patch model is the exact geometric model. There is no discretization error introduced by the model. Wilkes and Cha have proposed a simple and efficient basis function which is defined in terms of curvilinear cooordinates on the parametric surface in question and conforms to its exact curvature. The parametric element is a curved triangle in physical space and a flat triangle with straight edges in the parametric space. Wilkes and Cha's basis functions have all the properties that a good basis function should have. Namely, these basis functions are linearly independent and capable of accurately representing the equivalent current on the surface. They are also simple to work with. They are discussed in detail in Chapter 2. Some comparison of performance between the popular RWG technique with flat facets and ParaMoM is presented in Chapter 5.

The integral equation formulation for computational electromagnetics is considerably aided by use of the electromagnetic equivalence principle. Harrington [3] has shown that it may be used to derive most of the practical formulations for both conductors and penetrable objects [25][24][23][3]. It is now generally appreciated that use of the equivalence principle eliminates much of the tedium and possibility for error formerly associated with deriving integral equations from Green's theorems. Its many forms also suggest alternative formulations.

It is seen that a common thread in any moment method formulation is the computation of fields due to equivalent electric and magnetic current sources radiating in unbounded homogeneous regions. These fields are most conveniently represented in terms of magnetic vector, electric scalar, electric vector, and magnetic scalar potentials.

When the scatterer has axial symmetry, the MoM approaches can be simplified. The body of revolution method [30][29][24][23][28][27][8][7] is a good example.

Above, we have briefly reviewed the MoM approach and given a little more detail about surface modeling.



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Sat Dec 3 17:51:03 EST 1994