Next: Combined Field Integral Up: EM Scattering from Previous: Numerical Formulation for

Magnetic Field Integral Equation Formulation

In this section, the magnetic field integral equation (MFIE) is derived for a conducting scatterer. It is well-known that the MFIE applies only to closed bodies, so that throughout this section we assume that the object has no boundary edges. The vector basis functions given in Section 2.1, are used for both the expansion and testing functions in the numerical solution of the MFIE.

Let denote the surface of a perfectly conducting scatterer with unit normal vector . The incident magnetic field is due to an impressed source in the absence of the scatterer. The scatterer is in a homogeneous space characterized by a pair of parameters (), where is the inductivity or permeability and is the capacitivity or permittivity. The result of enforcing the boundary condition on the magnetic field is given by

where is an outward unit normal vector on , is the surface is just inside of , and and are the incident and scattered magnetic fields, respectively. The tangential component of the scattered magnetic field can be expressed as a limit for observation points not on an edge, (see [48] and [51])

where is the induced electric surface current on , and the integral on the right hand side in (), with the field point exactly on , is interpreted as the Cauchy principal value. is free space Green's function, and is the gradient operator on the primed coordinates. The Green's function and its gradient are given below as

and

where is the wave number as defined in the previous section, is the distance between the source point and the observation point which is , and approaches from the interior. Substituting () into (), we obtain the magnetic field integral equation:

In order to apply the method of moments, the surface of the scatterer is decomposed into a set of curved triangular patches using a parametric description of the surface. The procedure of the parametric surface model generation has been described in Section 2.1. The next step after surface modeling is to define a set of basis functions which are used to approximate the surface current. Here, the basis functions defined in () are to be used as expansion functions on the parametric surface. Then, the electric surface current can be approximated as a linear combination of the expansion functions with a set of unknown coefficients as in (). Substituting () into () gives an integral equation with unknown coefficients. The method of moments allows one to select a set of testing functions which are used to test () with a symmetric product which is defined in (). When the testing functions are the same as the expansion functions, the procedure is called the Galerkin procedure. This procedure gives

where is the testing function and . is the unknown coefficient associated with the basis function .

The linear equations for unknowns in () can be rewritten in a matrix equation as

where and are column vectors and is an square matrix. The element of is the unknown coefficient associated with the expansion function. The element of the row and the column of the matrix and the element of the row of the vector are given by

and

From the definition of the symmetric product in (), () can be rewritten as

Note that when the testing function and the expansion function reside on the same patch the singularity contribution is the same for both the flat patch and the curved patch. In other words, there is a term when for both the flat patch and the curved patch. However, the contribution for the principal value integral is different. In the flat patch case, the current vector is always on the same plane as , so that (see [16]). However, it is not the case for a curved patch. Due to the surface curvature the current vector on the patch is not always in the same plane as , so that , as shown in Figure 2.4. Thus, the principal value integral has a singular integrand when the testing patch is also the source patch. That may be the only disadvantage of this model for MFIE. Fortunately, this integral is easily evaluated..

Following the procedure in the previous section, we intend to compute the elements of sequentially by source-field patch pairs for all integrals. It will avoid the costly and inefficient recomputation of an identical integral up to nine times which would result if the elements of were computed sequentially by basis functions. As with the EFIE, it is convenient to write all the required integrals in terms of integrals over a source-field patch pair, as

where is the computation from testing over the magnetic field due to the part of the electric current on , and are either + or signs, and is given by

For convenience, let be the inner integral of the second term on the right-hand side of (). It can be expressed as

where the vector function is defined after substituting () and () into the integral, as

Here , and is when is + and when is . We transform the parametric coordinates to a local system of area coordinates within , so () can be rewritten as

where and are given by (). Directly applying the numerical integration technique for a triangular region in [49] to () gives

where is the number of points where the integrand is sampled on the triangle , and is the weight corresponding to the integration point (). , and are given in (Table 8.2, [49]).

Similarly, the testing integral in () can also be treated in the same way as above. The final numerical equation for when is then given by

where is given in () and is given in (). , , , and have the same meanings as in (), and and are given in ().

When , the integrand of the integral over in () is singular at the testing point. There are two ways to treat this singularity. One is, as with EFIE, to subtract and add an analytically integrable function with the same singularity as the integrand. Another is to divide the patch into three sub-patches for which the testing point is a common vertex and then apply numerical integration directly on these sub-patches. Numerically integrating over these sub-patches, we obtain

where , , and are shown in Figure 2.5. Since the testing point is a vertex of these three sub-patches and vertices are never picked as integration points, the result of numerical integration over the sub-patches will be finite and, hopefully, accurate. The numerical integration technique used in the previous section and this section is applied to (). To avoid repeating work, the same notation as in the previous section will be used. Thus, for , the is given by

where is defined on in the same way that is defined on . Because , is of () with the vertex replaced by the vertex of opposite the common edge of the two triangles where exists. For , is the contribution to of () due to integration over .



Next: Combined Field Integral Up: EM Scattering from Previous: Numerical Formulation for


xshen@
Sat Dec 3 17:51:03 EST 1994