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Numerical Solution of EFIE

There are two equations for two unknowns. The surface currents will be obtained after solving the equations together. Using ( ) the curl of the electric vector potential () can be expressed as

where is the divergence of the Green's function with respect to the primed coordinates. Since is on , the integral on the right-hand side of the above equation must be interpreted in the Cauchy principal value sense.

The surface current will be computed using the parametric method of moments described in the previous chapter. To do so, first the surface is decomposed into a set of curved triangular patches. Secondly, the unknown current on S is approximated as a linear combination of the basis functions in () with a set of unknown coefficients in (). Using the symmetric product defined in () to test the result of substituting () into () with gives

where is defined in () for . The electric charge is obtained from the electric current according to the continuity equation (). Substituting the current expansion () into () yields the following matrix equation for the unknown electric current coefficients

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

The element of the row and the column of the matrix is given by:

where and are given in () and (), respectively. is the magnetic current related to the electric current expansion by ():

Applying the same technique as stated in Chapter 2, we intend to compute sequentially by faces all the vector and scalar potential integrals associated with each observation-face and source-face combination, to avoid the costly and inefficient recomputation of identical integrals which would result if the elements of were computed sequentially by edges. The numerical integration for a triangular region in [49] is applied to evaluate the integrals in () after transforming coordinates to a local system of area coordinates. To implement this idea, it is necessary to rewrite the element of in () in terms of each pair of faces as

where is the contribution from testing over the electric field due to the parts of and on , and p and q are either + or signs. can be further divided into two parts, as

where , the part of originally contributed by the electric surface current, arises from the second and third terms on the right-hand side of equation (). It is given by

where and are given in () and (). Similarly, , the part of originally contributed by the magnetic surface current, arises from the first and fourth terms on the right-hand side of equation . It is given by

Comparing the expression of in () with the expression in (), they are exactly the same. Therefore, the numerical evaluation of in () is going to be exactly the same as that of () described in Chapter 2. Therefore, we will only use the result produced in Chapter 2 and leave the derivation out here. The comparison between the expression of in () and that of in (), shows that there is only a little bit of difference in the inner integral of the second term on the right-hand side of them. Here, we only focus on the numerical evaluation of the second term on the right-hand side of ().

As with MFIE for a perfect conductor in the previous chapter, let be the inner integral of the second term on the right hand side of (). It can be written in terms of the coordinates, as:

where is the Jacobian defined in (), and the vector function is defined as

where

and

where denotes the area of the triangle , is a + or sign, is if is + and if is (see Fig. ). If we transform the parametric coordinates to a local system of area coordinates within in , () can be rewritten as:

where and are given in (), and the details of the coordinates transformation are in Appendix A.

Directly applying the numerical integration technique for a triangular region in [49] to () gives

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

When , all the integrals in are well-behaved, so can be numerically evaluated as

where the vector function and the scalar function are defined in (), the vectors and are given by () and (), respectively, is given in (), and and are given in ().

For the case of , and have a singularity at . To remove the singularity of in , the treatment described in Section 2.2.2 of Chapter 2 will be applied. The singularity of in will be treated by the method used in Section 2.3 of Chapter 2.

When , numerical formulation of is given by

where and are defined in (), and

Furthermore, , , , and are given in (), (), (), and (), respectively. , , and are defined in () with the integral domain , , and , respectively.



Next: Parallel Implementation Up: EM Scattering from Previous: Impedance Boundary Conditions


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Sat Dec 3 17:51:03 EST 1994