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Numerical Formulation for Surface Current

Let a basis function as defined in the previous section be associated with the non-boundary edge of the curved triangulated structure. Surface current exists on both sides of the structure for an open surface. The unknown current , being solved for in the integral equation, is the vector sum of the currents on opposite sides of . At boundaries of , the component of this vector sum normal to the boundary must vanish due to continuity of the current; therefore, we need not define basis functions associated with boundary edges. The current on can be approximated as a linear combination of the basis functions with a set of unknown coefficients.

where is the total number of non-boundary edges, and is the unknown coefficient associated with the basis function . Since there may be up to three non-boundary edges in a triangle, so there will be up to three non-zero basis functions within each triangular face. To convert () into a matrix equation, we choose the expansion functions as testing functions. The symmetric product for any two vector functions and as given by Harrington [3] is

Testing () with yields

where is defined in (), . Utilizing a surface calculus identity [48], the last term of left hand side in () can be rewritten as

The continuity equation () is substituted into () to make the current be the only unknown for the scattered field. Inserting the current expansion () into eqs () and () then into () yields N linear equations for unknown coefficients. It may be written in a matrix form as:

where is an square matrix which is called the ``moment matrix" or the ``generalized impedance" matrix. and are column vectors. is called the ``generalized current" vector, and is named the ``generalized voltage" vector. The element of is . The element of is given by:

The evaluation of the generalized impedance matrix elements follows from equations () through (). Thus, utilizing the approximation (),

where denotes the domain of , and and are given by:

and

where and are the magnetic vector potential and the scalar potential due to the current on the edge, Substituting the expansion function into () and (), () and () can be rewritten as:

and

where denotes the domain of and is the surface divergence operator on the primed variables.

We note that each matrix element of is associated with a pair of non-boundary edges and . However, the domains of the integrals and locations of the observation points are associated with the faces attached to these edges. For each pair of triangular patches, contributions to the interactions between up to nine different combinations of source and field basis functions must be computed. Each source-field basis function interaction corresponds to a single matrix element. Much of the information required to compute the interaction between the source and field basis functions is only related to geometry. This information is the same regardless of which basis function is currently considered. Once computed, the geometry information between a pair of patches may be used to obtain the contributions to a maximum of nine different matrix elements.

To evaluate these surface integrals, we first transfer the curved triangular domain to a flat triangular parametric domain. Secondly, we transform the (u, v) parametric space to a local system of area coordinates (see Appendix ) with the corresponding triangle. Finally, the numerical integration formula for a triangular region in (see Chapter 8, [49]) are used to evaluate these surface integrals. Implementing this idea, we can rewrite in terms of each pair of faces as:

where is the contribution from testing over on the electric field due to the electric current on , and p and q are either + or signs.

where and are given by:

and

where and are, respectively, the magnetic vector potential and the electric scalar potential produced by the part of on the patch . Both and can be numerically computed after mapping the curved physical space to the parametric space with parametric description in () and Jacobian in (). Then, we transform the space to a local coordinate system. After substituting the basis function and the expression for the surface element, is given by

where is if is + and if is (see Fig. ), and is the area of the triangle . Since and are, respectively, and where is given by (), can be rewritten as:

where , a vector function, is defined by

where () is used to replace by and by .

Similarly, can be written as

where , a scalar function of , is defined by

Equations () and () can be evaluated by Gaussian quadrature after transforming the coordinates to a local system of area coordinates within . The details of the local system definition are given in Appendix A. Then, and are given by:

where only and appear in the above equations because and are given in () where is a linear combination of and . Note that direct application of a technique for numerical technique over a triangular region [49] allows equations () and () to be evaluated as:

where is the total number of integration points, is the weight for the integration point , where and are given in Table 8.2, [49], and and are given in (). The testing integrals over in () can also be evaluated numerically in the same way. For , both and are well behaved. Substituting () and () into () and evaluating the testing integrals using the same procedure as for the potential integrals, the numerical representation of is given, for , by

where is the total number of integration points on chosen according to the accuracy requirement, and is the weight associated with the integration point at . The quantities and are given in Table 8.2, [49], and are given in (), is either + or , and is the area of the triangle . The vector function and the function are defined by

For , the integrands of both the vector and scalar potentials integrals are singular. A term will be added and subtracted from each integrand. The term to be added and subtracted must have the same singular behavior when approaches to and can be analytically evaluated. Thus, the result of the magnetic vector potential in () is presented after adding and subtracting a selected term.

Adding and subtracting a term which has a singularity when approaches zero in the integraand of the scalar potential integral , one has

where is either + or , and is the distance between the testing point and the source point . is an approximation to when is very close to , which can be expressed in terms of a Taylor series approximation. Here, and are given by

Now, has the same behavior as near the singularity. Hence, the integrand of the first integral on the right-hand side of each of () and () is well-behaved on entire region, so these integrals can be evaluated numerically with the technique in [49]. The second integral on the right-hand side of each of () and () can be evaluated analytically, and is discussed in Appendix C of [16] and [50]. Therefore, for is given by

where is either + or depending on whether the patch of the testing function that contributes to resides on or . The vector function and the scalar function are defined in (). , , , and are given by

and

where is the total number of integration points and is the weight for the integration point . Here and are given in (Table 8.2 [49]). Furthermore, and can be found in ().

The vector functions and in () and () are defined by

and

where is if is + and if is -. And the scalar functions in () and () are defined by

and

The analytical evaluations of the integrals in and are given in [50].



Next: Magnetic Field Integral Up: Electric Field Integral Previous: Derivation of the


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Sat Dec 3 17:51:03 EST 1994