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# Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems

DOI link for Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems

Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems book

# Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems

DOI link for Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems

Implementation of SIBCs for the Boundary Integral Equation Method: High-Frequency Problems book

## ABSTRACT

In Chapter 6, we have shown how the SIBC concept can be coupled with boundary integral equations to form an efﬁcient method of solution for low-frequency applications. The concepts developed are now extended to include high-frequency applications. We start, by reintroducing the displacement currents in Maxwell’s equations. This means that the basic derivations in Chapter 3, including the surface impedance functions must be generalized to take into account the displacement currents in Maxwell’s equations. Using the vector Green’s function we ﬁrst develop the electric ﬁeld integral equation (EFIE) and the magnetic ﬁeld integral equation (MFIE) representation of the ﬁelds based on the incident electric and magnetic ﬁelds. These equations are then coupled with the SIBCs previously developed to implement formulations of various orders of approximations in the time and frequency domains in a manner analogous to that in Chapter 6. The shift from low-frequency to high-frequency carries both advantages

and penalties. Of course, in conductors, the skin depths tend to be thinner, resulting in more accurate representation of ﬁelds. In that sense, one of the main differences is that at high frequencies, it is also possible to treat lossy dielectrics-that is, the surface impedance may now be applied at the interfaces between lossy dielectrics and dielectrics (such as free space). On the other hand, the introduction of displacement currents in Maxwell’s equations results in negation of one of the most important of the advantages we found in the low-frequency transient representation-that of separation of variables. We saw in Chapter 6 that the variables in the surface impedance function given in Equations 3.82 through 3.84 admitted separation into time and space variables only by neglecting displacement currents and treating each term of Equations 3.82 through 3.84 independently. In the high-frequency case we cannot do that and the time convolution integrals cannot be precomputed and tabulated in advance. Yet, direct computation of the convolution integral in the time-domain integral equations is impractical due to the large computational time and storage requirements. To mitigate this difﬁculty we also include a discussion of methods for efﬁcient

implementation of the time convolution integrals in the formulations. More speciﬁcally, we show in Appendix 7.A.1 how Prony’s method may be incorporated to approximate the impedance by decaying exponentials and by doing so we arrive at a simple recursive evaluation of the convolution integrals. As in previous chapters we use an example to illustrate the theory developed. In this case, the problem of scattering from a cylinder is used as an example.