The major limitation of all ray optical techniques is the failure in predicting fields at the caustics of the ray tubes running the divergence factor of the ray optical fields to infinity. This was described for ray optical field expressions in Section 4.1. For the case of UTD, this means that diffracted fields cannot be predicted for observation points placed directly at the edge. Fields at caustics are computed using source based asymptotic methods like PO or PTD [27], [33]-[35]. Another shortcoming of all ray optical methods, is that they can predict fields only in the ray directions dictated by Fermat’s principle. These directions are given by Snell’s law for the GO fields and Keller’s law for the UTD fields as described in Sections 4.1 and 4.2, respectively. This restricts the applicability of ray optical methods in some cases as e.g. in monostatic computations.
Further, UTD fails to predict the fields at the Edge Diffraction Shadow Boundary (EDSB), which is considered as the Keller’s cone at the corners of a finite edge [28]. Across these
boundaries the UTD field abruptly falls to zero. However, this failure is compensated by corner diffraction, which enforces UTD fields to be continuous across EDSBs, in the same way as UTD ensures the continuity of GTD fields across their shadow boundaries [67]. In addition, inside the transition regions UTD fields are not purely ray optical because of the transition functions. In particular, the transition functions depend on specific distance parameters and this dependence results in an amplitude variation which is no more smooth but rather rapid. Consequently, within transition regions the amplitude variation of UTD fields is no more governed by the conservation of power flux within the ray tube as described in Section 4.1.1 and in this sense UTD fields are not purely ray optical within transition regions. However, outside the transition regions the transitions functions become equal to one and the diffracted fields reduce to GTD fields, which are purely ray optical.
Finally, the asymptotic solution of the edge diffraction problem provides a limitation to the distance parameter in the argument of the transition functions. According to this, the source and observation point must be sufficiently away from the edge. For detailed expression of this limitation the reader is referred to the literature [30], [147]. This condition can be violated if s, s become to small or if β0 is to small, i.e. the incident rays are in the paraxial region
close to the edge. For these cases, additional coefficients that extend the validity of UTD in these regions have been developed like in [154].
In this work, the numerical methods presented in Chapters 2 to 4 are fully combined using novel hybrid formulations as described in detail in the following chapters.
Hybridization of MLFMM with UTD
5.1
Introduction
In the present thesis, the hybrid FEBI-MLFMM method described in Chapter 2 and Chap- ter 3 is combined with UTD, which was discussed in Chapter 4. The hybridization is per- formed in both the BIM part of the hybrid method for CFIE formulation by modifying the Green’s functions of the problem and the incident field as well as within the matrix-vector multiplications in the various levels of the MLFMM part resulting in a far-field approximation of the translation operator for ray optical contributions. In each case, the Green’s function and the incident field are modified according to superposition of all received contributions at an observation point for a given source point. The hybrid formulations provide full coupling between large and composite metallic/dielectric arbitrarily shaped FEBI objects and elec- trically very large UTD objects within the same environment. Thereby, dielectric regions of FEBI objects are handled conventionally through the efficient combination of BIM with FEM discussed in Section 2.4, which does not affect the hybridization with UTD.
The resulting hybrid method is referred to as hybrid FEBI-MLFMM-UTD method combin- ing for the first time a fast and powerful integral equation method with ray optical techniques [70]-[73]. This provides wide and powerful modeling capabilities especially compared to ex- isting hybridizations of UTD with conventional BI formulations without fast IE solution, which is the case in the hybrid FEBI-UTD method developed in [69]. In that method, mod- eling flexibility is not good and UTD must often be applied to moderately large objects that should be modeled within the BI part. The hybrid FEBI-MLFMM-UTD method developed in this thesis provides a significant extension to the hybrid FEBI-UTD presented in [69] through novel hybrid formulations, which combine the CFIE as well as the multipole field representations of MLFMM with the ray optical fields of UTD. Thereby, double diffracted ray optical fields at arbitrarily oriented straight metallic edges are included using scalar diffraction coefficients of standard UTD [85], [86]. It is noticed, that to the knowledge of the author the hybrid BIM-UTD field formulations for the MFIE part along with the hybrid MLFMM-UTD formulations are reported in the scientific community for the first time. In the following, the hybrid field formulations for the FEBI-MLFMM-UTD method are presented and details of the numerical implementation will be discussed.