Contribution to the improvement of integral equation methods for penetrable scatterers
Authors:   , Úbeda Farré Eduard 
Published: October 2023

The study of the electromagnetic phenomena along the last two centuries has brought about outstanding contributions for the human progress. The electromagnetism represents still now, at the beginning of the third millenium, a very important research area. The radiation pattern of particular types of antennas -for example, fractal or microstrip-, the analysis of the effect of the cellular communications on human beings or the detection of buried mines represent specific examples of the wide variety of problems of great interest nowadays. <br/>The study of such a variety of problems relies on the application of the Maxwell equations, which rule all the electromagnetic behaviour. Since the analytical solution can only be obtained for very particular cases of canonical forms, to tackle the analysis of an arbitrary problem, one makes use of the numerical methods. The discretization of electromagnetic integral equations by the Method of Moments -MoM- excels as a powerful and reliable tool for analysing bodies composed of locally homogeneous regions -penetrable or perfectly conducting- immerse in a wide and nearly uniform medium -typically the ground or the free-space-. These integral methods result from the surface equivalence theorem, which allows in general two different formulations, the Electric Field Integral Equation (EFIE) and the Magnetic Field Integral Equation (MFIE). For the case of penetrable bodies, the Poggio, Miller, Chang, Harrington and Wu (PMCHW) formulation, that results from the subtraction of the EFIE and MFIE at both sides of the surfaces, can also be employed.<br/>The Method of Moments is based on the full expansion of the physical magnitudes, field and current, over the interface surfaces between the regions. In consequence, the solution of the problem is obtained through the inversion of a full-matrix, which, for electrically large problems, requires excessive memory resources and computation time. That is why the MoM is widely considered a brute-force method. The expansion of the magnitudes is carried out through the discretization of the surface; that is, patches spreading over the interface. <br/>The first half of this dissertation Thesis tackles the development of the MoM applied to problems with bodies with symmetry of revolution -BoR-. Since in this case the physical magnitudes present an azimuthal periodicity, they can be expressed as a Fourier series. The orthogonality between the different modes enables to obtain separately each azimuthal mode of the solution. It is thus only required to spread the patches along the generating arc of the bodies for each mode, which is very advantageous because the electromagnetic analysis can be carried out indeed for dimensionally large problems. A well-known PeC-EFIE BoR formulation is developed. Accordingly, PeC-MFIE and PMCHW formulations are developed from scratch. Furthermore, it is commented in detail and corrected to some extent the numerical error associated to the fastest-varying part of the PeC-MFIE BoR operator. The BoR-codes are particularly useful in modelling the electromagnetic behaviour of buried mines, which very often show revolution symmetry. <br/>The most outstanding contribution of this dissertation Thesis is the study of the appropriate conditions to develop correctly the 3D operators so as to yield accurate results for any structure. Since the discretization implies a break on the continuity properties of the physical magnitudes -field and current- the valid 3D-operators must ensure the physical electromagnetic requirements in the discretized surface. In mathematical terms, these requirements set the rank -field- and domain- -current- spaces, which essentially require the enforcement of the continuity across the edges of either the tangential or the normal component of the expanded magnitudes.<br/>For the case of an arbitrary perfectly conducting -PeC- body, it is recommended in this work the use of the divergence-conforming and of the curl-conforming functions respectively in the development of the PeC-EFIE and the PeC-MFIE operators. Low-order sets over triangular facets -RWG and unxRWG- are chosen to develop the PeC-operators. Furthermore, it is reasoned theoretically the inherent misbehaviour of the PeC-MFIE in case the current expansion relies on a divergence-conforming set. A heuristic correction is provided. The better behaviour of PeC-EFIE(RWG) and PeC-MFIE(unxRWG) is confirmed with examples. In view of the results, it is reasoned the suitability of PeC-EFIE(RWG) for the analysis of physical polyhedrons, which makes PeC-MFIE(unxRWG) excel as a more appropriate operator for curved bodies. A procedure for improving the performance of PeC-EFIE(RWG) for coarsely meshed spheres is given.<br/>For the case of arbitrary penetrable bodies, the same low-order sets are used to expand the operators EFIE, MFIE and PMCHW. It is shown their compatibility with the combination of the right PeC-operators. In the dielectric case, in addition to the required continuity of the magnitudes across the edges at each region, the fields at both sides of the surface must satisfy the interface continuity, which is ignored in the conducting case -the fields are null inside the conductor-. The impossibility of meeting both continuity requirements at the same time justifies the apparition of inherent and different errors in the dual EFIE-MFIE and in PMCHW. It is thoroughly reasoned and confirmed with examples the suitability of PMCHW for problems with only penetrable regions. It is also shown and discussed in detail the robustness of EFIE-MFIE since its behaviour is appropriate for electrically not too small structures with perfectly conducting or penetrable regions. The analysis of composite structures -very useful to model microstrip antennas- can be considered as a group of disjoint bodies with null distances of separation. For this type of problems, it is recommended in this work the use of EFIE-MFIE since, unlike PMCHW, they can ensure the continuous transition to zero of a distance of separation increasingly small. <br/>Finally, efficient methods -IE-MEI and MLFMM- relying on the previous 3D-operators. The development of the PeC 3D IE-MEI cannot maintain the advantages present in the 2D case since the harmonic metrons are not valid in the 3D general case. A new set of metrons that ensures little discontinuity of the current across the edges is presented. It is confirmed with examples how these metrons, so-called quasi-continuous, reduce the number of required coefficients per row for a certain current error. Some examples of penetrable spheres with moderate electrical dimensions analysed under a MLFMM implementation are shown and commented.