Scattering and Radiation Analysis of Three-Dimensional Cavity Arrays...: Finite Element Modeling of Patch Antenna and Cavity Sources.pdf

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Scattering and radiation analysis of
three-dimensional cavity arrays via a hybrid finite
element method
Jian-Ming Jin and John L. Volakis
Radiation Laboratory
Department of Electrical Engineering and Computer Science
The University of Michigan
Ann Arbor, Michigan 48109-2122





Abstract
A hybrid numerical technique is presented for a characterization of the scattering and radiation properties of three-dimensional cavity arrays recessed in a ground plane. The technique combines the finite element and boundary integral methods and invokes Floquet's representation to formulate a system of equations for the fields at the apertures and those inside the cavities. The system is solved via the conjugate gradient method in conjunction with the fast Fourier transform (FFT) thus achieving an 0(N) storage requirement. By virtue of the finite element method, the proposed technique is applicable to periodic arrays comprised of cavities having arbitrary shape and filled with inhomogeneous dielectrics. Several numerical results are presented, along with new measured data, which demonstrate the validity, efficiency and capability of the technique.

1 Introduction
Recently, a hybrid finite element technique (FE-BI) was proposed for a characterization of the scattering and radiation properties of several three-dimensional cavity-backed structures including microstrip patch antennas and arrays [1]—[3]. The technique combines the finite element method with the boundary integral equation to formulate a system suitable for solution via the conjugate or biconjugate gradient method in conjunction with the fast Fourier transform (FFT). By virtue of the finite element method, the proposed technique is applicable to complex structures such as those involving inhomogeneous dielectrics, conducting and resistive patches, feed probes and impedance loads. Accurate results have already been obtained for scattering and radiation by cavities, slots and microstrip patch antennas and these have demonstrated the method's capability.
In this paper, we develop the aforementioned finite element—boundary integral technique for scattering and radiation by infinite cavity arrays. Although, at least in principle, the technique is suitable for this application, there are new issues which must be addressed. For example, the application of Floquet's theorem leads to a system which is non-symmetric. In addition, the matrix elements are dependent on the incidence angle in the case of scattering or the scan angle if the array is treated as a radiator. Consequently, for such a system an iterative solver is preferable over the usual direct solvers. Another issue addressed in this paper concerns the approximation of the finite array by a truncated infinite array. This kind of approximation has been adopted by necessity to make practical use of the infinite array analysis. However, its validity has not been examined for three-dimensional arrays and the two-dimensional studies are not conclusive [4]. Herewith we present three-dimensional comparisons of measured and calculated patterns which are perhaps the first to provide a measure of this approximation.
The paper begins with a short description of the standard equivalence principle for subdividing the exterior and interior computational region. Using Floquet's theorem the exterior fields are formulated and expressed by an infinite sum involving the spectral representation of the exterior region's Green's function. This gives a set of discrete equations for the fields at the cavity interface. The interior fields are formulated via the finite element method leading to a sparse system of equations. By invoking tangential field continuity, this system is combined with the discrete set of equations developed for the exterior region. The final system is solved via the conju-ate gradient method and certain restructuring is carried out to achieve an
0(N) memory demand. Finally, a number of numerical computations are
presented which demonstrate the capability and accuracy of the technique.
2 Formulation
The geometry under consideration is illustrated in Fig. 1, where a periodic array of cavities is recessed in an infinite ground plane. Each cavity is identical but may be of arbitrary shape and filled with inhomogeneous material. Also, the entire geometry may be covered with a thin dielectric layer. For scattering computations the excitation is assumed to be a plane wave and particular attention is given to the case of transverse electric (TE) and transverse magnetic (TM) incidences. For the radiation problem, the assumed excitation is a distributed current or current filament placed inside the cavities.

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