Hybrid Finite Element-Fast Spectral Domain Multilayer Boundary Integral Modeling of Doubly Periodic Structures Page: 3 of 18
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(IE) formulations. Typically, the spectral domain (SD) of the Green's function is used [1] cou-
pled with cascading for dealing with multilayer structures. The spectral domain formulation
has also been extended to multilayered planar structures such as aperture coupled microstrip
patches [2, 3]. More recently, the free periodic Green's functions have been incorporated into
hybrid finite element (FE)-boundary integral (BI) methods [4, 5, 6, 7, 8, 9] for analysis of
full three-dimensional (3D) structures (FSS and antennas) which may include inhomogeneous
sections. In this context, the FE method is employed to model a unit cell representing the
array and the BI provides for a rigorous mesh truncation at the upper and/or lower surfaces
of the discretized unit cell.
Many array configurations can be analyzed by employing the appropriate half-space pe-
riodic Green's functions for the BI. However, modern array designs often require complex
substrates and superstrate configurations. In the case of thick, possibly multilayered sub-
strates/superstrates, it is not efficient to use the FE method to model the dielectric region.
Instead, it is more appropriate to employ the multilayer Green's function in the context of the
FE-BI method.
In this manner (see Fig. 1), the FE method is only used to model the inhomogeneous
section of the domain which may involve metallizations or imperfect surfaces whereas the
thick multilayer substrates/superstrates can be modeled using the multilayer spectral Green's
functions. When compared with the standard implementations, the key difference in the
proposed hybridization is the placement of the BI at the interface separating the multilayer
region with the finite element domain. In previous FE-BI formulations, the BI had been
placed at the interface of the FE domain with the free space region. The multilayer Green's
function has been used in the context of the FE-BI in (10, 11] but not for periodic array
applications. When dealing with periodic structures in the presence of multilayered layers,
the SD form of the Green's function is particularly attractive. Of particular importance is the
use of the SD representation of the multilayer Green's function in the recently introduced fast
spectral domain algorithm (FSDA) [9] to attain O(N) CPU and memory requirements. In
this manner, very thick substrates can be modeled accurately even though they span several
hundred wavelengths as is the case with millimeter wave and infrared filters.
Below, we begin by presenting the FE-BI formulation in a manner that incorporates the
multilayered Green's function. This is followed by the FSDA implementation for carrying out
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Eibert, T.F.; Volakis, J.L. & Erdemli, Y.E. Hybrid Finite Element-Fast Spectral Domain Multilayer Boundary Integral Modeling of Doubly Periodic Structures, report, March 3, 2002; Schenectady, New York. (https://digital.library.unt.edu/ark:/67531/metadc782040/m1/3/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.