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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 1 /31 Local Sub-cell Models & Non-uniform Grids (1 sessions)

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 2 /31 A fundamental problem arises in any grid-based numerical modeling tool. Modeler options: 1. a variable lattice of space cells to precisely model shape and features of a structure. 2. a simpler and much more uniform mesh than option 1. Simplest Contour-path Sub-cell Models: These is a effective approach to dealing material structures whose bounding surfaces do not conform to grid planes in a uniform Cartesian mesh. There are two simple sub-cell models as: Local Sub-cell Models Staircase model, TE grid, Diagonal split-cell model for PEC surface, average-properties model for material interface.

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 3 /31 Others are updated using normal Yee algorithm. Contour-path Model of Narrow Slot: Slot is assumed to provide a sub cell air gap. Faraday's law contour paths C 1, C 2 and C 3 are used to derive special FDTD updates: Contour C 1 : Contour C 2 : Contour C 3 : Local Sub-cell Models Taflove et al., IEEE Trans. Antennas and Propagation, 1988. pp. 247-257, 1988 IEEE.

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 4 /31 Test: Contour-path Model of Narrow Slot Comparison of contour-path FDTD and frequency-domain moment-method results for gap electric field distribution in a slotted PEC screen for broadside TE, illumination: Magnitude phase

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 5 /31 Faraday's law contour path for thin wire. Umashankar, IEEE Trans. Antennas and Propagation, 1987, pp. 1248 1257, 1987 IEEE Contour-path Model of Thin Wire: It permits incorporation of near-field physics, yielding special purpose time-stepping expressions that are not obvious from pure FDM. An excellent example involves interaction of an EM with a PEC wire having a sub cell diameter [2,5]. Looping H Components: Where: Radial E Components: Where: For others components refer [2,5] in: Contour-path Model of Thin Wire

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 6 /31 Test: Contour-path Model of Thin Wire

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 7 /31 Locally Conformal Models of Curved Surfaces: Contour-path technique was proposed in [3,4] to implement conformal models of structures having curved surfaces within a Cartesian FDTD lattice. Yu-mitra Technique for PEC Structure: A TE z cut-plane in a 3D-FDTD lattice is shown in: Free space is assumed to be to the left of PEC. Results for PEC Structures: Twisted Elliptical Cross Section Waveguide Cavity: [9] Using Ansoft HFSS: Locally Conformal Models of Curved Surfaces Contour path for conformal FDTD PEC surface model. Yu and Mitra, IEEE Antennas and Propagation Magazine, Oct. 2000, pp. 28-39, 2000 IEEE. Contour-path FDTD yields an error at a mesh resolution of =/8 which is comparable to that obtained using staircased FDTD at =/32. Therefore a very significant storage and running- time reduction of approximately (32/8) 3 :1 or 64:1 and (32/8) 4 :1 or 256:1, respectively.

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 8 /31 Results for Wing like Object: Mono-static RCS at 10GHz of a wing like aluminum plate having acute 11 o angles for its leading and trailing edges [10]. Target specifications: 1. Length= 30.48cm (10 o ) in vertical (z) direction, width=25.4cm (8.47 o ); 2. Cross section shape (in x-y plane) of an isosceles triangle with base angles=11.31 o ; 3. Triangle sides opposite to base are smoothly joined by a 15.24cm (5.1 o ) radius cylindrical chamfer; 4. Base has a vertical triangular slot centered in its span. Slot dimensions: 1.27cm (0.42 o ) depth, and 2.54cm (0.85 o ) width. Results: Validation of 2D contour-path FDTD model Source: Taflove, Report to General Dynamics, 1990. Locally Conformal Models of Curved Surfaces

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 9 /31 Results for Pair of PEC Spheres: Discussion: Locally conformal FDTD models of PEC structures with curved surfaces provide clear advantages relative to use of stair casing. In a number of simulations ranging from studies of cavity resonances to calculations of RCS, conformal FDTD models achieve accuracy levels at mesh resolutions of o /20 that staircased models might achieve at mesh resolutions of o /80 or finer. Locally conformal FDTD models may also provide advantages relative to use of unstructured and partially structured space lattices. Away from PEC surface, locally conformal models globally propagate numerical modes in uniform and well-characterized Cartesian Yee lattice. In contrast, unstructured-lattice models propagate numerical modes in non-uniform, non-Cartesian meshes which have numerical dispersion and stability properties that vary with spatial position, and from problem to problem. Validation of 3D contour-path FDTD model of bistatic RCS of two 1 0 diameter PEC spheres separated by a 1 0 air gap, oblique incidence case. solid line =FDTD model; dots =generalized multi pole technique. Source: Jurgens and Taflove, IEEE Trans. Antennas and Propagation, 1993, pp. 1703-1708, 1993 IEEE. Locally Conformal Models of Curved Surfaces

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 10 /31 Yu-Mitra Technique for Material Structures: Second conformal FDTD technique reported by Yu and Mittra [12], which provides a simple and efficient treatment of dielectric structures with curved surfaces. This technique is much easier to use than contour-path model or effective-dielectric-constant method of [13]. It shows a significant improvement in accuracy relative to stair-casing that is comparable to either of two previous approaches. Space-cell geometry for Yu-Mittra model in a TE z cut-plane in a 3D-FDTD space lattice is shown: Embedded a dielectric structure whose surface intersects this cut plane such that a triangular part of cell is filled with 2 whereas remainder of cell is filled with 1 An effective permittivity eff is assigned as: Yu-Mittra technique is simpler because it does not require area or volume calculations pertaining to how space cells are cut in 3D by arbitrarily oriented surfaces. Hence, mesh generation for this conformal technique is quite simple. Yu-MiUra Technique for Material Structures Red walls have not any problem Red-blue walls have a problem

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Numerical Electromagnetics LN11_Local Subcell zakeri@nit.ac.ir 11 /31 Maloney-Smith Technique For Thin Material Sheets Maloney-Smith Technique For Thin Material Sheets: It is a contour-path method to model planar material sheets (dielectric and conducting) of sub-cell thickness, where sheet is perpendicular to one of major axes of FDTD space lattice [14]: By embedding a material sheet of thickness d