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1 Ann Arbor, Monday May 22, 2006 IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dynamic AMR-FDTD: Method Microwave Circuit Example Conclusion Intro Dynamic AMR-FDTD: Optical Structure Example Dynamic AMR-FDTD: Error analysis/ control Efficient Finite-Difference Time- Domain Modeling of Driven Periodic Structures Dongying Li and Costas D. Sarris The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Centers of Excellence.

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2 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Outline Introduction Motivation of the research Previous work Theory Floquet's theorem and sine-cosine periodic boundary condition (PBC) Array scanning method Numerical Examples Printed structure on PBG substrates Transmission-line (TL) metamaterials Conclusions and future work

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3 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Motivation Metamaterials Simultaneous negative Split-ring resonator (SRR), strip-wire, transmission- line (TL) grid Design of metamaterials closely related to periodic structure modeling periodic structures modeling

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4 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Finite-Difference Time-Domain (FDTD) Intro Example : FDTD discretization of Marching in time scheme FDTD: Domain decomposition in “Yee cells”; marching in time Yee’s cell

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5 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Mesh Refinement in FDTD Local mesh refinement schemes: Embedding a locally dense mesh into a coarse mesh. Mesh refinement guided by physical intuition; statically defined at the start of the simulation. Side-effect: stability. Sample references: Example: Non-uniform mesh for microstrip I. S. Kim and W. J. R. Hoefer, MTT-T, June S. S. Zivanovic, K. S.Yee, and K. K. Mei, MTT-T, Mar M. W. Chevalier, R. J. Luebbers, and V. P. Cable, AP-T, Mar M. Okoniewski, E. Okoniewska, and M. A. Stuchly, AP-T, Mar Intro

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6 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Why Dynamic Mesh Refinement Time-domain methods register the evolution of a source pulse and its retro-reflections in a given domain. Edges, high-dielectric regions etc. are not continuously illuminated during an FDTD simulation; local mesh refinement around them is NOT always necessary. Absorbing boundary Simulated Structure Wideband source Intro

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7 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Previous Work Adaptive Mesh Refinement [Berger, Oliger, J. Comput. Physics,1984]: – Computational fluid dynamic tool for hyperbolic PDEs. – Performs selective mesh refinement by factors of 2. – Allows for the dynamic re-generation of coarse/dense mesh regions. Moving-Window FDTD (MW-FDTD, Luebbers et al., Proc. IEEE AP-S, June 2003): –Single moving window of fixed width, velocity tracking a forward wave in a wireless link. Intro

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8 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dynamically AMR-FDTD: Overview Key features of this work on Dynamically Adaptive Mesh Refinement (AMR)-FDTD –Combination of the FDTD technique with the AMR algorithm. –Implementation of a three-dimensional adaptive, moving mesh. Dynamic AMR-FDTD: Method

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9 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Root/Child Meshes The AMR-FDTD starts by covering the computational domain in a coarse mesh (called root mesh), of Yee cell dimensions Every N AMR time steps, checks whether mesh refinement is needed at any part of the domain. Dynamic AMR-FDTD: Method

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10 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Root/Child Meshes (cont-d) Clustering together cells that have been “flagged” for refinement, it generates a child mesh that covers them, with cell sizes Recursively, child meshes can be refined by a factor of two if flagged at a later check. Dynamic AMR-FDTD: Method

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11 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Root / Child Meshes (cont-d) Mesh generation corresponds to a tree structure. MESH TREE Yee cells Level 1 Level 2 Level 3 A: Level 1 (Root) Mesh B1, B2,…, B5: Level 2 Meshes C1, C3: Level 3 Meshes Dynamic AMR-FDTD: Method

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12 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Time Stepping Stability condition for the root mesh: Courant number s < 1. Keeping the same Courant number in all meshes, the time step of level M mesh is: Note Note: Minimum cell size affects the time step of the corresponding mesh level only (asynchronous updates). Field Updates in AMR-FDTD Dynamic AMR-FDTD: Method

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13 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Each N AMR time steps, the mesh tree is regenerated. Method: Calculate energy in each Yee cell and then gradient throughout the domain. Adaptive Mesh Refinement Dynamic AMR-FDTD: Method

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14 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun both If both of the following conditions are met : Adaptive Mesh Refinement (cont-d) : predefined thresholds cell (i, j, k) of the root mesh is flagged for refinement First criterion: Captures energy gradient peaks. Second criterion: Prevents numerical noise (at later stages) from triggering spurious refinements. Dynamic AMR-FDTD: Method

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15 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Mesh Refinement is extended at a distance D around a flagged cell: This accounts for wave propagation within the mesh refinement time window of N AMR time steps. Flagged cells are clustered in rectangular regions following the algorithm of [Berger and Rigoutsos, IEEE Trans. Systems, Man, Cybernetics, Sept. 1991]. Adaptive Mesh Refinement: Clustering Flagged cells : “spreading” factor Dynamic AMR-FDTD: Method

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16 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Application: Microstrip Low-Pass Filter* A=40mm, B 1 =2mm, B 2 =21mm, W=3mm, 0.8mm substrate of r =2.2 Vertical electric field magnitude Time = 0 Number of Child Meshes = 1 Refined volume/total volume = *From Sheen et al, IEEE MTT-T, July Dynamic AMR-FDTD: Microwave Circuit Example

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17 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Application: Microstrip Low-Pass Filter Time = 100 t Number of Child Meshes = 1 Refined volume/total volume = A=40mm, B 1 =2mm, B 2 =21mm, W=3mm, 0.8mm substrate of r =2.2 Vertical electric field magnitude Dynamic AMR-FDTD: Method Conclusion Intro Error analysis/ control Dynamic AMR-FDTD: Microwave Circuit Example

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18 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Application: Microstrip Low-Pass Filter Time = 200 t Number of Child Meshes = 1 Refined volume/total volume = Vertical electric field magnitude A=40mm, B 1 =2mm, B 2 =21mm, W=3mm, 0.8mm substrate of r =2.2 Dynamic AMR-FDTD: Microwave Circuit Example

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19 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Application: Microstrip Low-Pass Filter Time = 500 t Number of Child Meshes = 3 Refined volume/total volume = A=40mm, B 1 =2mm, B 2 =21mm, W=3mm, 0.8mm substrate of r =2.2 Vertical electric field magnitude Dynamic AMR-FDTD: Microwave Circuit Example

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20 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Application: Microstrip Low-Pass Filter Time = 800 t Number of Child Meshes = 3 Refined volume/total volume = 0.28 A=40mm, B 1 =2mm, B 2 =21mm, W=3mm, 0.8mm substrate of r =2.2 Vertical electric field magnitude Dynamic AMR-FDTD: Microwave Circuit Example

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21 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Evolution of Child Meshes Coverage=Volume of child meshes / total volume of the domain Dynamic AMR-FDTD: Microwave Circuit Example In the long-time regime, AMR-FDTD is equivalent to a root-mesh based uniform mesh FDTD (reason for no late-time instability).

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22 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Late-Time Regime No late-time instability observed ! Dynamic AMR-FDTD: Microwave Circuit Example

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23 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Microstrip Low-Pass Filter: S-parameters 94.6% reduction in execution time Dynamic AMR-FDTD: Microwave Circuit Example

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24 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Optical applications: Power Splitter Dimensions are given in microns. Dielectric constants: Dynamic AMR-FDTD: Optical StructureExample

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25 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Power Splitter: Time-Domain Results Power Splitter: Time-Domain Results Port 2 AMR-FDTD with four levels Port 3 Dynamic AMR-FDTD: Optical StructureExample

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26 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Power Splitter: Numerical Results (cont-d) Error Metric: Dynamic AMR-FDTD: Optical StructureExample

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27 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Power Splitter: Wave front Tracking Power Splitter: Wave front Tracking Dynamic AMR-FDTD: Optical StructureExample

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28 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dielectric Ring Resonator* (4-level AMR) Dielectric Ring Resonator* (4-level AMR) *Hagness et al., IEEE J. Lightwave Tech., vol. 15, pp , Nov Port 2 Dynamic AMR-FDTD: Optical StructureExample

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29 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dielectric Ring Resonator (cont-d) Dielectric Ring Resonator (cont-d) Dynamic AMR-FDTD: Optical StructureExample

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30 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dielectric Ring Resonator: Late-time regime Dielectric Ring Resonator: Late-time regime No late-time instability observed ! Dynamic AMR-FDTD: Optical StructureExample

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31 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Determining the AMR Parameters Objective : Produce clear guidelines for the determination of the controlling parameters of the AMR-FDTD. Methodology: 2-D TE case studies run; error compared to reference simulation (densely gridded FDTD) was measured at probe points distributed throughout the computational domain, over time: This procedure is aimed at rendering the error bound estimates independent of the simulated structure. Dynamic AMR-FDTD: Error analysis/ control

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32 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Choice of thresholds e, g Dynamic AMR-FDTD: Error analysis/ control Errors from several numerical experiments as a function of the thresholds e, g are collected. Error curves are fitted with the function: Error bounds for the cases when q e =0 or q g =0 are derived along with the constants C 1 -C 4.

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33 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dynamic AMR-FDTD: Error analysis/ control Effect of window of mesh refinement N AMR Increasing N AMR reduces errors, but also increases simulation time (because of “spreading factor”). A value that compromises accuracy and efficiency is N AMR =10. Every N AMR time steps, cells that need mesh refinement are “flagged”. Mesh Refinement is extended at a distance D around a flagged cell to account for wave propagation within the mesh refinement time window : : “spreading” factor

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34 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Dynamic AMR-FDTD: Error analysis/ control Effect of number of mesh refinement levels Keeping the resolution constant, the effect of increasing AMR levels is tested (root mesh gets coarser). Eventually, as the number of levels increases (beyond typically 4), error and simulation time increases. Example: Corrugated waveguide simulation Dimensions in microns

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35 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Conclusions Conclusion The dynamically AMR-FDTD implements multiple, adaptively generated sub- grids in two/three-dimensional FDTD and achieves up to two-orders of magnitude computation time savings. The mesh generation in AMR-FDTD is a self-adaptive process, based on pre- defined accuracy parameters (CAD- oriented feature).

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36 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Conclusions (cont-d) Conclusion Guidelines for the choice of the AMR parameters were provided by studying their effect on time-domain error metrics. Future Research Refinement of mesh refinement criteria ! Closed-domain, evanescent-wave problems High-order finite-differences, conformal meshing

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37 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun References C.D. Sarris, Adaptive Mesh Refinement for Time-Domain Electromagnetics, Morgan&Claypool. Morgan&Claypool Y. Liu, C.D. Sarris, ``Fast Time-Domain Simulation of Optical Waveguide Structures with a Multilevel Dynamically Adaptive Mesh Refinement FDTD Approach'', IEEE/OSA J. Lightwave Technology, vol. 24, no. 8, pp , Aug Y. Liu, C.D. Sarris, ``Efficient Modeling of Microwave Integrated Circuit Geometries via a Dynamically Adaptive Mesh Refinement (AMR)-FDTD Technique'', IEEE Trans. on Microwave Theory Tech., vol. 54, no.2, Feb

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38 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Thank you ! Questions/Remarks?Questions/Remarks?

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39 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Field Update Flowchart: General Check the number of time steps; if it is an integer multiple of N AMR, re-generate the mesh tree. Update field grid points of the root mesh Copy fields from the root mesh to the child meshes. Update level M meshes 2 M-1 times. If maximum time step has been reached, terminate. Otherwise go back to (1) Copy fields of the child meshes back to the root mesh for the time steps of the root mesh (interpolating as needed). Field Updates in AMR-FDTD

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40 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Mesh boundary updates Types of boundaries Field Updates in AMR-FDTD

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41 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Mesh boundary updates Types of boundaries Segment ea: “Physical boundary” (PB) of a child mesh to a terminating boundary (ABC, PEC etc.). Field Updates in AMR-FDTD

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42 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Mesh boundary updates Types of boundaries Segment cd: “Sibling boundary” (SB) between child meshes of the same level (same Yee cell volume). Field Updates in AMR-FDTD

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43 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun AMR-FDTD: Mesh boundary updates Types of boundaries Segments ab, bc, ed: “Child-Parent boundaries” (CPB’s) between child and parent meshes. Field Updates in AMR-FDTD

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44 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Mesh boundary updates: CPBs Child/Parent grid points: Never collocated in space or time (always interleaved). Transfer of field values from the one mesh to the other involves spatial and temporal interpolations. : Child mesh : Parent mesh Field Updates in AMR-FDTD

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45 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Update H-field points in the parent mesh at (n+1/2) t. Update E-field points in the parent mesh at (n+1) t. For each child mesh, update H-field points at (n+1/4) t. For each child mesh, obtain non-boundary E- field points at (n+1/2) t Field Update Flowchart: Child/Parent Connection For each child mesh, obtain boundary E-field points at (n+1/2) t, invoking interpolated H- field values of the parent mesh. Field Updates in AMR-FDTD

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46 Floquet’s theorem & PBC Array scanning method Conclusion Intro PBG substrates example TL-meta- material example IEEE AP-S International Symposium, Honolulu Hawaii, Jun Repeat 3, 4, 5 to advance each child mesh to time step (n+1) t. At regions where child/parent meshes overlap, update parent grid points by interpolating child grid points. 6 7 Field Update Flowchart: Child/Parent Connection (cont-d) This algorithm is recursively applied for the interconnection of higher-level child/parent meshes (for example to connect level 2 to level 3 and so on). Field Updates in AMR-FDTD

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