Int ro Dynamic AMRFDTD Method Efficient FiniteDifference Time

Int ro Dynamic AMRFDTD: Method Efficient Finite-Difference Time. Domain Modeling of Driven Periodic Structures Dynamic AMR-FDTD: Microwave Circuit Example Dynamic AMR-FDTD: v. Dongying Li and Costas D. Sarris Optical Structure Example Dynamic AMRFDTD: Error analysis/ control Co ncl usi on v The Edward S. Rogers Sr. Department of v Electrical and Computer Engineering v University of Toronto Research supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Centers of Excellence. 1 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007 Ann Arbor, Monday May 22, 2006

Outline Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on v Introduction • Motivation of the research • Previous work v Theory • Floquet's theorem and sine-cosine periodic boundary condition (PBC) • Array scanning method v Numerical Examples • Printed structure on PBG substrates • Transmission-line (TL) metamaterials v Conclusions and future work 2 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Motivation Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 3 v Metamaterials • Simultaneous negative • Split-ring resonator (SRR), strip-wire, transmission-line (TL) grid • Design of metamaterials closely related to periodic structure modeling v periodic structures modeling IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Finite-Difference Time-Domain (FDTD) Yee’s cell FDTD: Domain decomposition in “Yee cells”; marching in time Example : FDTD discretization of TLmetamateria l exampl e Co ncl usi on Marching in time scheme 4 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Mesh Refinement in FDTD Int ro ro Floquet ’s theore m& PBC • Local mesh refinement schemes: Embedding a locally dense mesh into a coarse mesh. Example: Non-uniform mesh for microstrip Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on • Mesh refinement guided by physical intuition; statically defined at the start of the simulation. I. S. Kim and W. J. R. Hoefer, MTT-T, June 1990. • Side-effect: stability. S. S. Zivanovic, K. S. Yee, and K. K. Mei, MTT-T, Mar. 1991. M. W. Chevalier, R. J. Luebbers, and V. P. Cable, AP-T, Mar. 1997. M. Okoniewski, E. Okoniewska, and M. A. Stuchly, AP-T, Mar. 1997. • Sample references: 5 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Why Dynamic Mesh Refinement Int ro ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on • Time-domain methods register the evolution of a source pulse and its retro-reflections in a given domain. Wideband source Simulated Structure Absorbing boundary • Edges, high-dielectric regions etc. are not continuously illuminated during an FDTD 6 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Previous Work Int ro ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on v 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. v 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. 7 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 8 Dynamically AMR-FDTD: Overview v. 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. IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Root/Child Meshes Int ro Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e v. The AMR-FDTD starts by covering the computational domain in a coarse mesh (called root mesh), of Yee cell dimensions TLmetamateria l exampl e Co ncl usi on v. Every NAMR time steps, checks whether mesh refinement is needed at any part of the domain. 9 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e AMR-FDTD: Root/Child Meshes (cont-d) v. Clustering together cells that have been “flagged” for refinement, it generates a child mesh that covers them, with cell sizes TLmetamateria l exampl e Co ncl usi on 10 v. Recursively, child meshes can be refined by a factor of. Honolulu two if. Hawaii, flagged IEEE AP-S International Symposium, Jun. 14 at 2007

AMR-FDTD: Root / Child Meshes (cont-d) Int ro Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC v Mesh generation corresponds to a tree structure. A: Level 1 (Root) Mesh B 1, B 2, …, B 5: Level 2 Meshes C 1, C 3: Level 3 Meshes Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on MESH TREE Yee cells Level 1 Level 2 Level 3 11 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Time Stepping Int ro v Stability condition for the root mesh: Floquet Dynamic Field ’s AMRUpdate theore FDTD: s in m& AMRMethod PBC FDTD Array scanni ng method PBG substra tes exampl e v Courant number s < 1. TLmetamateria l exampl e Co ncl usi on v Keeping the same Courant number in all meshes, the time step of level M mesh is: v Note: Note Minimum cell size affects the time step of the corresponding mesh level only (asynchronous updates). 12 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Adaptive Mesh Refinement Int ro Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e v Each NAMR time steps, the mesh tree is regenerated. v Method: Calculate energy in each Yee cell and then gradient throughout the domain. TLmetamateria l exampl e Co ncl usi on 13 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro Adaptive Mesh Refinement (cont -d) v If both of the following conditions are met : both Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e : predefined thresholds TLmetamateria l exampl e Co ncl usi on cell (i, j, k) of the root mesh is flagged for refinement Ø First criterion: Captures energy gradient peaks. Ø Second criterion: Prevents numerical noise (at later stages) from triggering spurious refinements. 14 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro Adaptive Mesh Refinement: Clustering v Mesh Refinement is extended at a distance D around a flagged cell: Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on : “spreading” factor v This accounts for wave propagation within the mesh refinement time window of NAMR time steps. v Flagged cells are clustered in rectangular regions following the algorithm of [Berger and Rigoutsos, IEEE Trans. Systems, Man, Cybernetics, Sept. 1991]. Flagged cells 15 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Application: Microstrip Low-Pass Filter* Int ro Vertical electric field magnitude Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on A=40 mm, B 1=2 mm, B 2=21 mm, W=3 mm, 0. 8 mm substrate of er=2. 2 *From Sheen et al, IEEE MTT-T, July 1990. Time = 0 Number of Child Meshes = 1 Refined volume/total volume = 0. 043 16 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Application: Microstrip Low-Pass Filter Int ro ro Vertical electric field magnitude Floquet Dynamic ’s AMRtheore FDTD: m& Method PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi. AMRDynamic FDTD: on A=40 mm, B 1=2 mm, B 2=21 mm, W=3 mm, 0. 8 mm substrate of er=2. 2 Error analysis/ control Co ncl usi on 17 Time = 100 Dt Number of Child Meshes = 1 Refined volume/total volume = 0. 134 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Application: Microstrip Low-Pass Filter Int ro Vertical electric field magnitude Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on A=40 mm, B 1=2 mm, B 2=21 mm, W=3 mm, 0. 8 mm substrate of er=2. 2 Time = 200 Dt Number of Child Meshes = 1 Refined volume/total volume = 0. 525 18 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Application: Microstrip Low-Pass Filter Int ro Vertical electric field magnitude Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on A=40 mm, B 1=2 mm, B 2=21 mm, W=3 mm, 0. 8 mm substrate of er=2. 2 Time = 500 Dt Number of Child Meshes = 3 Refined volume/total volume = 0. 442 19 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Application: Microstrip Low-Pass Filter Int ro Vertical electric field magnitude Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on A=40 mm, B 1=2 mm, B 2=21 mm, W=3 mm, 0. 8 mm substrate of er=2. 2 Time = 800 Dt Number of Child Meshes = 3 Refined volume/total volume = 0. 28 20 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Evolution of Child Meshes Int ro Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on Coverage=Volume In the long-time regime, of child AMR-FDTD meshes / total is equivalent volume of tothe a root-mesh domain based uniform mesh FDTD (reason for no late-time instability). 21 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Late-Time Regime Int ro Floquet ’s theore m& PBC Array scanni Dynamic ng AMR-FDTD: method Microwave Circuit PBG Example substra tes exampl e TLmetamateria l exampl e Co ncl usi on No late-time instability observed ! 22 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Optical applications: Power Splitter Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on v Dimensions are given in microns. v Dielectric constants: 24 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Power Splitter: Time-Domain Results Int ro v AMR-FDTD with four levels Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on Port 2 Port 3 25 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Power Splitter: Numerical Results (cont-d) Int ro Floquet ’s theore m& PBC v Error Metric: Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on 26 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Int ro Power Splitter: Wave front Tracking Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on 27 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Dielectric Ring Resonator* (4 -level AMR) Int ro Floquet ’s theore m& PBC Array scanni ng method Port 2 PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on *Hagness et al. , IEEE J. Lightwave Tech. , vol. 15, pp. 2154 -2165, Nov. 1997. 28 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Dielectric Ring Resonator (cont-d) Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on 29 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Dielectric Ring Resonator: Late-time regime Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Dynamic AMR-FDTD: TLmeta. Optical materia Structure l Example exampl e Co ncl usi on No late-time instability observed ! 30 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Determining the AMR Parameters Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi AMR Dynamic -FDTD: on • Objective: Produce clear guidelines for the determination of the controlling parameters of the AMRFDTD. • 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: Error analysis/ control This procedure is aimed at rendering the error bound estimates independent of the simulated structure. 31 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Choice of thresholds qe, qg Int ro Floquet ’s theore m& PBC Array scanni ng method Errors from several numerical experiments as a function of the thresholds qe, qg are collected. Error curves are fitted with the function: PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi AMR Dynamic -FDTD: on Error analysis/ control 32 • Error bounds for the cases when qe=0 or qg=0 are derived along with the constants C 1 -C 4. IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Effect of window of mesh refinement NAMR Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi AMR Dynamic -FDTD: on Error analysis/ control • Every NAMR 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 • Increasing NAMR reduces errors, but also increases simulation time (because of “spreading factor”). • A value that compromises accuracy and efficiency is NAMR=10. 33 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Effect of number of mesh refinement levels Floquet ’s theore m& PBC • 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. Array scanni ng method • Example: Corrugated waveguide simulation Int ro PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi AMR Dynamic -FDTD: on Error analysis/ control Dimensions in microns 34 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Conclusions Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on Co nc lu si on 35 v. The dynamically AMR-FDTD implements multiple, adaptively generated subgrids in two/three-dimensional FDTD and achieves up to two-orders of magnitude computation time savings. v. The mesh generation in AMR-FDTD is a self-adaptive process, based on predefined accuracy parameters (CADoriented feature). IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Conclusions (cont-d) Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on Co nc lu si on 36 v. Guidelines for the choice of the AMR parameters were provided by studying their effect on time-domain error metrics. v. Future Research • Refinement of mesh refinement criteria ! • Closed-domain, evanescent-wave problems • High-order finite-differences, conformal meshing IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

References Int ro Floquet ’s theore m& PBC v C. D. Sarris, Adaptive Mesh Refinement for Time-Domain Electromagnetics, Morgan&Claypool. Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on v 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. 3235 -3247, Aug. 2006. v Y. Liu, C. D. Sarris, ``Efficient Modeling of Microwave Integrated Circuit Geometries via 37 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Questions/Remarks? Int ro Floquet ’s theore m& PBC Array scanni ng method PBG substra tes exampl e Thank you ! TLmetamateria l exampl e Co ncl usi on E-mail: cds@waves. toronto. edu 38 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Field Update Flowchart: General Int ro Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 39 1 Check the number of time steps; if it is an integer multiple of NAMR, re-generate the mesh tree. 2 Update field grid points of the root mesh 3 Copy fields from the root mesh to the child meshes. Update level M meshes 2 M-1 times. 4 Copy fields of the child meshes back to the root mesh for the time steps of the root mesh (interpolating as needed). 5 If maximum time step has been reached, terminate. Otherwise go back to (1). IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Mesh boundary updates Int ro v Types of boundaries Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 40 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Mesh boundary updates Int ro v Types of boundaries Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on Segment ea: “Physical boundary” (PB) of a child mesh to a terminating boundary (ABC, PEC etc. ). 41 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Mesh boundary updates Int ro v Types of boundaries Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on Segment cd: “Sibling boundary” (SB) between child meshes of the same level (same Yee cell volume). 42 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

AMR-FDTD: Mesh boundary updates Int ro v Types of boundaries Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on Segments ab, bc, ed: “Child-Parent boundaries” (CPB’s) between child and parent meshes. 43 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Mesh boundary updates: CPBs Int ro Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method v Child/Parent grid points: Never collocated in space or time (always interleaved). : Child mesh : Parent mesh PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on v Transfer of field values from the one mesh to the other involves spatial and temporal interpolations. 44 IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Field Update Flowchart: Child/Parent Connection Int ro Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 1 2 Update E-field points in the parent mesh at (n+1)Dt. 3 For each child mesh, update H-field points at (n+1/4)Dt. 4 For each child mesh, obtain non-boundary Efield points at (n+1/2)Dt. 5 45 Update H-field points in the parent mesh at (n+1/2)Dt. For each child mesh, obtain boundary E-field points at (n+1/2)Dt, invoking interpolated Hfield values of the parent mesh. IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007

Field Update Flowchart: Child/Parent Connection (cont-d) Int ro Floquet Field ’s Update theore s in m& AMRPBC FDTD Array scanni ng method PBG substra tes exampl e TLmetamateria l exampl e Co ncl usi on 46 6 7 Repeat 3, 4, 5 to advance each child mesh to time step (n+1)Dt. At regions where child/parent meshes overlap, update parent grid points by interpolating child grid points. v 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). IEEE AP-S International Symposium, Honolulu Hawaii, Jun. 14 2007