faster than fdtd pushing the boundaries of time-domain modeling … · 2006-05-29 · time-domain...

Dynamic Dynamic AMRAMR--FDTD:FDTD:

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Wireless Wireless Channel Channel ModelingModeling

GPUGPU--basedbased

TimeTime--DomainDomain

ModelingModeling

Radiation Radiation from NRIfrom NRI--TLTL

structuresstructures

ConclusionConclusion

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Faster than FDTDFaster than FDTD: : Pushing the boundaries of TimePushing the boundaries of Time--Domain Domain

Modeling for Wireless and Optical Modeling for Wireless and Optical Propagation Problems Propagation Problems

Costas D. SarrisCostas D. SarrisAssistant ProfessorAssistant Professor

The Edward S. Rogers Sr. Department ofThe Edward S. Rogers Sr. Department ofElectrical and Computer EngineeringElectrical and Computer Engineering

University of TorontoUniversity 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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AcknowledgementsAcknowledgements

•• Graduate studentsGraduate students– Abbas Alighanbari, Gerard Baron, Titos Kokkinos

•• PostPost--doctoral fellowdoctoral fellow– Dr. Yaxun Liu

•• ColleaguesColleagues–– Prof. George Eleftheriades, Prof. Eugene FiumeProf. George Eleftheriades, Prof. Eugene Fiume

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OutlineOutline

•• IntroductionIntroduction– Finite-Difference Time-Domain (FDTD) – Current challenges; how our work addresses them

•• Dynamically Adaptive Mesh Refinement FDTDDynamically Adaptive Mesh Refinement FDTD– Overview of the algorithm– Applications: Microwave and optical structures– Performance evaluation

•• FullFull--Wave Indoor Channel ModelingWave Indoor Channel Modeling–– The SThe S--MRTD technique applied to channel modelingMRTD technique applied to channel modeling–– GPU Accelerated SGPU Accelerated S--MRTDMRTD

•• Fast FDTD Analysis of Periodic LeakyFast FDTD Analysis of Periodic Leaky--Wave StructuresWave Structures–– FloquetFloquet--based methodology and validationbased methodology and validation–– Application to NRIApplication to NRI--TL based leakyTL based leaky--wave antennaswave antennas

•• ConclusionConclusion

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Introduction: TimeIntroduction: Time--Domain ModelingDomain Modeling

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Example :

FDTD discretization of

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂

∂

μ=

∂∂

yE

zE1

tH zyx

⎥⎥⎦

⎤

⎢⎢⎣

⎡

Δ

−−

Δ

−

μΔ

+= ++++++++−+++ y

EEz

EEtHHz

2/1k,j,inz

2/1k,1j,iny

k,2/1j,iny

1k,2/1j,inx2/1k,2/1j,i2/1n

x2/1k,2/1j,i2/1n

Marching in time scheme

In FDTD, the computational domain is divided in “Yee’s cells” and Maxwell’s equations are solved by marching in time.

Yee’s cell

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FDTD: Successful, yet..FDTD: Successful, yet..

IntroIntro Simple to implement, versatile, direct update equations (no

matrix assembly, storage).

Pronounced “numerical dispersion” necessitates use

of dense grid.

• “High-order” methods can do much better

Small time step enforced by stability criterion

• CFL criterion:

s<=1

• Maintaining large cell size through “adaptive meshing”

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Mesh Refinement in FDTDMesh 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: Late-time instability.• Sample references:

Example: Non-uniform mesh for microstrip

I. S. Kim and W. J. R. Hoefer, MTT-T, June 1990.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.

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Dynamic Mesh Refinement: Motivation Dynamic Mesh Refinement: Motivation

• Time-domain methods register the evolution of a source pulse and its retro-reflections in a given domain.

• Edges, high-dielectric regions etc. are notcontinuously illuminated during an FDTD simulation; local mesh refinement around them is NOT always necessary.

Absorbing boundary

Simulated Structure

Widebandsource


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Dynamic Mesh Refinement: Previous WorkDynamic Mesh Refinement: 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.

– Window is terminated into absorbing boundaries (no possibility for reflected wave modeling).


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Dynamically AMRDynamically AMR--FDTD: OverviewFDTD: 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.

– Evaluation of efficiency and accuracy for electromagnetic problems, considering realisticrealisticapplications.


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AMRAMR--FDTD: Root/Child MeshesFDTD: Root/Child Meshes

The AMR-FDTD starts by covering the computational domain in a coarse mesh (called root mesh), of Yee cell dimensions

Every NAMR time steps, checks whether mesh refinement is needed at any part of the domain.

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.

.zyx Δ×Δ×Δ

.2z

2y

2x Δ

×Δ

×Δ


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AMRAMR--FDTD: Root / Child Meshes (contFDTD: Root / Child Meshes (cont--d)d)

Mesh generation corresponds to a tree structure.

A

B1 B2 B3 B4 B5

C1 C2

A

B1 B2

B3B4

B5

C1 C2

zyx Δ×Δ×Δ

2z

2y

2x Δ

×Δ

×Δ

4z

4y

4x Δ

×Δ

×Δ

MESH TREEMESH TREE Yee cellsYee cells

Level 1

Level 2

Level 3

A: Level 1 (Root) MeshB1, B2,…, B5: Level 2 MeshesC1, C3: Level 3 Meshes


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AMRAMR--FDTD: Time SteppingFDTD: Time SteppingStability 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:

NoteNote: Minimum cell size affects the time step of the corresponding mesh level only (asynchronous updates).

Field Field Updates in Updates in AMRAMR--FDTDFDTD


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Field Update Flowchart: GeneralField Update Flowchart: General

Check the number of time steps; if it is an integer multiple of NAMR, 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 2M-1 times.

If maximum time step has been reached, terminate. Otherwise go back to (1).

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22

33

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Copy fields of the child meshes back to the root mesh for the time steps of the root mesh

(interpolating as needed) .


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Mesh boundary updates: Mesh boundary updates: CPBsCPBsChild/Parent grid points: Never collocated in space Child/Parent grid points: Never collocated in space or time (always or time (always interleavedinterleaved). ).

Transfer of field values from the one mesh to the Transfer of field values from the one mesh to the other involves other involves spatialspatial and and temporaltemporal interpolationsinterpolations. .

: Child mesh: Parent mesh


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Each NAMR time steps, the mesh tree is regenerated.

Method: Calculate energy in each Yee cell and then gradient throughout the domain.

Adaptive Mesh RefinementAdaptive Mesh Refinement


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If bothboth of the following conditions are met :

Adaptive Mesh Refinement (contAdaptive Mesh Refinement (cont--d)d)

: : predefinedpredefined thresholdsthresholds

cell (i, j, k) of the root mesh is flagged for refinementcell (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.


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Mesh Refinement is extended at a distance D around a flagged cell:

This accounts for wave propagation within the mesh refinement time window of NAMR time steps.

Flagged cells are clustered in rectangular regions following thealgorithm of [Berger and Rigoutsos, IEEE Trans. Systems, Man, Cybernetics, Sept. 1991].

Adaptive Mesh Refinement: ClusteringAdaptive Mesh Refinement: Clustering

Flagged cells

: “spreading” factorDynamic Dynamic

AMRAMR--FDTD:FDTD:

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Application: Microstrip LowApplication: Microstrip Low--Pass Filter*Pass Filter*

A=40mm, B1=2mm, B2=21mm, W=3mm,

0.8mm substrate of εr=2.2

Vertical electric field magnitude

Time = 0 Number of Child Meshes = 1 Refined volume/total volume = 0.043

*From Sheen et al, IEEEMTT-T, July 1990.

Dynamic AMR-FDTD:Examples

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Application: Microstrip LowApplication: Microstrip Low--Pass FilterPass Filter

Time = 100Δt Number of Child Meshes = 1 Refined volume/total volume = 0.134

A=40mm, B1=2mm, B2=21mm, W=3mm,




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A=40mm, B1=2mm, B2=21mm, W=3mm,



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A=40mm, B1=2mm, B2=21mm, W=3mm,




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A=40mm, B1=2mm, B2=21mm, W=3mm,




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Microstrip LowMicrostrip Low--Pass Filter: Evolution of Child MeshesPass Filter: Evolution of Child Meshes

Coverage=Volume of child meshes / total volume of the domain


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Microstrip LowMicrostrip Low--Pass Filter: Evolution of Child MeshesPass Filter: Evolution of Child Meshes

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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LateLate--Time RegimeTime Regime


No late-time instability observed !

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Microstrip LowMicrostrip Low--Pass Filter: SPass Filter: S--parametersparameters

94.6% reduction in execution time


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Application: Microstrip Branch Coupler*Application: Microstrip Branch Coupler*A=40 mm, B1=7 mm,

B2=11 mm ,B3=9 mm, W1=2 mm,W2=3 mm,


94.8% reduction in execution time

*From Sheen et al, IEEE MTT-T, July 1990.


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Application: Microstrip Spiral InductorApplication: Microstrip Spiral Inductor

82% reduction in execution time

A1=60 mm, A2=40 mm, B1=24 mm, B2=20 mm ,B3=18 mm, B4=4 mm ,

W1=W2=2 mm,0.8mm substrate of εr=2.2


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Optical applications: Power SplitterOptical applications: Power Splitter

• Dimensions are given in microns. • Dielectric constants: 4lens

r =ε 9guider =ε


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Power Splitter: TimePower Splitter: Time--Domain ResultsDomain Results

Port 2

• AMR-FDTD with four levels

Port 3


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Power Splitter: Numerical Results (contPower Splitter: Numerical Results (cont--d)d)

• Error Metric:


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Power Splitter: Wave front TrackingPower Splitter: Wave front Tracking


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Dielectric Ring Resonator* (4Dielectric Ring Resonator* (4--level AMR)level AMR)

*Hagness et al., IEEE J. Lightwave Tech., vol. 15, pp. 2154-2165, Nov. 1997.


Port 2

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Dielectric Ring Resonator (contDielectric Ring Resonator (cont--d)d)


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Dielectric Ring Resonator: LateDielectric Ring Resonator: Late--time regimetime regime


No late-time instability observed !

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ReferencesReferences

1. Y. Liu and C.D. Sarris, “Efficient Modeling of Microwave Integrated Circuit Geometries via a Dynamically Adaptive Mesh Refinement (AMR) - FDTD Technique”, IEEE Trans. Microwave Theory and Tech., vol. 54, no. 2, pp. 689-703, Feb. 2006.

2. Y. Liu and C.D. Sarris, “A Multilevel Dynamically Adaptive Mesh Refinement (AMR)-FDTD Technique Applied to Dielectric Waveguide Structures”, IEEE/OSA Journal of Lightwave Tech., to appear 2006.

3. Y. Liu and C.D. Sarris, “Numerical Error Analysis and Control in a Dynamically Adaptive Mesh Refinement (AMR) - FDTD Technique”, Proc. 2006 IEEE MTT-S International Microwave Symposium, to appear.


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Indoor Wireless Channel ModelingIndoor Wireless Channel Modeling

Wireless Channel Modeling

– Advances in computing enable the application of differential methods to large-scale problems.

–– Recent papersRecent papers: • Hybridize FDTD with UTD [Bernardi et al., IEEE MTT-

T, Dec. 2003, Ray-Tracing [Wang et al., IEEE AP-T, May 2000].

• Demonstrate physical effects (e.g. wall attenuation) that information-theoretic models do not accurately account for [Yun et al., IEEE AP-T, April 2004].

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What does What does ““highhigh--orderorder”” mean? mean? – Spatial and/or temporal partial derivatives

approximated within an error ~ Δp.

SecondSecond--order finite order finite difference difference

(FDTD)(FDTD)

HighHigh--order finite order finite differencedifference

(MRTD(MRTD--exceptexcept Haar, Haar, PSTD, spectral methods PSTD, spectral methods

etc) etc) “connection coefficients”

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What can highWhat can high--order methods do for largeorder methods do for large--scale problems?scale problems?– Coarser mesh: Memory savings.– Potential for execution time economy.

Disadvantages/Potential bottlenecksDisadvantages/Potential bottlenecks– Boundary conditions.– Material properties.– Increased inter-processor communication in

domain-decomposition.

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HighHigh--Order SOrder S--MRTD Technique*MRTD Technique*


S-MRTD discretization

:“connection coefficients”

*M. Krumpholz, L. Katehi, IEEE MTT-T, April 1996.

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HighHigh--Order SOrder S--MRTD Technique (contMRTD Technique (cont--d)d)


Dispersion analysis for Coifman SDispersion analysis for Coifman S--MRTD*MRTD*

*A. Alighanbari, C.D. Sarris, IEEE T*A. Alighanbari, C.D. Sarris, IEEE T--AP, Aug. 2006, to appear.AP, Aug. 2006, to appear.

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Case study: 27m x 27m indoor floor plan.Bi-orthogonal Deslariers-Dubuc functions used.

source

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TimeTime--domain field waveforms (contdomain field waveforms (cont--d)d)

• S-MRTD accuracy at 5 cells per wavelength comparable to FDTD accuracy at 20 cells per wavelength.

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TimeTime--domain field waveforms (contdomain field waveforms (cont--d)d)

• S-MRTD accuracy at 5 cells per wavelength comparable to FDTD accuracy at 20 cells per wavelength.

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Vertical electric field magnitude at 900 MHzVertical electric field magnitude at 900 MHz


• Wall conductivity 0.002 S/m

FDTD: 52 hrs 36 mins

MRTD: 12 hrs 44 mins

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Vertical electric field magnitude at 900 MHzVertical electric field magnitude at 900 MHz



FDTD: 52 hrs 36 mins

MRTD: 12 hrs 44 mins

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SystemSystem--level fading modelslevel fading models


•Wall conductivity 0.002 S/m, LOS points

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SystemSystem--level fading models (contlevel fading models (cont--d)d)


•Wall conductivity 0.05 S/m, LOS points

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Wall attenuation model Wall attenuation model



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Wall attenuation model (contWall attenuation model (cont--d)d)



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General Purpose Computing on Graphics HardwareGeneral Purpose Computing on Graphics Hardware

• Commodity Graphics Hardware– Inexpensive– Fast

• Intel Pentium 4 3GHz– 6 GFLOPS (Theoretical)– 5.96 GB/sec peak (Theoretical)

• NVIDIA GeForce 6800 Ultra– 53 GLOPS (Observed)– 35.2 GB/sec peak (Observed)

– Faster• CPU

– 1.5x Annual Growth– 60x Per Decade Growth

• GPU– Greater than 2x Annual Growth– Greater than 1000x Per Decade

Growth

Values from “GPGPU: General-Purpose Computing on Graphics Hardware” Short Coarse Slides, SIGGRAPH ‘04

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General Purpose Computing on Graphics HardwareGeneral Purpose Computing on Graphics Hardware• Real-Time 3D Graphics

– Interactivity over physical accuracy

– Assumptions for parallelization

• Geometric primitives processed independently

• Interdependencies (i.e. shadows or reflections) only approximated

– Hardware acceleration by

• Fast local memory• Multiple pipelines

– Latest Innovation• Programmable Pipelines

(Shaders)• High-Precision Floating

Point

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Graphics Accelerated FDTDGraphics Accelerated FDTD

– S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Graphics Processor Unit (GPU) Acceleration of Finite-Difference Time-Domain (FDTD) Algorithm,” in IEEE International Symposium on Circuits and Systems, May 2004.

– G. S. Baron, C. D. Sarris, and E. Fiume, “Fast and Accurate Time-Domain Simulation with Commodity Graphics Hardware,” in Proceedings of IEEE Antennas andPropagation Society International Symposium, July 2005.

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Graphics Accelerated SGraphics Accelerated S--MRTDMRTD

– Reformulate update equations; consider them as discrete spatial convolutions:

– Time-stepping is an iterative sequence of filters.

GPU-basedTime-

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kernels

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Implementation: DetailsImplementation: Details

• Step one: Field and Material Property Meshes as Images– Pixel Shader constructs for

sampling 2D and 3D single-precision float pointing arrays or textures

– Textures: input and output streams

• 4 color channels • Up to 16 texture inputs• Up to 4 texture outputs

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Implementation: DetailsImplementation: Details

• Step two: Realize update equations as sums of Finite Impulse Response filters– Update equations

combination of a filter and blend

– Enforcing source is a so-called screen or overlay

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Evaluation: PerformanceEvaluation: Performance

– Per Scheme Speedup• GPU consistently faster than CPU• FDTD scheme approaches 10x speedup• S-MRTD schemes approach 30x speedup !!

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EvaluationEvaluation• Overall Speedup

– How does the performance equivalent (in terms of accuracy) FDTD and S-MRTD compare?

– Test: All schemes• coarse to very-fine

discretizations • λ/3 to λ/24 • 64x64 to 512x512 cell meshes

– Reference: FDTD• 1024x1024 cell mesh at λ/48

– Simulation Parameters:• 1.0 GHz Gaussian source• 6.4x6.4m2 cavity• 7.55 ns absolute time

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Evaluation: Overall SpeedupEvaluation: Overall Speedup

– Error

– No practical difference between CPU and GPU (e.g. non-standard arithmetic not an issue)

• Observation: Very-Fine (λ/24) FDTD comparable to Medium (λ/6) DD4 simulation

– Medium discretization 4x coarser (e.g. 16x fewer cells)– Demonstration of S-MRTD’s superior dispersion

characteristics

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Evaluation: Overall SpeedupEvaluation: Overall Speedup– Execution Times (ms): GPU and CPU

– Medium S-MRTD/Very-Fine FDTD Speedups• GPU: 6.67x• CPU: 5.95x• Across Architectures: 78.43x

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Application: Wireless Channel ModelingApplication: Wireless Channel Modeling

• Indoor Wireless Channel Simulation– 1.0 and 2.4 GHz sources– 15.36x15.36 m2 floor plan– 6-8cm walls

(σ = 0.002 Ω-1 and εr=2.89)– 386.44 ns absolute time– Test: DD4 GPU

• Δx,Δz = 2 cm(λ/15 and λ/6.25)

• 728x728 cell mesh• 16 layer UPML ABC• 17.25 minutes

– Reference: FDTD CPU• Δx,Δz = 0.5 cm

(λ/60 and λ/25)• 3072x3072 cell mesh• 64 layer UPML ABC• 38.53 hours

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GPUGPU--Based SBased S--MRTD: NotesMRTD: Notes

– The use of S-MRTD enables the reduction of the memory requirements of a given problem; can make an otherwise GPU-unsolvable problem (due to memory) “fit” in the GPU !

– If so, sustainable 30x speed-ups are possible (much larger than any reported FDTD speed-ups).

– What was long perceived as a disadvantage of S-MRTD, i.e. increased operations per cell, is actually advantageous for GPU implementations.

– Possibility of “real-time” full-wave channel modeling provided by GPU-based S-MRTD.

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IntroIntro Floquet theorem : TwoFloquet theorem : Two--dimensional casedimensional case

( ) ( ) pkj perE~prE~rrrrrrr

−=+

ykxkk yxp ˆˆ +=r

Lattice vectorLattice vectorydxdp yx ˆˆ +=r

Bloch wavevectorBloch wavevector

(x)

(y)

dy

dx

Modeling of Infinite Periodic StructuresModeling of Infinite Periodic Structures

Radiation from NRI-TLstructures

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Modeling of unit cell through Periodic Boundary Conditions Modeling of unit cell through Periodic Boundary Conditions (PBCs)(PBCs)

dy

dx

• Application of Floquet’ s conditions between the boundaries in the directions of periodicity, enables the dispersion analysis of the structure.

PBC

PBC

Modeling of Infinite Periodic Structures (contModeling of Infinite Periodic Structures (cont--d)d)

(x)

(y)

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FDTD Modeling of Periodic Structures*FDTD Modeling of Periodic Structures*

• Dispersion analysis is carried out in two-steps:

– Enforce the Floquet boundary condition between boundaries for a fixed real real Bloch wavevector kp.

– Sample the electric field inside the periodic structure; determine resonant frequencies kp(ω), from the Fourier transform.

*A. Taflove, “Computational Electrodynamics: The FDTD method”, ArtechHouse, 1995


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Motivation: LeakyMotivation: Leaky--Wave StructuresWave Structures

• Leaky-Wave Antennas: – Periodic structures supporting leaky-wave

radiation.– Renewed interest stemming from recent

advances in metamaterials.

z

0z k<β

x

LWAβx

k0

Air

θ

0sin k

xβθ =

Fast (leaky) waves

Forward Endfire: βx=k0

Broadside: βx=0

emerging at angle θ,

z

|βx|/k0 <1:


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IntroIntro

Problem StatementProblem Statement

• FDTD analysis of LWAs relies on the simulation of truncated periodic structures, long enough to achieve convergence of α, β (computationally expensive).

• Is it possible to use periodic FDTD analysis to extract the attenuation constant of leaky-wave/lossy structures (complex Bloch wavevector) ?


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Example: 2DExample: 2D--NRINRI--TL Medium*TL Medium*

*G.V. Eleftheriades et al, IEEE Trans. MTT, vol. 50, Dec. 2002.


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Zo 141 Ohm

Lo 5.6 H

Co 1 pF

Stop Band

Light cone (β=ko)

FastFast

WaveWave

RegionRegion

FDTD Results from: T. Kokkinos, C.D. Sarris, G.V. Eleftheriades, IEEE Trans. MTT, vol. 53, Apr. 2005

Example: 2DExample: 2D--NRINRI--TL Medium (contTL Medium (cont--d)d)

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Periodic FDTD Analysis of 2DPeriodic FDTD Analysis of 2D--TL NRI MediumTL NRI Medium

[4] Harms et al, IEEE Trans. AP, vol. 42, Sept. 1994.

• Phase progression between periodic boundaries enforced via the “sine-cosine” method*.

Propagation along x-axis (kxd=λ/2)

x-axisy-axisRadiation

from NRI-TLstructures

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FDTD Field Resonances for Slow/Fast WavesFDTD Field Resonances for Slow/Fast Waves

Slow wave: Perfect resonance

Fast wave: Decaying resonance

• In both cases, the real part of the Bloch wavenumber was enforced.

Light cone

0k=β

Fast Waveregion

SlowWaveregion

SlowWaveregion


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Periodic FDTD Modeling of LeakyPeriodic FDTD Modeling of Leaky--Wave StructuresWave Structures

• Question:

– Does the enforcement of the phase progression of a wave within a periodic FDTD mesh still allow for the extraction of the attenuation constant ?

• Answer: Inspection of the implementation of PBCs in FDTD.Radiation


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Implementation of PBCs in FDTDImplementation of PBCs in FDTD

unit cell

2D-Yee cell


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2D-Yee cell

unit cell


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2D-Yee cell

unit cell


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2D-Yee cell

PBC updates are

bi-directional !

unit cell


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Periodic Lossy/Radiating Structure

Periodic FDTD Modeling of LWPeriodic FDTD Modeling of LW--Structures, revisitedStructures, revisited

Unit Cell


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Periodic FDTD Modeling: Conclusion Periodic FDTD Modeling: Conclusion

• The enforcement of a PBC involving a real Bloch wavenumber β in the FDTD through the sine-cosine method does notnot in generalin general result in:

• Numerical investigation and analytical inspection of the method shows that the condition enforced is:

•• Spatial attenuation can be calculated in a PBCSpatial attenuation can be calculated in a PBC--terminated terminated unit cellunit cell.

dje)r(E)zdr(E β−=+

d)r(E)zdr(E β−=∠−+∠


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Attenuation Constant: CalculationAttenuation Constant: Calculation• Complex propagation constant can be calculated

using two samples of the fields E(zi,t) and E(zj,t), assuming:

))t,z(E())t,z(E(ln

zzj)(

j

i

ij FF

−=ωγ

Distance between the sampling points should be close to one period of the structure.

)(Re)( ωγωβ =)(Im)( ωγω =a

))zz()j(exp())t,z(E())t,z(E( ijij −

γ

β+α−=43421

FF


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MetalMetal--StripStrip--Loaded Dielectric LWA*Loaded Dielectric LWA*

Operating Freq. 80 GHz

Periodicity d 2.5 mm

Strip length 1.25 mm

Strip width w 3 mm

* K. L. Klohn et al, IEEE Trans. MTT, vol. 26, Oct. 1978

Unit cell

• Metal strips are used as perturbations on a dielectric waveguide

• Operated at microwaves.• The n=-1 spatial harmonic is

radiated


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MetalMetal--StripStrip--Loaded Dielectric LWALoaded Dielectric LWAPeriodic FDTD Analysis vs. Ansoft HFSS (truncated structure)


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Leaky CPWLeaky CPW--based slot antenna arrays*based slot antenna arrays*

Operating Freq. 30 GHz

Periodicity d 2.093 mm

• Capacitively loaded CPW lines.• Operated at millimeter-wave

region.• Radiates its fundamental spatial

harmonic (n=0).

Computational Domain

*A.Grbic, G.V. Eleftheriades, IEEE Trans. AP, vol. 50, Nov. 2002


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Leaky CPWLeaky CPW--based slot antenna arraysbased slot antenna arrays

• Dispersion Analysis: β/ko = 0.72 at 30 GHz

Periodic FDTD vs. Agilent Momentum (truncated structure)


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Complex Propagation Constant of 2DComplex Propagation Constant of 2D--NRINRI--TL based TL based LWAsLWAs

Open Stop Band

Stop Band

Stop Band

Zo 141 Ohm

Lo 5.6 nH

Co 1 pF

Dispersion Diagram ( β < ko)

Attenuation Constant ( α/ko)

When β→ko fast waves are coupled to non-radiating

surface waves.


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Backward Radiation Patterns

Directivity

Complex Propagation Constant of 2DComplex Propagation Constant of 2D--NRINRI--TL based TL based LWAsLWAs (cont(cont--d)d)

Open Stop Band

Forward Radiation Patterns


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Zo 141 Ohm

Lo 10 nH

Co 1 pF



o

oo

CLZ

=2


Closed Stop Band*

*Condition for closed stop-band discussed, derived in: G.V. Eleftheriades et al, IEEE Trans. MTT, vol. 50, Dec. 2002, eq. 29.


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Closed Stop Band

Backward Radiation Patterns Forward Radiation Patterns

Directivity


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SynopsisSynopsisOpen Stop Band

Closed Stop Band

DirectivityRadiation Patterns

Radiation Patterns DirectivityRadiation


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Equivalent Circuit for 2DEquivalent Circuit for 2D--NRINRI--TL TL LWAsLWAs

Equivalent Circuit for NRI-TL (operating outside the light coneoperating outside the light cone)

• Propagation constants can be calculated via the periodic analysis of each circuit.

• Values of Lo,Co, Lx, Cx are known (loading elements, hosting TL).• Rrad = unknown.

Equivalent Circuit for NRI-TL (operating within the light coneoperating within the light cone)


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Equivalent Circuit for 2DEquivalent Circuit for 2D--NRINRI--TL TL LWAsLWAs


Closed Stop Band


•Rrad=2950 Ohms•Calculated by fitting the FDTD results with the equivalent circuit dispersion curve.


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ConclusionsConclusionsAMR-FDTD is a technique that implementsmultiple, dynamically/adaptively generated subgrids in the three-dimensional FDTD.

The mesh generation in AMR-FDTD is a self-adaptive process, based on pre-defined accuracy parameters (CAD-oriented feature).

Microwave/optical applications have shown the excellent computational performance of AMR-FDTD.

Conclusion

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Conclusions (contConclusions (cont--d)d)For indoor wireless problem, the feasibility of the S-MRTD technique was demonstrated.

S-MRTD combined with GPU programming delivers a fast modeling tool, further enhanced by the rapid advances in GPU technology.

Fast periodic simulations of radiating structures were achieved with a periodic FDTD technique.

Conclusion

faster than fdtd pushing the boundaries of time-domain modeling … · 2006-05-29 · time-domain...

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