prospects for finite-difference time-domain (fdtd ... presentation.pdf · idev (t) v dev (t) + ......

Prospects for Finite-DifferenceTime-Domain (FDTD)

Computational Electrodynamics

Allen Taflove

Department of Electrical and Computer EngineeringNorthwestern University, Evanston, IL 60208

Presented at:

IEEE Antennas and Propagation / Microwave Theory and Techniques SocietiesChicago SectionOctober 24, 2002

The Classic FDTD Algorithm

• 2nd-order accurate central space differences

• 2nd-order accurate leapfrog time-stepping

• Absorbing boundary condition at edge ofthe space lattice

Kane Yee, IEEE Trans. Antennas and Propagation,May 1966.

Yearly Number of FDTD Publications

Yee (1966)

Source: J. Schneider and K. Shlager (1998)

FDTD Literature Database* www.fdtd.org

As of Oct. 22, 2002, the total number of entries in thisNSF/ONR - sponsored database was 4793.

Breakdown:

— Books: 9— Ph.D. dissertations: 162— Masters theses: 68— Journal articles: 2549— Conference proceedings: 1951— Technical reports: 15— Miscellaneous publications: 39

*Maintained by John Schneider, Washington State University

At Least 17 Commercial FDTD Codes areFound on the Web

APLAC http://www.aplac.hut.fi/aplac/general.htmlApollo Photonics http://www.apollophoton.com/Applied Simulation Technology http://www.apsimtech.com/CFD Research http://www.cfdrc.com/datab/software/maxwell/maxwell.htmlCray http://lc.cray.com/Empire http://www.empire.de/EMS Plus http://www.ems-plus.com/ezfdtd.htmlETH http://www.iis.ee.ethz.ch/research/bioemc/em_simulation_platform.en.htmlOptima Research http://www.optima-research.com/Software/Waveguide/fullwave.htmOptiwave http://www.optiwave.com/Quick Wave http://www.ire.pw.edu.pl/ztm/pmpwtm/qw3d/Remcom http://www.remcominc.com/html/index.htmlRSoft http://www.rsoftinc.com/fullwave_info.htmSchmid http://www.semcad.com/solver_performance.htmlVector Fields http://www.vectorfields.com/concerto.htmVirtual Science http://www.virtual-science.co.uk/celia/Celia_code/celia_home.htmZeland Software http://www.zeland.com/fidelity.html

Why FDTD is Popular

• It is conceptually simple and systematic.

• It is accurate and robust.

• It uses no linear algebra.

• It treats impulsive behavior naturally.

• It treats nonlinear behavior naturally.

• It readily allows multi-physics simulations.

• Personal computer capabilities have caught up withthe requirements of FDTD for a wide range of important engineering and physics modeling problems.

Goals of This Presentation

• Review key FDTD applications and validations in engineering and physics

• Discuss emerging modeling areas

• Forecast the state of computational electrodynamics modeling by FDTD andits offspring in the time frame of 2015

Review of Key FDTD Applications andValidations

Topic 1: Electromagnetic WaveScattering and Radar Cross Section

Surface Currents on a λ/3 Metal Cube

Taflove and Umashankar,IEEE Trans. ElectromagneticCompatibility, 1983.

Monostatic RCS of a 9×3 - λ T-Shape Metal Target

Taflove and Umashankar, Proc. IEEE, 1989.

Bistatic RCS of Two 1-λ Diameter PEC Spheres

FDTD

• • • Generalized multipole techniqueJurgens and Taflove, IEEE Trans. Antennas

and Propagation, 1993.

Visualization of Surface Currents andMutual Interaction of the Two Spheres

Monostatic RCS of VFY-218 Jet Fighter at 500 MHz

Monostatic angle (degrees)

Taflove, ComputationalElectrodynamics: The Finite-Difference Time-DomainMethod, 1995.R

adar

cro

ss s

ectio

n (d

Bsm

)


Topic 2: Electromagnetic WavePenetration and Coupling

Penetration into a Circular Cylinder Below Cutoff

A. Taflove, IEEE Trans. ElectromagneticCompatibility, 1980.

300 MHz plane wave axiallyincident upon a hollowmetal right circular cylinderhaving a waveguide cutofffrequency of 900 MHz

FDTD

Freq. domainintegral equation

Coupling to Wires Within the LLNL PLUTO

Umashankar, Taflove, et al., IEEE Trans. Antennas and Propagation, 1987.

Microwave Penetration into a Missile Radome

Maloney and Smith in Taflove and Hagness, Computational Electrodynamics: TheFinite-Difference Time-Domain Method, 2nd ed., 2000.


Topic 3: Antennas and Radiation

Cylindrical Monopole Antenna Above aFinite Ground Plane

Reflected voltage pulsein the coaxial feedline

Maloney et al., IEEE Trans. Antennas and Propagation, 1994.

Standard Gain Horn Antenna

Radiation Patterns and Gain of Horn Antenna

Maloney and Smith in Taflove andHagness, ComputationalElectrodynamics: The Finite-Difference Time-Domain Method,2nd. ed., 2000.

Boresight gain

Radiation patterns

8-Element Array of Vivaldi Quads

E-Plane Co-Polarized Radiation Patterns of8-Element Vivaldi Quad Array

6 GHz, 0o beam steer 12 GHz, 45o beam steer

Thiele and Taflove, IEEE Trans. Antennas and Propagation, 1994.


Topic 4: Interactions with Human Tissues

Microwave Irradiation of the Human Eye

Taflove and Brodwin, IEEE Trans. Microwave Theory and Techniques, Nov. 1975.

Calculated SAR in Human Eye Model at 1.5 GHz

Horizontal symmetry plane

Vertical symmetry plane

Taflove and Brodwin, IEEE Trans. MicrowaveTheory and Techniques, Nov. 1975.

Experimental Validation for aBrain-Equivalent Phantom

Yu et al., IEEE Trans. Electromagnetic Compatibility, 1999.

Half-wavelength dipoleradiating 0.5W at 1900MHz located at d=5, 15,or 25 mm from the brainphantom.

Cellphone Interaction With The Human Head

Maps of the E-field and SAR within the cut plane.Relative intensities are shown in dB.

Source: Remcom Inc. website: http://www.remcominc. com/html/index.html

Cut plane throughthe cellphone

Ultrawideband Plane-Wave Pulse Illuminating aHighly Detailed, Frequency-Dispersive Model of

the Human Head

Source:

Remcom Inc. website:http://www.remcominc.com/html/index.html

dB scale

Emerging Modeling Areas

Topic 1: High-Speed Electronic Circuits

Coupling and Crosstalk of a High-SpeedLogic Pulse Within a Conventional Dual In-Line

Integrated Circuit Package

Source: Melinda Piket-May, University of Colorado-Boulder

Embedding of Nonlinear andActive Circuits Within the Space Grid:

Interface with SPICE

IN (t) CN

Idev (t)

Vdev (t)

+

–Idis (t)

Embedded circuit device∆

Norton Equivalent Circuit “Looking Into” the FDTD Grid

Thomas et al., IEEE Microwave and Guided Wave Lett., 1994.

MESFET Transistor Example

Mounting in amicrostrip circuit

Large-signal model of theMESFET integrated withthe Thevenin equivalentcircuits for the FDTD gridat its gate and drainterminals

Kuo et al., IEEE Trans. Microwave Theory and Techniques, 1997.

Validation Relative to HP-MDS

6-GHz amplifier in packaging box Large-signal harmonic generationwithout the packaging box

Kuo et al., IEEE Trans. Microwave Theory and Techniques, 1997.


Topic 2: Particle Accelerator Cavities.Design Enabled by Improved

Mesh-Generation Techniques.

New Locally Conformal Mesh Generator

Staircase FDTD

D-FDTD

Faceted surface generatedby a standard CAD tool isimported into the FDTD grid.

FDTD grid resolution can be relaxedby 4:1 for comparable accuracy incalculating resonant frequencies.

Waldschmidt and Taflove, IEEE Trans. Antennas and Propagation, submitted.

Twisted Waveguide Slow-Wave Structure

Interior of Twisted Waveguide

Details

• Twisted waveguide was designed with ProE andimported into the D-FDTD mesh generator.

• Typical mesh for a 4-period twisted waveguideincluded 50,000 modified FDTD grid edges, andwas created in 5 minutes.

• Provided error detection for meshing irregularities,and a C++ visualization tool.

• HFSS™ required 500 MB of memory and 4 hoursfor the solution of a 3-period twisted waveguide.

• D-FDTD required 20 MB of memory and 30 minutes for the same solution.


Topic 3: Propagation of ElectromagneticWaves and Beams in Dispersive and

Nonlinear Dispersive Media

Propagation in a Linear Dispersive Medium

Permittivity of a Lorentz mediumhaving three resonances in theoptical range

Reflection coefficient for a planewave normally incident upon a half-space composed of this medium

Taflove, Computational Electrodynamics: The Finite-Difference Time-DomainMethod, 1995.

Calculation of the Sommerfeld Precursor in aLinear Single-Lorentz-Resonance Medium

Joseph, Hagness, and Taflove, Optics Letters, 1991.

“Braided” Co-Phased Spatial Solitons

Joseph and Taflove, IEEE Photonics Technology Letters, 1994.

Soliton Braiding Transitions to DivergenceWhen the Beamwidth Approaches 1 λd

Joseph and Taflove, IEEE Photonics Technology Letters, 1994.

Light Bullet

Goorjian andSilberberg,

JOSA B, 1997.


Topic 4: Micro-Optical Structures

Photonic Bandgap Waveguides

Mingaleev and Kivshar, Optics andPhotonics News, July 2002.

Prather Optics and PhotonicsNews, June 2002.

Photonic Bandgap Defect Cavities

Fabricated device:membrane microresonator

in InGaAsP

Images of degenerate microcavity modes in2-D thin-film photonic crystal defect cavities

Source: E. Yablonovitch, UCLA

Photonic Bandgap Defect Mode Lasers

Painter et al., Science, June 11, 1999.

Waveguides Coupled to Disks and Rings

1st- and 2nd-order radial whisperinggallery mode resonances

λ = 1.55 µm(off resonance)

S. C. Hagness, D. Rafizadeh, S. T. Ho, and A. Taflove, IEEE J. Lightwave Tech., 1997.

fabricated device

Lasing in a Random Clump of ZnO Particles

382380

E.I.

(a.u

.)

Wavelength (nm)

size ~ 1.7 µm

Contains ~ 20,000 particles

Wavelength (nm)

E.I.

(a.

u.)

3.2 µmMeasured

2-D FDTDmodel

H. Cao et al.,Phys Rev Lett., 2000


Topic 5: Multi-Level Atomic States

Four-Level, Two-Electron Model for ZnO

[ ]ENNkPdt

dP

dt

Pdaaa

aa

a03

22

2

−=++

[ ]ENNkPdt

dP

dt

Pdbbb

bb

b12

22

2

−=++

( ) ( )dt

dPE

NNNN

dt

dN a

a

?+−−−−=h

111

30

03

32

233

( ) ( )dt

dPE

NNNN

dt

dN b

b

?+−−−=h

111

21

12

32

232

( ) ( )dt

dPE

NNNN

dt

dN b

b

?−−−−=h

111

10

01

21

121

( ) ( )dt

dPE

NNNN

dt

dN a

a

?−−+−=h

111

10

01

30

030

EC

EV

N0

N3

N2

N1

N0

N3

N1

N2

Optical pumping

e

e

32

21

10

30

PPaa

PPbb

.

.

.

.

Chang, Cao, and Taflove (in progress)

Initial Results

0.0 5.0x10-12 1.0x10-11 1.5x10-11

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

n

Time(sec)

n1 n2 n3 n0

8.0x10-121.0x10-111.2x10-111.4x10-111.6x10-11

0.4950.4960.4970.4980.4990.5000.5010.5020.5030.5040.5050.506

n

Time (sec)

n1 n2

0.0 5.0x10-12 1.0x10-11 1.5x10-11

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

n

Time(sec)

n1 n2 n3 n0

8.0x10-121.0x10-111.2x10-111.4x10-111.6x10-11

0.4950.4960.4970.4980.4990.5000.5010.5020.5030.5040.5050.506

n

Time (sec)

n1 n2

0.0 2.0x10-10 4.0x10-10 6.0x10-100.0

2.0x1011

4.0x1011

6.0x1011

8.0x1011

Inte

nsity

Time (sec)

1.E+04

1.E+06

1.E+08

1.E+101.E+12

1.E+10 1.E+11 1.E+12 1.E+13

pump I

Out

put I

Lasing threshold

Populations n(t)Pumping vs. lasingintensity

Pump intensity

Out

put


Topic 6: ELF Propagation PhenomenaInvolving the Entirety of the

Earth-Ionosphere Waveguide

Whole-Earth Models of ELF Propagation

• There is a rich history of investigationof ELF and VLF electromagnetic wavepropagation within the Earth-ionosphere waveguide.

• Applications:– Submarine communications– Remote-sensing of lightning and

sprites– Global temperature change– Subsurface structures– Potential earthquake precursors

Grid Layout for Whole-Earth ELF Models

South pole

North poleWrap-aroundto east side

Wrap-aroundto west side

Isosceles trapezoidal gridcells in rows j = 2through j = m–1

Isosceles triangular grid cells inrows j = 1 and j = m

Grid row j = m

Gridcolumn i = 2m

Gridcolumn i = 1

Grid row j = 1

J. Simpson and A. Taflove, IEEE Antennas and Wireless Propagation Lett., in press.

2-D Whole-Earth Model: 1024 × 512 Cells(40 × 40 km resolution at Equator)

J. Simpson and A. Taflove, IEEE Antennas and PropagationSociety Int. Symp., San Antonio, TX, June 2002.

3-D Whole-Earth Model: 1024 × 512 × 40 Cells(40 × 40 × 5 km resolution at Equator; continents + oceans + ionosphere)

J. Simpson and A. Taflove, IEEE Antennas and PropagationSociety Int. Symp., San Antonio, TX, June 2002.


Topic 7: Biomedical Imaging

FDTD Modeling of Novel Utrawideband RadarBreast Cancer Detection Technology

5:1

17.5:1

Breast TissueDielectric properties

X. Li and S. C. Hagness, IEEEMicrowave and WirelessComponents Lett., March 2001.

Example: Simulated Detection of a 2-mm Tumor

Image reconstructedfrom FDTD-calculatedbackscatteredwaveforms. ColorsIndicate relative signalstrength in decibels.

• Permittivity contrastbetween malignant andnormal tissues = 5:1

• variability in normaltissue: ±10%

S/C=16 dB

Numerical breast phantom

S. Davis, E. Bond, X. Li, S. C. Hagness, and B. Van Veen,J. Electromagnetic Waves and Applications, in press.

Prospects for the Year 2015

Topic 1: Algorithm Advances

High-Order / Low-Dispersion Algorithms

Spectral time-domain methods are becoming ofgreat interest for modeling complex, electrically largeproblems:

• Applied to regular grids (possibly with multipleregions) — Q. H. Liu, Duke University

• Applied to unstructured grids — J. S. Hesthavenand T. Warburton, Brown University

Multiresolution Time-Domain (MRTD) Methods

Wavelet-based MRTD techniques provide anothermeans to attack complex problems having a widerange of characteristic length scales:

• Battle-Lemarie scaling and wavelet functions —L. Katehi, Purdue University

• Haar scaling and wavelet functions — L. Carin,Duke University

Algorithms for Time-Stepping Beyond theUsual Courant Limit

Recent alternating-direction implicit (ADI) algorithmspresent possibilities for FDTD modeling over a widerange of time scales:

• T. Namiki, Fujitsu

• Z. Chen, Dalhousie University

Algorithms for Time-Stepping Beyond theUsual Courant Limit

Very recently, a “one-step” method based upon theChebyshev polynomial expansion approximation ofa quantum-mechanics-like time-evolution operatorhas been proposed:

• H. De Raedt, K. Michielsen, J. S. Kole, andM. T. Figge, University of Groningen, The Netherlands

Additional Algorithm Advances

• PML absorbing boundary conditions, especially fornon-Cartesian and unstructured grids

• Multigrid / subgrid techniques

• Digital signal postprocessing, especially to analyzetime-windowed data for resonances of high-Q structures

• Numerical hybrids linking FDTD to other computational electromagnetics techniques

• Multiphysics modeling


Topic 2: Implications ofTechnology Advances in

Off-the-Shelf Personal Computers

What Happened During the 1990’s

Consider first the increase in personal computer(PC) capabilities in the 1990’s in clock speed andrandom access memory (RAM):

1990: 16-MHz clock, 4 MB of RAM

2000: 1-GHz clock, 256 MB of RAM

This is a 60:1 increase in both clock speed and RAMover a 10-year period, representing an averagedoubling time of 20 months.

Extrapolation to 2015

If this trend continues through 2015, we will havePCs having an effective clock rate of 460 GHz and120 GB of RAM. Very likely, these capabilities willbe achieved primarily by employing many parallelprocessors.

Even today (2002), this capability is availableusing a Beowulf cluster of approximately 300Pentium IV processors clocked at 2.2 GHz.The price for such a capability will probably drop toless than $20K by 2015.

Extrapolation to 2015, continued

The FDTD performance of such an equivalent300 Pentium-IV processor computer is roughly:

• 3–billion Yee cells (1.8E10 unknown field

vector components) in RAM, equivalent to a3-D space grid spanning 1400 × 1400 × 1400

cells

• 1–hour running time for marching this grid

through 10,000 time steps

Year-2015 Modeling Capabilities Using PC’sRunning Standard FDTD Algorithms

CompleteStructure Modeled

Uniform VolumetricSpace Resolution

Jet fighter 1 cm

Human body 0.5 mm

Human head 0.2 mm

Cellphone 30 µm

Microchip 1 µm

Implication

Thus, even without any improvements in FDTDalgorithms, continuation of present trends inpersonal computing capabilities should permiteveryone to routinely generate highly detailedelectromagnetic wave models of a number ofvolumetrically complex structures of greatengineering and scientific importance.


Topic 3: Implications ofTechnology Advances in

High-End Computers

At the High End

Assuming high-end systems will always have100–1000 times the power of the typical PC, then by2015 such systems will be able to process3E11 – 3E12 Yee cells containing 1.8E12 – 1.8E13unknown field vector components.

3-D FDTD grids spanning as many as 14,000cells in each dimension would be processed.Dimensional dynamic ranges would thus exceed4 orders-of-magnitude.

Conclusion

Computer and algorithmic advances are pushingFDTD and its derivatives to the forefront of modeling21st-century electromagnetics technologies fromELF through optical frequencies.

The accuracy, flexibility, and scalability of FDTDare compelling advantages. We expect FDTD tosolve a wide variety of complex physics andengineering problems having significant societalimpact, especially in high-speed computing andcommunications, biomedicine, and defense.

prospects for finite-difference time-domain (fdtd ... presentation.pdf · idev (t) v dev (t) + ......

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