Prospects for Finite-Difference Time-Domain (FDTD) Computational - - PowerPoint PPT Presentation

Prospects for Finite-Difference Time-Domain (FDTD) Computational Electrodynamics

Allen Taflove

Department of Electrical and Computer Engineering Northwestern University, Evanston, IL 60208

Presented at:

IEEE Antennas and Propagation / Microwave Theory and Techniques Societies Chicago Section October 24, 2002

The Classic FDTD Algorithm

• 2nd-order accurate

central space differences

• 2nd-order accurate

leapfrog time-stepping

• Absorbing boundary

condition at edge of the 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 this NSF/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 are Found on the Web

APLAC http://www.aplac.hut.fi/aplac/general.html Apollo Photonics http://www.apollophoton.com/ Applied Simulation Technology http://www.apsimtech.com/ CFD Research http://www.cfdrc.com/datab/software/maxwell/maxwell.html Cray http://lc.cray.com/ Empire http://www.empire.de/ EMS Plus http://www.ems-plus.com/ezfdtd.html ETH http://www.iis.ee.ethz.ch/research/bioemc/em_simulation_platform.en.html Optima Research http://www.optima-research.com/Software/Waveguide/fullwave.htm Optiwave http://www.optiwave.com/ Quick Wave http://www.ire.pw.edu.pl/ztm/pmpwtm/qw3d/ Remcom http://www.remcominc.com/html/index.html RSoft http://www.rsoftinc.com/fullwave_info.htm Schmid http://www.semcad.com/solver_performance.html Vector Fields http://www.vectorfields.com/concerto.htm Virtual Science http://www.virtual-science.co.uk/celia/Celia_code/celia_home.htm Zeland 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 with

the 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 and its offspring in the time frame of 2015

Review of Key FDTD Applications and Validations

Topic 1: Electromagnetic Wave Scattering and Radar Cross Section

Surface Currents on a λ/3 Metal Cube

Taflove and Umashankar, IEEE Trans. Electromagnetic Compatibility, 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

technique Jurgens and Taflove, IEEE Trans. Antennas and Propagation, 1993.

Visualization of Surface Currents and Mutual Interaction of the Two Spheres

Monostatic RCS of VFY-218 Jet Fighter at 500 MHz

Monostatic angle (degrees)

Taflove, Computational Electrodynamics: The Finite- Difference Time-Domain Method, 1995. Radar cross section (dBsm)

Review of Key FDTD Applications and Validations Topic 2: Electromagnetic Wave Penetration and Coupling

Penetration into a Circular Cylinder Below Cutoff

• A. Taflove, IEEE Trans. Electromagnetic

Compatibility, 1980.

300 MHz plane wave axially incident upon a hollow metal right circular cylinder having a waveguide cutoff frequency of 900 MHz FDTD

• Freq. domain

integral 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: The Finite-Difference Time-Domain Method, 2nd ed., 2000.

Review of Key FDTD Applications and Validations Topic 3: Antennas and Radiation

Cylindrical Monopole Antenna Above a Finite Ground Plane

Reflected voltage pulse in 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 and Hagness, Computational Electrodynamics: 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 of 8-Element Vivaldi Quad Array

6 GHz, 0

• beam steer

12 GHz, 45

• beam steer

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

Review of Key FDTD Applications and Validations 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. Microwave Theory and Techniques, Nov. 1975.

Experimental Validation for a Brain-Equivalent Phantom

Yu et al., IEEE Trans. Electromagnetic Compatibility, 1999. Half-wavelength dipole radiating 0.5W at 1900 MHz located at d=5, 15,

• r 25 mm from the brain

phantom.

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 through the cellphone

Ultrawideband Plane-Wave Pulse Illuminating a Highly 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-Speed Logic Pulse Within a Conventional Dual In-Line Integrated Circuit Package

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

Embedding of Nonlinear and Active 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 a microstrip circuit Large-signal model of the MESFET integrated with the Thevenin equivalent circuits for the FDTD grid at its gate and drain terminals Kuo et al., IEEE Trans. Microwave Theory and Techniques, 1997.

Validation Relative to HP-MDS

6-GHz amplifier in packaging box Large-signal harmonic generation without the packaging box Kuo et al., IEEE Trans. Microwave Theory and Techniques, 1997.

Emerging Modeling Areas Topic 2: Particle Accelerator Cavities. Design Enabled by Improved Mesh-Generation Techniques.

New Locally Conformal Mesh Generator

Staircase FDTD

D-FDTD

Faceted surface generated by a standard CAD tool is imported into the FDTD grid. FDTD grid resolution can be relaxed by 4:1 for comparable accuracy in calculating 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 and

imported into the D-FDTD mesh generator.

• Typical mesh for a 4-period twisted waveguide

included 50,000 modified FDTD grid edges, and was created in 5 minutes.

• Provided error detection for meshing irregularities,

and a C++ visualization tool.

• HFSS™ required 500 MB of memory and 4 hours

for the solution of a 3-period twisted waveguide.

• D-FDTD required 20 MB of memory and

30 minutes for the same solution.

Emerging Modeling Areas Topic 3: Propagation of Electromagnetic Waves and Beams in Dispersive and Nonlinear Dispersive Media

Propagation in a Linear Dispersive Medium

Permittivity of a Lorentz medium having three resonances in the

• ptical range

Reflection coefficient for a plane wave normally incident upon a half- space composed of this medium Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 1995.

Calculation of the Sommerfeld Precursor in a Linear 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 Divergence When the Beamwidth Approaches 1 λd

Joseph and Taflove, IEEE Photonics Technology Letters, 1994.

Light Bullet

Goorjian and Silberberg, JOSA B, 1997.

Emerging Modeling Areas Topic 4: Micro-Optical Structures

Photonic Bandgap Waveguides

Mingaleev and Kivshar, Optics and Photonics News, July 2002. Prather Optics and Photonics News, June 2002.

Photonic Bandgap Defect Cavities

Fabricated device: membrane microresonator in InGaAsP Images of degenerate microcavity modes in 2-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 whispering gallery 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

382 380 E.I.(a.u.) Wavelength (nm)

size ~ 1.7 µm Contains ~ 20,000 particles

Wavelength (nm) E.I. (a.u.)

3.2 µm Measured 2-D FDTD model

• H. Cao et al.,Phys Rev Lett., 2000

Emerging Modeling Areas Topic 5: Multi-Level Atomic States

Four-Level, Two-Electron Model for ZnO

[ ]

E N N k P dt dP dt P d

a a a a a a 3 2 2 2

− = + +

[ ]

E N N k P dt dP dt P d

b b b b b b 1 2 2 2 2

− = + +

( ) ( )

dt dP E N N N N dt dN

a a

? + − − − − = h 1 1 1

30 3 32 2 3 3

( ) ( )

dt dP E N N N N dt dN

b b

? + − − − = h 1 1 1

21 1 2 32 2 3 2

( ) ( )

dt dP E N N N N dt dN

b b

? − − − − = h 1 1 1

10 1 21 1 2 1

( ) ( )

dt dP E N N N N dt dN

a a

? − − + − = h 1 1 1

10 1 30 3

E

C

E

V

N N

3

N

2

N

1

N0 N3 N1 N2

Optical pumping

e e

32 21 10 30

P Pa

a

P Pb

b

. . . .

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

0.495 0.496 0.497 0.498 0.499 0.500 0.501 0.502 0.503 0.504 0.505 0.506

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

0.495 0.496 0.497 0.498 0.499 0.500 0.501 0.502 0.503 0.504 0.505 0.506

n Time (sec) n1 n2

0.0 2.0x10

• 10

4.0x10

• 10

6.0x10

• 10

0.0 2.0x10

11

4.0x10

11

6.0x10

11

8.0x10

11

Intensity Time (sec)

1.E+04 1.E+06 1.E+08 1.E+10 1.E+12 1.E+10 1.E+11 1.E+12 1.E+13

pump

I Output I

Lasing threshold Populations n(t) Pumping vs. lasing intensity Pump intensity Output

Emerging Modeling Areas Topic 6: ELF Propagation Phenomena Involving the Entirety of the Earth-Ionosphere Waveguide

Whole-Earth Models of ELF Propagation

• There is a rich history of investigation
• f ELF and VLF electromagnetic wave

propagation 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 pole Wrap-around to east side Wrap-around to west side Isosceles trapezoidal grid cells in rows j = 2 through j = m–1 Isosceles triangular grid cells in rows j = 1 and j = m

Grid row j = m Grid column i = 2m Grid column 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 Propagation

Society 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 Propagation

Society Int. Symp., San Antonio, TX, June 2002.

Emerging Modeling Areas Topic 7: Biomedical Imaging

FDTD Modeling of Novel Utrawideband Radar Breast Cancer Detection Technology

5:1 17.5:1

Breast Tissue Dielectric properties

• X. Li and S. C. Hagness, IEEE

Microwave and Wireless Components Lett., March 2001.

Example: Simulated Detection of a 2-mm Tumor

Image reconstructed from FDTD-calculated backscattered

• waveforms. Colors

Indicate relative signal strength in decibels.

• Permittivity contrast

between malignant and normal tissues = 5:1

• variability in normal

tissue: ±

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 of great interest for modeling complex, electrically large problems:

• Applied to regular grids (possibly with multiple

regions) — Q. H. Liu, Duke University

• Applied to unstructured grids — J. S. Hesthaven

and T. Warburton, Brown University

Multiresolution Time-Domain (MRTD) Methods

Wavelet-based MRTD techniques provide another means to attack complex problems having a wide range 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 the Usual Courant Limit

Recent alternating-direction implicit (ADI) algorithms present possibilities for FDTD modeling over a wide range of time scales:

• T. Namiki, Fujitsu
• Z. Chen, Dalhousie University

Algorithms for Time-Stepping Beyond the Usual Courant Limit

Very recently, a “one-step” method based upon the Chebyshev polynomial expansion approximation of a quantum-mechanics-like time-evolution operator has been proposed:

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

Netherlands

Additional Algorithm Advances

• PML absorbing boundary conditions, especially for

non-Cartesian and unstructured grids

• Multigrid / subgrid techniques
• Digital signal postprocessing, especially to analyze

time-windowed data for resonances of high-Q structures

• Numerical hybrids linking FDTD to other

computational electromagnetics techniques

• Multiphysics modeling

Prospects for the Year 2015 Topic 2: Implications of Technology 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 and random 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 RAM

• ver a 10-year period, representing an average

doubling time of 20 months.

Extrapolation to 2015

If this trend continues through 2015, we will have PCs having an effective clock rate of 460 GHz and 120 GB of RAM. Very likely, these capabilities will be achieved primarily by employing many parallel processors. Even today (2002), this capability is available using a Beowulf cluster of approximately 300 Pentium IV processors clocked at 2.2 GHz. The price for such a capability will probably drop to less than $20K by 2015.

Extrapolation to 2015, continued

The FDTD performance of such an equivalent 300 Pentium-IV processor computer is roughly:

• 3–billion Yee cells (1.8E10 unknown field

vector components) in RAM, equivalent to a 3-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’s Running Standard FDTD Algorithms

Complete Structure Modeled Uniform Volumetric Space 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 FDTD algorithms, continuation

• f present trends in

personal computing capabilities should permit everyone to routinely generate highly detailed electromagnetic wave models of a number of volumetrically complex structures

• f

great engineering and scientific importance.

Prospects for the Year 2015 Topic 3: Implications of Technology Advances in High-End Computers

At the High End

Assuming high-end systems will always have 100–1000 times the power of the typical PC, then by 2015 such systems will be able to process 3E11 – 3E12 Yee cells containing 1.8E12 – 1.8E13 unknown field vector components. 3-D FDTD grids spanning as many as 14,000 cells in each dimension would be processed. Dimensional dynamic ranges would thus exceed 4 orders-of-magnitude.

Conclusion

Computer and algorithmic advances are pushing FDTD and its derivatives to the forefront of modeling 21st-century electromagnetics technologies from ELF through optical frequencies. The accuracy, flexibility, and scalability of FDTD are compelling advantages. We expect FDTD to solve a wide variety of complex physics and engineering problems having significant societal impact, especially in high-speed computing and communications, biomedicine, and defense.