UIUC MURI Review

J.-M. Jin, A. C. Cangellaris, and W. C. Chew

Center for Computational ElectromagneticsDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, Illinois 61801-2991

Program Director: Dr. Arje Nachman, AFOSR

June 19, 2006

Time-Domain Finite Element Method for Analysis of Broadband Antennas

and Arrays

J.-M. Jin

Center for Computational ElectromagneticsDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, Illinois 61801-2991

Acknowledgment:

This work is sponsored by AFOSR via a MURI grant (Program Director: Dr. Arje Nachman)

Year 1:Year 1:

TDFEM Antenna Analysis

Novel, Highly Efficient Domain DecompositionNovel, Highly Efficient Domain Decomposition– Large antennasLarge antennas

– Finite array antennasFinite array antennas

Periodic TDFEMPeriodic TDFEM– Infinite periodic phased-array antennasInfinite periodic phased-array antennas

Year 2:Year 2:

Truncation of Open Free SpaceTruncation of Open Free Space– Absorbing boundary condition (ABC)Absorbing boundary condition (ABC)

– Perfectly matched layers (PML)Perfectly matched layers (PML)

Feed ModelingFeed Modeling– Simplified fSimplified feed eed mmodel: odel: eelectric lectric pprobe robe ffeedeed

– Waveguide Waveguide pport ort bboundary oundary cconditioondition (WPBC)n (WPBC)

dVM j

V

iij NN

dVB j

V

iij NJN 0

dVdVS j

V

ij

V

iij NKNNN 20

dVhj

jj

V

ii NuLN30

dVgj

jjj

V

ii NuNuMN30

)(0/ teeu jt

j

02

2

gheSt

eB

t

eM

Time-Domain Discretization

{e} and {u} are discretized in time domain according to Newmark-Beta method

Resultant system is stable for time marching

Convolution

2

11

2

2 2

t

eee

t

te nnn

t

ee

t

te nn

2

11

4

2 11

nnn eeete

4

2 11

nnn

tuuu

u

Perfectly Matched Layers (PML)

Spatial FEM discretization:Spatial FEM discretization:

Time-Domain WPBC

inc)(ˆ UEE Pn

1

TMTM

TE

1

TETEM0

TEM0

)(

)()()(

m S

mm

S

mm

m

S

dS

dSdSP

Eee

EeeEeeE

G

HL

inc inc TEM TEM inc0 0

TE TE inc

1

TM TM inc

1

ˆ ( )

( )

( )

S

m mm S

m mm S

n dS

dS

dS

U E e e E

e e E

e e E

L

H

G

Time-Domain Formulation:

Assume dominant modeincidence:

incidence TMdominant )(2

incidence TEdominant )(2

incidence TEM)(2

incTM1

incTE1

incTEM0

inc

f

f

f

G

H

L

e

e

e

U

Monopole Antenna

mm 1.0a

mm 2.3b

mm 32.8h

Measured data:Measured data: J. Maloney, G. Smith, and W. Scott, “Accurate computation of the radiation J. Maloney, G. Smith, and W. Scott, “Accurate computation of the radiation from simple antennas using the finite difference time-domain method,” from simple antennas using the finite difference time-domain method,” IEEE Trans. A.PIEEE Trans. A.P., vol. ., vol. 38, July 1990.38, July 1990.

Logarithmic Spiral Antenna

Probe FeedG. Deschamps, “Impedance properties of G. Deschamps, “Impedance properties of complementary multiterminal planar complementary multiterminal planar structures,” structures,” IRE Trans. Antennas PropagatIRE Trans. Antennas Propagat., ., vol. AP-7, Dec. 1959.vol. AP-7, Dec. 1959.

Antipodal Vivaldi AntennaReflection at the TEM port

““The 2000 CAD benchmark unveiled,”The 2000 CAD benchmark unveiled,”Microwave Engineering OnlineMicrowave Engineering Online, July 2001, July 2001

Radiation patterns at 10 GHz

Antipodal Vivaldi Antenna

H-plane

E-plane

Domain Decomposition

Traditional Methods:Traditional Methods: Schwartz MethodsSchwartz Methods

Schur Complement (Substructuring) MethodSchur Complement (Substructuring) Method

Finite Element Tearing and Interconnecting Finite Element Tearing and Interconnecting

(FETI):(FETI): Use Lagrange multiplier to formulate the interface Use Lagrange multiplier to formulate the interface

problem, usually solved by an iterative solverproblem, usually solved by an iterative solver

Subdomain problems can be solved independently Subdomain problems can be solved independently

based on interface solutionsbased on interface solutions

Time-domain FETI is less efficient since the interface Time-domain FETI is less efficient since the interface

problem needs to be solved repeatedly at each time stepproblem needs to be solved repeatedly at each time stepTime-Domain Dual-Field Domain Decomposition (DFDD):Time-Domain Dual-Field Domain Decomposition (DFDD):

Does not require solving the interface problemDoes not require solving the interface problem

Computes both electric and magnetic fieldsComputes both electric and magnetic fields

Employs a leapfrog time-marching scheme similar to the FDTD Employs a leapfrog time-marching scheme similar to the FDTD

1V

2V

3V

4V

Interfaces

Two-Domain DFDD

tttcimmm

rmr

JEEE 002

2

20

11

Second-Order Vector Wave Equation (m=1,2):

im

r

m

r

mrm

r ttcJ

HHH

111

2

2

20

Boundary Conditions:

SM metallic surface

SA impedance surface

0En̂

0 Hn̂

01

EE nn

tn e ˆˆˆ

01

EH nn

tn h ˆˆˆ

BAmm

m

SS

mr

i

V

imi

V

mi

mi

rmi

r

dSndVt

dVttc

ENJ

N

EN

ENEN

1

1

0

02

2

20

ˆ

Weak-Form Representation:

BAm SS

mr

i

V

imir

V

mi

r

mi

rmi

r

dSndV

dVttc

HNJN

HN

HNHN

11

12

2

20

ˆ

Bm

Amm

S

si

V

imi

S

mi

V

mi

mi

rmi

r

dSt

nndVt

dSt

nnYdVttc

JN

JN

EN

EN

ENEN

ˆˆ

ˆˆ

00

002

2

20

1

B Bm

Amm

S S

sir

si

V

imir

S

mi

V

mi

r

mi

rmi

r

dSnndSt

nndV

dSt

nnZdVttc

MNM

NJN

HN

HN

HNHN

ˆˆˆˆ

ˆˆ

0

02

2

20

1

1

Bs Sn on HJ ˆ

Equivalent Surface Currents:

Bs Sn on EM ˆ

Two-Domain DFDD

Temporal Discretization

ne11

1ne 1

1ne

2

1

2

nh3

2

2

nh 2

1

2

nh 2

3

2

nh

t

t

tnt

n tee

tnt

nthh

2

12

1

2121

011

011

110

11220

2

2

1

2

1

4

1

2

1

4

1

//

nnnm

nm

nm

nm

nmem

nm

nmemem

nm

nm

nmem

jjtcfftceeeM

eeABtceeeStc

Leapfrog on subdomain interfaces

Newmark-Beta method inside each subdomain

nnnnnm

nm

nm

nmhm

nm

nmhmhm

nm

nm

nmhm

mmtcmmtcgtchhhM

hhABtchhhStc

1220

10

220

2/12/12/3

2/12/30

2/12/12/3220

2

12

2

1

2

1

4

1

2

1

4

1

Computational Performance (Serial) Tested on SGI-Altix 350 system with Intel

Itanium II 1.5GHz processor

Subdomain problems are solved in serial on a single processor

Each subdomain system is pre-factorized using direct solver before time marching

Factorization Time CPU Time per Step

Peak Memory

stepfacttotal TNTT Total CPU Time:

Computational Performance (Parallel)

Tested on SGI-Altix 350 system with multiple Intel Itanium II 1.5GHz processors

Each subdomain is assigned to a different processor

Speedup

10-by-10 Vivaldi Array

2.8 million unknowns

Distributed on 72

processors

Solving time per step: 0.3 s

Dipole Radiating in Photonic BandGap

Air

r = 11.56

Photonic Bandgap:a

c

a

c44303020 .. ~

FEM Discretization

– Element size: 0.08 ~ 0.25 m

– 208,747 mixed-2nd order tets

– 1.4 million unknowns

– Partitioned into 9 subdomains

9 m

9 m

1 m

f = 0.25 c/a f = 0.35 c/a f = 0.50 c/a

Reflection at Coaxial PortNumber of Unknowns 155,000 X 9

Memory Requirement 1.5 GB

Preprocessing Time 516.7 s

Solving Time per Step 1.95 s

Total Solving Time

(3000 steps)1.7 hr

Dipole Radiating in Photonic BandGap

A Generic Periodic Phased Array

Technical challenges:Technical challenges:

1.1. Enforcement of periodic boundary conditionsEnforcement of periodic boundary conditions

2.2. Mesh truncation in the non-periodic directionMesh truncation in the non-periodic direction

Periodic boundary conditions in the frequency domainPeriodic boundary conditions in the frequency domain

Introduce a transformed field variableIntroduce a transformed field variable

Transformed Field Variable

M. E. Veysoglu, R. T. Shin, M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-and J. A. Kong, “A finite-difference time-domain difference time-domain analysis of wave scattering analysis of wave scattering from periodic structures: from periodic structures: oblique incidence case,” oblique incidence case,” J. J. Electromag. Waves Appl.Electromag. Waves Appl., , vol. 7, pp. 1595-1607, Dec. vol. 7, pp. 1595-1607, Dec. 1993.1993.

Transformed Field Variable

wherewhere

Second-order vector wave equation:Second-order vector wave equation:

Solved via a Galerkin method in space using vector Solved via a Galerkin method in space using vector testing functions residing in a tetrahedral mesh and testing functions residing in a tetrahedral mesh and time-integration via thetime-integration via the Newmark- Newmark- method method

Higher-Order Floquet ABC

Floquet expansion for the transformed field variableFloquet expansion for the transformed field variable

Time-domain expressionTime-domain expression

A very accurate truncation condition can be constructedA very accurate truncation condition can be constructed

Reflection from an Array of Spheres

M. Inoue, “Enhancement of local field by a two-dimensional M. Inoue, “Enhancement of local field by a two-dimensional array of dielectric spheres placed on a substrate,” array of dielectric spheres placed on a substrate,” Physical Physical Review BReview B, vol. 36, pp. 2852-2862, Aug. 1987., vol. 36, pp. 2852-2862, Aug. 1987.

i = 20o with the ABC a small distance from the surface of the sphere

TM-polarization TE-polarization

A Vivaldi Phased-Array Antenna

D. T. McGrath and V. P. Pyati, “Phased array antenna D. T. McGrath and V. P. Pyati, “Phased array antenna analysis with the hybrid finite element method,” analysis with the hybrid finite element method,” IEEE IEEE Trans. Antennas PropagatTrans. Antennas Propagat., vol. 42, pp. 1625-1630, Dec. ., vol. 42, pp. 1625-1630, Dec. 1994.1994.

Summary

A complete TDFEM modeling of broadband antennas and arrays involving complex geometry A complete TDFEM modeling of broadband antennas and arrays involving complex geometry and material and material

A highly effective PML formulation to emulate a free-space environmentA highly effective PML formulation to emulate a free-space environment A highly accurate waveguide port boundary condition for a physical modeling of antenna A highly accurate waveguide port boundary condition for a physical modeling of antenna

feedsfeeds A novel, highly efficient dual-field domain decomposition technique to handle large-scale A novel, highly efficient dual-field domain decomposition technique to handle large-scale

simulationssimulations TDFEM analysis of infinite phased arraysTDFEM analysis of infinite phased arrays

Work in Progress:Work in Progress: Hybridization of TDFEM and ROM to interface antenna feeds Hybridization of TDFEM and ROM to interface antenna feeds

and feed network and feed network Hybridization of TDFEM and TDIE (TD-AIM & PWTD) to Hybridization of TDFEM and TDIE (TD-AIM & PWTD) to

model antenna/platform interactionmodel antenna/platform interaction

Achievements:Achievements:

Finite Element Based, Broadband Macro-modeling

of Antenna Array Feed Networks

H. Wu and A. C. Cangellaris

Center for Computational Electromagnetics & EM LabDepartment of Electrical and Computer Engineering

University of Illinois at Urbana-ChampaignUrbana, Illinois 61801-2991

cangella@uiuc.edu

Objectives Generate compact, multi-port macromodels for

antenna feed network – Broadband macro-models

Generated directly from FEM model using Krylov subspace model order reduction methods

– Compatible with both frequency-domain and time-domain EM solvers

Cast in terms of generalized impedance matrix for the electromagnetic multiport

Both waveguide mode-based ports and lumped-circuit ports supported

Impedance matrix elements in terms of rational function of frequency

Frequency interpolation for use with frequency-domain solvers Computationally-efficient interfacing with time-domain solvers

Multi-Layered Feed Network Radiating Elements Customization of

supporting substrate for improved array performance (patterned substrate, embedded EM band-gap structures, …)

Spacer - Custom patterning for enhanced array performance; layer for integration of active electronics

Slots Feed network - Extended to multiple

layers to support biasing network for any active electronics

Dispersive Attributes of Feed Network

Conductor loss Dielectric loss Dispersive (macroscopic)

properties of artificially-designed substrates

Network matrix abstraction of a portion of the feed network in terms of frequency-dependent multiport

Macromodeling of FEM Models With Dispersion

Krylov subspace-based model order reduction Surface impedance boundary conditions

Skin effect in lossy conductors Generalized surface impedance boundary conditions

Frequency-dependent permittivity and permeability Debye media, Lorentz media, Drude media,…

Incorporation of frequency-dependent electromagnetic multi-ports in FEM models

Frequency-dependent, multi-port macromodel abstractions of sub-domains

The Finite Element Model

1 2

,

, ,

, ,1

2

Discretization of Vector Helmholtz Equation:

0

Field expansion using edge elements:

;

Useful property: 1

FEM System: 0

e

t m n m n n m

m n m n

N

e i e ii

e

E s E s E

W e m n

e

E x w

Y sZ s T x

1

2

3

4

Skin-effect Surface Impedance

1

2

, ,,

ˆ ˆ ˆ

(1 )

Modified E-field FEM model:

= ,

ˆ ˆ=c

s

sc c

p e

cp e i e ji j S

n H Z n n E

fZ j s

Y sZ sZ s T x sFI

Z n w n w ds

n̂

Frequency-dependent Permittivity

2

, , ,

0

( )

( )

Example: Debye medium

( )1

e

i j e i e j

S sZ s T s x sFI

T w s w dv

ss

Two-Port Macro-model of Metal Plates

11 121 1 1 1 1

21 222 2 2 2 2

111 22

112 21

( ) ( )ˆ ˆ ˆ

( ) ( )ˆ ˆ ˆ

( ) ( ) / tanh

( ) ( ) / sinh

,

Y s Y sn H n n E

Y s Y sn H n n E

Y s Y s d

Y s Y s d

ss s

s

1̂n

d2n̂

2

( )

( 1)

2

( )(Order ):

(Order ): Assume exists.

Define the , , as .

( )

e

s ee H

e

N n

e n e e

H Hs e

He

Y sZ s T H s Z x sFIN

V L x

n X

x x Xx

X Y sZ s T H s Z Xx X FI

V L Xx

Original Model

Reduced Model

reduced - order state vector

2

2

( )

( ):

H H H H Hs e

HHe

s e

He

X YX s X ZX s X TX H s X Z X x X F I

V X L x

Y sZ s T H s Z x sFI

V L x

Reduced - order Model

Definition of Reduced-Order Model

FEM Model for Dispersive Media

2

1

( ) ( ) ( )

( ) ( )

types of media with frequency-dependent electromagnetic

behavior: ( ), 1, 2, , exhibit general frequency dependence

, , , , , are independ

K

k k ek

e

k

k

S sZ s T H s Z x s sFI s

y s Lx s

K

H s k K

S Z T Z F L

ent of frequency

Rational function fit of H(s)

0 11

2

12

Impedance Matrix ( )

( ) ( )

( ) ( )

( )

( ) ( )

( ) ( ) ( )

k

k k e

e

Ni

ki

k

ki

k

GZ s

S sZ s T H s Z x s sFi s

y s Lx s

y

rH

s sL S sZ s T H

s H s h h ss

s Z i s

p

F

Moments of ZG(s) (1)

20 1 0 2 0

120 0 0 0

1 1 0

2 1 1 2 0

1

( ) ( ) ( )

The moments are computed recursively as follows:

( )

, 3

G

i

k k

n

n i n ii

Z s sL R R s s R s s

R

R R S s Z s T H s Z F

R A R

R A R A R

R AR n

Moments of ZG(s) (2)

0

0

0

121 0 0 0 0

212

2 0 0 0 2

120 0 0

( ) 2 ( )

( ) ( )

1( ) ( ) , 3

!

k k k ks s

k k k k

s s

n

n k k k kn

s s

dA S s Z s T H s Z Z s T H s Z

ds

dA S s Z s T H s Z T H s Z

ds

dA S s Z s T H s Z H s Z n

n ds

Moments of ZG(s) (3)

121 0 0 0

0 1 21 0

122 0 0 0 3

1 0

120 0 0 1

1 0

( )

2

( )

( ) 1 , 3

k

k

k

k k

Ni

ki i

Ni

k k ki i

Nn i

n k k kni i

A S s Z s T H s Z

rZ s T h Z

s p

rA S s Z s T H s Z T Z

s p

rA S s Z s T H s Z Z n

s p

Construction of the Krylov Subspace

1 2 0 1

1 2

1 2

columns

( ; , ,...) , , , ,

Orthogonalization of ( ; , ,...)

orthogonal basis: , , ,

Use as the projection matrix for the development

of the reduced-order model

q n

q

q

q

K R A A span R R R

K R A A

X span X X X

X

Generation of Reduced-order Model

12

12

Original Model: ( ) ( )

Projection through congruence transformation:

, ,

, ,

Reduced-order Model: ( ) ( )

G k k

H H Hk k

H H

G k k

Z s sL S sZ s T H s Z F

S X SX Z X ZX Z X Z X

T X TX L LX F X F

Z s sL S sZ s T H s Z F

Example 1: Microstrip On Debye substrate

6 cm-long line terminated at a 60-Ohm load– Substrate relative permittivity: εr = 2 +10(1+ jω2×10-10)-1

– All dimensions in mm

0.250.6 0.6

0.5

0.2

0.02

Example 1: Microstrip on Debye substrate

Example 2: Microstrip band-pass filter

Substrate relative permittivity = 9.8

Example 2: Microstrip band-pass filter

Example 3: Coupling through lossy ground

Example 3: Coupling through lossy ground

Summary

Krylov subspace, equation preserving, model order reduction of FEM models that include frequency-dependent features

– Hybrid distributed-lumped element models

– Dispersive media

Cost of reduction dominated by the solution of E-field finite element equation at the expansion frequency

Generated reduced-order model provides for:– Fast frequency interpolation of system’s electromagnetic response

– Rational function matrix macro-modeling of complex, passive multiports for computationally efficient interfacing with time-domain solvers

W. C. ChewCenter for Computational Electromagnetics

and Electromagnetics Laboratory

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, Urbana, IL 

OSU MURI Review

June 19, 2006

MURI REVIEW MURI REVIEW 20062006

Acknowledgements

A. Hesford, P. Atkins, M. Saville, C. Davis, J. Xiong, I.-T. Chiang, M.K. Li

Progress to Date

An equivalence principle algorithm (EPA) for domain decomposition with integral equation.

A novel thin dielectric sheet model. Novel formulation of layered medium Green’s

function. Multilevel multipole-free algorithm. Frequency independent scattering algorithm. Inverse problem.

Leveraged off other funding sources.Leveraged off other funding sources.

Equivalence Principle Operator (EPO)

EPA on Non-Connected Regions (Cont’d)

–W. C. Chew and C. C. Lu, The use of W. C. Chew and C. C. Lu, The use of Huygens' equivalence principle for Huygens' equivalence principle for solving the volume integral equation solving the volume integral equation of scattering,. of scattering,. IEEE Trans. Antennas IEEE Trans. Antennas Propagat.Propagat., vol. 41, no. 7, pp. 897.904, , vol. 41, no. 7, pp. 897.904, July 1993July 1993.

–Jensen etal

–Chen etal

–Jandhyala etal

–Lee etal

EPA on Non-Connected Regions (Cont’d)

Advantages

– Reduction of the Number of Unknowns (Fine Details only contribute to the near field)

– Reduction of the Memory Usage for Problems with Identical Domains

– Good for Solving Problems with Fine Features, Random Antenna Arrays, Periodical Structures with Defects

Tap Basis Scheme

EPA on Connected Regions ( Cont’d )

2x2 XM Antenna Arrays

Preliminary Research Results

– Equivalent Currents

Two Connected Conductors

Preliminary Research Results ( Cont’d )

Novel TDS

Model a thin dielectric sheet with 1/3 the unknowns compared to volume integral equation.

Can be improved to include lamination and anisotropy.

Proven to work with microstrip antennas and radomes of antennas.

Include both normal and tangential components to capture the physics better.

Scattering by a Thin Dielectric Plate

Scattering by a 1:0mx1:0mx0:02m inhomogeneous dielectric plate with epsilon= 2:2, 5:7, 4:1, and 7:3. The frequency is at 0:1 GHz and the incident wave is vertically polarized from (phi; theta) = (1200; 300). (a) RCS at Á = 00. (b) Tangential current. (c) Normal current.

Some Examples of Novel TDS

More Examples of Novel TDS—Microstrip Antenna

MoM friendly formulation for Layered Medium Green’s

Function

• Succinct and

elegant derivation

• MOM friendly implementation

• Weaker singularity involved

• Easy extension to complex case ( straddling objects)

Advantages:Advantages:

S1

1

N-1

3

2

N

1r

2r

3r

rN1rN

...

General Dyadic Green’s Function

where

• Two Sommerfeld integrals to evaluate

• Only zero-th order Bessel function used

Layered Medium Green’s Function

(numerical result)

f= 300 MHZ0.3m

10m

R =1m

1 1.0r

2 2.56r

3 6.5 0.6r i

60

0

oinc

oinc

Decreasing Computation Time

Tabulate two basic Green's function integrals:– Calculate Green's Function integrals over a grid of possible

source and observation locations.– Interpolate between precomputed integral values to

approximate the Green's functions and their derivatives at a given set of points.

Multilevel Multipole-Free Fast Algorithm for Layered Media

Heritage – Developed at CCEML-UIUC• Fast Steepest Descent Path Algorithm1, • Fast Inhomogeneous Plane Wave Algorithm2 (FIPWA)

Advantages• Scales as multilevel fast multipole algorithm (MLFMA) – O(N logN)• Simpler than multipole expansion of FIPWA and MLFMA• Controllable accuracy– no approximation of reflected terms,

surface wave or pole contributions

1 E. Michielssen and W. C. Chew, Radio Science, vol. 31, no. 5, pp. 1215{1224,Sept.-Oct. 1996. 2 B. Hu and W. C. Chew, Radio Science, vol. 35, no. 1, pp. 31{43, Jan.-Feb. 2000.

NOTE: Poles and branch cuts are computed in a similar fashion

MMFFA Approach

Diagonalize Green’s Function in Multilevel Architecture Cast into nested Sommerfeld Integrals Accelerate integration via steepest descent path (SDP) integrals Factorize into radiation/receiving patterns and translators Diagonalize translator with interpolation/extrapolation

2-D Illustration

MMFFA vs. FIPWA

Benchmark case– PEC cylinder over two-layered medium

– 600 MHz, N = 9708, inc = 30 deg

Scaling

Ansatz-Based Methodsfor Frequency Independent

Scattering

Three significant challenges– Generating the ansatz (APEx, travel-time function, analytical)

– Order-1 numerical integration

– Uniqueness / error controllability

xik

qq

qeJ

Slowly varying amplitude Slowly varying amplitude

function – easily function – easily

discretizeddiscretized

Order 1 Quadrature

1

0

)1(0 1 dxxkHI

We can also do some integrations on quadratic patches: branch We can also do some integrations on quadratic patches: branch

cuts, stationary phase points.cuts, stationary phase points.

Uniqueness/Error Controllability

Implementation notes:– flat strip, TEz, normal incidence, ka = 9π

– Moment method, quadratic basis functions

– Two-variable, frequency independent integration

610 10

The Multiple-Frequency Solution

If an initial image is made at a lower frequency, the object is electrically smaller.

The local-minimum problem is less significant at lower frequencies.

The low-frequency image may be used as an initial guess for the final, high-frequency image.

Since the initial guess is close to the actual solution, local minima are avoided even at high frequencies.

Original

Reconstruction

The Fréchet Derivative Operator

The Fréchet derivative operator (red) produces fields at the receivers (R) due to currents in the object.

Currents are a product of excitation by the transmitters.

The adjoint operator (green) correlates fields produced by the transmitter with fields produced by simultaneous excitation of all receivers.

Each of these operators may be computed through calls to a forward solver.

MM

†

Parallel Acceleration

Parallel acceleration is crucial to make DBIM competitive with faster imaging methods.

Fields (and currents) produced by each transmitter are independent of the others.

By passing forward-solver calls (for the Fréchet derivative and its adjoint) for distinct transmitters to distinct processors, near-ideal parallel efficiency is possible.

Future Work

Applying novel TDS to antennas with complex feeds.

Extending EPA to solve large arrays. Incorporate novel layered medium

formulation for antenna feeds.