uiuc muri review
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UIUC MURI Review. J.-M. Jin, A. C. Cangellaris, and W. C. Chew Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801-2991 Program Director: Dr. Arje Nachman, AFOSR June 19, 2006. - PowerPoint PPT PresentationTRANSCRIPT
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
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
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.