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Domain Decomposition Methods
for FEM Modeling of Large-Scale
Phased Arrays
J.-M. Jin
Center for Computational Electromagnetics
Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign
Urbana, Illinois 61801-2991
July 30, 2012
Antennas Mini-Symposium
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AN/FPS-85 Spacetrack Radar Ballistic Missile Early Warning
System
• Extremely high directivity/resolution
• Rapid electronic scanning over 120o
• Simultaneous tracking of multiple
targets
13 stories high
5,134 transmitters/5,928 antennas
4,660 receivers/19,500 antennas
Phased-Array Antennas
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X-band Phased-Array Antenna Phased-Array Antenna with
Distributed Feed Network
MMIC phase-shifters & T/R modules
256 elements
Applied Radar, Inc.
New applications:
• Wireless communications
• Video games
Phased-Array Antennas
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Very large-scale inhomogeneous problem with non-uniform excitation
Requires an excessively long computation time and extensive resources
Can only handle problems with a limited size on a single computer
Requires parallel computers to handle relatively large problems
Phased-Array Antennas
Computational Challenges
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FEM Basics
Vector wave equation:
Boundary conditions:
on
1imp00
2
0 JEE Zjkk r
r
Dn on0ˆ E
NN
r
nnjk
n
on)ˆ(ˆ
1ˆ 0 KEE
2
0
00 0 imp
1 1ˆ( ) ( ) ( ) ( )
ˆ ˆ( ) ( )
D
N
r
r r
N
k d n d
jkn n d jk Z d
W E W E W E
W E W K W J
Weak-form representation:
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FEM Basics
Spatial discretization:
edge
1
N
i
iiENE
FEM matrix equation:
where
[ ]{ } { }K E b
2
0
0
1( ) ( )
1ˆ ˆ( ) ( )
N
ij i j r i j
r
i j
K k d
jk n n d
N N N N
N N
0 0 imp
N
i i i Nb jk Z d d
N J N K
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Decompose the computational domain
into many small subdomains
Reduce a 3D global problem to an explicit
interface problem for a dual unknown
with a much smaller size
Construct a coarse grid problem defined
at the subdomain corner edges
Suitable for general 3D problems using
parallel computing
Utilize geometrical redundancies in
finite array-like structures
The ElectroMagnetic Dual-Primal Finite Element Tearing &
Interconnecting method (FETI-DPEM)
FETI-DPEM Overview
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Advantages of FETI-DPEM
1. Able to solve very large 3-D problems because it reduces
a 3-D global problem to a much smaller explicit interface
problem.
2. Extremely suitable for parallel computation because
subdomains are fully decoupled.
3. Fast convergence rate for solving the interface problem by
enforcing the continuity at the corner edges.
4. Able to solve finite periodic problems with billions of
unknowns very efficiently on a single workstation.
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Domain Decomposition Methods
i j
i j
FETI-DPEM
w/ weak BC
FETI-DPEM
w/ strong BC
FETI-DPEM
w/ mixed BC
• Corner unknowns
belong to interfaces
• Field continuity is
enforced implicitly
• Interface meshes
don’t have to match
• Corner and interface
unknowns are separated
• Field continuity is
enforced explicitly
• Interface meshes
have to match
• Corner and interface
unknowns are separated
• Field continuity is
enforced explicitly at the
corners and implicitly at
the interfaces
c
c
c
c
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FETI-DPEM1 Formulation
Computational domain
partitioned in a check-board
fashion.
ˆ in
ˆ jnij
c
c
c
c c
c
c
c
c
All degrees of freedom related
to the corner-edges are global
primal variables: 1[ ... ]sN T
r r cE E E E
[ ] [ ]s s s s T s s T
V I c r cE E E E E E
In the sth subdomain the primal
variables are grouped as:
ji
Introducing a Boolean matrix ,
which extracts the corner dofs of
the sth subdomain as:
s
cB
s s
c c cB E E
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FETI-DPEM1 Formulation
s 2 s
0 0 0
1in s s
r imps
r
E k E jk Z J
Vector wave equation:
FEM matrix equation: s s s sK E f
s
c
ˆ in
s
I
s
impJ
s
I
s
I
c
c
c
1 1ˆ ˆ ˆ ˆ on s s q q s s q q sq
Is q
r r
n E n E n E n E
Boundary conditions:
0
ˆ 0 on
ˆ 0 on
1ˆ ˆ ˆ 0 on
s s s
PEC
s s s
PMC
s s s s s s
ABCs
r
n E
n E
n E jk n n E
Interface condition:
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Analysis of subdomain
problem.
s
c
ˆ in
s
I
s
impJ s s 2 s s
0
s s
0
1( ) ( )
ˆ ˆ( ) ( )
T
s
sI
s T
r
r
T
K N N k N N dV
jk n N n N dS
s
0 0 s
s s
impf jk Z N J dV
Subdomain FEM matrix :
Subdomain excitation
vector:
Interface contribution:
1ˆ
T
I
s s s s s
rs
r
N n E dS B
is a signed Boolean matrix that
extracts the interface dofs.
s
rB
s
I
s
I
c
c
c SubdomainFEM matrix equation:
s s s sK E f
FETI-DPEM1 Formulation
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Using the “r” and “c” notations:
T
s s
rr rcs
s s
rc cc
K KK
K K
s
rs
s
c
ff
f
The subdomain problem in the matrix form: T
T
s s s s srr rc r r r
ss s scrc cc c
K K E f B
EK K f
Note: λ is unknown
1
( )Ts s s s s s s s
r r r rr r r rc cB E B K f B K E
The electric field at the interfaces of the sth subdomain:
The subdomain level corner dofs related system:
1 1 1
( )T T T Ts s s s s s s s s s s s
cc rc rr rc c c rc rr r rc rr rK K K K E f K K f K K B
FETI-DPEM1 Formulation
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Subdomain system equations are assembled using
the electric field continuity equation:
1
0sN
s s
r r
s
B E
The corner-related dofs are assembled globally as a
super finite element system, which couples the
whole computational domain:
1 1 1
1 1 1
( ) ( ) ( )s s s
T T T TN N N
s s s s s s s s s s s s s s s T
c cc rc rr rc c c c c rc rr r r rr rc c
s s s
B K K K K B E B f K K f B K K B
T
cc c c rcK E f F
FETI-DPEM1 Formulation
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where
By eliminating and , the interface equation for
solving for the dual variable λ is given as:
1 1T
rr rc cc rc r rc cc cF F K F d F K f
1
1
1 1
1 1
s s
s s
N Ns s s sT
rr rr r rr r
s s
N Ns s s s s
rc rc r rr rc c
s s
F F B K B
F F B K K B
1
1
1 1
1 1
( )
s s
s sT T T
N Ns s s s
r r r rr r
s s
N Ns s s s s s s
c c c c c rc rr r
s s
d d B K f
f f B f B K K f
1
1 1
( ) ( )s s
TN N
s s s s s s T s s s
cc cc c cc c rc c rr rc c
s s
K K B K B K B K K B
rE cE
FETI-DPEM1 Formulation
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( ) ( ) ( ) ( )
( ) ( )
1 1ˆ ˆi i j j
i j
r r
n n
E E Λ
( ) ( ) ( ) ( )ˆ ˆi i j jn n E E
Subdomain interface conditions:
Vector wave equation:
FETI-DPEM1 Formulation
2
0 0 0 imp
1r
r
k jk Z
E E J
i j
c
c
c
c
Y. Li and J. M. Jin, “A vector dual-primal finite element tearing and interconnecting method for solving 3-
D large-scale electromagnetic problems,” IEEE Trans. Antennas Propagat., vol. 54, no. 10, pp. 3000-
3009, Oct. 2006.
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Final interface system for dual unknowns:
1 1
rr rc cc cr r rc cc cK K K K f K K f
Global interface system:
rr rc c rK K E f
Global corner system:
cc c c crK E f K
FETI-DPEM1 Formulation
i j
c
c
c
c
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31x31 Vivaldi Antenna Array
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Convergence history for solving
the interface problem of various
Vivaldi antenna arrays at 3 GHz.
Numerical scalability Radiation patterns
Radiation patterns of the 100 x
100 Vivaldi antenna array at 3
GHz for θ= 45º, φ= 0º scan.
Finite Vivaldi Antenna Array
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Size of array
Total number of unknowns
(Million)
Total number of
dual unknowns
Memory
(MByte)
Interface solving time
(h:min:s)
Iteration number
Total time
(h:min:s)
10 x 10 5.7 0.1M 500 00:00:10 19 00:06:36
31 x 31 39.7 0.7M 541 00:01:21 19 00:11:44
50 x 50 100.9 1.9M 685 00:03:02 18 00:20:23
75 x 75 224.2 4.4M 1300 00:06:01 17 00:40:11
100 x 100 396.2 7.8M 2100 00:10:54 17 01:12:30
150 x 150 887.3 17.6M 3800 00:28:58 20 02:18:42
200 x 200 1572.2 31.5M 6700 01:09:36 20 04:16:35
250 x 250 2452.3 49.4M 10000 02:35:07 20 08:34:19
300 x 300 3528.4 70.8M 15000 05:27:15 20 12:27:14
Computation time & memory usage of the FETI-DPEM method for
various Vivaldi antenna array simulations
Finite Vivaldi Antenna Array
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Numerical Scalability
Numerical scalability of the FETI-DPEM1 method. (a) Numerical
scalability with respect to frequency for the 100-by-100 array. (b)
Numerical scalability with respect to the array size at different
frequencies.
(a) (b)
100x100 array
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( ) ( ) ( ) ( ) ( ) ( )
0( )
1ˆ ˆ ˆj j j j j j
j
r
n jk n n
E E Λ
( ) ( ) ( ) ( ) ( ) ( )
0( )
1ˆ ˆ ˆi i i i i i
i
r
n jk n n
E E Λ
Subdomain interface conditions:
Equivalent interface conditions: ( ) ( ) ( ) ( ) ( )
0
( ) ( ) ( ) ( ) ( )
0
ˆ ˆ2
ˆ ˆ2
i j j j j
i j i i i
jk n n
jk n n
Λ Λ E
Λ Λ E
Vector wave equation:
FETI-DPEM2 Formulation
2
0 0 0 imp
1r
r
k jk Z
E E J
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Final interface system for dual unknowns:
1 1
rr rc cc cr r rc cc cK K K K f K K f
Global interface system:
rr rc c rK K E f
Global corner system:
cc c c crK E f K
FETI-DPEM2 Formulation
Y. Li and J. M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of
3D large-scale electromagnetic problems,” IEEE Trans. Antennas Propagat., vol. 55, no. 10, pp. 2803-
2810, Oct. 2007.
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Numerical Scalability
Numerical scalability of the FETI-DPEM2 method. (a) Numerical
scalability with respect to frequency for the 100-by-100 array. (b)
Numerical scalability with respect to the array size at different
frequencies.
(a) (b)
100x100 array
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Mutual Coupling Between Arrays
Geometry of two 9 x 9 patch antenna
arrays recessed in dielectric cavities
on a planar platform
Normalized received power
patterns
23.9 million primal unknowns
1.2 GB RAM, 5 hours on a
single 1.5-GHz processor
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Photonic Crystal Nanocavity
x
y
Number of layers
Number of subdomains
Number of unknowns
Memory
(MByte)
Simulation time
14 919 7.2M 1100 14m57s
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Photonic Crystal Nanocavity
Geometry of the PhC nanocavity analyzed (H. Y. Ryu, 2002).
The yellow region is the dielectric slab with the refractive index
of n = 3.4 and the red circles represent the air holes. The slab
thickness t = 0.4a.
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Energy stored in the cavity as a function of
frequency. The locations of the energy
peaks represent the resonant frequencies of
the resonant modes.
Dipole Hexapole
Quadrupole
Monopole
Photonic Crystal Nanocavity
Q ~ 100,000
Q ~ 5,000
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Improved Waveguide Bend
Problem analyzed
Number of subdomains
Number of unknowns
Memory
(GByte)
Simulation time
Original 487 11.7M 1.7 47m34s
Modified 487 11.7M 1.8 47m51s
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Patch Antenna on a Platform
Normalized radiation patterns in the
H-plane at 3.3 GHz for a patch
antenna on a cylinder with a wing.
Speedup = 4 ×
T4/TNp
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Non-realistic Scattering Example
Original object
Subdomain decomposition
Parallel speedup
Current distribution
at 300MHz
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Geometry of a 31x31 cavity-backed patch antenna array on an infinitely
long PEC cylinder. (a) The patch antenna array on platform. (b) The
geometry of the array element.
(a) (b)
31-by-31 Patch Array on Curved Surface
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Radiation patterns and active
reflection coefficients (mid-row)
for the 31x31 cavity-backed patch
antenna array on various
platforms.
31-by-31 Patch Array on Curved Surface
E-plane
H-plane
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Patch Antenna Arrays on Battleship
Phased Array on a Platform
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Nonconformal FETI-DPEM
(1) Lagrange-multiplier (LM)-based nonconformal FETI-DPEM
(2) Cement-element (CE)-based nonconformal FETI-DPEM
Nonconformal FETI-DP Methods for Large-Scale Electromagnetic
Simulation
Features:
• Both methods implement the Robin-type transmission condition at the subdomain
interfaces;
• Both methods formulate a global coarse problem related to the degrees of freedom
at the subdomain corner edges to propagate the residual error to the whole
computational domain in the iterative solution of the global interface equation;
• The first method extends the conformal FETI-DP algorithm, which is based on two
Lagrange multipliers, to deal with nonconformal interface and corner meshes;
• The second method employs cement elements on the interface and combines the
global primal unknowns with the global dual unknowns.
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LM-based Nonconformal FETI-DPEM
s ss s si iii ib ic
s s s s s s s
bi bb bc b b bb
s s s s sci cb cc c c
E fK K K
K K K E f B
K K K E f
1ˆ ˆ ˆ( )s s s s s s s
s
r
n n n
E E Λ
sΛ
{ } { }s s T s
b Λ N
[ ]s
bbB
Note: Different from the conformal FETI-DPEM method, the dual unknown
expanded in terms of a set of curl-conforming vector basis functions as
Therefore, is no longer a Boolean matrix.
Vector wave equation:
Subdomain interface conditions:
2
0 0 0 imp
1 s s s
r
r
k jk Z
E E J
Subdomain matrix equation:
here is explicitly
Equivalent interface conditions: ˆ ˆ( ) ( )
ˆ ˆ( ) ( )
s q s q s s s
q s q
s q s q q q q
q s s
n n
n n
Λ Λ E
Λ Λ E
Note: The global system matrix related to the global interface and corner unknowns is
quite similar to that of the conformal FETI-DPEM scheme, except for some Boolean
matrices are replaced by real-valued sparse matrices.
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CE-based Nonconformal FETI-DPEM
Subdomain interface conditions:
Subdomain matrix equation:
Equivalent interface conditions:
0 0
1 1ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )s s s s s q q q q q
s q
r r
n jk n n n jk n n
E E E E
Cement auxiliary variable:
ˆ ( )s s sn b E00
0
0 0 0
0 0
s s ss s si i iii ib ic
s ss s s s sb bbi bb bb bc b
s s ss sbb bb
s s s ss sci cb cc cc c
E f fK K K
E fK K B K f
D C gb g
K K K fE f
neighbor( )
{ } [ ]
q
bs q q
s s qq s
Eg U V
b
[ ]{ } [ ]{ } { }rr rc c rK u K E f
[ ]{ } { } [ ]{ }cc c c crK E f K u
Note: The global system matrix equation is written in the following form, which can be solved
by eliminating the global corner unknowns first.
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Nonconformal Global Corner System
slave
cNmaster
cN
ccorner
slavemastercorner
cornerslave
cornerslave
slave
cNslave
cN
slave master
c cN N
Master and slave corners associated with one
shared cross point, and we choose
master slave
t tE = E
slave
master
slave slave slave
, ,
1
master master master
, ,
1
c
c
N
t c n c n
n
N
t c n c n
n
E
E
E N
E N
slv-slv slave slv-mst master[ ]{ } [ ]{ }cc c cc cG E H E
slv-slv slave slave
, , ,
slv-mst slave master
, , , .
c
c
cc mn c m c n
cc mn c m c n
G dl
H dl
N N
N N
slave slv-slv 1 slv-mst master{ } [ ] [ ]{ }c cc cc cE G H E
In order to remove the conformal-mesh restriction on the geometrical crosspoints, we
impose the Dirichlet continuity condition at the corner in a weak sense as
Basis expansion:
Galerkin testing:
where
Represent slave corner unknowns by
master ones:
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Eigenspectra and Convergence Test
Conformal interface and corner
meshes Nonconformal interface and
conformal corner meshes Nonconformal interface and
corner meshes
0 20 40 60 80 10010
-10
10-5
100
Number of Iterations
Resid
ue
LM FETI-DP
CE FETI-DP
0 20 40 60 80 10010
-10
10-5
100
Number of Iterations
Re
sid
ue
FETI-DPEM2
LM FETI-DP
CE FETI-DP
0 20 40 60 80 10010
-10
10-5
100
Number of IterationsR
esid
ue
LM FETI-DP
CE FETI-DP
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Finite Vivaldi Array Revisit
Simulation of the 100-by-100 Vivaldi antenna
array at 3 GHz. (a) Broadside scan E-plane
relative pattern. (b) Broadside scan H-plane
relative pattern. (c) Convergence history.
-90 -60 -30 0 30 60 90-60
-50
-40
-30
-20
-10
0
10
(degrees)
Re
lative
Pa
tte
rn (
dB
)
E-plane Relative Pattern
FETI-DPEM2
LM FETI-DP(w/ nonconformal mesh)
CE FETI-DP(w/ nonconformal mesh)
-90 -60 -30 0 30 60 90-60
-50
-40
-30
-20
-10
0
10
(degrees)
Re
lative
Pa
tte
rn (
dB
)
H-plane Relative Pattern
FETI-DPEM2
LM FETI-DP(w/ nonconformal mesh)
CE FETI-DP(w/ nonconformal mesh)
0 10 20 30 40 5010
-4
10-3
10-2
10-1
100
Number of Iterations
Re
sid
ue
FETI-DPEM2
LM FETI-DP(w/ nonconformal mesh)
CE FETI-DP(w/ nonconformal mesh)
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105
106
107
108
109
101
102
103
104
105
Total Number of Unknowns
Com
puta
tion T
ime (
Second)
Interface Time
Total Time
105
106
107
108
109
101
102
103
104
105
Total Number of Unknowns
Co
mp
uta
tio
n T
ime
(S
eco
nd
)
Interface Time
Total Time
Scalability of LM-based Nonconformal FETI-DPEM
Vivaldi Antenna Array Photonic Crystal Cavity
9 subs
100 subs
961 subs
10000 subs
81 subs
841 subs
8281 subs
40401 subs
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Conclusion
• Domain decomposition methods offer a most
promising approach to modeling large-scale and
multi-scale EM and multi-physics problems
• FETI-DPEM is a unique method, which achieves a
faster convergence through the formulation of a
coarse grid system
• FETI-DPEM is highly parallelizable and has an
excellent scalability
• Both conformal and nonconformal FETI-DPEM have
been developed to model large phased arrays with a
fully exploitation of geometry repetition
• FETI-DPEM has been validated numerically and
experimentally using NRL’s dual-polarized Vivaldi
phased array
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Conclusion (cont’d)
• Good agreement has been achieved for the radiation
patterns for all the frequencies and scan angles
considered
• FETI-DPEM has been shown to be a powerful
numerical method for simulating large phased-array
antennas
• FETI-DPEM is also applicable to many other array-
type problems and general problems including
antenna-platform interaction analysis