Finite Element Analysis of Antennas and Phased Arrays

in the Time Domain

Jian-Ming Jin

Center for Computational Electromagnetics

and Electromagnetics Laboratory

Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Urbana, Illinois 61801

July 30, 2012

Antennas Mini-Symposium

Time vs. Frequency Domain

Multiple excitation / angular sweep

Dispersive material modeling

Steady state phenomena

Frequency domain:

Broadband simulation / frequency sweep

Nonlinear material/device modeling

Transient phenomena (coupling paths,

resonances, multiple bounces)

Physical insight useful to design engineers

Time domain:

Basic TDFEM formulations and unconditionally stable algorithm

Perfectly matched layers (PML) for mesh truncation

Hybrid FDTD-TDFEM for efficient mesh truncation

Time-domain FE-BI for accurate mesh truncation

Time-domain waveguide port boundary condition (WPBC)

Highly efficient domain decomposition methods

Periodic TDFEM with Floquet ABC for mesh truncation

Modeling of general dispersive, lossy, and anisotropic materials

Explicit TDFEM (Discontinuous Galerkin and Huygens’ methods)

Tree-cotree splitting to fix low-frequency breakdown problems

Hybrid field-circuit-network simulation based on TDFEM

Applications to scattering, antennas, antenna arrays, microwave

devices, RF circuits, FSS, photonic crystals, etc.

Progress in TDFEM

Maxwell’s equations:

TDFEM Basics

imp

( )( ) ( )

tt t

t

HE M

imp

( )( ) ( ) ( )

tt t t

t

EH E J

Curl-curl wave equation:

0 o

0

1ˆ ˆ ˆ 0 on n Y n n S

t

E E

2imp imp

2

1 ( ) ( )( )

t tt

t t t

J ME EE

Radiation boundary condition:

Weak-form representation:

TDFEM Basics

Spatial discretization:

2nd-order ordinary differential equation:

o

2

02

imp imp

1ˆ ˆ( ) ( ) ( )

V S

V

dV Y n n dSt t t

dVt

E E ET E T T T

J MT

edge

1

( , ) ( ) ( )

N

i i

i

t E t

E r N r

2

2

{ } { }[ ] [ ] [ ]{ } { }

d E d ET R S E f

dt dt

Unconditionally Stable TDFEM

( 1) ( 1){ } { } { }

2

n nd E E E

dt t

2 ( 1) ( ) ( 1)

2 2

{ } { } 2{ } { }

( )

n n nd E E E E

dt t

( 1) ( ) ( 1){ } { } (1 2 ){ } { }n n nE E E E

Unconditionally stable when ! 1 4

Newmark-beta method:

1910 – 1981

Civil Engineering

University of Illinois

Temporal discretization via Newmark- method:

TDFEM Basics

Unconditionally stable time-marching equation:

( 1) ( )

2 2

( 1) ( 1) ( ) ( 1)

2

1 1 1 2 1[ ] [ ] [ ] { } [ ] [ ] { }

( ) 2 4 ( ) 2

1 1 1 1 1 1[ ] [ ] [ ] { } { } { } { }

( ) 2 4 4 2 4

n n

n n n n

T R S E T S Et t t

T R S E f f ft t

1

( 1) ( )

2 2

( 1) ( 1) ( ) ( 1)

2

1 1 1 2 1{ } [ ] [ ] [ ] [ ] [ ] { }

( ) 2 4 ( ) 2

1 1 1 1 1 1[ ] [ ] [ ] { } { } { } { }

( ) 2 4 4 2 4

n n

n n n n

E T R S T S Et t t

T R S E f f ft t

Major Challenges:

TDFEM Challenges

Modeling of Large Computational

Domains and Finite Arrays

Modeling of Infinite Periodic

Structures

Truncation of Open Free Space

Modeling of Waveguide Ports

Modeling of Dispersive Material

J. M. Jin and D. Riley, Finite Element Analysis of

Antennas and Arrays. Wiley, 2009.

TDFEM Mesh Truncation

Absorbing boundary conditions (ABC)

Easy and highly efficient, yet approximate

Perfectly matched layers (PML)

Direct implementation in TDFEM

Complicated formulation, uses a single mesh

Implementation via FEM-FDTD hybrid

Highly robust, uses a hybrid mesh

Boundary integral equations (BIE)

Most accurate and most expensive

Total-Field / Scattered-Field Interface

Near-Field to Far-Field Transformation Boundary

Perfectly Matched Layer

E inc

Finite-Difference Region

(Structured Hexahedra, Explicit)

Gri

d T

erm

inati

on

Finite-Element Region

(Unstructured, Implicit)

FEM-FDTD Hybrid

Implicit – Explicit Solution Scheme

Time-Domain Finite Element Approach (Hybridized with FDTD)

RCS Validation: Standard Benchmark

incEincE

Elevation (degrees)

Mo

no

sta

tic

RC

S(d

Bsm

)

-90 -60 -30 0 30 60 90-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

Measurement (VV)

FETD-FDTD (VV)

Elevation (degrees)

Mo

no

sta

tic

RC

S(d

Bsm

)

-90 -60 -30 0 30 60 90-60

-55

-50

-45

-40

-35

-30

-25

-20

-15

-10

Measurement (HH)

FETD-FDTD (HH)

VV-Polarization HH-Polarization

(a) Double Ogive Geometry

Surface Current with 30 GHz

RF Illumination (b) Monostatic RCS at 9 GHz.

FEM-FDTD Hybrid Example

J. M. Jin and D. Riley, Finite Element Analysis of Antennas and Arrays. Wiley, 2009.

Input impedance of a cavity-backed

microstrip patch antenna

Detailed parameters: See J. Jin, The Finite

Element Method in Electromagnetics (2nd edition),

New York: Wiley,2002.

TDFEM-PML Example

Waveguide Port Modeling

Simplified feed model: electric probe feed

Simplified feed model: voltage gap

Waveguide port boundary condition (WPBC)

Coaxial Cable Rectangular Waveguide Microstrip Line

inc)(ˆ UEE Pn

1

TMTM

TE

1

TETEM

0

TEM

0

)(

)()()(

m S

mm

S

m

m

m

S

dS

dSdSP

Eee

EeeEeeE

inc inc TEM TEM inc

0 0

TE TE inc

1

TM TM inc

1

ˆ ( )

( )

( )

S

m m

m S

m m

m S

n dS

dS

dS

U E e e E

e e E

e e E

Time-Domain Formulation:

Assume dominant mode

incidence:

incidence TMdominant )(2

incidence TEdominant )(2

incidence TEM)(2

incTM

1

incTE

1

incTEM

0

inc

f

f

f

e

e

e

U

Waveguide Port Model

Z. Lou and J. M. Jin, “Modeling and

simulation of broadband antennas

using the time-domain finite element

method,” IEEE Trans. Antennas

Propagat., vol. 53, no. 12, pp. 4099-

4110, Dec. 2005.

Monopole Antennas

mm 1.0a

mm 2.3b

mm 32.8h

mm 1.0a

mm 2.3b

23.1 mmh

' 2.0 mmh

o30

Measured data: J. Maloney, G. Smith, and W. Scott, “Accurate computation of the radiation from simple

antennas using the finite difference time-domain method,” IEEE Trans. A.P., vol. 38, July 1990.

Microwave Resonator Filter

Measured data:

J. R. Montejo-Garai and J. Zapata,

“Full-wave design and realization of

multicoupled dual-mode circular

waveguide filters,” IEEE T-MTT,

vol. 43, pp. 1290-1297, June 1995

Dispersive Modeling

Constitutive relations: 0 0( ) ( ) ( ) ( )et t t t D E E

0 0( ) ( ) ( ) ( )mt t t t B H H

Weak-form solution:

2

1

0 2

0

2imp

0 2

1 ( ) ( )( ) ( ) ( ) ( )

( )( ) ( )ˆ( )

e

V

e

S V

t tt t

t t

tt tt dV n dS dV

t t t

E ET E T Q T T

JE HT T T

1

0

( ) 1( ) ( )

tt t

t

HQ E

0 0 imp

( ) ( )( ) ( ) ( ) ( )m m

t tt t t t

t t

H HE H M

Dispersion Modeling Example

Frequency (GHz)

r,

r

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12

-r''

r'

r'

-r''

r() = 1 + 9/(1+ j 510

-11)

r() = 2 + 4/(1+ j 210

-10)

Frequency (GHz)

|

|,|T

|

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

Exact - Reflection

Exact - Transmission

FETD - Reflection

FETD - Transmission

Dispersive Electric & Magnetic Slab

Transmission Coefficient

Normal Incidence

Reflection Coefficient

( ), ( )

incE

10 cm

Dispersion Modeling Example

( ), ( )

incE

10 cm

Frequency (GHz)

r

0.6 0.8 1 1.2 1.4-40

-20

0

20

40

60

80

100

-r''

r() = 1 +

r

2/ (

r

2+ j 2

r-

2)

r= 2 10

9

r= 5 10

7

r'

Frequency (GHz)

|

|,|T

|(d

B)

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4-140

-120

-100

-80

-60

-40

-20

0

20

FETD (Reflection)

FETD (Transmission)

Exact (Reflection)

Exact (Transmission)

Normal Incidence

Reflection Coefficient

TransmissionCoefficient

Four-Arm Sinuous Antenna

20-mil Dielectric Substrate

Frequency (GHz)

Inp

ut

Active

Imp

ed

an

ce

(Oh

ms)

4 5 6 7 8-20

-10

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Predicted Resistance (Volmax)

Predicted Reactance (Volmax)

Measured Average Resistance (80 Ohms 4-12 GHz)

Measured Average Reactance (0 Ohms 4-12 GHz)

Predicted Average Resistance (84.4 Ohms 4-8 GHz)

Predicted Average Reactance (4.4 Ohms 4-8 GHz)

Measured (average)

Predicted (average)

4-Arm Sinuous

20-mil Substrate

Thin Substrate Drops Theoretical Free-Space Resistance of 133.3

Ohms Down To Approximately 80 Ohms

Sinuous Antenna Etched on 20-mil Substrate*

*N. Montgomery and D. Riley, “Broadband antenna predictions using state-of-the-art hybrid FETD-FDTD,”

URSI Symposium Digest, July 2001.

Coaxial

Feeds

D. J. Riley and C. D. Tuner,

“Volmax: A solid-model-based,

transient, volumetric Maxwell

solver using hybrid grids,”

IEEE AP. Mag., vol. 39, Feb. 1997.

The entire antenna structure

is partitioned into three

sections

Adjacent sections are

connected by WPBC

TDFEM result is obtained by

cascading S-parameters of

three sections

Vlasov Antenna (Return Loss)

Measured

TDFEM

q

Vlasov Antenna (Gain pattern)

Five-Monopole Array (Geometry)

unit: inch

Finite Ground Plane:

• 12’’ X 12’’

• Thickness: 0.125’’

SMA Connector:

• Inner radius: 0.025’’

• Outer Radius: 0.081’’

• Permittivity: 2.0

Monopole Array (Impedance Matrix)

1 2 3 4 5

5

4

3

2

1

Domain Decomposition

Time-Domain Dual-Field Domain

Decomposition (DFDD):

Decomposes the computational

domain into small subdomains

Computes both the electric and

magnetic fields

Employs a leapfrog time-marching

scheme similar to the FDTD

Couples subdomains by exchanging

surface fields at the interfaces

Physical interpretation as application

of Huygens’ principle or equivalence

principle

1V

2V

3V

4V

Interfaces

E, H

No global interface problems to solve!!!

2

111 02 2

0

1 ir

r c t t

JEE

subdomain 1

1 1

2 21 1 1

ˆn n

s n

J H

subdomain 2

2

222 02 2

0

1 ir

r c t t

JEE

interface 1n̂

2n̂

2

11 12 2

0

1 1ri

r rc t

HH J

1 1 1ˆn n

s n M E

2 2 2ˆn n

s n M E

2

22 22 2

0

1 1ri

r rc t

HH J

Time step n Time step n+1/2 Time step n+1

1 1

2 22 2 2

ˆn n

s n

J H

subdomain 1

subdomain 2

interface 1n̂

2n̂

2

111 02 2

0

1 ir

r c t t

JEE

subdomain 1

1 1

2 21 1 1

ˆn n

s n

J H

subdomain 2

2

222 02 2

0

1 ir

r c t t

JEE

interface 1n̂

2n̂

1 1

2 22 2 2

ˆn n

s n

J H

Dual-field domain decomposition method (DFDD):

DFDD Flowchart

• Break down the entire FEM system into many smaller subsystems to

significantly speed up FEM analysis and reduce memory requirements

Physical Interpretation

E, H

E, H

E, H E, H

PMC

Js

E, H

PEC

Ms

E, H

E, H

E, H

Equivalence principle #1

Original problem Equivalent problem

Equivalence principle #2

E-field calculation

at time step n

H-field calculation

at time step n+1/2

Stability Analysis

Eigenvalue Analysis

TMct

10

21

2

1

0

0

h

e

M

MM

0

0

Q

PT

dSnnnjiP j

S

i

B

NN ˆˆˆ,

dSnnnjiQ i

S

j

B

NN ˆˆˆ,

TDFEM

(central difference)

FDTD 222

0 111

11

zyx

ct

SMc

t1

0

21

TDFEM

(Newmark-Beta)

Method

t

TMct

10

21

DFDD-TDFEM

Stability Condition

Only depend on local mesh

Example: Vlasov Antenna

Number of Subdomains

1 8

Total Number of Unknowns

964,826 8 X

135,414

Peak Memory (MB)

15,000 990

Factorization Time (s)

3,582 105

CPU Time per Step (s)

10.9 0.88

Volmax data: Courtesy of D. Riley.

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

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

8× 8 Vivaldi Phased Array

E-plane

H-plane

o o45 , 90s sq

8× 1 Vivaldi Phased Array

Single-Stage Divider

8× 1 Vivaldi Phased Array

Wilkinson Divider

Multi-Stage Divider

Explicit TDFEM

m

For element m:

tttc

immmrm

r

JEEE 002

2

2

0

11

im

r

m

r

mrm

r ttcJ

HHH

1112

2

2

0

Weak-Form Representation:

Vector Wave Equation:

2

02 2

0

0 0

1

ˆ

m

m m

m mri m i i

rV

im mi i

V S

dVc t t

dV n dSt t

E EN E N N

J HN N

2

2 2

0

0

1

1ˆ

m

m B

m mri m i i

r rV

i im i m

r rV S

dVc t t

dV n dSt

H HN H N N

N J N EDFDD-ELD

Explicit TDFEM

m

1 12 2

11 1n e e n e n

m m m m m m

n ne e n

m l m l m

e A B e C e

D h E h f

3 1 12 2 2

1

1

n n nh h h

m m m m m m

h n h n n

m l m l m

h A B h C h

D e E e g

For element m:

Pass to the

adjacent elements

Receive from

the adjacent elements

1n

me 1n

le

Pass to the

adjacent elements

Receive from

the adjacent elements

32

n

mh

32

n

lh

2n

me DFDD-ELD

Numerical Example

Mono-conical antenna

DGTD Methods

• J. S. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured

grids,” J. Comput. Phys., vol. 181, pp. 186–211, 2002.

• T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin method for dispersive and

lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys.,

vol. 200, pp. 549–580, 2004.

• L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability

of a discontinuous Galerkin time-domain method for the 3D heterogeneous

Maxwell equations on unstructured meshes,” ESAIM: M2AN, vol. 39, no. 6, pp.

1149–1176, 2005.

• T. Xiao and Q. H. Liu, “Three-dimensional unstructured-grid discontinuous

Galerkin method for Maxwell’s equations with well-posed perfectly matched

layer,” Microwave Opt. Tech. Lett., vol. 46, no. 5, pp. 459–463, 2005.

• S. Gedney, C. Luo, B. Guernsey, J. A. Roden, R. Crawford, and J. A. Miller,

“The discontinuous Galerkin finite-element time-domain method (DGFETD): A

high order, globally-explicit method for parallel computation,” IEEE Int. Symp.

Electromagn. Compatibility, Honolulu, HI, pp. 1–3, July 2007.

• N. Godel, S. Lange, and M. Clemens, “Time domain discontinuous Galerkin

method with efficient modeling of boundary conditions for simulations of

electromagnetic wave propagation,” APEMC, Singapore, pp. 594-597, 2008.

DGTD Methods

• E. Montseny, S. Pernet, X. Ferriéres, and G. Cohen, “Dissipative terms and local

time-stepping improvements in a spatial high order discontinuous Galerkin

scheme for the time-domain Maxwell’s equations,” J. Comput. Phys., vol. 227,

no. 14, 2008.

• S. Dosopoulos and J.-F. Lee, “Interior penalty discontinuous Galerkin finite

element method for the time-dependent first order Maxwell’s equations,” IEEE

T-AP, vol. 58, pp. 4080-4090, Dec. 2010.

DGTD --- An extension from FVTD and FEM

• Adopts the idea of basis and testing functions from FEM

• Integrates over each element (instead of the entire

computational domain), as in FVTD

• Couples all the elements through fluxes at the element

interfaces, as in FVTD

DGTD Methods

DGTD-Central

1

2( ) t n t

Element 1

Element 2

Interface1̂n

2n̂

Solve H-Eq For

1

nE E

2

nE E

Element 1

Element 2

Interface1̂n

2n̂

1

21

n

H H

1

22

n

H H

Solve E-Eq For 1

nE

Solve E-Eq For 2

nE

Element 1

Element 2

Interface1̂n

2n̂

1

21

n

H H

1

22

n

H H

Solve E-Eq For 1

1

nE

Solve E-Eq For 1

2

nE

1

21

n

H

Solve H-Eq For1

22

n

H

1ˆ ( )

2

ET H T H H

V S

dV n dSt

1ˆ ( )

2

HT E T E E

V S

dV n dSt

E-Eq:

H-Eq:

( 1)t n t t n t

DGTD Methods

DGTD-Upwind

Element 1

Element 2

Interface1̂n

2n̂

51

1

n c E E

Solve Both Eqs

For 1 1,

n n E H

1 1ˆ ˆ ˆ( ) ( )E

T H T H H T E E

V S S

dV Z Z n dS Z n n dSt

1 1ˆ ˆ ˆ( ) ( )H

T E T E E T H H

V S S

dV Y Y n dS Y n n dSt

E-Eq:

H-Eq:

t n t ( ) , 1,...,5

it n c t i

( 1)t n t 0 1i

c

51

1

n c H H

51

2

n c E E

51

2

n c H H

Solve Both Eqs

For 2 2,

n n E H

Element 1

Element 2

Interface1̂n

2n̂

1

1in c

E E

Solve Both Eqs

For 1 1,i in c n c E H

1

1in c

H H

1

2in c

E E1

2in c

H H

Solve Both Eqs

For2 2

,i in c n c E H

Element 1

Element 2

Interface1̂n

2n̂

5

1

n cE E

Solve Both Eqs

For 1 1

1 1,

n n E H

5

1

n cH H

5

2

n cE E

5

2

n cH H

Solve Both Eqs

For 1 1

2 2,

n n E H

Convergence Rate

2.752.100.91DGTD-Central (p)

Polynomial Order (p) 1 2 3

DFDD (p) 1.35 1.98 3.18

DGTD-Upwind (p+1) 1.96 3.01 3.83

2.752.100.91DGTD-Central (p)

Polynomial Order (p) 1 2 3

DFDD (p) 1.35 1.98 3.18

DGTD-Upwind (p+1) 1.96 3.01 3.83

DGTD vs DFDD-ELD

DGTD vs DFDD-ELD

Conclusions:

DFDD-ELD, DGTD-Central, and DGTD-Upwind have a similar performance

However, DFDD-ELD can be implemented easily in a hybrid explicit-implicit algorithm

Hybrid Explicit-Implicit Scheme

Numerical scheme:

1. Group small elements and

apply the implicit TDFEM

2. Apply the explicit TDFEM to

large elements

Advantages:

1. Use very small elements to

model fine features

2. No penalty on the time step

size

Example:

Explicit TDFEM: t < 0.25 fs

Hybrid algorithm: t < 1.5 fs

Implicit region: 3906 tets

Explicit region: 25936 tets

Example: Differential Via Pairs

Reference: E. Laermans et al., “Modeling

complex via hole structures,” IEEE Trans. Adv.

Packag., vol. 25, no. 2, pp. 206-214, May 2002.

Hybrid Explicit-Implicit Scheme

A Generic Periodic Phased Array

Technical challenges & solutions:

1. Enforcement of periodic boundary conditions

Transformed field variable

2. Mesh truncation in the non-periodic direction

Floquet absorbing boundary condition

Periodic boundary condition:

Floquet absorbing boundary condition:

Weak-form vector wave equation:

Infinite Phased Array

( )

( , , ) ( , , )s s

x yx yj mk T nk T

x yx mT y nT z x y z e

E E

0sin coss

x s sk k q 0sin sins

y s sk k q

( )ˆ ˆ ˆ( ) ( , ) ( ) xp yqj k x k y

xp yq pq

p q

z z k k z e

E G E

uc uc

uc

1 2

0 0

1

0 0 imp imp

ˆ( ) ( ) ( )

( )

r r

V S

r

V

k dV j n dS

jk Z dV

T E T E T H

T J M

Frequency-Domain Analysis

Z. Lou and J. M. Jin, “Finite element

analysis of phased array antennas,”

Microwave Opt. Tech. Lett., vol. 40,

no. 6, pp. 490–496, March 2004.

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

s sx yj k x

e

k yx y z x y z e

P E

( , , ; )( , , ; )

( , , ; )

e x

e y

e

x T y zx y z

x y T z

PP

P

Transform E and H to remove the phase variation

on periodic surfaces

Such that

Infinite Phased Array

Time-Domain Analysis

Second-order vector wave equation:

221

02 2

im

1

2 2

1 1

p imp

1(

1 ˆ ˆ

1 1ˆ

)

( , )ˆ

e es s et r t

s se er t

r

t r

r ec t tc t

c t c t

P PP

g J

Pk k

P Pk Mk

L. E. R. Petersson and J. M. Jin, “Analysis of

periodic structures via a time-domain finite

element formulation with a Floquet ABC,”

IEEE Trans. Antennas Propagat., vol. 54, no. 3,

pp. 933–944, March 2006.

Periodic Boundaries

Coaxial

Feeds (2)

Radiator

Ground Plane

Periodic Boundaries

Dispersive Magnetic Substrate

Frequency (GHz)

Perm

eabi

lity

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

r'

r''

Dispersive Permeability Profile

Dispersive Permeability

20:1 with 1 dB Insertion Loss

Bandwidth (Broadside)

Frequency (GHz)

Inse

rtio

nL

oss

(dB

)

VS

WR

0 0.4 0.8 1.2 1.6 2 2.4 2.8

0.0

0.5

1.0

1.5

2.0

2.5

3.0 1.0

1.5

2.0

2.5

3.0

3.5

4.0

VSWR

INSERTION LOSS

HFSS

FETD

HFSS

FETD

Broadside Scan

Insertion Loss & VSWR

Ultra-Wideband Phased Array

D. Riley and J. M. Jin, “Finite-element time-domain

analysis of electrically and magnetically dispersive

periodic structures,” IEEE Trans. Antennas

Propagat., vol. 56, no. 11, November 2008.

Hybrid Field/Circuit Systems

Symmetric Field/Circuit Coupling

Lumped

Circuit

FEM

kV FEM

kVˆkl

FEM

Lumped

Circuit

FEM

kV FEM

kVˆkl

FEM

CKT,nl CKT CKTCKT

CP

{ }{ }

{ }{ }

n n nn

T

nn

Y B VV

C eIB

I I

0 0

EM-to-Circuit Coupling Circuit-to-EM Coupling

Lumped

Circuit

CKT

kI

CP CKT

k kI I

ˆkl

CKT

kI

FEM

Lumped

Circuit

CKT

kI

CP CKT

k kI I

ˆkl

CKT

kI

FEM

ˆk

kj j kl

C l dl N

CKT

P

,

1

C

1ˆ}

){

(k

i

n

i k k nl

l dV It

b

t

N

FEMFor 1,2,...,i N

CKT

T CP CP

2

1

{ } 1{ } { }

2n n

n

bC I I

t t

0 1 1 2 2

FEM2

0 0

1

CKT

1

{ } { } { }

{ } { }

n

n n n

n

E e E e E e

bc t Z

t

b

t

CKT CKT,nl CKT CKT{ } { }n n n nY V V I I

Contains

1’s and 0’s

only

Coupling

Matrix

Field/Circuit Global System

( )n nF x b

T

CKT CP{ } { } { }n n n ne V Ix

T

0

CKT CKT,nl CKT

T CP

{ }

( ) [ ] [ ] { } { }

[ ] { }

n

n n n n

n

E C e

Y B V V

C B I

00

F x 0 I

0 0

T

1 1 2 2 2

CKT

{ } { } { }

n n n

n n

E e E e C I

0

b I 0 0

0 0 0

T

1 1 2 2

CKT CKT CKT

1 2

CP CP

1 2

{ } { }

{ } { }

{ } { }

n n

n n n

n n

E e E C e

V V

I I

0 0 0 0

I 0 0 0 0 0 0

0 0 0 0 0 0 0

Define 0 0* ( ) 2c tZ

Symmetric

MESFET Amplifier: Large Signal Analysis

Summary

TDFEM has been maturing for EM analysis

• Difficulties (mesh truncation , port modeling, low-frequency

breakdown, dispersion modeling, periodic BC and Floquet

ABC) have been successfully resolved

TDFEM has unparalleled modeling capabilities

• Excellent modeling of complex structures & materials

• Excellent modeling of wave ports, networks, and circuits

• Large-scale simulation via domain decomposition

TDFEM has been successfully demonstrated for

simulating a variety of EM problems

• Antennas, phased arrays, microwave devices, high-speed

circuits, electronic packaging and interconnecting, EMC,

scattering, photonic crystals, metamaterials, etc.

TDFEM has a great potential in tackling multi-scale and

multi-physics problems

Acknowledgment

Dr. Douglas Riley, Northrop Grumman Aerospace Systems;

Prof. Andreas Cangellaris, University of Illinois at Urbana-

Champaign, and Eric Michielssen, formerly with the University

of Illinois at Urbana-Champaign

Drs. Dan Jiao, Zheng Lou, Rickard Petersson, Thomas

Rylander, Ali Yilmaz, and Rui Wang, formerly with the

University of Illinois at Urbana-Champaign

MURI/AFOSR, HPCMP, Northrop Grumman, Sandia