Dynamic Phase Boundary Estimation using Electrical Impedance

Tomography

By Umer Zeeshan Ijaz,

Control Engineering Lab, Department of Electronic Engineering, Cheju National University, Cheju 690-756, Korea

Thesis Defense(Supervised by Professor Kyung Youn Kim)

Dated: 13.11.2007

2

CONTENTS

• Introduction• Electrical Impedance Tomography• Boundary Representation

– Fourier Coefficients– Front Points

• Extended Kalman Filter• Kinematic Models• Interacting Multiple Model Scheme• Unscented Kalman Filter• Gauss-Newton Unscented Kalman Filter

3

INTRODUCTIONChemical engineers frequently encounter the flow of a mixture of two fluids in

Liquid-gas or liquid-vapor mixtures condensers and evaporatorsgas-liquid reactors combustion systemstransport of some solid materialsslurry of the solid particles in a liquid, and pumping the mixture through a pipeLiquid-liquid mixtures in emulsions as well as liquid-liquid extraction.

Types of Flows

4

Electrical Impedance Tomography (EIT) is a imaging modality in which the internal resistivity distribution is reconstructed based on the measured voltages on the surface object.

COMPUTERCOMPUTER

ReconstructionAlgorithm

ReconstructionAlgorithm

Interface withInstrument

Interface withInstrument

VI

Concept of electrical impedance tomography

WHAT IS EIT?

5

• The forward problem calculates the voltages on the electrodes by using the injected current and assumed resistivity distribution.• The inverse problem reconstructs the resistivity distribution by using the voltage measurements on the electrodes.

Forward vs. inverse problem for EIT

FORWARD SOLVER VS INVERSE SOLVER

Inverse Solver

Forward Solverk calculatedV

1 somethingk k

0 measuredV

An iterative inverse solver

6

Governing Equation derived from Maxwell Equation

Boundary Conditions: Complete Electrode Model

MATHEMATICAL MODEL: FORWARD SOLVER

1.( ) 0u

1

1

1

on , 1, 2,...,

, 1, 2,...,

0 on

l

l l l

le

uu z U e l L

nudS I l L

nu

n

1

L

ll

e

1

0L

ll

I

1

0L

ll

U

Between electrodes, no current crosses the boundary if the impedance outside the imaged volume is much greater than that inside

There is an existence of a thin, high-impedance layer beneath electrodes delivering current. This layer may be modelled as the limit of a thin layer of thickness d and impedance z/d as d goes to zero. (use ohm’s law)

Beneath electrodes, neither potential nor the current crossing the boundary is known. Net current crossing the boundary beneath an electrode is equal to the current being delivered to it by tomograph electronics

Constraints: For the solution to be unique

7

FEM DISCRETIZATION OF FORWARD PROBLEM

Ab = I

1

1( , ) , , 1, 2,...,

l

L

i j i jell

i j dr dS i j Nz

B

11 1

1 1( , ) , 1, 2,..., ,

j

i ije e

i j dS dS i Nz z

C

1

1

11

1 1

| |

( , ) , , 1, 2,..., 1| || | j

j

ei j

zi j i j L

eei j

z z

D

| |le Electrode area i Basis function

( 1) ( 1)

( 1)

( 1)

N L N L

N L P

N L P

A

b

I

1, 2,..., 1j L

N-Nodes, L-Electrodes, m-elementsP-Patterns

1 2

34

N

1

L

2

DNNCN

CNBA TT

β

αb

IN

0I ˆ~

T

PNα PL )1(β

100

010

001

111

,,, 121

LnnnN

N

iii

h yxyxuu1

,,

1

1

L

jjj

hU n

Potential inside:

Potential on electrodes:

8

CURRENT INJECTION PROTOCOL

Current frame

0

0

cos( ), 1, 2,..., / 2

sin( ), 1, 2,..., / 2 1lp

ll

I p p LI

I p L

2 /l l L

Trigonometric (L-1 Curent Patterns)

cos( )l cos(2 )l sin( )l sin(2 )lcos(3 )l cos(4 )l sin(3 )l

Opposite (L/2 Current Patterns)

Adjacent (L Current Patterns)

0I

0I

0I0I

9

1

( ) ( )( )

( ) ( )

l

l

x xNl n n

l y yl n n n

x s sC s

y s s

1,2,...,l S

0 1

2 1 sin(2 )n n s

2 cos(2 )n n s

1,2,...n

1,2,...n

1 1 1 1 21 1 1,..., , ,..., ,..., , ,...,x x y y y y y N

N N N N

Truncated Fourier Coefficients Approach(Close Boundary)

BOUNDARY INTERFACE REPRESENTATION 1/2

A 0

A 1

A 3 A 2

C 1

C 3C 2

10

1 11 1( , ) cos , sin

d dX Y R R

R R

( , ) ( , )X Y x d

( , ) cos , sind d

X Y R RR R

11 2( , ,..., )Td d d d

x 0 x 1 x kxK

d 0

d 1 d kd K

fron t po in tsd k pa ram ete rs to be estim a ted

BOUNDARY INTERFACE REPRESENTATION 2/2Front Points Approach (Open Boundary)

2,..., 1

σ=σ0

σ=σ1

A0

A1

C

11

Inverse Solver

Forward Solverk calculatedV

1 somethingk k

0 measuredV

( )kJ

Inverse Solver

Forward Solverk calculatedV

1 somethingk k

0measuredV

k k

( )kJ

BOUNDARY INTERFACE FORWARD SOLVER 1/2

Changes required

( )kJ ( )kJ d

Analytical Jacobians

Boundary to Resistivity Profile Mapping (Forward Solver)

( )k kV ( )k kV d

0 1

1,

k

K L

k i j i jk A e

i j d dSz

B

Nji ,,2,1,

12

x 0 x 1 x kxK

d 0

d 1 d kd K

fron t po in tsd k pa ram e te rs to be estim a ted

s1

s2

Al

Ar

σl

σr

Sl

Sr

Cl(s)

Ni

Nj

( )l l r r

ee l r

S S

S S S

l

r

r

u

BOUNDARY INTERFACE FORWARD SOLVER 2/2

(a) description of interface with front points /fourier coefficients b) mesh elements above the interface/inside the target are assigned one conductivity value ; (c) mesh elements below the interface/outside the target are assigned second conductivity value ; (d) mesh elements lying on the interface are assigned area average conductivity values assigned using equation ; and (e)

final conductivity values at the end of assignment.

(a) (b) (c) (d) (e)

e

e

13

1 1 1k k k kd F d w

( )k k k kU V d v

| 1

| 1 | 1( ) ( )

k k

kk k k k k k k k

k d

VU V d d d v HOT

d

| 1 | 1( )k k k k k k k k k k ky U V d J d J d v

State Space Model

Random-walk modelNonlinear measurement equation

Linearizing the measurement equation about the predicted mean in the previous step

kF

[ ]Tk k kE w w Q

[ ]Tk k kE v v R

[ ] 0kE w [ ] 0kE v

Regularization

*

kk

R

yy

L d

| 1( )k k kk

R

J dH

L

k k k ky H d

[ ]Tk k kE v v R

[ , ]k kBlockdiag R I

INVERSE SOLVER

14

| 1 1 1| 1k k k k kd F d

| 1 1 1| 1 1 1T

k k k k k k kP F P F Q

1| 1 | 1( )T T

k k k k k k k k kK P H H P H

| | 1 | 1( )k k k k k k k k kd d K y H d

| 1 | 1 | 1( ) ( )k k k k k k k k k ky U V d J d d

*

kk

R

yy

L d

| 1( )k k kk

R

J dH

L

| | 1( )k k k k k kP I K H P

kUMeasurement Update

Time Update

|k kP|k kd

1| 1k kd 1| 1k kP 0|0d 0|0P

Jacobian

Forward Solver

,kQ[ , ]k kBlockdiag R I

,kF ,RL,,kR

Predefined

EXTENDED KALMAN FILTER (Front Points)

0|0 0|0,d P

1|0 1|0,d P

1|1 1|1,d P

2|2 2|2,d P

3|3 3|3,d P

4|4 4|4,d P

2|1 2|1,d P

3|2 3|2,d P

4|3 4|3,d P

1U

2U

3U

4U

15

EXTENDED KALMAN FILTER (Front Points) Results

3% Noise

10-Front Points, Contrast Ratio of 1:100, Moving every 4 Current Patterns (First two modes of cosine and sine with additional cosine in image reconstruction)

|| ||

|| ||estimated true

dtrue

d dRMSE

d

|| ||

|| ||estimated true

Utrue

U URMSE

U

16

Bubble moving with constant velocity Bubble moving with constant acceleration

Bubble expanding with constant velocity Bubble expanding with constant acceleration

KINEMATIC MODEL (Fourier Coefficients)

17

2

, ,12

1

.

Lmeas l homo ll

Lll

U UN D

I

, ,1 1

,1

.

L Lhomo l l meas l ll l

Lhomo l ll

U I U IP D

U I

Distinguishability can be defined as a measurement ability to differentiate between homogeneous and inhomogeneous conductivities inside the domain. Power distinguishability is defined as the measured power change between the homogeneous and inhomogeneous cases, divided by the power applied in homogeneous case.

OPTIMAL CURRENT PATTERN (Front Points)

-15 -10 -5 0 5 10 15

-15

-10

-5

0

5

10

15

e1

e2

e3

e4e5e6

e7

e8

e9

e10

e11

e12 e13

e14

e15

e16

1. Trigonometric method with first 2 modes of cosine and sine (4 injections; 5 EKF states with repeated use of the first cosine)2. Opposite method with e1-e9 and e5-e13 pairs (2 injections; 5 states with repeated use of e1-e9, e5-e13, e1-e9)3. Cross method with e3-e7, e5-e13 pairs (2 injections; 5 states with repeated use of e3-e7, e5-e13, e3-e7)4. Opposite method with e3-e11, e7-e15 pairs (2 injections; 5 states with repeated use of e3-e11, e7-e15, e3-e11)5. Opposite method with e3-e11, e7-e15, e5-e13 pairs (3 injections; 5 states with repeated use of e3-e11, e7-e15).

1 2 3 4 5

1% Noise

18

a) Bubble moving with constant velocity b) Bubble expanding with constant velocity c) Bubble moving with constant acceleration d) Bubble expanding with constant acceleration

Solid Line: True BoundaryDotted Line: Estimated Boundary

Solid Line: Kinematic ModelDotted Line: Random-Walk Model

KINEMATIC MODEL RESULTS (Fourier Coefficients)6-Fourier Coefficients, Contrast Ratio of 1:106 , Moving every current Pattern

19

1 ,...kQ

T.U

M.U

EKF12 ,...kQ

T.U

M.U

EKF23 ,...kQ

T.U

M.U

EKF3

IMM SCHEME (Fourier Coefficients) 1/3

1 1 1 11 1 1 1|k k k k ke y H

2 2 2 21 1 1 1|k k k k ke y H

3 3 3 31 1 1 1|k k k k ke y H

1 1 1 11 1 1| 1 1( )Tk k k k k kS H P H

2 2 2 21 1 1| 1 1( )Tk k k k k kS H P H

3 3 3 31 1 1| 1 1( )Tk k k k k kS H P H

1 1 1 11 1 1 1

11

1 1exp[ ( ) ( ) ]

22 | |

Tk k k k

k

L e S eS

2 2 2 2

1 1 1 12

1

1 1exp[ ( ) ( ) ]

22 | |

Tk k k k

k

L e S eS

3 3 3 3

1 1 1 13

1

1 1exp[ ( ) ( ) ]

22 | |

Tk k k k

k

L e S eS

11 1 1

1 1 2 31 1 1 2 1 3

kk

k k k

L c

L c L c L c

22 1 2

1 1 2 31 1 1 2 1 3

kk

k k k

L c

L c L c L c

33 1 3

1 1 2 31 1 1 2 1 3

kk

k k k

L c

L c L c L c

*As error decreases, modelling probability increases

20

IMM SCHEME (Fourier Coefficients) 2/3

11k 2

1k 3

1k

1k 2

k 3k

1|1 1|2 1|3, ,k k k 2|1 2|2 2|3, ,k k k 3|1 3|2 3|3, ,k k k 31 32 33, ,

21 22 23, ,

EKF2 EKF3EKF1

11 12 13, ,

Transition Probability

| 1i j iij kk

jc

1

Mi

j ij ki

c

0 |||

1

Mj i ji

k kk k ki

0 0 0 |

| | || | |1

[ ( )( ) ]M

j j j i ji i i Tk k k k k kk k k k k k k

i

P P

11| 1

11| 1

k k

k kP

21| 1

21| 1

k k

k kP

31| 1

31| 1

k k

k kP

Predefined

Mixing of estimates and error covariances

21

IMM SCHEME (Fourier Coefficients) 3/3

Interacting/Mixing of the Estimates

Filter 1 Filter M

Linearization

State Estimation Combination

Model Probability

Update

1|k k 1

|k kP |Mk k |

Mk kP

01|k k 01

|k kP 0|M

k k 0|M

k kP

11ke

11kS

11| 1k k

11| 1k kP 1| 1

Mk k 1| 1

Mk kP

11 1

Mk k

11

1

k

Mk

1ky

1| 1k k

1| 1k kP

1Mke 1

MkS

1ku

One-cycle flow diagram of the inverse solver with the IMM scheme.

EKF1

EKF2

EKF3

1k

2k

3k

22

5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Patterns

Mod

el P

roba

bilit

y k

EKF1

EKF2

EKF3

EKF1 EKF2 EKF3 IMM

IMM SCHEME RESULTS (Fourier Coefficients)6-Fourier Coefficients, Contrast Ratio of 1:106, Moving after 8 current patterns

23

++

+ + + ++

+++

++

+ + + ++

+++

++

+ + + ++

++

+

++

+ + + ++

++

+

++

+ + + ++

+++

++

+ + + ++

+++

++

+ + + ++

+++

++

+ + + ++

+++

++

+ + + ++

++

+

+ +

+ +

++

+ +

+ +

++

+ +

+ +

++

+ ++ +

++

+ ++ +

++

+ ++ +

+++ +

+ ++

++

+ ++

+

+

+ +

++

+

+

+

++ ++ +

+ +

++

++ +

+++

+ ++

++

+ ++

+++ +++

++ +

++ ++

++ ++ ++++ ++++++ +++

+++ +

++++ +

++ + ++

+++ +

+++ ++

++

+++ +

+++ ++++

+++

+++ ++

++ +

Actual (Sampling)

true mean

mean

covariance

Linearized (EKF) UT

UT mean

UT covariance

sigma points

transformed sigma points

EKF mean

EKF covariance

UNSCENTED TRANSFORM

An example of unscented transform for mean and covariance propagation: a) actual; (b) first-order linearization (EKF); and (c) unscented transform

(a) (b) (c)

( )f ( )f

( )if

24

UNSCENTED KALMAN FILTER (1/4)

0 0ˆ ˆ[ ] [ ]a a T TE x x x 0 0

0

0 0 0 0 0

0 0

ˆ ˆ[( )( ) ] 0 0

0 0

a a a a a TE

P

P Q

R

x x x x

1 1 1 1 1 1ˆ ˆ ˆ( ) ( )a a a a a ak k k k k kM M

P Pχ x x x

Generate 2n+1 sigma points where n is the size of augmented vector

1ˆ ak x

Each point is the augmented vector

| 1 1 1.x xk k k k

Iχ χ χ

Run the state equation

2( )

, | 10

ˆM

mk i i k k

i

W

x χ

2( )

, | 1 , | 10

ˆ ˆ[ ][ ]M

c x x Tk i i k k k i k k k

i

W

P χ x χ x

Calculate predicted mean and covariance

State Space Model: 1 1.k k k Ix x w( )k k k kV U x v

[ ]Tk kE Qw w

[ ]Tk kE Rv v

[ ] 0kE w[ ] 0kE v

25

| 1 | 1 1( )xk k k k k kV

ψ χ χ2

( ), | 1

0

ˆM

mk i i k k

i

W

U ψ

Run the measurement equation and find the mean

Time update complete

Create covariance matrices2

( ), | 1 , | 1

0

ˆ ˆ[ ][ ]k k

Mc T

i i k k k i k k kU Ui

W

P ψ U ψ U

2( )

, | 1 , | 10

ˆˆ[ ][ ]k k

Mc T

x U i i k k k i k k ki

W

P χ x ψ U

The sigma points should move towards the mean and at the same time, the sigma points on x domain should move towards the mean

ˆkU

ˆkx

UNSCENTED KALMAN FILTER (2/4)

26

Calculate the gain and update the estimates and error covariance matrices

UNSCENTED KALMAN FILTER (3/4)

1k k k k

k x U U UΚ P P

ˆˆ ˆ ( )k k k k kU Κx x U

k k

Tk k k kU U

P P Κ P Κ

Actual measurement

True value

( )0 /( )mW M

( ) 20 /( ) (1 )cW M ( ) ( ) 1/{2( )}m ci iW W M

1,..., 2i M

Define weights

2M N L 2 ( )M M

where

Composite scaling parameter

Spread of sigma points, usually1e-3

Usually zero

Usually 2 for Gaussian distribution

Measurement update complete

27

ˆ ax Pa

.M

{ }aiχ

+ -

State Equation

Nonlinear Measurement Equation [FEM Forward Solver]

ˆ -x

P-

-U

{ }iψ

( ){ }miW

Weighted Mean

( ){ }ciW

Weighted Covariance

UUP

xUP

Kalman Gain

ˆ ax aP

k=k+1

Uk

k

UNSCENTED KALMAN FILTER (4/4)

Block diagram of unscented Kalman filter for phase boundary estimation

28

solid line : true, dotted line : EKF, dashed line : UKF

Phantom

Plastic Target

UKF RESULTS (Fourier Coefficients)32 Electrodes, 6-Fourier Coefficients, Contrast Ratio of 1:106, Moving after 6 current patterns

|| ||

|| ||estimated true

true

RMSE

29<EKF: -x- UKF : -o- >

2% Noise

3% Noise

Rippled surface

UKF RESULTS (Front Points)10-Front Points, Contrast Ratio of 1:100, moving every current pattern

30

1 1.k k k Ix x w( )k k k kV U x v

M

( )mow M

( ) 2(1 )cow M

( ) ( ) 1 1,..., 2

2( )m ci iw w i M

M

2 ( )M M

0 1kx 11 ( ) 1,...,

ki k x ix i M P

11 ( ) 1,..., 2ki k x ix i M M P

*( ) . iik I

2*( )( )

0

ˆM

imk i k

i

x w

2

*( ) *( )( )

0

ˆ ˆ ˆ[ ][ ]k

Mi ic T

x i k kk ki

w x x

P Q

ˆ ˆˆ ˆ ˆk kk k k x k xx x x

P P

2( )( )

0

ˆ ( )M

imk i k k

i

U w V

2

( ) ( )( )

0

ˆ ˆ ˆ[ ( ) ][ ( ) ]k

Mi ic T

U i k k k kk ki

w V U V U

P

2( ) ( )( )

0

ˆ ˆˆ[ ][ ( ) ]k k

Mi ic T

x U i k kk ki

w x h U

P

1 1

1

ˆ ˆˆ ( ) ( ( )

ˆ ˆ ˆ ( )) 1, 2,...

k k k

k k k

j jk x U U kk k

jTx U U k k

x x U V x

x x j

P P R

P P

State Equation

1ˆ ˆ ˆ ˆ( )k k k k k k k

Tx x x U U x U

P P P P R Pˆ jk kx x

Gauss-Newton Measurement Update

GAUSS-NEWTON UNSCENTED KALMAN FILTER

Offline

Online

kU

State Space Model

[ ]Tk kE Qw w

[ ]Tk kE Rv v

[ ] 0kE w[ ] 0kE v

31

2% Noise1% Noise 3% Noise

GNUKF RESULTS (Front Points)

32

-Analytical Jacobian used

-successful till 16 Frontpoints

-Contrast ratio of 1:10000

-3% Relative Noise

-Current patterns reduced to 4 / target remains static with 16 electrodes configuration based on distinguishability analysis for EKF

-Extended Kalman Filter and Unscented Kalman filter (recent) formulation for online monitoring

-Gauss Newton Unscented Kalman filter formulation for improvement over unscented Kalman filter

-With Unscented Kalman Filter and Gauss Newton Unscented Kalman Filter, image reconstruction using 1 current pattern is also possible.

Front points (open boundary)Fourier coefficients (close boundary)-Analytical Jacobian used

-6 coefficients to represent an elliptic object, can go for more, however, higher coefficients are quite sensitive

-Contrast ratio of 1:1000000

-3% Relative Noise

-Current patterns reduced to 6 / target remains static in experiments with 32 electrodes configuration.

-Extended Kalman Filter and Unscented Kalman filter (recent) formulation for online monitoring

-Interacting Multiple Model Scheme for time-varying process noises

-Kinematic models (velocity, acceleration) done for movement of air bubbles, void fractions

RESEARCH MILESTONES

33

Any Questions?

34

APPENDIX: Derivation of Jacobian 1/10In some cases, the voltages are measured only at some selected electrodes, not every electrode. Also, the selected electrodes may be different at each current pattern. The measured voltages at the measurement electrodes can be obtained asU

PEThT NβMUMU

where, is the number of the measurement electrodes and is the measurement matrix. The element is set to ‘1’ if the -th electrode is measured at the -th current pattern and otherwise set to zero. Furthermore, can be extracted directly from by introducing the extended mapping matrix

E ELM),( pM p

hU bN~

)1(),(~ LNLN0N and bNU

~h

where NL0 . Therefore, we have

bMbNMUMU~~ˆ ThT

where the extended measurement matrix is defined as

)1(~~ LNETNMM

If the pseudo-resistance matrix defined as

ELNT )1(1 ~~MAR or TMRA

~~

is given we can calculate the Jacobian matrix. The pseudo-resistance matrix can be easily obtained during the solution of the system equation

IMbRA~~~ T

INMN

00

βR

αRA ˆ~

~

2

1TT

or where ENN :),:1(~~

1 RR and ( 1)2 ( 1: 1,:) L EN N L R R

35

APPENDIX: Derivation of Jacobian 2/10

1 1ˆ

TU A AMA A I R b

d d d

Jacobian:

Front Points Approach

11 2

2

ˆ 0

0 0 0

T

T T

B BRU

d dR Rd R

1

ˆTU B

Rd d

1 1 1 1

1 1 1 1 1 1 1

X Y Y XB B B B B

d X d Y d R X R Y

B B

d Y

Y XB B B

d R X R Y

2,..., 1

36

, 1

1( , ) . ,

r l

L

r i j i jA elr l u l

B i j d dSz

, 1, 2,...,i j N

0

( , ) ( , )limd

B X X Y Y B X YB

d d

0 | ( )

( )lim .

u mm i j

l ui jAd m supp

dd

APPENDIX: Derivation of Jacobian 3/10

Since we are considering the stratified flow of two immiscible liquids therefore, the matrix B will be

37

12

( ) ( ) ( , , )C x S x x X X

Assuming that the interface is represented by a set of linear piecewise interpolation functions:

1[ , ]x X X,

11 1

1

( ) ( )Y Y

S x x X YX X

unit pulse defined for 1[ , ]x X X

Any small perturbation of results in small perturbation in and in

d X X Y Y

21 2 1 2( ) ( ) ( , , ) ( )C x S x x X X O

21 1 1( ) ( ) ( , , ) ( ) ( , , ) ( )C x S x x X X S x x X X O

2,..., 1 21( ) ( ) ( , , ) ( )C x S x x X X O

1 1

1 1

( )Y Y x X

S x X YX X X X

1 11

1 1

( )Y Y X x

S x X YX X X X

where

1[ , ]x X X

1[ , ]x X X

APPENDIX: Derivation of Jacobian 4/10

38

0

1lim ( , )

u mAdf x y d

d

APPENDIX: Derivation of Jacobian 5/10Considering the interface for mesh crossing elements ( )m i jsupp

( , ) .i jf x y where

For a small perturbation in only and will changed ( )P x 1( )P x

0

1lim ( , )

u mAdf x y d

d 1

10

1lim ( , )

X C C

X Cdf x y dydx

d

The function can be expanded about the interface ( , )f x y ( )C x

2( , ) ( , ) ( ) ( )y C

ff x y f x C y C O

y

Finally, we have

0

1lim ( , )

u mAdf x y d

d

2

1

2 1 21 1

2 1 2 1

1( , )

X

X

Y Y X xY X f x C dx

R X X X X

1

1

1 1

1 1

( , ) ( , )X X

X X

x X X xf x C dx f x C dx

X X X X

1

1 1

1 1

1( , )

X

X

Y Y x XY X f x C dx

R X X X X

1

2,..., 1

39

(X k ,Y k )

(X k -1,Y k -1)

(X k +1,Y k +1)

(X k ,Y k+Y k )

P k (x)P k +1(x)

TY P E 1TY P E 2

TY P E 2

TY P E 3

TY P E 4

TY P E 4

TY P E 5

Five types of interface-crossing elements in case of an arbitrarily small perturbation of kY in kY

.

There are five types of interface-crossing elements when kY is perturbed by an arbitrarily small perturbation of kY . Assume that there are only two intersections of the interface and the mesh faces and the intersections

),( 11 yx and ),( 22 yx where 21 xx . Recalling that jiyxf ),(

the integration for each type will be evaluated as

are denoted as is constant in a certain mesh,

TYPE 1: 1

212

1

1

2

),(),(

2

1

kk

kx

X kk

k

XX

XxCxfdx

XX

XxCxf

k

1

121112

1

1

2

),(),(

2

1

kk

kkx

x kk

k

XX

xxXxXxCxfdx

XX

XxCxfTYPE 2:

kk

kkkk

kk

kkkk

x

X kk

kX

x kk

k

XX

XxXXxX

XX

xXXxXXCxf

dxXX

xXdx

XX

XxCxf

k

k

1

2121

1

1111

1

1

1

1

2

),(

),(2

1TYPE 3:

kk

kkx

x kk

k

XX

xxxXxXCxfdx

XX

xXCxf

1

121121

1

1

2

),(),(

2

1

TYPE 4:

kk

kX

x kk

k

XX

xXCxfdx

XX

xXCxf

k

1

211

1

1

2

),(),(

1

1

TYPE 5:

APPENDIX: Derivation of Jacobian 6/10

40

Fourier Coefficients Approach

IA

MIAMb

MU ~~~~~ˆ 1

1

kkkknnnn

yxkn ,,

bA

AIAA

AIA

kkknnn

1111 ~~

bA

MAbA

MAbA

AMU

kkkkn

TT

n

TTT

nn

~~)(

~ˆ111

The derivative of the stiffness matrix with respect to the coefficient is

00

0B

Ak

k nn

)(|

0

0

)(lim

jim mk

k m Aji

kxn

d

supp

B

APPENDIX: Derivation of Jacobian 7/10

41

In order to obtain the Jacobian, now, let us consider the evaluation of the expression

mkA

dyxf

),(1

lim0

We define a new coordinate system where is the positively oriented coordinate along the closes curve , and is the coordinate outward normal from the region

),( ps s kC pkA

)(

)(

)(

)(

)(

)(

sy

sx

s

sp

sy

sx

s

p

k

k

The perturbed boundary will bekC~

)(~

)(~)(~

)(

)(

)(

)(sC

sy

sx

s

s

sy

sxs k

k

k

k

k

s1s2

C k

C k~

s

pTherefore,

2

1 000 ),(

),(),(

1),(

1limlim

s

ss pA

dpdssp

yxyxfdyxf

mk

APPENDIX: Derivation of Jacobian 8/10

42

The Jacobian for the transformation of the coordinate will be

pxpy

ds

dp

ds

dyds

dp

ds

dx

s

y

p

ys

x

p

x

sp

yx

),(

),(

The function can be expanded about the boundary ),( yxf kC

)(),0(),(),( 2

0

pOpp

fsfspfyxf

p

We have

2

1

2

1

),0(

)(),0(1

),(1

02

00

0

lim

lim

s

ss

s

ss pp

A

dsxysf

dpdspxpypOpp

fsf

dyxf

mk

APPENDIX: Derivation of Jacobian 9/10

43

In this, is evaluated at the boundary . When differentiating with respect to , that is perturbing , we have and .On the other hand when differentiating with respect to , we have and . Finally, the derivative of the matrix with respect to the coefficients becomes

),0( sf kCkx

n )(sxn )(sx

n 0 ky

n 0 )(syn

)()(|,0

2

1)()()(

jikmkk

CBm

s

sxnkCyxjikx

n

dsssy

supp

B

)()(|,0

2

1)()()(

jikmkk

CBm

s

synkCyxjiky

n

dsssx

supp

B

where

0|)( kmmk CCBdenotes the set of elements crossing kC

If , and constant in each element, we have ji )(sxk )(syk

)()(| 12

120

2

1)(

)()()(

jikmmk CBm

s

sxnjikx

n

dssss

sysy

supp

B

)()(| 12

120

2

1)(

)()()(

jikmmk CBm

s

synjiky

n

dssss

sxsx

supp

B

APPENDIX: Derivation of Jacobian 10/10