12_PhdDefense

Isogeometric Analysis and Shape Optimization

in Electromagnetism

Nguyen Dang Manh

Department of Mathematics, Technical University of Denmark

.

Supervisors: Jens Gravesen, Anton Evgrafov, Allan R. Gersborg (Feb 2009-Jun 2010).

Introduction Hearing the shape of a drum Magnetic resonators An iterative procedure Conclusion

Collaborators Sergey I. Bozhevolnyi, Morten Willatzen and their group at University of SouthernDenmark;

Domenico Lahaye and his group at Delft University of Technology.


Outline

1 Introduction

2 Hearing the shape of a drum

3 Shape optimization of sub-wavelength antennas

4 An iterative procedure for shape optimization using isogeometricanalysis

5 Conclusion


Motivation: Electromagnetic field enhancement

TaskMaximize the field strength of an incoming uniform plane wave in a specific region



Inspiration[N. Aage, N. Mortensen, O. Sigmund, 2005] suggested the use of two symmetriccopper antennas.


Goal: What is the best" shape?

Should the shapes of the antennas be:




0 0.5 1

Y

XZ

An earlier design[N. Aage et al., 2010]




0 0.5 1

Y

XZ

or or?

An earlier design[N. Aage et al., 2010]


Method: Systematic design

4 2 0 2 44

3

2

1

0

1

2

3

4



4 2 0 2 44

3

2

1

0

1

2

3

4

-Numerical method



4 2 0 2 44

3

2

1

0

1

2

3

4

-Numerical method

?

Gradient-driven optimization



4 2 0 2 44

3

2

1

0

1

2

3

4

-Numerical method

?


Optimal shape



4 2 0 2 44

3

2

1

0

1

2

3

4

-Numerical method

Isogeometric analysis

?


Optimal shape


Isogeometric analysis: B-splines

B-splines are piecewise polynomials of a degree p, typically continuously differentiableup to p 1 times at joint points,


Isogeometric analysis: Basis function

Define a basis function on a physical domain Parametrize the physical domain: F Pull F back to the parameter domain: F1

Compose the inverse map F1 with a bivariate B-spline:





-F





F1





F1

R

7

Rk





F1

R

7

Rk

SSSSSSSo k=F

1Rk


Isogeometric analysis Approximate a physical quantity

f : R

Utilize the basis functions defined in

k : R

Approximation of the function f is

fh = c11 + c22 + . . .+ cNN ,

where c1, c2, . . . , cN are unknowns.

nothing but for balancing the spacenothing but for balancing the spacenothing but for balancing the spacenothing but for balancing the space


Finite element analysis (FEA)

Triangulate the domain . The boundary is not exact. Use piecewise linear functions as basisfunctions (or higher order polynomials).

Globally only C0.

Isogeometric analysis (IGA)

Parametrize the domain . The boundary is exact. Use tensor product B-splines of degree nas basis functions".

Globally Cn1 for a simply connecteddomain.



Triangulate the domain .

The boundary is not exact. Use piecewise linear functions as basisfunctions (or higher order polynomials).

Globally only C0.


x

Parametrize the domain .

The boundary is exact. Use tensor product B-splines of degree nas basis functions".




Triangulate the domain . The boundary is not exact.

Use piecewise linear functions as basisfunctions (or higher order polynomials).

Globally only C0.


x

Parametrize the domain . The boundary is exact.

Use tensor product B-splines of degree nas basis functions".




Triangulate the domain . The boundary is not exact. Use piecewise linear functions as basisfunctions (or higher order polynomials).

Globally only C0.


Parametrize the domain . The boundary is exact. Use tensor product B-splines of degree nas basis functions".



Outline

1 Introduction




5 Conclusion


History


History

1966, Mark KacCan one hear the shape of a drum?


History

1992, Gordon, C.; Webb, D.; Wolpert, S.One can not hear the shape of a drum.


History

In 1972, Hutchinson-Niordson: "hear" the shape of a "harmonic" drum.Harmonic drum: The ratio of the first four eigenfrequencies:

f1 : f2 : f3 : f4 = 2 : 3 : 3 : 4

@@@I

play C@

@@@

@@I

G@@

@@@

@@I

C


Related workIn 1972, Hutchinson-Niordson:

In 1995, Kane-Schoenauer used ge-netic algorithms:


Related workIn 1972, Hutchinson-Niordson: In 1995, Kane-Schoenauer used ge-

netic algorithms:


Our work

Method Systematic gradient driven method:fmincon (MATLAB), IPOPT

Isogeometric analysis + shapeoptimization

Issues Non-unique solutions Double eigenvalues

Results

Harmonic drums Other drums

( Hutchinson-Niordson)

How??


Our work




Results

Harmonic drums

Other drums


How??


Our work




Results

Harmonic drums Other drums


How??


Harmonic drum: Shape optimization problem

Given prescribed eigenvalues 0k , k = 1, . . . , N .

minimizeboundary control points

shape perimeter

s.t.

k = 0k if

0k has multiplicity one,

k + k+1 =

0k +

0k+1

kk+1 = 0k

0k+1

if 0k = 0k+1

with Kuk = kMuk

for k = 1, . . . , N

triple control points at corners are colinear.


Harmonic drum: A first try

?: Wrong sensitivities when 2 3.Reason: 2 and 3 are NOT dif-

ferentiable when 2 3.



?: Wrong sensitivities when 2 3.

Reason: 2 and 3 are NOT dif-ferentiable when 2 3.



?: Wrong sensitivities when 2 3.Reason: 2 and 3 are NOT dif-

ferentiable when 2 3.


Double eigenvalues: Imposing a 120 symmetry

The multiplicities of 2, 3 do not change

fixed boundaryhorizontal parameter boundary correspondencehorizontal parameter line correspondencesvertical parameter boundary correspondencevertical parameter line correspondences

free boundary control pointfree corner control pointinner control pointfixed boundary control pointfixed corner control point


Double eigenvalues: Using the functions 2 + 3 and 23

Proposition:C(t): a smooth family of sym. matrices.Eigenvalues: 1(t) < 2(t) 3(t) < 4(t). If w1 and w2 areeigenvectors with eigenvalues 2(t) and 3(t), then we have

(2 + 3) = w1, C w1+ w2, C w2

(23) = 3w1, C w1+ 2w2, C w2.


Harmonic drum: Optimization with correct sensitivity


Harmonic drum: Validating the resulting B-spline parametrization

Jacobian determinant = det(J) =M,N

k,`=1 ck,`M2p1k (u)N

2q1` (v),

If the coefficients ck,` > 0 = the determinant > 0

positive control pointnegative control pointcorner control point

0

5

10

15

@@@I


0

5

10

15

after refinement


Harmonic drum: Further steps

initial-

minimizeperimeter

shape: 91 control net

-

opt.eigs

shape: 91 control net

-

refine+opt. eigs

-after 20 iterations

control net


Harmonic drum: Improve the B-spline parametrization

Original Max.mink,`(ck,`)

Min.Winslow func.


0

5

10

15

20


0

5

10

15

20


0

5

10

15

20

@@I


Harmonic drum: Results

Maximize the minimal coefficients Minimize the Winslow functional


Eigenmodes

Mode 1 Mode 2

Mode 4 Mode 7


Outline

1 Introduction




5 Conclusion


Motivation: Concentration of the magnetic energy

H = (0, 0, ejk0x)


Motivation: Concentration of the magnetic energy

H = (0, 0, ejk0x) maximize H2


Motivation: Why do we use shape optimization?

0 0.5 1

Y

XZ 0 0.5 1

Y

XZ

TopOpt attempt: Aage, Mortensen, Sigmund, IJNME, 2010.

Fabrication: uncertain boundary. Analysis: domain dependence. Shape optimization overcomes the issues.



0 0.5 1

Y

XZ 0 0.5 1

Y

XZ

TopOpt attempt: Aage, Mortensen, Sigmund, IJNME, 2010. Fabrication: uncertain boundary.

Analysis: domain dependence. Shape optimization overcomes the issues.



0 0.5 1

Y

XZ 0 0.5 1

Y

XZ

TopOpt attempt: Aage, Mortensen, Sigmund, IJNME, 2010. Fabrication: uncertain boundary. Analysis: domain dependence.

Shape optimization overcomes the issues.



0 0.5 1

Y

XZ 0 0.5 1

Y

XZ

TopOpt attempt: Aage, Mortensen, Sigmund, IJNME, 2010. Fabrication: uncertain boundary. Analysis: domain dependence. Shape optimization overcomes the issues.


Shape optimization problem: IGA design

Shape optimization problem

maximize"boundaries" of the scatterers

Wm =

W

|Hz|2 dV W

How to parametrize the physical domain and what should W be such that Wm andits sensitivity can be easily computed?

Our choice: Freeze parametrization inthe energy harvesting area and choose the image of one knot span as W .





Wm =

W

|Hz|2 dV W


Our choice: Freeze parametrization inthe energy harvesting area

and choose the image of one knot span as W .





Wm =

W

|Hz|2 dV W


Our choice: Freeze parametrization inthe energy harvesting area and choose the image of one knot span as W .


Shape optimization problem: IGA knot vectors

Knot vectors for design variables

Knot vectors for para.

and for parametrization

and for analysis

: Design variable

Analysis mesh lines

: Fixed control points

(local refinement")

,: inner control points


Shape optimization problem: IGA knot vectors

Knot vectors for design variables Knot vectors for para.and for parametrization and for analysis

: Design variable Analysis mesh lines: Fixed control points (local refinement"),: inner control points


Shape optimization problem: Without regularity conditions

maximize ck

log10(Wm) = log10(

W

|Hz|2 dV )

s.t. yck d0

Vol. =1

2

Cdet(r, r) d r20

c, = (MN), dr,t(u, v) 0

Metallicscatterers

Wk

H

E

.

sr d

air

: Design variable


, : Inner control points


Shape optimization: A first try



|Hz|2



|Hz|2

Regularity conditions



|Hz|2

Regularity conditions Constraining the scatterers volume,



|Hz|2

Regularity conditions Constraining the scatterers volume, Preventing the scatterers boundary from self intersection.


Shape optimization problem: With regularity conditions

maximize ck

log10(Wm) = log10(

W

|Hz|2 dV )

s.t. yck d0

Vol. =1

2

Cdet(r, r) d r20

c, = (MN), d2r,t(u, v) 0, d2r,t(ui, vj) 2

Metallicscatterers

Wk

H

E

.

sr d

air

: Design variable


, : Inner control points


Shape optimization process


Results


Magnetic energy vs. frequency

90 100 115 120 130 140

4

3

2

1

0

1

2

3

4

5

6

frequency (MHz)

ener

gy

optimizing frequency

114.96 115115 115.02 115.041

0

1

2

3

4

5

6

frequency (MHz)en

ergy

optimizing frequency

log10(W)

log10(W)

log10(W)

log10(W)

log10(W) of Aage et al.9

log10(W) of Aage et al.9

When comparing note that in top. opt. (Aage et al.), the two antennas areconfined in two circular domains only.

The model corresponding to the peak has the energy at a factor of one milliontimes better than the top. opt. result.


Results: Resonance

Eigen-problem

Perfect electric conductor

Hz=0


Results: Resonance

Eigenmode

Solution


Results: Resonance

Eigenmode Solution

f13 = 1.1498 108 [Hz] f = 1.15 108 [Hz]


Results: Resonance

Eigenmode Solution

f13 = 1.1498 108 [Hz] f = 1.15 108 [Hz]

98% of the L2-energy of the solution is contained in the mode 13.


Nano-antenna: Motivation

H = (0, 0, ejk0x) maximize H(field enhancement)


Nano-antenna: IGA setup

Multiple patch layout Control net

4 2 0 2 41

0

1

2

3

4

5

F2F4

F1

F3

F5

4 2 0 2 41

0

1

2

3

4

5

Solving inside the antennas,


Nano-antenna: IGA setup

Multiple patch layout Control net

4 2 0 2 41

0

1

2

3

4

5

F2F4

F1

F3

F5

4 2 0 2 41

0

1

2

3

4

5

Solving inside the antennas, In nano-scale.


Theory

log10(|Hz/H0|2

)

The antennas are infinitely long. Incoming wave propagates in thex-direction (left to right).

The surrounding material is AIR(r = 1).

MeasurementOptimized antennas

Antenna gap

40 nm

log10(|Hz/H0|2

)

The antennas are 1/7.5-wavelength long. Incoming wave propagates in thez-direction (outward from the slide).

The surrounding material is OIL(r = 2.1025).


Theory

log10(|Hz/H0|2

)

The antennas are infinitely long.

Incoming wave propagates in thex-direction (left to right).


Measurement

log10(|Hz/H0|2

)

The antennas are 1/7.5-wavelength long.

Incoming wave propagates in thez-direction (outward from the slide).



Theory

log10(|Hz/H0|2

)



Measurement

log10(|Hz/H0|2

)




Theory



Measurement




Theory

800 1000 1200 1400 1600

0

1

2

3

4

5

6

wavelength [nm]

log 1

0 of

nor

mal

ized

mag

netic

ene

rgy

FEM: p=6; Q = 3645.35FEM: p=7; Q = 3644.71



Measurement

500 600 700 800

125

150

175

200 x50_ypol_D12_40nm_gl_oil

x50_ypol_D12_56nm_gl_oil


RR

el_

gl

(%)

Wavelength (nm) Raw data

Reflection from array relative to glass, y-polarized Gap=40, 56, 72 nm

y

NB! No significant difference observed between the three different gap sizes! The antennas are 1/7.5-wavelength long. Incoming wave propagates in thez-direction (outward from the slide).



Theory

800 1000 1200 1400 1600

0

1

2

3

4

5

6

wavelength [nm]

log 1

0 of

nor

mal

ized

mag

netic

ene

rgy

FEM: p=6; Q = 3645.35FEM: p=7; Q = 3644.71



Measurement

500 600 700 800

125

150

175

200 x50_ypol_D12_40nm_gl_oil



RR

el_

gl

(%)

Wavelength (nm) Raw data

Reflection from array relative to glass, y-polarized Gap=40, 56, 72 nm

y

NB! No significant difference observed between the three different gap sizes! The antennas are 1/7.5-wavelength long. Incoming wave propagates in thez-direction (outward from the slide).


Try it again!


Outline

1 Introduction




5 Conclusion


A test problem

Let Az be the solution to

Az = 0 in D

Az = c on b

Az = u0 cos(x

)ey on l,r,t


minimize

D

(B2x

)2dV

whereB = (Az

y,Azx

, 0)

space

t

l

r

b

D

design control pointfixed control pointlinear generated control point


Control points of the Jacobian determinant

Determinant of the Jacobian of F

det(J) =

m,ni,j=1

m,nk,`=1

det[di,j , dk,`]dMpi (u)

duNqj (v) M

pk (u)

dNq` (v)

dv

Spline representation

det(J) =

M,Nk,`=1

ck,`M2p1k (u)N2q1` (v)

ck,` = dTQk,`d

test


Shape optimization using IGA: An iterative procedure

Shape optimization problemminimizedd f(d)

Iterative process Start with a guess d0 int(d).

Assume: d(d0) = Ad0d0 + Bd0corresponds to a valid parametrization,and

ck,`(d0) .

space

d0.ck,l


Shape optimization using IGA: An iterative procedure

Shape optimization problemminimizedd f(d)

Iterative process Start with a guess d0 int(d). Assume: d(d0) = Ad0d0 + Bd0corresponds to a valid parametrization,and

ck,`(d0) .

space

d0.ck,l


Iterative procedure: Find optima in a neighborhood of d0

We find d1 as the solution to the problem

minimizedd

f(d),

such that ck,`(d) .

with initial variable d0.

space

d0.ck,l


Iterative procedure: Find optima in a neighborhood of d0

We find d1 as the solution to the problem

minimizedd

f(d),

such that ck,`(d) .


space

d0.ck,l d0. .d1

ck,l


Iterative procedure: Repeat the first step

Update the reference parametrization

Find d2 as the solution to the problem

minimizedd

f(d),

such that ck,`(d) .


space

d0. .

d1

ck,l



Update the reference parametrization Find d2 as the solution to the problem

minimizedd

f(d),

such that ck,`(d) .


space

d0. .

d1

ck,l



Update the reference parametrization Find d2 as the solution to the problem

minimizedd

f(d),

such that ck,`(d) .


space

d0. .

d1

.d2


Iterative procedure: Validate the resulting shape

initial shape

optimized shape

analytical shape


Iterative procedure: Validate the resulting shape

initial shapeoptimized shapeanalytical shape

best L2approximation of analytical shape


Iterative procedure: Advantages

FlexibilityParametrization extension methods (Ad0 and Bd0) that can be used in the iterativeprocess Mean value coordinate; Linearized Winslow functional; (Using the same inner control points).




Advantages

The Jacobian Impossible domains

The Jacobian and impossible domains

The Jacobian of the parametrisation is J = (xu, xv ) and the determinant is

det J = det(xu, xv ) =

xu

xv

yu

yv

.

We need det J > 0.

Both partial derivatives are determined in the corners so det J isdetermined too. Thus there are impossible domains.

+

+ +

Jens Gravesen (DTU Mathematics) Parametrisation in Isogeometric Analysis Dagstuhl, 23 May 2011 9 / 22(impossible domain, Jens Gravesen)




Advantages Containing constraints on the validity"of patch angles

Containing constraints on the localregularity of the domain boundaries.

The Jacobian Impossible domains

The Jacobian and impossible domains

The Jacobian of the parametrisation is J = (xu, xv ) and the determinant is

det J = det(xu, xv ) =

xu

xv

yu

yv

.

We need det J > 0.

Both partial derivatives are determined in the corners so det J isdetermined too. Thus there are impossible domains.

+

+ +

Jens Gravesen (DTU Mathematics) Parametrisation in Isogeometric Analysis Dagstuhl, 23 May 2011 9 / 22(impossible domain, Jens Gravesen)


Outline

1 Introduction




5 Conclusion


Conclusions

Isogeometric analysis (IGA) fits well with shape optimization.

Shape optimization using IGA is recommended over finite element method-basedshape optimization.

Throughout our study, traditionally unphysical restrictions on the variations ofdesign variables have been significantly avoided.

Shape optimization using isogeometric analysis is a promising design tool forrealistic applications in electromagnetics.


Future work

Re-do the shape optimization of the nano-antennas with correct electromagneticparameters.

Incorporate a local refinement method for isogeometric analysis for the current code.The combination is expected in the design problem of the nano-antennas.

A 3D shape optimization problem of the nano-antennas may be considered.


PublicationsThe following research papers and manuscripts have been writtenduring the three years of the Ph.D. study1 Nguyen D. M., A. Evgrafov, A.R. Gersborg, J. Gravesen, Isogeometric shapeoptimization of vibrating membranes, Computer Methods in Applied Mechanicsand Engineering, vol. 200, pp. 1343-1353, 2011.

2 J. Gravesen, A. Evgrafov, Nguyen D. M., On the sensitivities of multipleeigenvalues, Structural and Multidisciplinary Optimization, vol. 44, pp. 583-587,2011.

3 Nguyen D. M., A. Evgrafov, J. Gravesen, Shape optimization of sub-wavelengthantenna using isogeometric analysis, in submission.

4 Nguyen D. M., A. Evgrafov, J. Gravesen, D. Lahaye, Systematic designs ofmagnetic density separators using isogeometric analysis and shape optimization, inpreparation.


Acknowledgments

2:3:3:4(CGC) 4:5:5:6(CEG)

Jens Gravesen, Anton Evgrafov, Allan Roulund Gersborg, Peter Nrtoft Nielsen,Niels Aage

Sergey I. Bozhevolnyi, Morten Willatzen, University of Southern Denmark

Domenico Lahaye, Delft University of Technology

THANK YOU!


Acknowledgments

2:3:3:4(CGC) 4:5:5:6(CEG)

Jens Gravesen, Anton Evgrafov, Allan Roulund Gersborg, Peter Nrtoft Nielsen,Niels Aage

Sergey I. Bozhevolnyi, Morten Willatzen, University of Southern Denmark

Domenico Lahaye, Delft University of Technology

THANK YOU!

Introduction

Hearing the shape of a drum

Shape optimization of sub-wavelength antennas

An iterative procedure for shape optimization using isogeometric analysis

Conclusion

0.0: 0.1: 0.2: 0.3: 0.4: 0.5: 0.6: 0.7: 0.8: 0.9: 0.10: 0.11: 0.12: 0.13: 0.14: 0.15: 0.16: 0.17: 0.18: 0.19: 0.20: 0.21: 0.22: 0.23: 0.24: 0.25: 0.26: 0.27: 0.28: 0.29: anm0: 1.0: 1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 1.8: 1.9: 1.10: 1.11: 1.12: 1.13: 1.14: 1.15: 1.16: 1.17: 1.18: 1.19: 1.20: 1.21: 1.22: 1.23: 1.24: 1.25: 1.26: 1.27: 1.28: 1.29: anm1: 2.0: anm2: 3.0: 3.1: 3.2: 3.3: 3.4: 3.5: 3.6: 3.7: 3.8: 3.9: 3.10: 3.11: 3.12: 3.13: 3.14: 3.15: 3.16: 3.17: 3.18: 3.19: 3.20: 3.21: 3.22: 3.23: 3.24: 3.25: 3.26: 3.27: 3.28: 3.29: anm3: 4.0: 4.1: 4.2: 4.3: 4.4: 4.5: 4.6: 4.7: 4.8: 4.9: 4.10: 4.11: 4.12: 4.13: 4.14: 4.15: 4.16: 4.17: 4.18: 4.19: 4.20: 4.21: 4.22: 4.23: 4.24: 4.25: 4.26: 4.27: 4.28: 4.29: anm4: 5.0: anm5: 6.0: anm6: 7.0: 7.1: 7.2: 7.3: 7.4: 7.5: 7.6: 7.7: 7.8: 7.9: 7.10: 7.11: 7.12: 7.13: anm7: 8.0: 8.1: 8.2: 8.3: 8.4: 8.5: 8.6: 8.7: 8.8: 8.9: 8.10: 8.11: 8.12: 8.13: anm8: 9.0: 9.1: 9.2: 9.3: 9.4: 9.5: 9.6: 9.7: 9.8: 9.9: 9.10: 9.11: 9.12: 9.13: 9.14: 9.15: 9.16: 9.17: 9.18: 9.19: 9.20: 9.21: 9.22: 9.23: 9.24: 9.25: 9.26: 9.27: 9.28: 9.29: anm9: 10.0: 10.1: 10.2: 10.3: 10.4: 10.5: 10.6: 10.7: 10.8: 10.9: 10.10: 10.11: 10.12: 10.13: 10.14: 10.15: 10.16: 10.17: 10.18: 10.19: 10.20: 10.21: 10.22: 10.23: 10.24: 10.25: 10.26: 10.27: 10.28: 10.29: anm10: 11.0: anm11: 12.0: 12.1: 12.2: 12.3: 12.4: 12.5: 12.6: 12.7: 12.8: 12.9: 12.10: 12.11: 12.12: 12.13: 12.14: 12.15: 12.16: 12.17: 12.18: 12.19: 12.20: 12.21: 12.22: 12.23: anm12: 13.0: anm13: 14.0: 14.1: anm14: 15.0: anm15: 16.0: anm16: 17.0: anm17: 18.0: anm18: 19.0: anm19: 20.0: 20.1: 20.2: 20.3: 20.4: 20.5: 20.6: 20.7: 20.8: 20.9: 20.10: 20.11: 20.12: 20.13: 20.14: 20.15: 20.16: 20.17: 20.18: 20.19: 20.20: 20.21: 20.22: 20.23: 20.24: 20.25: 20.26: 20.27: 20.28: 20.29: 20.30: 20.31: 20.32: 20.33: 20.34: anm20: 21.0: 21.1: 21.2: 21.3: 21.4: 21.5: 21.6: 21.7: 21.8: 21.9: 21.10: 21.11: 21.12: 21.13: 21.14: 21.15: 21.16: 21.17: 21.18: 21.19: 21.20: 21.21: 21.22: 21.23: 21.24: 21.25: 21.26: 21.27: 21.28: 21.29: 21.30: 21.31: 21.32: 21.33: 21.34: anm21: 22.0: 22.1: 22.2: 22.3: 22.4: 22.5: 22.6: 22.7: 22.8: 22.9: 22.10: 22.11: 22.12: 22.13: 22.14: 22.15: 22.16: 22.17: 22.18: 22.19: 22.20: 22.21: 22.22: 22.23: 22.24: 22.25: 22.26: 22.27: 22.28: 22.29: 22.30: 22.31: 22.32: 22.33: 22.34: anm22: 23.0: 23.1: 23.2: 23.3: 23.4: 23.5: 23.6: 23.7: 23.8: 23.9: 23.10: 23.11: 23.12: 23.13: 23.14: 23.15: 23.16: 23.17: 23.18: 23.19: 23.20: 23.21: 23.22: 23.23: 23.24: 23.25: 23.26: 23.27: 23.28: 23.29: 23.30: 23.31: 23.32: 23.33: 23.34: 23.35: 23.36: 23.37: 23.38: 23.39: 23.40: 23.41: 23.42: 23.43: 23.44: 23.45: 23.46: 23.47: 23.48: 23.49: 23.50: 23.51: 23.52: 23.53: 23.54: 23.55: 23.56: 23.57: 23.58: 23.59: 23.60: 23.61: 23.62: 23.63: 23.64: 23.65: 23.66: 23.67: 23.68: 23.69: 23.70: 23.71: 23.72: 23.73: 23.74: 23.75: 23.76: 23.77: 23.78: 23.79: 23.80: 23.81: 23.82: 23.83: 23.84: 23.85: 23.86: 23.87: 23.88: 23.89: 23.90: anm23: 24.0: 24.1: 24.2: 24.3: 24.4: 24.5: 24.6: 24.7: 24.8: 24.9: 24.10: 24.11: 24.12: 24.13: 24.14: 24.15: 24.16: 24.17: 24.18: 24.19: 24.20: 24.21: 24.22: 24.23: 24.24: 24.25: 24.26: 24.27: 24.28: 24.29: 24.30: 24.31: 24.32: 24.33: 24.34: 24.35: 24.36: 24.37: 24.38: 24.39: 24.40: 24.41: 24.42: 24.43: 24.44: 24.45: 24.46: 24.47: 24.48: 24.49: 24.50: 24.51: 24.52: 24.53: 24.54: anm24: 25.0: 25.1: 25.2: 25.3: 25.4: 25.5: 25.6: 25.7: 25.8: 25.9: 25.10: 25.11: 25.12: 25.13: 25.14: 25.15: 25.16: 25.17: 25.18: 25.19: 25.20: 25.21: 25.22: 25.23: 25.24: 25.25: 25.26: 25.27: 25.28: 25.29: 25.30: 25.31: 25.32: 25.33: 25.34: 25.35: 25.36: anm25: 26.0: 26.1: 26.2: 26.3: 26.4: 26.5: 26.6: 26.7: 26.8: 26.9: 26.10: 26.11: 26.12: 26.13: 26.14: 26.15: 26.16: 26.17: 26.18: 26.19: 26.20: 26.21: 26.22: 26.23: 26.24: 26.25: 26.26: 26.27: anm26: 27.0: 27.1: 27.2: 27.3: 27.4: 27.5: 27.6: 27.7: 27.8: 27.9: 27.10: 27.11: 27.12: 27.13: 27.14: 27.15: 27.16: 27.17: 27.18: 27.19: 27.20: 27.21: 27.22: anm27: 28.0: 28.1: 28.2: 28.3: 28.4: 28.5: 28.6: 28.7: 28.8: 28.9: 28.10: 28.11: 28.12: 28.13: 28.14: 28.15: 28.16: 28.17: 28.18: 28.19: 28.20: 28.21: 28.22: 28.23: 28.24: 28.25: 28.26: 28.27: 28.28: 28.29: anm28: 29.0: 29.1: 29.2: 29.3: 29.4: 29.5: 29.6: 29.7: 29.8: 29.9: 29.10: 29.11: 29.12: 29.13: 29.14: 29.15: 29.16: 29.17: 29.18: 29.19: 29.20: 29.21: 29.22: 29.23: 29.24: 29.25: 29.26: 29.27: 29.28: 29.29: anm29: 30.0: anm30: 31.0: 31.1: 31.2: 31.3: 31.4: 31.5: 31.6: 31.7: 31.8: 31.9: 31.10: 31.11: 31.12: 31.13: 31.14: 31.15: 31.16: 31.17: 31.18: 31.19: 31.20: 31.21: 31.22: 31.23: 31.24: 31.25: 31.26: 31.27: 31.28: 31.29: anm31: 32.0: 32.1: 32.2: 32.3: 32.4: 32.5: 32.6: 32.7: 32.8: 32.9: 32.10: 32.11: 32.12: 32.13: 32.14: 32.15: 32.16: 32.17: 32.18: 32.19: 32.20: 32.21: 32.22: 32.23: 32.24: 32.25: 32.26: 32.27: 32.28: anm32: 33.0: anm33: 33.EndLeft: 33.StepLeft: 33.PlayPauseLeft: 33.PlayPauseRight: 33.StepRight: 33.EndRight: 33.Minus: 33.Reset: 33.Plus: 34.0: 34.1: 34.2: 34.3: 34.4: 34.5: 34.6: 34.7: 34.8: 34.9: 34.10: 34.11: 34.12: 34.13: 34.14: 34.15: 34.16: 34.17: 34.18: 34.19: 34.20: 34.21: 34.22: 34.23: 34.24: 34.25: 34.26: 34.27: 34.28: 34.29: 34.30: 34.31: 34.32: 34.33: 34.34: 34.35: 34.36: 34.37: 34.38: 34.39: 34.40: 34.41: 34.42: 34.43: 34.44: 34.45: 34.46: 34.47: 34.48: 34.49: 34.50: 34.51: 34.52: 34.53: 34.54: 34.55: 34.56: 34.57: 34.58: 34.59: 34.60: 34.61: 34.62: 34.63: 34.64: 34.65: 34.66: 34.67: 34.68: 34.69: 34.70: 34.71: 34.72: anm34: 34.EndLeft: 34.StepLeft: 34.PlayPauseLeft: 34.PlayPauseRight: 34.StepRight: 34.EndRight: 34.Minus: 34.Reset: 34.Plus: 35.0: 35.1: 35.2: 35.3: 35.4: 35.5: 35.6: 35.7: 35.8: 35.9: 35.10: 35.11: 35.12: 35.13: 35.14: 35.15: 35.16: 35.17: 35.18: 35.19: 35.20: 35.21: 35.22: 35.23: 35.24: 35.25: 35.26: 35.27: 35.28: 35.29: 35.30: 35.31: 35.32: 35.33: 35.34: 35.35: 35.36: 35.37: 35.38: 35.39: 35.40: 35.41: 35.42: 35.43: 35.44: 35.45: 35.46: 35.47: 35.48: 35.49: 35.50: 35.51: 35.52: 35.53: 35.54: 35.55: 35.56: 35.57: 35.58: 35.59: 35.60: 35.61: 35.62: 35.63: 35.64: 35.65: 35.66: 35.67: 35.68: 35.69: 35.70: 35.71: 35.72: anm35: 35.EndLeft: 35.StepLeft: 35.PlayPauseLeft: 35.PlayPauseRight: 35.StepRight: 35.EndRight: 35.Minus: 35.Reset: 35.Plus: 36.0: anm36: 37.0: anm37: 38.0: anm38: 39.0: 39.1: 39.2: 39.3: 39.4: 39.5: 39.6: 39.7: 39.8: anm39: 40.0: anm40: 41.0: anm41: 42.0: anm42: 43.0: 43.1: 43.2: 43.3: 43.4: 43.5: 43.6: 43.7: 43.8: 43.9: 43.10: 43.11: 43.12: 43.13: 43.14: 43.15: 43.16: 43.17: 43.18: 43.19: 43.20: 43.21: 43.22: 43.23: 43.24: 43.25: 43.26: 43.27: 43.28: 43.29: anm43: 44.0: 44.1: 44.2: 44.3: 44.4: 44.5: 44.6: 44.7: 44.8: 44.9: 44.10: 44.11: 44.12: 44.13: 44.14: 44.15: 44.16: 44.17: 44.18: 44.19: 44.20: 44.21: 44.22: 44.23: 44.24: 44.25: 44.26: 44.27: 44.28: anm44: 45.0: anm45: 46.0: 46.1: 46.2: anm46: 47.0: 47.1: 47.2: anm47: 48.0: 48.1: 48.2: anm48: 49.0: 49.1: anm49: 50.0: 50.1: 50.2: 50.3: 50.4: 50.5: 50.6: 50.7: 50.8: 50.9: 50.10: 50.11: 50.12: 50.13: anm50: 51.0: 51.1: 51.2: 51.3: 51.4: 51.5: 51.6: 51.7: 51.8: 51.9: 51.10: 51.11: 51.12: 51.13: anm51: 52.0: 52.1: 52.2: 52.3: 52.4: 52.5: 52.6: 52.7: 52.8: 52.9: 52.10: 52.11: 52.12: 52.13: anm52: 53.0: 53.1: 53.2: 53.3: 53.4: 53.5: 53.6: 53.7: 53.8: 53.9: 53.10: 53.11: 53.12: 53.13: 53.14: 53.15: 53.16: 53.17: 53.18: 53.19: 53.20: 53.21: 53.22: 53.23: 53.24: 53.25: 53.26: 53.27: anm53: 54.0: 54.1: 54.2: 54.3: 54.4: 54.5: 54.6: 54.7: 54.8: 54.9: 54.10: 54.11: 54.12: 54.13: 54.14: 54.15: 54.16: 54.17: 54.18: 54.19: 54.20: 54.21: 54.22: 54.23: 54.24: 54.25: 54.26: 54.27: anm54: 55.0: 55.1: 55.2: 55.3: 55.4: 55.5: 55.6: 55.7: 55.8: 55.9: 55.10: 55.11: 55.12: 55.13: 55.14: 55.15: 55.16: 55.17: 55.18: 55.19: 55.20: 55.21: 55.22: 55.23: 55.24: 55.25: 55.26: 55.27: anm55:

12_PhdDefense

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