Finding damage in Space Shuttle foamsites.science.oregonstate.edu/~gibsonn/reu07.pdf · 2007. 7....

Finding damage in Space Shuttle foam

Nathan L. [email protected]

In Collaboration with:Prof. H. T. Banks, CRSC

Dr. W. P. Winfree, NASA

OSU REU – p. 1

Motivating Application

The particular motivation for this research is the detection of defects in theinsulating foam on the space shuttle fuel tanks in order to help eliminate theseparation of foam during shuttle ascent.

OSU REU – p. 2

Picometrix T-Ray Setup

• Step-block can be turned upside down to sample varying gap sizes.• Receiver and transmitter can be repositioned at various angles.• Signal can be focused or collimated.

OSU REU – p. 3

THz Gap

OSU REU – p. 4

FFT of THz Signal

0 1 2 3 4 5 6 7

x 1011

10−2

10−1

100

101

102

FFT of waveforms through various thicknesses of foam

Frequency (Hz)

Spe

ctra

l Am

plitu

de (

arb.

uni

ts)

ReferenceL=0.5"L=1"L=2"L=3"L=4"

FFT of THz signal recorded after passing through foam of varying thickness,in a pitch-echo experiment.

OSU REU – p. 5

THz Signal Through Foam

1100 1150 1200 1250 1300 1350 1400 1450

L=0.0"

L=0.5"

L=1.0"

L=2.0"

L=3.0"

L=4.0"

Waveforms traversing various thicknesses of foam

Time (ps)

Ele

ctric

Fie

ld (

arb.

uni

ts)

THz signal recorded after passing through foam of varying thickness, in apitch-echo experiment.

OSU REU – p. 6

Time-of-flight

Bitmap of time-of-flight recordings from step-block foam. Method clearlyshows steep boundaries between foam and voids.

• Shows contrast, but does not accurately characterize damage.• Less effective on horizontal discontinuities.

OSU REU – p. 7

Outline

• 1D Gap Detection Inverse Problem• 2D Void Detection• Microstructure Modeling

OSU REU – p. 8

Outline 1D Gap Detection

• Model• Numerical Methods• Inverse Problem• Computational Results

OSU REU – p. 9

Gap Detection Problem

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� � � � � � ��

� � � � � � �

x

z

y

E(t,z)

δd

OSU REU – p. 10

Modelµ0�0�rË + µ0IΩP̈ + µ0σIΩĖ − E

′′ = −µ0J̇s in Ω ∪ Ω0τ Ṗ + P = �0(�s − �∞)E in Ω

[Ė − cE ′]z=0 = 0

[E]z=1 = 0

E(0, z) = Ė(0, z) = 0

P (0, z) = 0where

Js(t, z) = δ(z)sin(ωt)I[0,tf ](t)

and�r = (1 + (�∞ − 1)IΩ).

OSU REU – p. 11

Numerical Discretization• Second order FEM in space

• piecewise linear splines

• Second order FD in time• Crank-Nicholson (P )• Central differences (E)• en → pn → en+1 → pn+1 → · · ·

• E equation implicit, LU factorization used

OSU REU – p. 12

Sample Problem

0 0.005 0.01 0.015 0.02−200

−100

0

100

200

z (meters)

elec

tric

fiel

d (v

olts

/met

er)

Signal at t=0.1182 ns

0 0.005 0.01 0.015 0.02−200

−100

0

100

200

z (meters)

elec

tric

fiel

d (v

olts

/met

er)


0 0.005 0.01 0.015 0.02−200

−100

0

100

200

z (meters)

elec

tric

fiel

d (v

olts

/met

er)


0 0.005 0.01 0.015 0.02−200

−100

0

100

200

z (meters)

elec

tric

fiel

d (v

olts

/met

er)


Computed solutions at different times of a windowedelectromagnetic pulse at f=100GHz incident on aDebye medium with a crack δ=.0002m wide locatedd=.02m into the material.

OSU REU – p. 13

Sample Problem (Cont.)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−200

−150

−100

−50

0

50

100

150

200

t (ns)

elec

tric

fiel

d (v

olts

/met

er)

Signal received at z=0

N=2048 Nt=6122 Ns=6122 depth=0.02 delta=0.0002 f=1e+11Hz

Reflected signal received at z=0.

OSU REU – p. 14

Sample Problem (Cont.)

0.325 0.33 0.335 0.34 0.345 0.35 0.355

−100

−50

0

50

100

t (ns)

elec

tric

fiel

d (v

olts

/met

er)

Signal received at z=0

N=2048 Nt=6122 Ns=6122 depth=0.02 delta=0.0002 f=1e+11Hz

Close up look at reflected signal received at z=0Shows “important” parts of the signal.

OSU REU – p. 15

Gap Detection Inverse Problem• Assume we have data, Êi, recorded at z=0• Given d and δ we can simulate the electric field• Estimate d and δ by solving an inverse problem:

Find q=(d, δ) ∈ Qad such that the followingobjective function is minimized:

J1(q) =1

2S

S∑

i=1

|E(ti, 0; q) − Êi|2.

OSU REU – p. 16

J1(q) Surface Plot

0

1

2

3

4

x 10−3

0.0160.018

0.020.022

0.0240.026

0.0280

50

100

150

200

250

300

350

delta

J1

depth

J

Surface plot of the Ordinary Least Squares objectivefunction demonstrating peaks in J1, and exhibitingmany local minima.

OSU REU – p. 17

Improved Objective Function

Consider the following formulation of the InverseProblem:Find q=(d, δ) ∈ Qad such that the following objectivefunction is minimized:

J2(q) =1

2S

S∑

i=1

∣

∣

∣|E(ti, 0; q)| − |Êi|

∣

∣

∣

2

.

OSU REU – p. 18

J2(q) Surface Plot

0

1

2

3

4

x 10−3

0.0160.018

0.020.022

0.0240.026

0.0280

50

100

150

200

250

delta

J2

depth

J

Close up surface plot of our Modified Least Squaresobjective function demonstrating lack of peaks in J2,but still exhibiting many local minima.

OSU REU – p. 19

Final Estimates (d)δ

d .0002 .0004 .0008

.02 (N=1024) .0200022 .0200006 .0200002

.04 (N=2048) .0399974 .0400005 .0399999

.08 (N=4096) .0799987 .0800006 .0800003

.1 (N=8192) .0999974 .1 .0999999

.2 (N=16384) .200005 .2 .200001

OSU REU – p. 20

Final Estimates (δ)δ

d .0002 .0004 .0008

.02 (N=1024) .000196754 .000398642 .00079707

.04 (N=2048) .000203916 .000394204 .000793622

.08 (N=4096) .000202273 .000395791 .000794401

.1 (N=8192) .000203876 .000396203 .000795985

.2 (N=16384) .000191808 .00040297 .00080129

OSU REU – p. 21

Comments on 1D Gap Problem• Our modified Least Squares objective function

“fixes” peaks in J• Can test on both sides of detected minima to

ensure global minimization• We are able to detect a .2mm wide crack behind

a 20cm deep slab• Even adding random noise (equivalent to 20%

relative noise) does not significantly hinder ourinverse problem solution method

OSU REU – p. 22

2D Problem Outline

• Model• Equations• Boundary Conditions

• Computational Methods• Sample Forward Simulations• Inverse Problem

OSU REU – p. 23

Voids in Foam

The foam on the space shuttle is sprayed on in layers(thus the acronym SOFI). Voids occur between layers.

OSU REU – p. 24

Cured Layer

As the top of each layer cures, a thin knit line is formed which is of higherdensity (i.e., is comprised of smaller, more tightly packed polyurethane cells).

OSU REU – p. 25

Sample Domain with Void

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

x

y

Dashed lines represent knit lines, dot-dash is foam/air interface. Ellipticalpocket (5 mm) between knit lines is a void. “+” marks the signal receiver.Back wall is perfect conductor.

OSU REU – p. 26

Simplifications• Assume single-cycle pulse of fixed frequency

• Possibly only tracking peak frequency• Possibly solving broadband problem in

parallel• Maxwell’s equations reduce to wave equation

• Assume homogenized material• For low frequency, microstructure is

negligible• For fixed frequency, single wave speed• Possibly from homogenization method

• Assume 2D (uniformity in third)

OSU REU – p. 27

2D Wave EquationWe assume the electric field to be polarized in the zdirection, thus for ~E = (0, 0, E) and ~x = (x, y)

�(~x)∂2E

∂t2(t, ~x) −∇ ·

(

1

µ(~x)∇E(t, ~x)

)

= −∂Js∂t

(t, ~x)

where �(~x) and µ(~x) = µ0 are the dielectricpermittivity and permeability, respectively.

Js(t, ~x) = δ(x)e−((t−t0)/t0)

4

,

where t0 = tf/4 when tf is the period of theinterrogating pulse.

OSU REU – p. 28

Boundary ConditionsConsider Ω = [0, 0.1] × [0, 0.2]

• Reflecting (Dirichlet) boundary conditions (right)

[E]x=0.1 = 0

• First order absorbing boundary conditions (left)

∂E

∂t−

√

1

�(~x)µ0

∂E

∂x

∣

∣

∣

∣

∣

x=0

= 0

• Symmetric boundary conditions (top and bottom)[

∂E

∂y

]

y=0,y=0.2

= 0

We use homogeneous initial conditions E(0, ~x) = 0.

OSU REU – p. 29

Modeling Knit Lines/Void• The speed of propagation in the domain is

c(~x) =c0

n(~x)=

√

1

�(~x)µ0,

where c0 is the speed in a vacuum and n is theindex of refraction.

• We may model knit lines or a void by changingthe index of refraction, thus effectively the speedin that region.

OSU REU – p. 30

2D Numerical Discretization

• Second order (piecewise linear) FEM in space

• Second order (centered) FD in time

• Linear solve (sparse)• Preconditioned conjugate-gradient

(matrix-free)• LU factorization• Mass lumping (explicit)

• Stair-stepping for non-rectilinear interfacesOSU REU – p. 31

Plane Wave Simulation

Source located at x = 0, receiver at x = 0.03.

OSU REU – p. 32

Plane Wave Signal

0.1 0.2 0.3 0.4 0.5 0.6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (ns)

E

Signal received at center line (x=0.03)

0.42 0.43 0.44 0.45 0.46 0.47 0.48

−0.1

−0.05

0

0.05

0.1

0.15

Interrogating signal simulates a sine curve truncated after one half period.Reflections off void are shown in inset.

OSU REU – p. 33

Picometrix T-Ray Setup

Note the non-normal incidence and ability to focus.OSU REU – p. 34

Oblique Plane Wave Simulation

Source located at x = 0, receiver at x = 0.03, but raised to collect center ofplane wave reflection.

OSU REU – p. 35

Oblique Plane Wave Signal

0.1 0.2 0.3 0.4 0.5 0.6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (ns)

E


0.42 0.43 0.44 0.45 0.46 0.47 0.48

−0.1

−0.05

0

0.05

0.1

0.15

Nearly all of original signal returns even with an oblique angle of incidence.(Note: last knit line removed.)

OSU REU – p. 36

Focused Wave Simulation

Source modeled using scattered field formulation of point source reflectedfrom elliptical mirror. Receiver located at x = 0.03. Note top and bottomboundary conditions are now absorbing.

OSU REU – p. 37

Focused Wave Signal

0.1 0.2 0.3 0.4 0.5 0.6−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (ns)

E


0.45 0.46 0.47 0.48 0.49 0.5 0.51

−0.1

−0.05

0

0.05

0.1

0.15

Although reflections off of void are larger, the total energy that returns is lessthan the plane wave simulation.

OSU REU – p. 38

Oblique Focused Wave

Source modeled using scattered field formulation of point source reflectedfrom elliptical mirror. Receiver located at x = 0.03, but raised to collectcenter of focused wave reflection.

OSU REU – p. 39

Oblique Focused Wave Signal

0.1 0.2 0.3 0.4 0.5 0.6−1

−0.5

0

0.5

1

t (ns)

E

Signal received at x=0.03

0.46 0.48 0.5

−0.1

0

0.1

Data received from non-normally incident, focused wave. Reflection fromvoid is similar in magnitude to normal incidence focused wave.

OSU REU – p. 40

2D Void Inverse Problem• Assume we have data, Êi at times ti and x = x+• Given the width of an elliptical void w, we can

simulate the electric field• Estimate void width w by solving an inverse

problem:

Find w ∈ Qad such that the following objectivefunction is minimized:

J1(w) =1

2S

S∑

i=1

|E(ti,x+; w) − Êi|

2.

OSU REU – p. 41

J1 – Objective Function

0.005 0.01 0.015 0.020

1

2

x 10−4

Width (m)

J

LSQ error as a function of void width

20 points

Location of the initial guess is crucial to minimizing with a gradient-basedmethod.

OSU REU – p. 42

J1 – Objective Function

0.005 0.01 0.015 0.020

1

2

x 10−4

Width (m)

J

LSQ error as a function of void width

20 points5 points

If we had only sampled the landscape with five points, we would have chosenpoorly.

OSU REU – p. 43


0.005 0.01 0.015 0.020

0.5

1

1.5x 10

−4

Width (m)

J2

modified LSQ error as a function of void width

20 points

J2(w) =1

2S

S∑

i=1

∣

∣

∣|E(ti,x

+; w)| − |Êi|∣

∣

∣

2

.

OSU REU – p. 44


0.005 0.01 0.015 0.020

0.5

1

1.5x 10

−4

Width (m)

J2

modified LSQ error as a function of void width

20 points5 points

Better representation of overall agreement of signals; more forgiving inchoosing initial guess.

OSU REU – p. 45

2D Inverse Problem Results

• LM converges to minimum of J2 after 12iterations

• Each forward solve is 1.5 hours• This does not incorporate noise (SNR ≈ 100:1)

OSU REU – p. 46

SOFI Under 20X Magnification

• Wavelength is on the order of 1mm; microstructure is smaller.• Most loss in the material is due to scattering from faces.• Modeling geometry of microstructure will help understand bulk effects.

OSU REU – p. 47

Microstructure

Model random microstructures:• “Random raindrops” algorithm• Apollonius tessellation• Constant wave speed

Note: Distributions of statistical parameters yieldsheterogeneity

OSU REU – p. 48

Voronoi Graph

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Voronoi diagram for randomly packed disks.

x

z

OSU REU – p. 49

Apollonius Graph

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Apollonius graph for randomly packed disks.

x

z

OSU REU – p. 50

Random Raindrop

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

12 3

45 6

789

10 11 12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28 29

3031

32

33

3435

36

37

38

39

40

41

42

4344

45

46

47

48

49

50

51

5253

54

55

Random Raindrop Algorithm

x

z

An example of the random raindrop or “drop and roll” algorithm for generating randomly closepacked disks.

OSU REU – p. 51

Filled-in

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

12 3 45 6

789

10 11 12

13

1415

16

17

18

19

20

21

22

23

24

25

26

27

28 29

3031

32

33

3435

36

37

38

39

40

41

42

4344

4546

47

48

49

50

51

5253

54

55

56

57

58

59

60

61

62

x

z

Random Raindrop Algorithm Filled (55+)

The “drop and roll” algorithm filled-in with disks of diameter at least the mean value minus twostandard deviations (disks labeled 55 through 62 in decreasing order of size).

OSU REU – p. 52

Truncated Domain

−0.5 0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Apollonius Graph and Truncated Domain

x

z

Apollonius graph on randomly close packed disks with a truncated domain indicated.

OSU REU – p. 53

Indicator Matrix

10 20 30 40 50

5

10

15

20

25

30

35

40

45

50

Indicator matrix for cell wall structure

i

j

Indicator matrix for cell wall with thickness 2h where h is the meshsize.

OSU REU – p. 54

Model of SOFI Microstructure

Apollonius graph is truncated and stretched, then edges are given a thickness and discretized toresult in an indicator matrix as shown in the bitmap.

OSU REU – p. 55

Model of SOFI with Knit Lines

Different sizes of drops are used to give different sized cells resulting in the appearance of knitlines.

OSU REU – p. 56

Continuing Directions• Modeling Approaches

• Microscale scattering model• Match attenuation observed in data

• Computational Methods• Edge elements• ABC/PML• Faster time-marching

• Quantify Robustness wrt Uncertainty• Material properties• Geometry

OSU REU – p. 57

Related Courses• Linear Algebra• ODEs• PDEs• Numerical Methods for the above• Optimization• Prob/Stat• Interdisciplinary

OSU REU – p. 58

Upcoming Conferences

OSU REU – p. 59

Motivating ApplicationPicometrix T-Ray SetupTHz GapFFT of THz SignalTHz Signal Through FoamTime-of-flightOutlineOutline 1D Gap DetectionGap Detection ProblemModelNumerical DiscretizationSample Problem Sample Problem (Cont.)Sample Problem (Cont.)Gap Detection Inverse Problem$mathcal {J}_1(q)$Surface PlotImproved Objective Function$mathcal {J}_2(q)$Surface PlotFinal Estimates ($d$)Final Estimates ($delta $)Comments on 1D Gap Problem2D Problem OutlineVoids in FoamCured LayerSample Domain with VoidSimplifications2D Wave EquationBoundary ConditionsModeling Knit Lines/Void2D Numerical DiscretizationPlane Wave SimulationPlane Wave SignalPicometrix T-Ray SetupOblique Plane Wave SimulationOblique Plane Wave SignalFocused Wave SimulationFocused Wave SignalOblique Focused WaveOblique Focused Wave Signal2D Void Inverse Problem$mathcal {J}_1$ -- Objective Function$mathcal {J}_1$ -- Objective FunctionImproved Objective FunctionImproved Objective Function2D Inverse Problem ResultsSOFI Under 20X MagnificationMicrostructureVoronoi GraphApollonius GraphRandom RaindropFilled-inTruncated DomainIndicator MatrixModel of SOFI MicrostructureModel of SOFI with Knit LinesContinuing DirectionsRelated CoursesUpcoming Conferences

Finding damage in Space Shuttle foamsites.science.oregonstate.edu/~gibsonn/reu07.pdf · 2007. 7....

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Transcript of Finding damage in Space Shuttle foamsites.science.oregonstate.edu/~gibsonn/reu07.pdf · 2007. 7....