Thermal Detection of Small Cracks in a Platebryan/reu2011/aip2011talk.pdf · Kurt Bryan Thermal...

OutlinePhysical Setting

Forward ProblemInverse Problem Approach

Further Work

Thermal Detection of Small Cracks in a Plate

Kurt Bryan1

Rose-Hulman Institute of Technology

May 20, 2011

1With some helpful discussions with Chris Earls, Cornell University School ofCivil and Environmental Engineering.

Kurt Bryan Thermal Detection of Small Cracks in a Plate



Further Work

1 Physical Setting

2 Forward Problem

3 Inverse Problem ApproachGaussian Test FunctionsEstimating the Crack ParametersEstimating Crack LengthNoise and Quantization ErrorNear Crack Expansion

4 Further Work




Further Work

Motivation

The USS Independence—newhull design, aluminum, proneto cracks.

How can we find small cracksefficiently?




Further Work

Motivation

The USS Independence—newhull design, aluminum, proneto cracks.

How can we find small cracksefficiently?




Further Work

Physical Setting

Experimental set-up:




Further Work

Physical Setting

Typical parameters: aluminum plate, 10 cm by 10 cm, 1 mmthickness, 5 to 20 watt (or more) laser.

Illuminate sample plate with laser for 0 < t < T (T from 30to 180 seconds)

Measure temperature on 250× 250 pixel grid, 0.1 degree Cresolution, every few seconds.

Goal: find any cracks—-position, orientation, size.




Further Work

Physical Setting








Further Work

Physical Setting








Further Work

Physical Setting








Further Work

Heat Equation

Let x = (x1, x2) position, t time, Ω the (2D) “plate,” γ a crack inthe plate, u(x , t) denote plate temperature. Assume u satisfies

cρut − α4 u = f on Ω \ γ × (0,T )

α∂u

∂n= 0 on γ × (0,T )

α∂u

∂n= 0 on ∂Ω× (0,T )

u(x , 0) = 0.

Here f embodies the input heat source, possibly radiation orconvection losses through the plate face.




Further Work

Rescaled Heat Equation

After rescaling to Ω = (0, 1)2 and α/(cρ) = 1 we have

ut −4u = f on Ω \ γ × (0,T )

∂u

∂n= 0 on γ × (0,T )

∂u

∂n= 0 on ∂Ω× (0,T )

u(x , 0) = 0.




Further Work

A Typical Solution

The temperature u is discontinuous across the crack γ.




Further Work

What’s So Hard About Finding Cracks?

We want to find cracks that are small—on the order of a couplepixels long, or even less than one pixel!

There are 3 cracks in the field of view below:




Further Work

Gaussian Test FunctionsEstimating the Crack ParametersEstimating Crack LengthNoise and Quantization ErrorNear Crack Expansion

Integrate by Parts

Let φ(x , t) be a suitably smooth “test function”, and defineF = φt +4φ. A little integration by parts shows that∫ T

0

∫γ

[u]∂φ

∂nds dt =

∫Ω\γ

u(x ,T )φ(x ,T ) dx

−∫ T

0

∫Ω\γ

uF dx dt +

∫ T

0

∫∂Ω

u∂φ

∂nds dt

−∫ T

0

∫Ω\γ

f φ dx dt.

where [u] denotes the jump in u over γ in the direction n.

Everything on the right is computable from knowledge of u on(Ω \ γ)× (0,T ).




Further Work


Integrate by Parts

Let φ(x , t) be a suitably smooth “test function”, and defineF = φt +4φ. A little integration by parts shows that∫ T

0

∫γ

[u]∂φ

∂nds dt =

∫Ω\γ

u(x ,T )φ(x ,T ) dx

−∫ T

0

∫Ω\γ

uF dx dt +

∫ T

0

∫∂Ω

u∂φ

∂nds dt

−∫ T

0

∫Ω\γ

f φ dx dt.

where [u] denotes the jump in u over γ in the direction n.

Everything on the right is computable from knowledge of u on(Ω \ γ)× (0,T ).




Further Work


In Summary

We have the identity∫ T

0

∫γ

[u]∂φ

∂nds dt = q(φ)︸︷︷︸

computable from data

for any smooth function φ, where q(φ) is computable fromknowledge of u on (Ω \ γ)× (0,T ).

Can we use this, with suitable choices for φ, to deduce γ?




Further Work


In Summary

We have the identity∫ T

0

∫γ

[u]∂φ

∂nds dt = q(φ)︸︷︷︸


for any smooth function φ, where q(φ) is computable fromknowledge of u on (Ω \ γ)× (0,T ).

Can we use this, with suitable choices for φ, to deduce γ?




Further Work


Assumptions

In what follows we’ll assume that

The crack γ is a line segment with center p = (p1, p2), angleθ, length |γ|.

The crack is “short.”




Further Work


Assumptions

In what follows we’ll assume that

The crack γ is a line segment with center p = (p1, p2), angleθ, length |γ|.The crack is “short.”




Further Work


Gaussian Test Functions

Consider a two-parameter family of test functions

φ(x1, x2) =e−((x1−a)2+(x2−b)2)/β2

πβ2

for some fixed β > 0. We can write∫ T

0

∫γ

[u]∂φ

∂nds dt = q(a, b)︸︷︷︸


If γ is short we may approximate

q(a, b) ≈ ∂φ

∂n(p)

∫ T

0

∫γ

[u] ds dt.




Further Work


Observations

From

q(a, b) ≈ ∇φ(p) · n∫ T

0

∫γ

[u] ds dt.

we see thatq(a, b) ≡ 0

if no crack is present.

In fact, since φ effectively vanishes outside a ball of radius 2β,q(a, b) ≈ 0 if there’s no crack in a ball of radius 2β around (a, b).




Further Work


Observations

From

q(a, b) ≈ ∇φ(p) · n∫ T

0

∫γ

[u] ds dt.

we see thatq(a, b) ≡ 0

if no crack is present.

In fact, since φ effectively vanishes outside a ball of radius 2β,q(a, b) ≈ 0 if there’s no crack in a ball of radius 2β around (a, b).




Further Work


Raw Temperature Data and Function q

Time T = 3, raw temperature on the left, function q(a, b) on theright.




Further Work


Close Up View of Function q near Crack




Further Work


More Observations

From

q(a, b) ≈ ∇φ(p) · n∫ T

0

∫γ

[u] ds dt.

we see that

q ≡ 0 on the line containing the crack.

q attains its maximum/minimum orthogonally off the crack

center p, at points p±√

22 βn.

We can thus recover the crack center and angle (hence normal n)from q(a, b) for values of (a, b) near the crack.




Further Work


More Observations

From

q(a, b) ≈ ∇φ(p) · n∫ T

0

∫γ

[u] ds dt.

we see that

q ≡ 0 on the line containing the crack.

q attains its maximum/minimum orthogonally off the crack

center p, at points p±√

22 βn.

We can thus recover the crack center and angle (hence normal n)from q(a, b) for values of (a, b) near the crack.




Further Work


Crack Length

One can prove that∫ T

0

∫γ

[u] ds dt =π

4|γ|2

∫ T

0∇u0(p, t) · n dt + O(|γ|3).

where u0 is the solution to the heat equation on Ω (no crack).

Asa result

q(a, b) ≈ π

4|γ|2(∇φ(p) · n)

∫ T

0∇u0(p, t) · n dt.

Since we now know p and n, we can estimate |γ|.




Further Work


Crack Length

One can prove that∫ T

0

∫γ

[u] ds dt =π

4|γ|2

∫ T

0∇u0(p, t) · n dt + O(|γ|3).

where u0 is the solution to the heat equation on Ω (no crack). Asa result

q(a, b) ≈ π

4|γ|2(∇φ(p) · n)

∫ T

0∇u0(p, t) · n dt.

Since we now know p and n, we can estimate |γ|.




Further Work


Close Up View of Function q

True crack center is (0.7, 0.543), angle −60 degrees, length 0.01.Recovered parameters are (0.7, 0.544), angle −60.3 degrees, length0.0095 (no noise/quantization).




Further Work


Multiple Cracks

Power input 5 watts, real time 97.2 seconds, crack lengths0.02, 0.01, 0.005.




Further Work


Connection to Small Volume Expansions

The formula q(a, b) ≈ π4 |γ|

2(∇φ(p) · n)∫ T

0 ∇u0(p, t) · n dt can bewritten

q(a, b) ≈ π

4|γ|2

∫ T

0∇φ(p)tM∇u0(p, t) dt

with M = nnt and viewed as an extreme limiting case of thesmall-volume asymptotic expansions of Ammari, Kang, Vogelius,etc.

Physically interpretation: a crack looks like a “thermal dipole” atposition p, orientation n, dipole moment π

4 |γ|2∫ T

0 ∇u0 · n dt.




Further Work


Connection to Small Volume Expansions

The formula q(a, b) ≈ π4 |γ|

2(∇φ(p) · n)∫ T

0 ∇u0(p, t) · n dt can bewritten

q(a, b) ≈ π

4|γ|2

∫ T

0∇φ(p)tM∇u0(p, t) dt

with M = nnt and viewed as an extreme limiting case of thesmall-volume asymptotic expansions of Ammari, Kang, Vogelius,etc.

Physically interpretation: a crack looks like a “thermal dipole” atposition p, orientation n, dipole moment π

4 |γ|2∫ T

0 ∇u0 · n dt.




Further Work


Noise and Quantization Error

Infrared cameras used in this type of work have noise andquantization error, typically in the ballpark of 0.1 degrees C.





Further Work



If we boost the power input to 25 watts (using more of thedynamic range of the camera) the results are





Further Work



True crack parameters

center angle (degrees) length

(0.53, 0.42443) 0 0.005(0.75113, 0.4) 90 0.02(0.7, 0.54325) −60 0.01

Recovered parameters

center angle (degrees) length

(0.536, 0.429) −23.5 0.01(0.75, 0.4) 86.3 0.028(0.699, 0.543) −59.4 0.0168




Further Work


Near Field Expansion

The formula for q(a, b) was based on a “far field” approximation,∫ T

0

∫γ

[u]∂φ

∂nds dt ≈ ∇φ(p) · n

∫ T

0

∫γ

[u] ds dt

≈ ∇φ(p) · n(π

4|γ|2

∫ T

0∇u0(p, t) · n dt

)obtained by treating ∇φ as constant over the crack.

But we’re taking the Gaussian test function with center near thecrack. Is this approximation reasonable?




Further Work



The formula for q(a, b) was based on a “far field” approximation,∫ T

0

∫γ

[u]∂φ

∂nds dt ≈ ∇φ(p) · n

∫ T

0

∫γ

[u] ds dt

≈ ∇φ(p) · n(π

4|γ|2

∫ T

0∇u0(p, t) · n dt

)obtained by treating ∇φ as constant over the crack.

But we’re taking the Gaussian test function with center near thecrack. Is this approximation reasonable?




Further Work






Further Work


Advantages

Computationally very fast.

Robust against model departures—precise knowledge of powerinput/profile, emissivity, boundary conditions not necessary.Even convection/radiation makes little difference.

Flexibility—test functions can be tailored to incorporate mostrelevant data.

Physical insight—illuminates what variables/geometries aremore important.




Further Work


Advantages








Further Work


Advantages








Further Work


Advantages








Further Work

Further Work

Work out the near crack expansion—better length estimates?

For cracks near or below one pixel, would a careful model ofthe image capture process (point-spread function, sensorgeometry) be worth it?

A more realistic boundary condition on the crack is∂u/∂n = k[u]. In the elliptic case the asymptotics are∫

γ[u] ds =

π/4

1 + 83πk |γ|

|γ|2 + O(c(k)|γ|3).

This ought to hold in the parabolic case too.

Would other test functions yield better results?




Further Work

Further Work




γ[u] ds =

π/4

1 + 83πk |γ|

|γ|2 + O(c(k)|γ|3).






Further Work

Further Work




γ[u] ds =

π/4

1 + 83πk |γ|

|γ|2 + O(c(k)|γ|3).






Further Work

Further Work




γ[u] ds =

π/4

1 + 83πk |γ|

|γ|2 + O(c(k)|γ|3).




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