Thermal Detection of Small Cracks in a Plate

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Hector Patrick
5 years ago
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1 1 Rose-Hulman Institute of Technology May 20, With some helpful discussions with Chris Earls, Cornell University School of Civil and Environmental Engineering.

3 Motivation The USS Independence new hull design, aluminum, prone to cracks.

4 Motivation The USS Independence new hull design, aluminum, prone to cracks. How can we find small cracks efficiently?

5 Experimental set-up:

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

7 Typical parameters: aluminum plate, 10 cm by 10 cm, 1 mm thickness, 5 to 20 watt (or more) laser. Illuminate sample plate with laser for 0 < t < T (T from 30 to 180 seconds)

8 Typical parameters: aluminum plate, 10 cm by 10 cm, 1 mm thickness, 5 to 20 watt (or more) laser. Illuminate sample plate with laser for 0 < t < T (T from 30 to 180 seconds) Measure temperature on pixel grid, 0.1 degree C resolution, every few seconds.

9 Typical parameters: aluminum plate, 10 cm by 10 cm, 1 mm thickness, 5 to 20 watt (or more) laser. Illuminate sample plate with laser for 0 < t < T (T from 30 to 180 seconds) Measure temperature on pixel grid, 0.1 degree C resolution, every few seconds. Goal: find any cracks -position, orientation, size.

10 Heat Equation Let x = (x 1, x 2 ) position, t time, Ω the (2D) plate, γ a crack in the plate, u(x, t) denote plate temperature. Assume u satisfies cρu t α 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 or convection losses through the plate face.

11 Rescaled Heat Equation After rescaling to Ω = (0, 1) 2 and α/(cρ) = 1 we have u t u = f on Ω \ γ (0, T ) u n = 0 on γ (0, T ) u n = 0 on Ω (0, T ) u(x, 0) = 0.

12 A Typical Solution The temperature u is discontinuous across the crack γ.

13 What s So Hard About Finding Cracks? We want to find cracks that are small on the order of a couple pixels long, or even less than one pixel! There are 3 cracks in the field of view below:

14 Integrate by Parts Let φ(x, t) be a suitably smooth test function, and define F = φ t + φ. A little integration by parts shows that T 0 γ [u] φ ds dt = n Ω\γ T 0 T 0 u(x, T )φ(x, T ) dx Ω\γ Ω\γ T uf dx dt + u φ ds dt 0 Ω n f φ dx dt. where [u] denotes the jump in u over γ in the direction n.

15 Integrate by Parts Let φ(x, t) be a suitably smooth test function, and define F = φ t + φ. A little integration by parts shows that T 0 γ [u] φ ds dt = n Ω\γ T 0 T 0 u(x, T )φ(x, T ) dx Ω\γ Ω\γ T uf dx dt + u φ ds dt 0 Ω n 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 ).

16 In Summary We have the identity T 0 γ [u] φ ds dt = q(φ) n }{{} computable from data for any smooth function φ, where q(φ) is computable from knowledge of u on (Ω \ γ) (0, T ).

17 In Summary We have the identity T 0 γ [u] φ ds dt = q(φ) n }{{} computable from data for any smooth function φ, where q(φ) is computable from knowledge of u on (Ω \ γ) (0, T ). Can we use this, with suitable choices for φ, to deduce γ?

18 Assumptions In what follows we ll assume that The crack γ is a line segment with center p = (p 1, p 2 ), angle θ, length γ.

19 Assumptions In what follows we ll assume that The crack γ is a line segment with center p = (p 1, p 2 ), angle θ, length γ. The crack is short.

20 Consider a two-parameter family of test functions φ(x 1, x 2 ) = e ((x 1 a) 2 +(x 2 b) 2 )/β 2 for some fixed β > 0. We can write T 0 If γ is short we may approximate γ πβ 2 [u] φ ds dt = q(a, b) n }{{} computable from data q(a, b) φ T n (p) [u] ds dt. 0 γ

21 Observations From T q(a, b) φ(p) n [u] ds dt. 0 γ we see that if no crack is present. q(a, b) 0

22 Observations From T q(a, b) φ(p) n [u] ds dt. 0 γ we see that if no crack is present. q(a, b) 0 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).

23 Raw Temperature Data and Function q Time T = 3, raw temperature on the left, function q(a, b) on the right.

24 Close Up View of Function q near Crack

25 More Observations From T q(a, b) φ(p) n [u] ds dt. 0 γ we see that q 0 on the line containing the crack.

26 More Observations From T q(a, b) φ(p) n [u] ds dt. 0 γ 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 ± 2 2 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.

27 Crack Length One can prove that T 0 γ [u] ds dt = π T 4 γ 2 u 0 (p, t) n dt + O( γ 3 ). where u 0 is the solution to the heat equation on Ω (no crack). 0

28 Crack Length One can prove that T 0 γ [u] ds dt = π T 4 γ 2 u 0 (p, t) n dt + O( γ 3 ). 0 where u 0 is the solution to the heat equation on Ω (no crack). As a result q(a, b) π T 4 γ 2 ( φ(p) n) u 0 (p, t) n dt. 0 Since we now know p and n, we can estimate γ.

29 Close Up View of Function q True crack center is (0.7, 0.543), angle 60 degrees, length Recovered parameters are (0.7, 0.544), angle 60.3 degrees, length (no noise/quantization).

30 Multiple Cracks Power input 5 watts, real time 97.2 seconds, crack lengths 0.02, 0.01,

31 Connection to Small Volume Expansions The formula q(a, b) π 4 γ 2 ( φ(p) n) T 0 u 0(p, t) n dt can be written q(a, b) π T 4 γ 2 φ(p) t M u 0 (p, t) dt 0 with M = nn t and viewed as an extreme limiting case of the small-volume asymptotic expansions of Ammari, Kang, Vogelius, etc.

32 Connection to Small Volume Expansions The formula q(a, b) π 4 γ 2 ( φ(p) n) T 0 u 0(p, t) n dt can be written q(a, b) π T 4 γ 2 φ(p) t M u 0 (p, t) dt 0 with M = nn t and viewed as an extreme limiting case of the small-volume asymptotic expansions of Ammari, Kang, Vogelius, etc. Physically interpretation: a crack looks like a thermal dipole at position p, orientation n, dipole moment π 4 γ 2 T 0 u 0 n dt.

33 Infrared cameras used in this type of work have noise and quantization error, typically in the ballpark of 0.1 degrees C. Power input 5 watts, real time 97.2 seconds, crack lengths 0.02, 0.01,

34 If we boost the power input to 25 watts (using more of the dynamic range of the camera) the results are Power input 25 watts, real time 97.2 seconds, crack lengths 0.02, 0.01,

35 True crack parameters Recovered parameters center angle (degrees) length (0.53, ) ( , 0.4) (0.7, ) center angle (degrees) length (0.536, 0.429) (0.75, 0.4) (0.699, 0.543)

36 Near Field Expansion The formula for q(a, b) was based on a far field approximation, T 0 γ [u] φ T ds dt φ(p) n n 0 γ ( π φ(p) n 4 γ 2 [u] ds dt T obtained by treating φ as constant over the crack. 0 ) u 0 (p, t) n dt

37 Near Field Expansion The formula for q(a, b) was based on a far field approximation, T 0 γ [u] φ T ds dt φ(p) n n 0 γ ( π φ(p) n 4 γ 2 [u] ds dt T obtained by treating φ as constant over the crack. 0 ) u 0 (p, t) n dt But we re taking the Gaussian test function with center near the crack. Is this approximation reasonable?

38 Near Field Expansion

39 Advantages Computationally very fast.

40 Advantages Computationally very fast. Robust against model departures precise knowledge of power input/profile, emissivity, boundary conditions not necessary. Even convection/radiation makes little difference.

41 Advantages Computationally very fast. Robust against model departures precise knowledge of power input/profile, emissivity, boundary conditions not necessary. Even convection/radiation makes little difference. Flexibility test functions can be tailored to incorporate most relevant data.

42 Advantages Computationally very fast. Robust against model departures precise knowledge of power input/profile, emissivity, boundary conditions not necessary. Even convection/radiation makes little difference. Flexibility test functions can be tailored to incorporate most relevant data. Physical insight illuminates what variables/geometries are more important.

43 Work out the near crack expansion better length estimates?

44 Work out the near crack expansion better length estimates? For cracks near or below one pixel, would a careful model of the image capture process (point-spread function, sensor geometry) be worth it?

45 Work out the near crack expansion better length estimates? For cracks near or below one pixel, would a careful model of the image capture process (point-spread function, sensor geometry) 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/ π k γ γ 2 + O(c(k) γ 3 ). This ought to hold in the parabolic case too.

46 Work out the near crack expansion better length estimates? For cracks near or below one pixel, would a careful model of the image capture process (point-spread function, sensor geometry) 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/ π k γ γ 2 + O(c(k) γ 3 ). This ought to hold in the parabolic case too. Would other test functions yield better results?

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