Finding damage in Space Shuttle foam

Transcription

1 Finding damage in Space Shuttle foam Nathan L. Gibson In Collaboration with: Prof. H. T. Banks, CRSC Dr. W. P. Winfree, NASA OSU REU p. 1

2 Motivating Application The particular motivation for this research is the detection of defects in the insulating foam on the space shuttle fuel tanks in order to help eliminate the separation of foam during shuttle ascent. OSU REU p. 2

3 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

4 THz Gap OSU REU p. 4

5 FFT of THz Signal 10 2 FFT of waveforms through various thicknesses of foam 10 1 Reference L=0.5" L=1" L=2" L=3" L=4" Spectral Amplitude (arb. units) Frequency (Hz) x FFT of THz signal recorded after passing through foam of varying thickness, in a pitch-echo experiment. OSU REU p. 5

6 THz Signal Through Foam L=0.0" Waveforms traversing various thicknesses of foam L=0.5" Electric Field (arb. units) L=1.0" L=2.0" L=3.0" L=4.0" Time (ps) THz signal recorded after passing through foam of varying thickness, in a pitch-echo experiment. OSU REU p. 6

7 Time-of-flight Bitmap of time-of-flight recordings from step-block foam. Method clearly shows steep boundaries between foam and voids. Shows contrast, but does not accurately characterize damage. Less effective on horizontal discontinuities. OSU REU p. 7

8 Outline 1D Gap Detection Inverse Problem 2D Void Detection Microstructure Modeling OSU REU p. 8

9 Outline 1D Gap Detection Model Numerical Methods Inverse Problem Computational Results OSU REU p. 9

10 Gap Detection Problem x z y E(t,z) δ d OSU REU p. 10

11 Model µ 0 ɛ 0 ɛ r Ë + µ 0 I Ω P + µ0 σi Ω Ė E = µ 0 J s in Ω Ω 0 τ P + P = ɛ 0 (ɛ s ɛ )E [Ė ce ] z=0 = 0 [E] z=1 = 0 in Ω where E(0, z) = Ė(0, z) = 0 P (0, z) = 0 J s (t, z) = δ(z)sin(ωt)i [0,tf ](t) and ɛ r = (1 + (ɛ 1)I Ω ). OSU REU p. 11

12 Numerical Discretization Second order FEM in space piecewise linear splines Second order FD in time Crank-Nicholson (P ) Central differences (E) e n p n e n+1 p n+1 E equation implicit, LU factorization used OSU REU p. 12

13 Sample Problem 200 Signal at t= ns 200 Signal at t= ns electric field (volts/meter) electric field (volts/meter) z (meters) z (meters) 200 Signal at t= ns 200 Signal at t= ns electric field (volts/meter) electric field (volts/meter) z (meters) z (meters) Computed solutions at different times of a windowed electromagnetic pulse at f=100ghz incident on a Debye medium with a crack δ=.0002m wide located d=.02m into the material. OSU REU p. 13

14 Sample Problem (Cont.) Signal received at z=0 N=2048 Nt=6122 Ns=6122 depth=0.02 delta= f=1e+11hz electric field (volts/meter) t (ns) Reflected signal received at z=0. OSU REU p. 14

15 Sample Problem (Cont.) Signal received at z=0 100 N=2048 Nt=6122 Ns=6122 depth=0.02 delta= f=1e+11hz electric field (volts/meter) t (ns) Close up look at reflected signal received at z=0 Shows important parts of the signal. OSU REU p. 15

16 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) Q ad such that the following objective function is minimized: J 1 (q) = 1 2S S i=1 E(t i, 0; q) Êi 2. OSU REU p. 16

17 J 1 (q) Surface Plot J J depth 0.02 Surface plot of the Ordinary Least Squares objective function demonstrating peaks in J 1, and exhibiting many local minima delta x 10 3 OSU REU p. 17

18 Improved Objective Function Consider the following formulation of the Inverse Problem: Find q=(d, δ) Q ad such that the following objective function is minimized: J 2 (q) = 1 2S S E(t i, 0; q) Êi 2. i=1 OSU REU p. 18

19 J 2 (q) Surface Plot J J depth 0.02 Close up surface plot of our Modified Least Squares objective function demonstrating lack of peaks in J 2, but still exhibiting many local minima delta x 10 3 OSU REU p. 19

20 Final Estimates (d) δ d (N=1024) (N=2048) (N=4096) (N=8192) (N=16384) OSU REU p. 20

21 Final Estimates (δ) δ d (N=1024) (N=2048) (N=4096) (N=8192) (N=16384) OSU REU p. 21

22 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 our inverse problem solution method OSU REU p. 22

23 2D Problem Outline Model Equations Boundary Conditions Computational Methods Sample Forward Simulations Inverse Problem OSU REU p. 23

24 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

25 Cured Layer As the top of each layer cures, a thin knit line is formed which is of higher density (i.e., is comprised of smaller, more tightly packed polyurethane cells). OSU REU p. 25

26 Sample Domain with Void y x Dashed lines represent knit lines, dot-dash is foam/air interface. Elliptical pocket (5 mm) between knit lines is a void. + marks the signal receiver. Back wall is perfect conductor. OSU REU p. 26

27 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

28 2D Wave Equation We assume the electric field to be polarized in the z direction, thus for E = (0, 0, E) and x = (x, y) ( ) ɛ( x) 2 E 1 (t, x) E(t, x) = J s (t, x) t2 µ( x) t where ɛ( x) and µ( x) = µ 0 are the dielectric permittivity and permeability, respectively. J s (t, x) = δ(x)e ((t t 0)/t 0 ) 4, where t 0 = t f /4 when t f is the period of the interrogating pulse. OSU REU p. 28

29 Boundary Conditions Consider 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 E = 0 ɛ( x)µ 0 x x=0 Symmetric boundary conditions (top and bottom) [ ] E = 0 y y=0,y=0.2 We use homogeneous initial conditions E(0, x) = 0. OSU REU p. 29

30 Modeling Knit Lines/Void The speed of propagation in the domain is c( x) = c 0 n( x) = 1, ɛ( x)µ 0 where c 0 is the speed in a vacuum and n is the index of refraction. We may model knit lines or a void by changing the index of refraction, thus effectively the speed in that region. OSU REU p. 30

31 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 interfaces OSU REU p. 31

32 Plane Wave Simulation Source located at x = 0, receiver at x = OSU REU p. 32

33 Plane Wave Signal Signal received at center line (x=0.03) E t (ns) Interrogating signal simulates a sine curve truncated after one half period. Reflections off void are shown in inset. OSU REU p. 33

34 Picometrix T-Ray Setup Note the non-normal incidence and ability to focus. OSU REU p. 34

35 Oblique Plane Wave Simulation Source located at x = 0, receiver at x = 0.03, but raised to collect center of plane wave reflection. OSU REU p. 35

36 Oblique Plane Wave Signal Signal received at center line (x=0.03) E t (ns) Nearly all of original signal returns even with an oblique angle of incidence. (Note: last knit line removed.) OSU REU p. 36

37 Focused Wave Simulation Source modeled using scattered field formulation of point source reflected from elliptical mirror. Receiver located at x = Note top and bottom boundary conditions are now absorbing. OSU REU p. 37

38 Focused Wave Signal Signal received at center line (x=0.03) E t (ns) Although reflections off of void are larger, the total energy that returns is less than the plane wave simulation. OSU REU p. 38

39 Oblique Focused Wave Source modeled using scattered field formulation of point source reflected from elliptical mirror. Receiver located at x = 0.03, but raised to collect center of focused wave reflection. OSU REU p. 39

40 Oblique Focused Wave Signal Signal received at x=0.03 E t (ns) Data received from non-normally incident, focused wave. Reflection from void is similar in magnitude to normal incidence focused wave. OSU REU p. 40

41 2D Void Inverse Problem Assume we have data, Êi at times t i 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 Q ad such that the following objective function is minimized: J 1 (w) = 1 2S S i=1 E(t i, x + ; w) Êi 2. OSU REU p. 41

42 J 1 Objective Function x 10 4 LSQ error as a function of void width 20 points 2 J Width (m) Location of the initial guess is crucial to minimizing with a gradient-based method. OSU REU p. 42

43 J 1 Objective Function x 10 4 LSQ error as a function of void width 20 points 5 points 2 J Width (m) If we had only sampled the landscape with five points, we would have chosen poorly. OSU REU p. 43

44 Improved Objective Function 1.5 x 10 4 modified LSQ error as a function of void width 20 points 1 J Width (m) J 2 (w) = 1 2S S E(t i, x + ; w) Ê i 2. i=1 OSU REU p. 44

45 Improved Objective Function 1.5 x 10 4 modified LSQ error as a function of void width 20 points 5 points 1 J Width (m) Better representation of overall agreement of signals; more forgiving in choosing initial guess. OSU REU p. 45

46 2D Inverse Problem Results LM converges to minimum of J 2 after 12 iterations Each forward solve is 1.5 hours This does not incorporate noise (SNR 100:1) OSU REU p. 46

47 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

48 Microstructure Model random microstructures: Random raindrops algorithm Apollonius tessellation Constant wave speed Note: Distributions of statistical parameters yields heterogeneity OSU REU p. 48

49 Voronoi Graph 1 Voronoi diagram for randomly packed disks. z x OSU REU p. 49

50 Apollonius Graph 1 Apollonius graph for randomly packed disks. z x OSU REU p. 50

51 Random Raindrop 2 Random Raindrop Algorithm z x An example of the random raindrop or drop and roll algorithm for generating randomly close packed disks OSU REU p. 51

52 Filled-in 2 Random Raindrop Algorithm Filled (55+) z x The drop and roll algorithm filled-in with disks of diameter at least the mean value minus two standard deviations (disks labeled 55 through 62 in decreasing order of size) OSU REU p. 52

53 Truncated Domain 2 Apollonius Graph and Truncated Domain x Apollonius graph on randomly close packed disks with a truncated domain indicated. z OSU REU p. 53

54 Indicator Matrix 50 Indicator matrix for cell wall structure i Indicator matrix for cell wall with thickness 2h where h is the meshsize. j OSU REU p. 54

55 Model of SOFI Microstructure Apollonius graph is truncated and stretched, then edges are given a thickness and discretized to result in an indicator matrix as shown in the bitmap. OSU REU p. 55

56 Model of SOFI with Knit Lines Different sizes of drops are used to give different sized cells resulting in the appearance of knit lines. OSU REU p. 56

57 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

58 Related Courses Linear Algebra ODEs PDEs Numerical Methods for the above Optimization Prob/Stat Interdisciplinary OSU REU p. 58

59 Upcoming Conferences OSU REU p. 59