Conference presentation: Acquiring elastic properties of thin composite structure from vibrational testing data

1. Acquiring elastic properties of thin composite
structure from vibrational testing data
P. Chebyshev Mathematical Ideas and Their Applications to
Natural Sciences International Conference
V. Aksenov, A. Vasyukov, K. Beklemysheva
May 15, 2021
Moscow Institute of Physics and Technology

2. Table of contents
1. Introduction
2. Numerical method
3. Numerical experiments
1

3. Introduction

4. Problem overview
Figure 1: Example of the vibrational
testing stand
• Setup:
• Non-destructive testing of
thin composite structures
• Amplitutde-frequency
characteristic is measured
• Motivation:
• Lack of reliable data from
the manufacturers
• Elastic properties may
change during production
• Effective model with coarser
geometry
2

5. Governing equations
The model describes linear anisotropic elastic medium with viscous
damping. The transient load with constant frequency is applied on the
border
ρwtt + γ0wt − (D66wxx + D55wyy ) = 0 (x, y) ∈ G
w(x, y, t) = g(x, y) · eiωt
(x, y) ∈ ΓD
σn = (D66wx nx + D55wy ny ) = 0 (x, y) ∈ ΓN (ΓN ∪ ΓD = ∂G)
Variables
• w(x, y, t) – displacement in normal direction
• ρ – known density
• θ = (γ0, D55, D66) – unknown parameters
3

6. Governing equations
w = wpart +
∞
X
i=0
Ci wi
We suggest that due to damping, the terms describing free vibrations
fade exponentially, thus only consider the term
wpart = u(x, y) · eiωt
We arrive at the BVP for u:
(−ρω2
+ iγ0ω)u − (D66uxx + D55uyy ) = 0 (x, y) ∈ G
(1.1)
u(x, y) = g(x, y) (x, y) ∈ ΓD
(1.2)
σn = (D66ux nx + D55uy ny ) = 0 (x, y) ∈ ΓN (ΓN ∪ ΓD = ∂G)
(1.3)
4

7. Numerical method

8. FEM formulation of the direct problem
Equivalent form of the equation (1.1)
cω
Z
G
uv d x d y + D66
Z
G
ux vx d x d y + D55
Z
G
uy vy d x d y =
=
Z
ΓD
vσn d Γ +
Z
ΓN
vσn d Γ
for arbitrary continous function v(x, y). We solve it approximately with
Finite Element method on a triangular grid
u =
X
i∈I
ui hi +
X
k∈D
gk hk , v ∈ {hi , i ∈ I}
where hi are 1-st degree basis polynomials, hi (xi ) = 1, D – grid vertices
on ΓD, I – all other vertices, gk = g(xk , yk )
5

9. FEM formulation of the direct problem
Numerical solution of the direct problem
K̃(ω, θ)u = ˜
f (ω, θ) (2.1)
K̃(ω, θ) = cωK0 + D66Kx + D55Ky
[K0]ij =
Z
G
hi hj d x d y
[Kx ]ij =
Z
G
hi,x hj,x d x d y
[Ky ]ij =
Z
G
hi,y hj,y d x d y
˜
f (ω, θ) = cωf0 + D66fx + D55fy
[f0]i = −
X
k∈D
gk
Z
G
hi hk d x d y
[fx ]i = −
X
k∈D
gk
Z
G
hi,x hk,x d x d y
[fy ]i = −
X
k∈D
gk
Z
G
hi,y hk,y d x d y
cω = −ρω2
+ iγ0ω
6

10. Inverse problem
In the experiment, the amplitude u is sampled at a test point (xt, yt) for
different frequencies ωk . Thus we obtain the amplitude-frequency
characteristic uexp(ωk )
Figure 2: Sample AFC
The goal is to obtain values of the parameters from these data 7

11. Inverse problem
We propose the following optimization problem:
min
θ
L(θ) =
Nω
X
k=1
k[u(θ, ωk )]t − uexp(ωk )k
2
(2.2)
s.t. θi > 0
Here u(θ, ωk ) is the solution of K̃(θ, ωk )u = ˜
f (θ, ωk )
The evaluation of L(θ) requires only standard linear algebra, thus we can
use fast automatic differentiation to obtain derivatives efficiently
8

12. Trust-region method
Trust-region subproblem
pk = arg min
p
fk + gT
k p +
1
2
pT
Bk p (2.3)
s.t. kpk k ≤ ∆k
(fk = f (xk ), gk = ∇f (xk ))
Newton-Gauss method:
Bk = ∇2
f (xk )
BFGS:
y = gk+1 − gk
s = xk+1 − xk
Bk+1 = Bk +
yyT
sT y
−
sT
BT
k Bk s
sT Bk s
9

13. Trust-region method
Algorithm 1 Trust-region method
Require: x0 — initial guess, ∆max > 0, ∆0 ∈ (0, ∆max ), η ∈ [0, 1
4
)
1: repeat
2: Evaluate pk as the solution of (2.3)
3: ρk = f (xk )−f (xk +pk )
mk (0)−mk (pk )
{relative improvement}
4: if ρk < 1
4
then
5: ∆k+1 = 1
4
∆k
6: else
7: if ρk > 3
4
and kpk k = ∆k then
8: ∆k+1 = min(2∆k , ∆max )
9: else
10: ∆k+1 = ∆k
11: if ρk > η then
12: xk+1 = xk + pk
13: Update the quadratic model
14: else
15: xk+1 = xk
16: until checkStopCondition() or k ≥ kmax 10

14. Numerical experiments

15. Setup
(a) (b)
Figure 3: Two experimental geometries, used for calculations
• Simple geometry, ∼ 300 vertices, 201 frequencies
• Reference AFC is evaluated for some θbase. Initial guess θ0 with
every value about 30% different 11

16. Convergence
1000 1100 1200 1300 1400 1500
15
10
5
0
5
10
15
(u(
))
Initial
Optimized
Reference
1000 1100 1200 1300 1400 1500
30
25
20
15
10
5
0
(u(
))
Initial
Optimized
Reference
Figure 4
Figure 5: AFCs for reference, initial and optimized parameters, geometry (a)
12

17. Convergence
1000 1100 1200 1300 1400 1500
6
4
2
0
2
4
6
(u(
))
Initial
Optimized
Reference
1000 1100 1200 1300 1400 1500
10
5
0
5
10
(u(
))
Initial
Optimized
Reference
Figure 6
Figure 7: AFCs for reference, initial and optimized parameters, geometry (b)
13

18. Convergence
0 10 20 30 40 50
Iteration number, k
10 6
10 4
10 2
100
L(
k
)
(a)
0 10 20 30 40 50
Iteration number, k
10 7
10 5
10 3
10 1
101
L(
k
)
(b)
Figure 8: Behavior of the loss function during the iterations
14

19. Convergence
0 10 20 30 40
Iteration number, k
10 6
10 5
10 4
10 3
10 2
10 1
Relative
error
D66
D55
15

20. 0 10 20 30 40 50
Iteration number, k
10 7
10 6
10 5
10 4
10 3
10 2
10 1
Relative
error D66
D55
Figure 10: Relative error during the iterations, geometry (b) 16

21. Discussion of the results
Results:
• Possibility to use both first
and second order methods
• Code is just-in-time compiled
for GPU
• Fast convergence achieved for
a rather decent perturbation
(30%)
Problems:
• Global convergence only to
local minimum, which are
possibly multiple
• Convergence in D55 is by
order of magnitude worse
• Geometries with larger
number of DOF might
exhaust GPU memory
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22. Thank you for your attention!
aksenov.vv@phystech.edu
17