Ultrasonic TOFD Technique for Cracks Sizing and Locating Based on PSO
Yacef Nabil Jijel University
NDT Lab Jijel, Algeria [email protected]
Bouden Toufik Jijel University
NDT Lab Jijel, Algeria [email protected]
Grimes Morad Jijel University
NDT Lab Jijel, Algeria [email protected]
Abstract—Ultrasonic Non-destructive testing has been widely used in industry to detect, characterize and size defects in materials. In this paper, an ultrasonic testing technique and an ultrasonic signal processing method are used to size and locate flaws in materials. The ultrasonic testing technique is based on determination of the time of flight of diffracted echoes from the defect edges (time of flight diffraction TOFD). To improve the arrival time resolution of a TOFD signal, an estimation technique based on Particle Swarm Optimization (PSO) and Matching Pursuit decomposition (MP) is proposed. The finite element method (FEM), using the ABAQUS software package, is employed for modeling the TOFD technique in a two-dimensional geometry.
Simulation and experimental results proved the efficiency of the proposed method.
Keywords—Ultrasonic testing; Time of flight diffraction; Wave propagation; Finite element methode; Particle swarm optimization;
Crack.
I. INTRODUCTION
In ultrasonic non-destructive testing (NDT) many techniques have been used for the determination of defects in materials and components in many industrial areas. The operating principle of these methods is based on the measurement of the time of flight (ToF), which is, the time that took an ultrasonic wave to travel from the transmitter to the receiver after being reflected from the target. The methods that known of its capability in detecting and sizing cracks in a wide variety of locations and orientations in many materials used in engineering [1] is the TOFD technique.
In order to get the highest possible accuracy in determining the depth and the defects length, the TOFD method was used.
In this work, we study the TOFD technique for accurate cracks sizing in cement paste. Firstly, the principle of the TOFD method will be introduced. Then, for accurate crack size measurement, an arrival time estimation technique based on PSO and MP is proposed. After that, the finite element package ABAQUS is used to model TOFD technique [2]. Finally, simulation and experimental results are used to validate the proposed method.
II. TIME-OF-FLIGHT DIFFRACTION TECHNIQUE
The TOFD Diffraction technique is based on the
measurement of time of arrival of the echoes diffracted by the defect edges. The basic principle is illustrated in Fig. 1.
The transmitter Tx emits an ultrasonic pulse into the specimen.
This propagated wave spread out into a beam with a certain angle. When the wave encounters a defect (such as crack) it will be scattered by it. The scattering energy from the edges of the crack can be captured by the receiver Rx. If the crack length is sufficient, the two diffracted signals from the top and the bottom tips of the crack can be separated in time and can be recognized.
As well as these two signals, another two kinds of wave are generated. The lateral wave, which travels from the transmitter to the receiver directly along the surface of the specimen and the back wall echo, which travel through the specimen to the receiver.
Fig. 1. Basic principle of the TOFD technique Lateral wave
Backwall echo Transducer
beam waves
Diffracted waves θ
h
2S
a h
Tx Rx
d
Backwall echo Crack top
echo
Crack bottom echo Lateral wave
In Fig. 1, h denotes the specimen thickness, 2S the distance between the two probes (Tx and Rx), d the depth of the top of the crack from the inspection surface, a the crack length.
Supposing, that the crack is oriented in a plane perpendicular to the inspection surface and the crack is located in the middle between the transmitter and the receiver. The crack through-wall size and depth can be calculated using Pythagoras’s theorem [1]
𝑑 = 1
2√𝑉𝑝2(𝑡1− 2𝑡0) − 4𝑆2 (1) 𝑎 = 1
2√𝑉𝑝2(𝑡2− 2𝑡0) − 4𝑆2− 𝑑 (2) Where t1 and t2 are the arrival times of the signals diffracted by the extremities of the crack, t0 is the delay time caused by a wedge and Vp is the velocity of the ultrasonic wave in the specimen.
III. ARRIVAL TIME ESTIMATION
A. Gaussian Echo Model
In ultrasonic testing, the reflected echo from a single reflector can be modeled as [3]:
s(θ,t) = β e−α(t− τ)2cos (2πfc(t−τ) + φ) (3) θ = [α τ fc φ β]
Where θ represents the parameter vector including the parameters, bandwidth factor (α), time of arrival (τ), center frequency (fc), phase (φ), and amplitude (β).
In practical ultrasonic testing, the reflected signal usually does not merely include one echo, so the detected ultrasonic signal can be modeled as multiple superimposed Gaussian echoes y(t)=∑s(θm,t)
M
m=1
+ ν(t) (4)
M is the number of superimposed Gaussian echoes and ν(t) is the additive WGN process.
The objective is to estimate the vector parameters θ1, θ2, …θM
and the model order M from the noisy observation of echoes (4).
TOFD technique focus on determination of the arrival time (τ) of diffracted waves from the crack tips. Many estimation techniques have been used to measure the time of arrival of waves [4-6].
In this paper, MP based on PSO is proposed to overcome this estimation problem. MP can decompose the ultrasonic signal into a linear expansion of Gabor functions (Gabor dictionary).
The parameters vector of these functions can be estimated by implementing the MP algorithm using PSO.
B. Matching Pursuit Decomposition
Matching pursuit decomposition (MP) is an iterative algorithm, introduced by Mallat and Zhang [7], which can decompose any given signal into a set of waves taken from a dictionary.
The MP algorithm decomposes a signal S using a dictionary D belonging to the Hilbert space H. The MP offers an approximation of the signal S as a linear expansion in terms of functions gi called atoms chosen from a complete dictionary.
We define the complete dictionary as a family
D = {gi; i = 0, 1,…, L} of vectors in H, such as ||gi||=1.
The first step of the MP, the atom g0 which best matches the signal S is chosen from dictionary D. In each of the consecutive steps, the atom gm is matched to the signal RmS, which is the residual left after subtracting results of previous iterations:
{
R0S = S RmS = 〈RmS,gm〉gm+ Rm+1S gm= arg maxgi∈D|〈RmS,gi〉|
(5)
After M iterations, the Matching Pursuit decomposes the signal S into:
S =∑〈RmS,gm〉gm+ RmS
M-1
m=0
(6)
C. Particle Swarm Optimization
PSO was developed by Kennedy and Eberhart [8] and has been successfully applied to various optimization problems.
The algorithm is inspired from the nature social behavior and dynamic movements with communications of insects, birds and fish to combine self-experiences with social experiences. It uses a number of agents (particles) that constitute a swarm moving around in the search space looking for the best solution.
Each particle in the swarm has current position pi, the previous best position pbesti, and the velocity vi.
The PSO search for the optimal solution by updating generations using a fitness function. The update equation
vi = wvi + c1γ1(pbesti−pi) + c2γ2(gbest−pi) (7) pi = pi + vi (8)
The vectors pi and vi are the current position and velocity of the i-th particle in the swarm. The swarm consists of N particles, pbesti is the best position of the particle and ɡbest is the best position of the swarm. The parameters γ1,2 ϵ [0,1] are uniformly distributed random values. c1 is the weight of local information and c2 is the weight of global information. w is the inertia factor which gives rises to a certain momentum of the particles.
The condition for the stability of the swarm are [9]
c1 + c2 < 4 (9) And
c1 + c2
2 −1 < w < 1 (10) The original PSO algorithm is summarized by the following steps [10]
Step 1. Initialize a population array of particles with random positions and velocities on D dimensions in the search space.
Step 2. For each particle, evaluate the desired optimization fitness function in D variables.
Step 3. Compare particle’s fitness evaluation with its pbesti. If current value is better than pbesti, then set pbesti equal to the current value, and pi equal to the current location in D- dimensional space.
Step 4. Identify the particle in the neighborhood with the best success so far, and assign its index to the variable gbest.
Step 5. Change the velocity and position of the particle according to (7) and (8).
Step 6. If a criterion is met (usually a sufficiently good fitness or a maximum number of iterations), stop the algorithm, if not, go to Step 2.
D. Matching Pursuit using Particle Swarm Optimization The MP algorithm use Gabor dictionary atoms, which the best matching atoms to the signal are selected. Then we get the residue signal by subtracting the selected atoms from the original signal. The role of PSO is to estimate the parameters of these atoms, therefore we define the fitness function as the inner product between atoms of the dictionary and the residue signal [10].
fitness = <RmS, gm> (11) The atom owning the largest inner product with the residue signal will be selected. Gabor elementary function is defined as gm(t) = Kmβme− αm(t− τm)2cos(2πfcm(t−τm) + ϕm) (12) Where Km is such that ||gm||=1. Writing Km explicitly gives
gm(t) = βme− αm(t− τm)2cos(2πfcm(t−τm) + ϕm)
‖βme− αm(t− τm)2cos(2πfcm(t−τm) + ϕm)‖ (13) Using the trigonometric identity, we write the Gabor function as gm(t) = βm[g1cos(ϕm) −g2sin(ϕm)]
‖βm[g1cos(ϕm) −g2sin(ϕm)]‖ (14) Where
g1= e− αm(t− τm)2cos(2πfcm(t−τm)+ ϕm) g2 = e− αm(t− τm)2sin(2πfcm(t−τm)+ ϕm)
The following list summarizes the steps of the algorithm [11]
Step 0. Set the first residue signal equal to the original signal.
Step 1. Define fitness function and parameters.
Step 2. Initialize the positions and the velocity of particles in the Swarm.
Step 3. Compute and evaluate fitness function for every particle to find the best particle in the Swarm.
Step 4. Checking PSO conditions, if they are satisfied, go to Step 6, if they aren’t, continue.
Step 5. Update the particles velocity and position, go to Step 3.
Step 6. Store estimated parameters of Gabor function.
Step 7. Calculate the next residue signal.
Step 8. Calculate the residue’s energy, if it is less than setting value, then stop; if not, calculate the new fitness function, go to Step 2.
Updating the fitness function in Step 8 is based on (11).
This algorithm was implemented and evaluated in MatLab 8.5.0 (R2015a) using a laptop (Intel® Core™ i5-3230M CPU 2.6 GHz with 4 Gbytes of RAM).
IV. NUMERICAL SIMULATION AND EXPERIMENTAL RESULTS
A. Modeling of the TOFD technique
to simulate the ultrasonic wave propagation in the time of flight diffraction technique, the FEM package ABAQUS is used.
The piezoelectric transmitter was modelled by the following equation [12]:
F(t) = [1− cos(2πf
N t)]cos(2πft) for 0 ≤ t ≤N
f (15) where f is the excitation frequency and N is the number of cycles. In the simulation model f = 2 MHz and N = 3. The excitation pulse is shown in Fig. 2.
Fig. 2. Ultrasonic excitation wave
The probe wedge, make the propagation of the excitation wave inside the material at the require angle (steering angle). The steering angle θs can be made using a time delay between the adjacent elements of the phased array transducer [13]:
∆t = dsinθs
V (16) where, d is the distance between two adjacent elements, θs is the steering angle of the ultrasonic beam, V is the wave speed in the media, and Δt is the time delay between two adjacent elements.
The simulated specimen is a cement block of 40 mm thickness (h) with a vertical internal crack. The crack length is 10 mm and its width is 0.5 mm. The FEM model is shown in Fig. 3, where the material with defect is discretized into number of elements (nodes). The transducers (Tx, Rx) are modelled with an equivalent cosine force (15) and the wedges are modelled by applying linear time delayed between the transducers nodes (16).
The cement properties are Young’s module = 40 GPa, Poisson ration = 0.33 and mass density = 2200 Kg/m3.
The transducer length is 10 mm. 4-node, 2D plane strain elements are used for the model discretization (mesh) with the element size Δx = 0.14 mm and the time step is 0.01 μs.
Fig. 3. FEM model for TOFD technique
Fig. 4 shows snapshots taken from the specimen at 6 and 10 μs. In Fig. 4, the longitudinal and shear waves are generated inside the material and the lateral wave propagates along the surface. The diffracted waves are generated at the crack tip when the longitudinal wave interacts with the crack.
6 μs
10 μs
Fig. 4. Snapshots of wave interaction with vertical crack
The lateral wave, the diffracted wave and backwall echo can be detected by the receiving transducer. Fig. 5 shows the simulated A-scan signal where the detected waves can be spotted.
MP based on PSO is used to estimate the simulated signal and extract the time-of-flight of the different wave in this signal.
The optimal parameters for the estimation algorithm are the following: Swarm size = 30, Iteration number = 1000, c1 = 1.2, c2 = 0.012, w = 0.0004. Fig. 6 shows the estimation results of the proposed algorithm,
B. Experimental results
The TOFD experimental system is set up as shown in Fig. 7.
The center frequency of the transducers is 1 MHz; the wedge angle is 45°. The specimen is cement paste plate of 50 mm thickness (h).
Fig. 5. Simulation A-scan signal
Fig. 6. Simulation A-scan signal and its estimation signal
The system consists of an ultrasonic Pulser/Receiver which provide an electrical pulse to the transmitter, a digital
oscilloscope (Tektronix TDS 1002) for signal collection and visualization and PC with WaveStar software for data- acquisition.
Fig. 7. Experimental set up for TOFD
The collected A-scan signal from this setup is plotted in Fig. 9. The proposed method is used to estimate and extract the time of flight for each wave in this signal. The estimation results are displayed in Fig. 10.
2S = 66 mm
Tx Rx
60°
Tip diffracted waves wave Lateral wave
Backwall wave
Computer
Oscilloscope
Pulser/Receiver
Specimen with crack
Tip diffracted waves wave Lateral wave
Backwall wave Fig. 8. Schematic diagram of experiment
Fig. 9. A-scan signal for crack cement block.
Fig. 10. Experimental A-scan signal and its estimation signal.
To calculate the reel time of flight of ultrasonic waves diffracted by the crack, the delay time 2t0 caused by de wedge must be obtained using an experiment where the transmitter and the receiver are placer face to face (sensor-to-sensor) [14].
The A-scan signal collected from sensor-to-sensor setup is displayed in Fig. 11.
From Fig. 6 and Fig. 10, it can be seen that the proposed technique gives an accurate measurement of the time of flight.
Table I shows the estimation result of the crack size using the
Fig. 11. A-scan for the sensor to sensor directly.
time of flight obtained by the proposed method. The estimation errors for both simulation and experimental tests, prove that we can rely on the proposed method to provide good results and extract the time of flight automatically without using the conventional or the manual method.
TABLEI
CRACK SIZE MEASUREMENT RESULTS
V. CONCLUSION
In this study, a crack size estimation method based on TOFD combined with MP and PSO was presented. The estimation algorithm was used to get directly the time of flight of the ultrasonic signal diffracted by the crack tip.
The ABAQUS software package has been used to simulate the TOFD technique on cement paste with an internal defect. An experimental study was carried out on the same material. The simulation and experimental results confirm the good performance of the proposed method.
A further study could improve the accuracy of the crack sizing in different material with different crack location.
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