Citation: Wu, Y.; Pei, C.; Zhang, H.;

Liu, Y.; Jia, P. A Fast Finite Element

Simulation Method of Phased Array

Ultrasonic Testing and Its

Application in Sleeve Fillet Weld

Inspection. Appl. Sci. 2022, 12, 5384.

https://doi.org/10.3390/

app12115384

Academic Editors: Sunil

Kishore Chakrapani and

Yiming Deng

Received: 9 April 2022

Accepted: 20 May 2022

Published: 26 May 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2022 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

applied sciences

Article

A Fast Finite Element Simulation Method of Phased ArrayUltrasonic Testing and Its Application in Sleeve FilletWeld InspectionYuxuan Wu 1, Cuixiang Pei 1,*, Hongbo Zhang 2, Yan Liu 2 and Pengjun Jia 2

1 State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Engineering ResearchCenter of NDT and Structural Integrity Evaluation, Xi’an Jiaotong University, Xi’an 710049, China;wyxuan@stu.xjtu.edu.cn

2 CNPC Tubular Goods Research Institute, Xi’an 710077, China; zhanghongbo002@cnpc.com.cn (H.Z.);liuyan019@cnpc.com.cn (Y.L.); jiapengjun@cnpc.com.cn (P.J.)

* Correspondence: pei.cx@xjtu.edu.cn

Abstract: Numerical simulation can provide quantitative information on ultrasonic beam propagationand plays an important role in analyzing its detection ability and in optimizing the correspondingparameters of the phased array ultrasonic testing (PAUT). In this paper, a fast finite element simulationmethod of PAUT is developed using an improved explicit integration algorithm and a non-zeroelement compressed storage method. The new method is applied for the simulation of PAUT ofa type-B sleeve weld, and compared with the commercial finite element software and experimentresults. We found that the computation time and memory consumption of the new method is onlyabout a 15th and a 40th of the commercial finite element software, respectively.

Keywords: phased array ultrasonic testing; fast simulation method; finite element method; type-Bsleeve fillet weld

1. Introduction

Phased array ultrasonic testing (PAUT) technology can flexibly deflect and focusthe beam through electronic scanning, so as to realize the rapid imaging detection ofthe internal defects of materials [1,2]. Compared with the traditional ultrasonic testingtechnology, it has the advantages of a faster detection speed, higher resolution, and beingmore suitable for the detection of complex parts [3–6]. At present, it has been widely usedin industrial nondestructive testing fields such as pipeline welds. However, during theimplementation of type-B sleeve welding repair and subsequent pipeline operation, somewelding defects often appear at the fillet weld [7], which affects the application of type-Bsleeve repair technology in the defect repair project of large-diameter and high-strengthpipelines. Therefore, the integrity of the type-B sleeve fillet weld structure needs to be testedand evaluated using the phased array ultrasonic testing method [8,9]. As a result of thespecial welding structure of the type-B sleeve, it is still difficult to detect and quantitativelyevaluate the weld defects using phased array ultrasound testing. In order to further studythe detection ability of the phased array ultrasonic detection method for internal defects ofwelding structures such as the type-B sleeve weld, using the numerical simulation methodcan simulate the propagation characteristics of the phased array ultrasonic in this structure,analyze its detection ability, and optimize its relevant parameters, which can greatly reducethe blindness and workload of the test [10,11].

In the field of numerical simulation of phased array ultrasonic testing, CIVA basedon the wave line method is one of the most commonly used simulation softwares [12–14].Based on geometric acoustics, this simplified approximation method has the advantagesof having a shorter calculation time and easy implementation [15], while it ignores thefluctuation of sound, resulting in a low accuracy and reliability. Compared with the wave

Appl. Sci. 2022, 12, 5384. https://doi.org/10.3390/app12115384 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 5384 2 of 11

line method, the finite difference time domain method (FDTD) calculated using MATLABhas a higher accuracy [16]. This method obtains the approximate solution of the differentialequation by configuring the discrete grid nodes in a continuous region in a certain way,and directly expanding the differential operator with truncated Taylor series on the gridnodes [17]. However, the calculation accuracy of this method is still not high enough, andit is difficult to deal with the problems of the free interface and complex structure [18].Compared with the wave line method and FDTD, the finite element method (FEM) and thecommercial software based on it, such as COMSOL [19] and Abaqus [20,21], have a highercalculation accuracy and adaptability [22,23]. However, in the process of ultrasonic finiteelement numerical simulation, the element size is usually less than a 10th of the wavelength,resulting in a large amount of calculations and storage, which limits its practical applicationof the present ultrasonic wave simulation method based on FEM.

In order to solve the above problem, a fast FEM-based simulation method of ultrasonictesting is developed in this paper. Using a new explicit integration algorithm and non-zeroelement one-dimensional compressed storage of the coefficient matrixes, the amount ofcalculation and storage of the new fast FEM-based simulation method can be significantlyreduced. To verify the calculation efficiency and accuracy of the new method, it is appliedfor the simulation of PAUT of the type-B sleeve weld and compared with the commercialFEM software and experiment results. These studies are shown in detail in the followingthree sections: Fast Finite Element Simulation Method for PAUT, Simulation Results andComparison, and Conclusion.

2. Fast Finite Element Simulation Method for PAUT2.1. Fast Finite Element Simulation Method

In ultrasonic testing, the probe generates an impulse force on the surface of thespecimen under the excitation of pulse voltage, so as to generate an ultrasonic wave inthe specimen. According to the basic theory of elasticity, the basic control equation of anultrasonic wave in solid materials can be obtained as follows [24]:

µ∇2u + (λ + µ)∇(∇ · u) + f = ρ..u + γ

.u (1)

where λ and µ are the Lame elastic constant of the material, ρ is the density of the material,γ is the ultrasonic damping coefficient, u is the particle displacement vector, and f is theexcitation force vector.

According to the basic principle of the finite element method, the above differential con-trol equation can be transformed into the following discrete finite element control equation:

[M]{ ..

U}+ [D]

{ .U}+ [K]{U} = {F} (2)

where [M] is the mass matrix, [D] is the damping matrix, [K] is the stiffness matrix, {F} isthe node load vector, and {U} is the displacement vector of the node to be solved. In thecalculation of the wave problem, the consistent (coordinated) mass matrix of an elementcan be transformed into a diagonally agglomerated mass matrix to obtain more reliablecalculation results. It is assumed that the mass of the element is agglomerated at the nodeof the element; that is, the element mass matrix becomes a diagonal matrix [25]:

Meij =

8∑

k=1Me

ik (i = j)

0 (i 6= j)(i, j = 1, . . . , 8) (3)

Equation (2) is actually a discrete dynamic initial value problem. For the generaldynamic initial value problem, the time-domain step-by-step integration algorithm canbe used to solve it. The current time-domain integration algorithms are roughly di-vided into two categories, namely the implicit integration method and explicit integrationmethod [26,27]. The implicit integration method is characterized by a high computational

Appl. Sci. 2022, 12, 5384 3 of 11

stability and unconditional stability, but it needs to solve the coupled linear equations,which require a large amount of storage and calculations. The characteristic of the explicitintegration method is that it does not need to solve the coupled linear equations, and can becombined with one-dimensional compressed storage technology. Its storage and calculationcan be greatly reduced, but its calculation is conditionally stable. In order to further reducethe amount of storage and calculations in the process of ultrasonic numerical simulationand to meet the need for more than second-order calculation accuracy in engineeringapplications, a new and more concise explicit integration improved algorithm based on thecentral difference method [26] and Newmark average velocity method [27] was establishedas follows:

{ .U}

t+∆t={ .

U}

t−∆t− 2∆t

[C̃]{ .

U}

t− 2∆t

[K̃]{U}t + 2∆t[M]−1{F}t

{U}t+∆t = {U}t +

{ .U}

t+∆t+{ .

U}

t2 ∆t

(4)

where[K̃]= [M]−1[K],

[C̃]= [M]−1[C]. It can be seen that solving Equation (4) does

not need to solve the coupled linear equations, and the amount of calculations are greatlyreduced compared with the traditional implicit integration method. Compared withthe existing explicit integration method (Lee’s method) [28], the storage and calculationamounts in the calculation process are also reduced by about half.

All numerical algorithms must have more than second-order accuracy in order to meetthe needs of practical engineering applications [29]. Therefore, in order to verify that thecalculation accuracy of the improved explicit integration algorithm can meet the actualneeds, we used the single free system as an example to perform a preliminary analysisof the accuracy and stability of this method. Firstly, it is assumed that a single degree offreedom system satisfies the following dynamic equation:

..x(t) + 2ξω

.x(t) + ω2x(t) = f (t) (5)

According to Equation (4), the following relationship is obtained in the case of a singledegree of freedom:

.xt+∆t = −2∆tω2xt − 4∆tξω

.xt +

.xt−∆t + 2∆t ft + o1 (6)

xt+∆t = xt +∆t2(

.xt+∆t +

.xt) + o2 (7)

where o1 and o2 represent the calculation errors of velocity and displacement introducedby the difference schemes, respectively. According to the exact expression of the centraldifference method and Taylor expansion,

.xt+∆t −

.xt−∆t

2∆t=

..xt + O

(∆t3)

(8)

xt+∆t = xt + ∆t.xt +

∆t2

..xt + O

(∆t3)

(9)

Substitute Equation (8) into Equation (6) to obtain o1= O(∆t4), and substitute

Equation (9) into Equation (7) to obtain o2= O(∆t3). Therefore, the calculation cut-off

error of Equation (5) is between O(∆t3) and O

(∆t4) in the case of a single degree of free-

dom. Through the above theoretical analysis, it can be seen that the calculation accuracy ofthe improved time-domain explicit integration algorithm is between the second-order andthe third-order, and may even reach third-order calculation accuracy. It can be seen that,compared with the traditional explicit integration method (Lee’s method), this method notonly needs less storage and calculations, but also improves the calculation accuracy, to acertain extent.

Appl. Sci. 2022, 12, 5384 4 of 11

Moreover, this paper combines the non-zero element one-dimensional compressedstorage method with the new explicit integration algorithm to minimize the amount ofcalculations and storage, because the coefficient matrixes

[K̃]

and[C̃]

in Equation (4)are sparse matrixes. Most of the elements in the matrixes are zero elements. As shownin Figure 1, only the non-zero elements in the coefficient matrix are stored, and theircorresponding row and column numbers are recorded. Compared with the traditionalhalf bandwidth storage method, the storage capacity of this method is further reduced. Atthe same time, when solving Equation (4), only non-zero elements need to be calculated.Therefore, the number of calculations is also greatly reduced.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 13

Substitute Equation (8) into Equation (6) to obtain ( )41=Oo tΔ , and substitute Equa-

tion (9) into Equation (7) to obtain ( )32 =Oo tΔ . Therefore, the calculation cut-off error

of Equation (5) is between ( )3O tΔ and ( )4O tΔ in the case of a single degree of free-

dom. Through the above theoretical analysis, it can be seen that the calculation accuracy of the improved time-domain explicit integration algorithm is between the second-order and the third-order, and may even reach third-order calculation accuracy. It can be seen that, compared with the traditional explicit integration method (Lee’s method), this method not only needs less storage and calculations, but also improves the calculation accuracy, to a certain extent.

Moreover, this paper combines the non-zero element one-dimensional compressed storage method with the new explicit integration algorithm to minimize the amount of

calculations and storage, because the coefficient matrixes K and C

in Equation

(4) are sparse matrixes. Most of the elements in the matrixes are zero elements. As shown in Figure 1, only the non-zero elements in the coefficient matrix are stored, and their cor-responding row and column numbers are recorded. Compared with the traditional half bandwidth storage method, the storage capacity of this method is further reduced. At the same time, when solving Equation (4), only non-zero elements need to be calculated. Therefore, the number of calculations is also greatly reduced.

Figure 1. Schematic diagram of the non-zero element compressed storage method for sparse matrix.

2.2. Numerical Model of PAUT of Type-B Sleeve Fillet Weld In the process of welding repair and the application of the type-B sleeve, the fillet

weld has welding defects, which are tested using the phased array ultrasonic testing method. During detection, the phased array ultrasonic probe is placed on the sleeve, and the internal defects in the triangular area of the weld can be detected through phased array ultrasonic sector scanning. According to the basic principle of phased array ultrasonic testing and the phased array ultrasonic testing method of the type-B sleeve fillet weld, a two-dimensional numerical calculation model of the phased array ultrasonic testing of the type-B sleeve fillet weld defects was established, as shown in Figure 2. The phased array ultrasonic excitation source can be approximated as multiple impulse Pi(t) loads arranged according to a certain delay sequence, and the expression of the ith excitation load can be written as follows:

Figure 1. Schematic diagram of the non-zero element compressed storage method for sparse matrix.

2.2. Numerical Model of PAUT of Type-B Sleeve Fillet Weld

In the process of welding repair and the application of the type-B sleeve, the fillet weldhas welding defects, which are tested using the phased array ultrasonic testing method.During detection, the phased array ultrasonic probe is placed on the sleeve, and the internaldefects in the triangular area of the weld can be detected through phased array ultrasonicsector scanning. According to the basic principle of phased array ultrasonic testing and thephased array ultrasonic testing method of the type-B sleeve fillet weld, a two-dimensionalnumerical calculation model of the phased array ultrasonic testing of the type-B sleeve filletweld defects was established, as shown in Figure 2. The phased array ultrasonic excitationsource can be approximated as multiple impulse Pi(t) loads arranged according to a certaindelay sequence, and the expression of the ith excitation load can be written as follows:

Pi(t) =

{A · sin[2π f · (t− ∆ti)] 0 ≤ t− ∆ti ≤ 1/ f

0 Others(10)

where A is the peak value of the excitation signal amplitude, f is the excitation signalfrequency, and ∆ti is the delay time of the ith excitation signal. The size array elementposition xi, phased array ultrasonic focusing depth df, deflection angle θ, and ultrasonicwave velocity v can be written as follows:

∆ti =(√

d f2 + xi

2 − 2 ∗ d f ∗ xi ∗ cos(θ + π/2)− d f

)/v (11)

Appl. Sci. 2022, 12, 5384 5 of 11

Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 13

sin[2 ( )] 0 1/( )

0i i

i

A f t t t t fP t

Othersπ⋅ ⋅ − Δ ≤ − Δ ≤

=

(10)

where A is the peak value of the excitation signal amplitude, f is the excitation signal fre-quency, and itΔ is the delay time of the ith excitation signal. The size array element posi-tion xi, phased array ultrasonic focusing depth df, deflection angle θ, and ultrasonic wave velocity v can be written as follows:

( )2 2 2* * *cos( / 2) /i f i f i ft d x d x d vθ πΔ = + − + − (11)

Figure 2. Numerical model of PAUT of the type-B sleeve fillet weld defects.

The size of each array element of the phased array ultrasonic probe used in this paper was 0.6 mm, the spacing between each array element was 0.2 mm, the center frequency of the probe was 2 MHz, and the number of excited array elements was 16. The type-B sleeve and fillet weld were made of X70 pipeline steel, with an elastic modulus of 210 GPA, Pois-son’s ratio of 0.3, and density of 7.85 g/mm3. The chemical compositions of the X70 pipe-line steel are shown in the Table 1. The thickness of the pipe and sleeve was 12.8 mm and 20 mm, respectively, and the gap between them was 2 mm.

Table 1. Chemical compositions of the X70 pipeline steel and the weld.

Element C Mn P S Proportion of mass

fraction (%) 0.06 1.12 0.01 0.004

3. Simulation Results and Comparison 3.1. Simulation of Phased Array Ultrasonic Sound Field and Detection Signal

In order to study the reliability and efficiency of the fast finite element numerical simulation method, a numerical code based on this method was developed in Fortran Language. This fast FEM code was used to simulate the phased array ultrasonic testing of the internal defects of the type-B sleeve fillet weld, as shown in Figure 3. The material of the type-B sleeve and its weld was X70 stainless steel. The parameters of the simulation model are shown in Table 2. To validate the reliability and efficiency of the new fast FEM code, commercial finite element software was also used with the same simulation model for comparison.

Figure 2. Numerical model of PAUT of the type-B sleeve fillet weld defects.

The size of each array element of the phased array ultrasonic probe used in this paperwas 0.6 mm, the spacing between each array element was 0.2 mm, the center frequencyof the probe was 2 MHz, and the number of excited array elements was 16. The type-Bsleeve and fillet weld were made of X70 pipeline steel, with an elastic modulus of 210 GPA,Poisson’s ratio of 0.3, and density of 7.85 g/mm3. The chemical compositions of the X70pipeline steel are shown in the Table 1. The thickness of the pipe and sleeve was 12.8 mmand 20 mm, respectively, and the gap between them was 2 mm.

Table 1. Chemical compositions of the X70 pipeline steel and the weld.

Element C Mn P S

Proportion of mass fraction (%) 0.06 1.12 0.01 0.004

3. Simulation Results and Comparison3.1. Simulation of Phased Array Ultrasonic Sound Field and Detection Signal

In order to study the reliability and efficiency of the fast finite element numericalsimulation method, a numerical code based on this method was developed in FortranLanguage. This fast FEM code was used to simulate the phased array ultrasonic testing ofthe internal defects of the type-B sleeve fillet weld, as shown in Figure 3. The material ofthe type-B sleeve and its weld was X70 stainless steel. The parameters of the simulationmodel are shown in Table 2. To validate the reliability and efficiency of the new fast FEMcode, commercial finite element software was also used with the same simulation modelfor comparison.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 13

Figure 3. Specimen and numerical simulation model of the B-type sleeve.

Table 2. Parameters of the simulation model.

Parameters Value Material density 7.85 kg/m3

Material elastic modulus 2.1 × 10 N/m2 Material Poisson’s ratio 0.3

Grid size 0.1 mm Number of probe array elements 16

Array element width 0.6 mm Space between array elements 0.2 mm

Thickness of pipe 12.8 mm Thickness of sleeve 20.0 mm

Position of defect relative to probe (22 mm, 15 mm) Diameter of defect 1.0 mm

Figure 4 shows the simulated PA longitudinal wave focusing field with a focusing angle of 56° and a focusing depth of 27 mm at a different given time using the fast FEM code and commercial finite element software. It can be seen that the ultrasonic wave fields simulated by the fast FEM code and commercial finite element software were almost the same as each other. In the ultrasonic wave fields, both the longitudinal wave and the transverse wave could be observed. However, because the velocity of the longitudinal wave was used in the focusing rule of Equations (10) and (11), the longitudinal wave had a much larger amplitude and better focus. The actual focusing angle and depth were in good agreement with the set values. The PA longitudinal wave signal of the defect simu-lated using the fast FEM code and commercial finite element software is shown in Figure 5. It can also be seen that the waveforms of the two signals were very similar to each other.

Figure 3. Specimen and numerical simulation model of the B-type sleeve.

Appl. Sci. 2022, 12, 5384 6 of 11

Table 2. Parameters of the simulation model.

Parameters Value

Material density 7.85 kg/m3

Material elastic modulus 2.1× 1011 N/m2

Material Poisson’s ratio 0.3Grid size 0.1 mm

Number of probe array elements 16Array element width 0.6 mm

Space between array elements 0.2 mmThickness of pipe 12.8 mm

Thickness of sleeve 20.0 mmPosition of defect relative to probe (22 mm, 15 mm)

Diameter of defect 1.0 mm

Figure 4 shows the simulated PA longitudinal wave focusing field with a focusingangle of 56◦ and a focusing depth of 27 mm at a different given time using the fast FEMcode and commercial finite element software. It can be seen that the ultrasonic wave fieldssimulated by the fast FEM code and commercial finite element software were almost thesame as each other. In the ultrasonic wave fields, both the longitudinal wave and thetransverse wave could be observed. However, because the velocity of the longitudinalwave was used in the focusing rule of Equations (10) and (11), the longitudinal wave had amuch larger amplitude and better focus. The actual focusing angle and depth were in goodagreement with the set values. The PA longitudinal wave signal of the defect simulatedusing the fast FEM code and commercial finite element software is shown in Figure 5. Itcan also be seen that the waveforms of the two signals were very similar to each other.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 13

(a)

(b)

Figure 4. Simulated PA longitudinal wave focusing fields with a focusing angle of 56°and focusing depth of 27 mm using (a) the fast FEM code and (b) commercial finite element software.

Figure 4. Simulated PA longitudinal wave focusing fields with a focusing angle of 56◦and focusingdepth of 27 mm using (a) the fast FEM code and (b) commercial finite element software.

Appl. Sci. 2022, 12, 5384 7 of 11

Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 13

2 4 6 8 10 12

-1.0

-0.5

0.0

0.5

1.0

1.5

Phased array ultrasonic echo signal

Am

plitu

de

Time(μs)

2 4 6 8 10 12

-0.4

-0.2

0.0

0.2

0.4

0.6

Phased array ultrasonic echo signal

Am

plitu

de

Time(μs) (a) (b)

Figure 5. PA longitudinal wave signal of the defect simulated using (a) the fast FEM code and (b) commercial finite element software.

Figure 6 shows the simulated PA transverse wave focusing field with a focusing an-gle of 56° and focusing depth of 27 mm at different given times using the fast FEM code and commercial finite element software. The velocity of the transverse wave was used in the focusing rule of Equations (10) and (11). It was found that the longitudinal wave al-most disappeared in the ultrasonic field as the focusing angle of the transverse wave was larger than the critical refraction angle. There are only focused transverse waves in the tested specimen during the transverse wave phased array detection. Moreover, when the transverse wave propagated to the location of the defect, it mainly produced an obvious reflected transverse wave rather than an obvious mode converted longitudinal wave sig-nal. These reduced the interference to the identification of the defect transverse wave echo signal. It can also be seen that the ultrasonic wave fields simulated using the fast FEM code and commercial finite element software were almost the same as each other. The PA transverse wave signal of the defect simulated using the fast FEM code and commercial finite element software is shown in Figure 7. It can also be seen that the waveforms of the two signals were very similar to each other.

Figure 5. PA longitudinal wave signal of the defect simulated using (a) the fast FEM code and(b) commercial finite element software.

Figure 6 shows the simulated PA transverse wave focusing field with a focusing angleof 56◦ and focusing depth of 27 mm at different given times using the fast FEM codeand commercial finite element software. The velocity of the transverse wave was usedin the focusing rule of Equations (10) and (11). It was found that the longitudinal wavealmost disappeared in the ultrasonic field as the focusing angle of the transverse wave waslarger than the critical refraction angle. There are only focused transverse waves in thetested specimen during the transverse wave phased array detection. Moreover, when thetransverse wave propagated to the location of the defect, it mainly produced an obviousreflected transverse wave rather than an obvious mode converted longitudinal wave signal.These reduced the interference to the identification of the defect transverse wave echosignal. It can also be seen that the ultrasonic wave fields simulated using the fast FEMcode and commercial finite element software were almost the same as each other. The PAtransverse wave signal of the defect simulated using the fast FEM code and commercialfinite element software is shown in Figure 7. It can also be seen that the waveforms of thetwo signals were very similar to each other.

In the finite element model, the distance between the defect and the center of the PAUTprobe was 26.63 mm. Using the arrival time of the defect signals in Figure 6, the velocity ofboth the longitudinal wave and transverse wave simulated using the fast FEM method andcommercial finite element software could be calculated and compared with the theoreticalvalues, as shown in Table 3. It can be seen that ultrasonic wave velocity from the results ofboth the fast FEM code and commercial finite element software are in good agreement withthe theoretical values.

After verifying the accuracy of the fast finite element numerical simulation method,the rapidity of this method was verified by comparing the storage and calculation timeof the fast finite element simulation program and commercial finite element software. Itcan be seen from Figure 8 that, under the same grid size and time step, the program basedon the fast finite element numerical simulation method occupied less CPU and computermemory and had a higher computational efficiency compared with the commercial finiteelement software.

3.2. Simulation of PAUT Imaging and Experiment Validation

In practical detection, the accuracy of phased array ultrasound imaging is also veryimportant. Therefore, the type-B sleeve fillet weld specimen with defects (in Figure 3)was tested using 16-channel transverse wave phased array ultrasonic testing equipmentto obtain the phased array fan scan image of the actual defects (as shown in Figure 9a).According to the fast finite element numerical simulation program established above,the echo signals of the focused sound field at different angles were simulated, and the

Appl. Sci. 2022, 12, 5384 8 of 11

simulated phased array fan scan imaging was finally generated (as shown in Figure 9b).In the numerical calculation model, the position coordinates of the defect center relativeto the center of the rightmost array element were 22 mm and 15 mm, and in the fan scansimulation imaging results, the position coordinates of the defect imaging center were22.44 mm and 15.13 mm. The absolute error between the defect position of the fan scanimaging and the defect position set by the numerical calculation model was within 0.5 mmin the horizontal and vertical directions, and the relative error was within 2.0% in thesetwo directions. In addition, the fan scan images obtained through the simulation andexperiment in Figure 9 had a high consistency.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 13

(a)

(b)

Figure 6. Simulated PA transverse wave focusing fields with a focusing angle of 56°and focusing depth of 27 mm using (a) the fast FEM code and (b) commercial finite element software.

Figure 6. Simulated PA transverse wave focusing fields with a focusing angle of 56◦and focusingdepth of 27 mm using (a) the fast FEM code and (b) commercial finite element software.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 13

4 8 12 16 20

-1

0

1

2

Phased array ultrasonic echo signal

Am

plitu

de

Time(μs)

4 8 12 16 20

-0.4

0.0

0.4

0.8

Phased array ultrasonic echo signal

Am

plitu

de

Time(μs) (a) (b)

Figure 7. PA transverse wave signal of the defect simulated using (a) the fast FEM code and (b) commercial finite element software.

In the finite element model, the distance between the defect and the center of the PAUT probe was 26.63 mm. Using the arrival time of the defect signals in Figure 6, the velocity of both the longitudinal wave and transverse wave simulated using the fast FEM method and commercial finite element software could be calculated and compared with the theoretical values, as shown in Table 3. It can be seen that ultrasonic wave velocity from the results of both the fast FEM code and commercial finite element software are in good agreement with the theoretical values.

Table 3. Comparison of the results of the ultrasonic wave velocity using the different methods.

Theoretical

Value

Commercial Finite Element Software

Fast FEM Code

Value Error Value Error Longitudinal

wave 6020.18 m/s 5964.17 m/s 0.93% 5875.34 m/s 2.4%

Transverse wave 3217.92 m/s 3196.88 m/s 0.65% 3166.47 m/s 1.6%

After verifying the accuracy of the fast finite element numerical simulation method, the rapidity of this method was verified by comparing the storage and calculation time of the fast finite element simulation program and commercial finite element software. It can be seen from Figure 8 that, under the same grid size and time step, the program based on the fast finite element numerical simulation method occupied less CPU and computer memory and had a higher computational efficiency compared with the commercial finite element software.

Figure 7. PA transverse wave signal of the defect simulated using (a) the fast FEM code and(b) commercial finite element software.

Appl. Sci. 2022, 12, 5384 9 of 11

Table 3. Comparison of the results of the ultrasonic wave velocity using the different methods.

TheoreticalValue

Commercial FiniteElement Software Fast FEM Code

Value Error Value Error

Longitudinalwave 6020.18 m/s 5964.17 m/s 0.93% 5875.34 m/s 2.4%

Transverse wave 3217.92 m/s 3196.88 m/s 0.65% 3166.47 m/s 1.6%Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 13

Percentage ofoccupied CPU

Average occupiedmemory

Computing time0.0

0.5

1.091min

6min

3350MB71.1%

Nor

mal

ized

am

plitu

de

Fast finite element numerical simulation program Commercial finite element software

12.1%

88.8MB

Figure 8. Comparison of the numerical simulation calculation and storage between the fast finite element program and commercial finite element software.

3.2. Simulation of PAUT Imaging and Experiment Validation In practical detection, the accuracy of phased array ultrasound imaging is also very

important. Therefore, the type-B sleeve fillet weld specimen with defects (in Figure 3) was tested using 16-channel transverse wave phased array ultrasonic testing equipment to ob-tain the phased array fan scan image of the actual defects (as shown in Figure 9a). Accord-ing to the fast finite element numerical simulation program established above, the echo signals of the focused sound field at different angles were simulated, and the simulated phased array fan scan imaging was finally generated (as shown in Figure 9b). In the nu-merical calculation model, the position coordinates of the defect center relative to the cen-ter of the rightmost array element were 22 mm and 15 mm, and in the fan scan simulation imaging results, the position coordinates of the defect imaging center were 22.44 mm and 15.13 mm. The absolute error between the defect position of the fan scan imaging and the defect position set by the numerical calculation model was within 0.5 mm in the horizontal and vertical directions, and the relative error was within 2.0% in these two directions. In addition, the fan scan images obtained through the simulation and experiment in Figure 9 had a high consistency.

(a) (b)

Figure 9. Phased array fan scan image in (a) the real experiment and (b) the simulation.

Figure 8. Comparison of the numerical simulation calculation and storage between the fast finiteelement program and commercial finite element software.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 13

Percentage ofoccupied CPU

Average occupiedmemory

Computing time0.0

0.5

1.091min

6min

3350MB71.1%

Nor

mal

ized

am

plitu

de

Fast finite element numerical simulation program Commercial finite element software

12.1%

88.8MB

Figure 8. Comparison of the numerical simulation calculation and storage between the fast finite element program and commercial finite element software.

3.2. Simulation of PAUT Imaging and Experiment Validation In practical detection, the accuracy of phased array ultrasound imaging is also very

important. Therefore, the type-B sleeve fillet weld specimen with defects (in Figure 3) was tested using 16-channel transverse wave phased array ultrasonic testing equipment to ob-tain the phased array fan scan image of the actual defects (as shown in Figure 9a). Accord-ing to the fast finite element numerical simulation program established above, the echo signals of the focused sound field at different angles were simulated, and the simulated phased array fan scan imaging was finally generated (as shown in Figure 9b). In the nu-merical calculation model, the position coordinates of the defect center relative to the cen-ter of the rightmost array element were 22 mm and 15 mm, and in the fan scan simulation imaging results, the position coordinates of the defect imaging center were 22.44 mm and 15.13 mm. The absolute error between the defect position of the fan scan imaging and the defect position set by the numerical calculation model was within 0.5 mm in the horizontal and vertical directions, and the relative error was within 2.0% in these two directions. In addition, the fan scan images obtained through the simulation and experiment in Figure 9 had a high consistency.

(a) (b)

Figure 9. Phased array fan scan image in (a) the real experiment and (b) the simulation. Figure 9. Phased array fan scan image in (a) the real experiment and (b) the simulation.

4. Conclusions

1. In this paper, through combining an improved explicit integration algorithm withthe non-zero element one-dimensional compressed storage method, a fast finite el-ement numerical simulation method for PAUT was established. Based on this newnumerical method, a fast FEM simulation program for PAUT in Fortran Languagewas developed.

Appl. Sci. 2022, 12, 5384 10 of 11

2. To verify the reliability and efficiency of this new fast simulation method and program,a simulation of PAUT of the type-B sleeve weld was realized and compared withthe commercial finite element software and experiment measurements. Throughthe simulation results, it was found that both the ultrasonic fields and measureddefect signals from the fast FEM simulation method and the commercial finite elementsoftware were in good agreement with each other.

3. Furthermore, the computation time and memory consumption of the new method wasonly about a 15th and a 40th of the commercial finite element software, respectively.Because of its high efficiency and low storage space, this method could not onlygreatly shorten the calculation time, but could also be used for the simulation ofphased array ultrasonic testing of large structures, which is difficult to carry outthrough conventional numerical simulations.

4. Finally, the PA fan scan imaging of the defect in the type-B sleeve fillet weld wassimulated using this fast FEM simulation method and compared with the experimentresults. The reliability of this fast numerical simulation method was verified again.At present, this method has only been compiled into a preliminary program andthe operability of this program can be further optimized in the future. In brief,this method has broad application prospects in the field of phased array ultrasonictesting simulations.

Author Contributions: Conceptualization, C.P.; Data curation, H.Z. and P.J.; Formal analysis, C.P.and Y.L.; Funding acquisition, Y.W.; Investigation, Y.W., Y.L. and P.J.; Resources, H.Z.; Writing—original draft, Y.W.; Writing—review & editing, Y.W. and C.P. All authors have read and agreed to thepublished version of the manuscript.

Funding: This research was funded by [the National Key R & D Projects of China] grant number [No.2020YFB1901504].

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments: The authors would like to thank the National Key R & D Projects of China (No.2020YFB1901504) for funding.

Conflicts of Interest: The authors declare no conflict of interest.

References1. Cho, S.; Jeong, H.; Park, I.K. Optimal Design of Annular Phased Array Transducers for Material Nonlinearity Determination in

Pulse–Echo Ultrasonic Testing. Materials 2020, 13, 5565. [CrossRef] [PubMed]2. Zhou, L.Y.; Fu, M.H.; Xu, W.P. Research on phased array ultrasonic testing for the girth weld of 4 mm~10 mm austenitic stainless

steel pipeline. In Proceedings of the ASME Pressure Vessels and Piping Conference (PVP 2018), Prague, Czech Republic, 15–20July 2018.

3. Da Cruz Payão Filho, J.; Kimus Dias Passos, E.; Stohler Gonzaga, R.; Drumond Santos, D.; Pereira Maia, V.; Russo Juliano, D. TheInfluence of the Welding Process on the Ultrasonic Inspection of 9%Ni Steel Pipe Circumferential Welded Joints. Materials 2020,13, 961. [CrossRef] [PubMed]

4. Duan, J.X.; Luo, L.; Gao, X.R.; Peng, J.P.; Li, J.L. Hybrid Ultrasonic TOFD Imaging of Weld Flaws Using Wavelet Transforms andImage Registration. J. Nondestruct. Eval. 2018, 37, 23. [CrossRef]

5. Moura, H.; Silva, V.; Guarneri, G.A.; Guerreiro, M.; Pipa, D.R. Image-Based Ultrasound Speed Estimation in Isotropic Materials.IEEE Sens. J. 2020, 20, 12903–12913. [CrossRef]

6. Iyer, S.; Sinha, S.K.; Pedrick, M.K.; Tittmann, B.R. Evaluation of ultrasonic inspection and imaging systems for concrete pipe.Autom. Constr. 2012, 22, 149–164. [CrossRef]

7. Tomków, J.; Janeczek, A.; Rogalski, G.; Wolski, A. Underwater Local Cavity Welding of S460N Steel. Materials 2020, 13, 5535.[CrossRef]

8. Laaidi, N.; Belattar, S.; Elbaloutti, A. Pipeline Corrosion, Modeling and Analysis. J. Nondestruct. Eval. 2011, 30, 158–163. [CrossRef]9. Witek, M. Gas transmission pipeline failure probability estimation and defect repairs activities based on in-line inspection data.

Eng. Fail. Anal. 2016, 70, 255–272. [CrossRef]

Appl. Sci. 2022, 12, 5384 11 of 11

10. Shuai, Y.; Wang, X.H.; Cheng, Y.F. Modeling of mechanical behavior of corroded X80 steel pipeline reinforced with type-B repairsleeve. Thin Walled Struct. 2021, 163, 107708. [CrossRef]

11. Pei, C.X.; Demachi, K.; Zhu, H.T.; Koyama, K.; Uesaka, M. Numerical simulation of phased-array laser ultrasound and itsapplication for defect inspection. Int. J. Appl. Electromagn. Mech. 2012, 39, 405–411. [CrossRef]

12. Reverdy, F.; Ithurralde, G.; Dominguez, N. Advanced Ultrasonic 2D Phased-Array Probes. In Proceedings of the 18th WorldConference on Nondestructive Testing, Durban, South Africa, 16–20 April 2012.

13. Szabolcs, S.; Zoltán, B.; Judit, D.; István, M. Modelling of Phased Array Ultrasonic Inspection of a Steam Generator DissimilarMetal Weld. Procedia Struct. Integr. 2016, 2, 1015–1022.

14. Si, J.; Wei, D.; Yang, Y.; Tang, X. CIVA Simulation and Experiment Verification for Thin-Walled Small-Diameter Pipes by UsingPhased Array Ultrasonic Testing. In Proceedings of the ASME 2020 Pressure Vessels & Piping Conference, Minneapolis, MN,USA, 19–24 July 2020.

15. Mesnil, O.; Recoquillay, A.; Druet, T.; Serey, V.; Hoang, H.T.; Imperiale, A.; Demaldent, E. Experimental Validation of Tran-sient Spectral Finite Element Simulation Tools Dedicated to Guided Wave-Based Structural Health Monitoring. Journal ofNondestructive Evaluation. Diagn. Progn. Eng. Syst. 2021, 4, 041003.

16. Xu, N.; Zhou, Z. Numerical simulation and experiment for inspection of corner-shaped components using ultrasonic phasedarray. NDT E Int. 2014, 63, 28–34. [CrossRef]

17. Satyanarayan, L.; Mohan, K.V.; Krishnamurthy, C.V.; Balasubramaniam, K. Finite difference time domain simulation of ultrasonicphased array sector scan for imaging cracks in large pipes, elbows, and tee sections. Res. Nondestruct. Eval. 2008, 19, 61–86.[CrossRef]

18. Xie, Y.; Yin, W.; Liu, Z.; Peyton, A. Simulation of ultrasonic and EMAT arrays using FEM and FDTD. Ultrasonics 2016, 66, 154–165.[CrossRef]

19. Liu, Z.H.; Jia, J.W.; Xiong, G. Research of the design and simulation of ultrasonic focusing system with phased array. Appl. Mech.Mater. 2014, 620, 162–165. [CrossRef]

20. Li, W.; Zhou, Z.; Li, Y. Inspection of butt welds for complex surface parts using ultrasonic phased array. Ultrasonics 2019, 96,75–82. [CrossRef]

21. Xie, Y.; Zhou, L.; Zhang, X. Finite Element Simulation Analysis of Full-focus Ultrasonic Inspection of Steel Plate Weld Defects. InProceedings of the 2021 6th International Conference on Intelligent Information Processing, Bucharest, Romania, 29–31 July 2021;pp. 210–219.

22. Jaskowiec, J.; Milewski, S. Coupling Finite Element Method with Meshless Finite Difference Method in ThermomechanicalProblems. Comput. Math. Appl. 2016, 72, 2259–2279. [CrossRef]

23. Bu, W.P.; Xiao, A.G.; Zeng, W. Finite Difference/Finite Element Methods for Distributed-Order Time Fractional DiffusionEquations. J. Sci. Comput. 2017, 72, 422–441. [CrossRef]

24. Satyanarayan, L.; Sridhar, C.; Krishnamurthy, C.V.; Balasubramaniam, K. Simulation of ultrasonic phased array technique forimaging and sizing of defects using longitudinal waves. Int. J. Press. Vessel. Pip. 2007, 84, 716–729. [CrossRef]

25. Ghaboussi, J.; Wu, X.S. Numerical Methods in Computational Mechanics; CRC Press: Boca Raton, FL, USA, 2016.26. Ji, Y.; Xing, Y. A two-sub-step generalized central difference method for general dynamics. Int. J. Struct. Stab. Dyn. 2020,

20, 2050071. [CrossRef]27. Wang, Y.C.; Murti, V.; Valliappan, S. Assessment of the accuracy of the Newmark method in transient analysis of wave propagation

problems. Earthq. Eng. Struct. Dyn. 1992, 21, 987–1004. [CrossRef]28. Li, X.J.; Zhang, Y.C. An explicit difference method for dynamic problems of damped systems. Earthq. Eng. Eng. Vib. 1992, 12,

74–80.29. Liu, X.Q.; Liu, J.B. Comparative analysis of stability and accuracy of step-by-step time-integration methods. J. Rock Mech. Eng.

2007, 26, 3000–3008.