Beam generating and sound field modeling of flexible phased arrays for inspecting complex geometric components

Journal Pre-proof Beam generating and sound field modeling of flexible phased arrays for inspecting complex geometric components Yanfang Zheng, Xinyu Zhao, Sung-Jin Song, Jianhai Zhang

PII: DOI: Reference:

S0165-2125(19)30309-9 https://doi.org/10.1016/j.wavemoti.2019.102494 WAMOT 102494

To appear in:

Wave Motion

Received date : 23 August 2019 Revised date : 2 December 2019 Accepted date : 11 December 2019 Please cite this article as: Y. Zheng, X. Zhao, S.-J. Song et al., Beam generating and sound field modeling of flexible phased arrays for inspecting complex geometric components, Wave Motion (2019), doi: https://doi.org/10.1016/j.wavemoti.2019.102494. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof 1

Beam Generating and Sound Field Modeling of Flexible Phased Arrays for Inspecting Complex

2

Geometric Components

3

Yanfang Zhenga,b, Xinyu Zhaoc, Sung-Jin Songd, and Jianhai Zhanga,b, a

Key Laboratory of CNC Equipment Reliability, Ministry of Education, Jilin University, Changchun, 130022, China

5

b

School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130022, China

6

c

School of Material Science and Engineering, Dalian Jiaotong University, Dalian, 116028, China

7

d

School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu 440-746, Republic of Korea

8

Abstract

of

4

The flexible phased array ultrasonic technology has been developed to tackle the long-term challenge of

10

damage inspection in complex-profiled components. However, due to the influences of curved interfaces, it is

11

sometimes difficult to control the transmission signal of the sound field, thereby creating unreliable transducer

12

performances. This paper proposes the time delay laws for generating steering or focusing beams on curved

13

surfaces (concave, convex, concave/convex) based on the ray acoustics theory. Then, we derive the analytic

14

expression of the entire flexible array ultrasonic field based on the multiple line source model and the time

15

delay laws. Finally, the acoustic pressure distribution of curved structures is simulated to verify the feasibility

16

of the derived principles. The numerical results show that the beams can realize dynamic steering and focusing

17

without distortions or disorientations even when the steering angle reaches 45 degrees. Furthermore, the

18

influences of the specimen surface profile, steering angle and focusing distance on the acoustic field are also

19

analyzed by the axial sound pressure plots. These preliminary results represent an essential step in the

20

development of a NDT system for inspecting components with complex surfaces.

21

Key words: flexible phased array; complex geometric components; time delay law; beam steering and

22

focusing; pressure distribution.

23

1. Introduction

Jo

urn

al

Pr e-

p ro

9

Phased array ultrasonic technology has been studied in recent years for applications in medical diagnosis

24



Corresponding author at: People’s Avenue NO. 5988, School of Mechanical and Aerospace Engineering, Jilin University, Changchun, Jilin 130022, China. E-mail address: [email protected] (J. Z) 1

Journal Pre-proof or industrial detection due to its noninvasiveness, high resolution, deep penetration capability and great

26

flexibility [1]. Thanks to the time delays, PAUT has the attractive ability to generate beam steering at different

27

angles and focusing at any desired point, which allows a full coverage scanning even with limited accessibility

28

of detection areas. The time delay law of dynamic beam generation is fundamental for theoretical researches

29

and practical applications of PAUT. Wooh et al. [2] extensively studied beam steering and focusing on the

30

surface of a flat structure. Nevertheless, in order to control the beam steering angle or mode conversion [3],

31

and avoid near-field effects, a coupling wedge or water immersion method is commonly adopted. Aiming

32

at this problem, L.W. Schmerr systematically investigated beam steering and focusing on the single or two

33

media in 2-D [4] and 3-D space [5]. Based on these principles, PAUT has been successfully applied to inspect

34

the planar structural integrity with high accuracy and efficiency.

p ro

of

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Recently, the inspection of structures with complex surfaces remains a challenge due to the difficulties

36

in matching the surface tightly and determining the real-time delay law associated with the irregular surface

37

[6]. The poor coupling effects and unsuitable time delay laws will lead to degradation of ultrasonic signals

38

due to distortions and sound energy losses, which affect the sensitivity of flaw localization and characterization

39

[6, 7]. For detecting shape-complicated structures, traditional linear phased array probes have a variety of

40

detection strategies, such as water immersion method [8] or ice encapsulation strategy [9], specially profiled

41

[10] or conformable [11] wedge or flexible liquid-filled membrane [12, 13]. But these methods have some

42

limits to in-service testing or are constrained by customization for special surfaces. With the development of

43

sensor fabrication technologies, flexible phased array transducer is a wonderful alternative to inspect the parts

44

with complex surfaces [14, 15].

urn

al

Pr e-

35

Casula et al. [15, 16] developed 2-D and 3-D smart flexible phased array transducer prototypes that were

46

mechanically assembled from some independent array elements with springs pushed to couple the irregular

47

surfaces directly. An embedded profilometer was applied to measure the specimen profile and then the real-

48

time delay law was calculated by an autofocus algorithm. Ultimately, the flexible array transducers were

49

performed to inspect the defects in irregular components, such as butt weld, nozzle, and elbow. Lane [17] and

50

Nakahata et al. [18] utilized a flexible array transducer and combined total focusing method to inspect and

51

image defects in components with irregular surfaces. In general work in this area is in its infancy and

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45

2

Journal Pre-proof 52

somewhat limited by the feasible and accurate calculation of the time delay laws for curved components. Since

53

the array elements are distributed on the curved surfaces, the theories of planar time delay laws [2, 4, 5] for

54

rigid probes are not applicable. It is difficult to guarantee the transmission characteristics of the radiated sound

55

wave without the time delay laws. The detection and characterization performances of structural defects are highly dependent on transducer

57

beam characteristics and acoustic energy. Whereas, ultrasonic beam self-focusing and divergence phenomena

58

[19] are caused due to the influences of the target interface curvature during the propagation, and the sound

59

energy will be concentrated or dispersed in curved components spontaneously. The ultrasonic field calculation

60

of the entire array transducer is performed by superimposing contributions of all elements with appropriate

61

time delays. Therefore, it is necessary to determine the time delay laws and sound field distribution accurately

62

inside the structures with curved surfaces, then to optimize the inspection strategy [20] and improve detection

63

reliability and repeatability.

Pr e-

p ro

of

56

In response to the inspection of complex curved surface structures using flexible phased array technology,

65

we develop the time delay laws that the beams are radiated directly to a curved surface (concave, convex,

66

concave/convex) in section 2. Then, the entire flexible phased array ultrasonic pressure field of beam steering

67

and focusing is derived in section 3. Simulations are implemented in section 4 to achieve beam dynamic

68

steering and focusing on the structures with curved profiles and the beam characteristics along the steering

69

angle are discussed. Finally, the conclusions of this paper are summarized in section 5.

70

2. Time delay laws for single-layer components with complex surfaces

urn

al

64

Flexible phased array transducers are developed to overcome the inspection challenges of curved

72

components, but there are few reports on the corresponding time delay algorithms that control beam

73

propagation. Fortunately, most of the industrial curved components, such as pipes, artillery shells, rotor shafts

74

and turbine blades, whose surface profiles can be simplified to the combination of concave and convex

75

surfaces. Accordingly, the detection of these curved structures can be classified into concave, convex and

76

concave/convex cases. Therefore, we establish geometric models with concave, convex and concave/convex

77

surfaces to derive the time delay laws of a 1-D array in 2-D space. The symbols are listed in Table 1. A flexible

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3

Journal Pre-proof 78

phased array probe is tightly coupled to the testing surface, assuming that the array has 𝑀 elements, 𝑚

79

represents an arbitrary element of the array, then the time delay ( 𝜏𝑚 ) of 𝑚 th element is obtained

80

on the basis of ray acoustics theory 𝜏𝑚 =

𝑚𝑎𝑥(𝑙𝑚 ) − 𝑙𝑚 𝑐

(1)

where 𝑐 is longitudinal wave or shear wave speed, 𝑙𝑚 is wave path from the center of 𝑚th element.

82

Therefore, the task is to compute the value of 𝑙𝑚 to calculate the time delays of beam pure steering or

83

focusing.

p ro

of

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Firstly, the time delays for the flexible array transducers on the concave or convex surface of single-layer

85

components are calculated. Then, the time delay law for the concave/convex surface is derived based on the

86

above fundamental principles. Finally, we propose the time delay laws for the concave or convex surface of a

87

double-layer structure in Appendix A, considering the non-destructive testing requirements of multi-layer

88

complex-shaped structures.

89

Table 1

90

Symbols of single-layer components geometric models for computing the time delays

Basic parameters

Pr e-

84

Element number

𝐶

Array central point

𝑚

Arbitrary element

𝑠

Element pitch

𝑙0

Focal distance or central beam steering distance

𝐹

Any desired focus

𝜃𝑠

Steering angle (counterclockwise is positive)

urn

Jo

𝑙𝑚

al

𝑀

Wave path from the center of 𝑚th element

Concave surface 𝑂1

Center

4

Journal Pre-proof 𝑅1

Radius

𝑂1 𝐹

Distance from the center 𝑂1 to the focus 𝐹

𝛼

Angle between the line 𝑂1 𝐹 and the line 𝑂1 𝐶 Central angle between the center of 𝑚th element and the array center

𝜑𝑚

𝑂2

Center

𝑅2

Radius

𝛼 𝜑𝑚

Distance from the center 𝑂2 to the focus 𝐹

p ro

𝑂2 𝐹

of

Convex surface

Angle between the line 𝑂2 𝐹 and the line 𝑂2 𝐶

Central angle between the center of 𝑚th element and the array center

Pr e-

Concave/convex surface 𝐸

Intersection of the concave and convex surfaces

𝑑

The element number of the concave part

Angle between the line 𝑂1 𝐸 and the vertical line which connects the center of the 𝜎 concave 𝑥𝑚

positive)

al

Horizontal distance between the point 𝐸 and the center of 𝑚th element (convex part is

Vertical distance between the point 𝐸 and the center of 𝑚th element (concave part is positive)

urn

𝑧𝑚

Central angle between the centerline connecting the center of 𝑚th element and the vertical 𝜑𝑚

line connecting the center 𝑂1 or 𝑂2 Angle between the line 𝑂1 𝐹 and the centerline 𝑂1 𝑂2

𝛽

Angle between the line 𝑂2 𝐹 and the centerline 𝑂1 𝑂2

Jo

𝜀

5

Journal Pre-proof O

O

1

φ

1

α φ

m

m

C

M

1

θs

F (l ,θ ) 0

s

(b) Beam steering and focusing

Fig. 1. Schematic diagram for calculating the time delays of single concave components

91 92

s

Pr e-

(a) Beam pure steering

θ

M

m

C

p ro

1

of

m

2.1 The time delay law for the concave surface

As plotted in Fig. 1, the meanings of symbols are shown in Table 1. By setting the point 𝐶 as the origin,

93

the centerline 𝑂1 𝐶 as the axis, a polar coordinate system is established.

95

2.1.1 Beam pure steering

al

94

If the beam is pure steering illustrated in Fig. 1 (a), the length of all ray paths can be expressed as 𝑅1 𝑠𝑖𝑛(𝜑𝑚 ) 𝑡𝑎𝑛(𝜃𝑠 ) + 𝑅1 𝑐𝑜𝑠(𝜑𝑚 ) − 𝑅1 − 𝑙0 /𝑐𝑜𝑠(𝜃𝑠 )

urn

96

𝑙𝑚 = ||

2

||

(2)

√(𝑡𝑎𝑛(𝜃𝑠 )) + 1

𝑀+1

97

where 𝜑𝑚 = (𝑚 −

98

2.1.2 Beam steering and focusing

) 𝑠⁄𝑅1 .

Jo

2

99

If the beam is steering and focusing, as depicted in Fig. 1 (b), 𝑙0 represents focal distance, 𝐹 denotes

100

focus, the wave path 𝑙𝑚 from the center of 𝑚th element to the focus 𝐹 can be calculated based on cosine

101

law

6

Journal Pre-proof 𝑙𝑚 = √𝑅1 2 + 𝑂1 𝐹 2 − 2𝑅1 ∙ 𝑂1 𝐹 ∙ 𝑐𝑜𝑠⁡(𝜑𝑚 − α) 𝑙 𝑠𝑖𝑛𝜃𝑠

where 𝑂1 𝐹 = √𝑅1 2 + 𝑙0 2 + 2𝑅1 ∙ 𝑙0 ∙ 𝑐𝑜𝑠 𝜃𝑠 , α = 𝑎𝑠𝑖𝑛 ( 0𝑂

1𝐹

s

C

m 1

)

M

θs

s

C

m

1

θ

M s

of

102

(3)

φ

p ro

φ

m

m

F (l ,θ ) 0

s

O

2

(a) Beam pure steering 103 104

Pr e-

α O

2

(b) Beam steering and focusing

Fig. 2. Schematic diagram for calculating the time delays of single convex components

2.2 The time delay law for the convex surface

The calculation is the same as the concave, as given in Fig. 2, whose symbols are listed in Table 1. By

106

setting the central position 𝐶 of the array as the origin, the centerline 𝑂2 𝐶 as the axis, a polar coordinate

107

system is established.

108

2.2.1 Beam pure steering

urn

109

al

105

If the beam is pure steering (Fig. 2 (a)), the length 𝑙𝑚 of all ray paths can be given 𝑅2 𝑠𝑖𝑛(𝜑𝑚 ) 𝑡𝑎𝑛(𝜃𝑠 ) + 𝑅2 − 𝑅2 𝑐𝑜𝑠(𝜑𝑚 ) − 𝑙0 /𝑐𝑜𝑠(𝜃𝑠 )

Jo

𝑙𝑚 = ||

||

(4)

√(𝑡𝑎𝑛(𝜃𝑠 )) + 1

𝑀+1

110

where 𝜑𝑚 = (𝑚 −

111

2.2.2 Beam steering and focusing

112

2

2

) 𝑠⁄𝑅2 .

If the beam is steering and focusing, as depicted in Fig. 2 (b), the wave path 𝑙𝑚 from the center of 𝑚th 7

Journal Pre-proof 113

element to the focus 𝐹 can be calculated based on cosine law 𝑙𝑚 = √𝑅2 2 + 𝑂2 𝐹 2 − 2𝑅2 ∙ 𝑂2 𝐹 ∙ 𝑐𝑜𝑠(𝛼 − 𝜑𝑚 )

114

(5a)

where 𝑂2 𝐹 = √𝑅2 2 + 𝑙0 2 − 2𝑅2 ∙ 𝑙0 ∙ 𝑐𝑜𝑠 𝜃𝑠 , 𝛼 can be derived as follows

115

p ro

of

𝑙0 𝑠𝑖𝑛𝜃𝑠 𝑎𝑠𝑖𝑛 ( ) , 𝑙0 ≤ 𝑅2 𝑂2 𝐹 𝑅2 𝑠𝑖𝑛𝜃𝑠 𝛼 = 𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃𝑠 , 𝑙0 > 𝑅2 ⁡𝑎𝑛𝑑⁡𝜃𝑠 > 0⁡ 𝑂2 𝐹 𝑅2 𝑠𝑖𝑛𝜃𝑠 −𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃𝑠 , 𝑙0 > 𝑅2 ⁡𝑎𝑛𝑑⁡𝜃𝑠 ≤ 0 { 𝑂2 𝐹

It is worth mentioning that if the focus is the convex surface center (𝑂2), each element time delay is zero.

σ φm1

Pr e-

O1

m1

1

M

m2

d

E

θs

φm2

O2

al

(a) Beam pure steering

O1

urn

σ

ε

m2 d m1

M

E θs

Jo

1

β F (l0,θs)

O2

(b) Beam steering and focusing 116

(5b)

Fig. 3. Schematic diagram for calculating the time delays of single concave/convex components 8

Journal Pre-proof 117

2.3 The time delay law for the concave/convex surface The time delay calculation of the concave/convex surface can be divided into concave and convex parts,

119

as displayed in Fig. 3, whose symbols are listed in Table 1. Assuming that there are 𝑑 elements on the

120

concave part, we first figure out the time delays of the concave part and then compute the convex part. By

121

setting the intersection 𝐸 as the origin, the vertical line connecting the origin 𝐸 as the axis, a polar

122

coordinate system is established.

123

2.3.1 Beam pure steering

125

If the beam is pure steering, as shown in Fig. 3 (a), assuming that the path distance of the array center is 𝑙0 , then the length of all ray paths can be expressed as

127 128

(i)

𝑥𝑚 𝑡𝑎𝑛(𝜃𝑠 ) + 𝑧𝑚 − 𝑙0 /𝑐𝑜𝑠(𝜃𝑠 )

(6)

√(𝑡𝑎𝑛(𝜃𝑠 ))2 + 1

Pr e-

𝑙𝑚 = 126

p ro

124

of

118

the concave part

where 𝑥𝑚1 = 𝑅1 𝑠𝑖𝑛(𝜑𝑚1 ) − 𝑅1 𝑠𝑖𝑛(𝜎), 𝑧𝑚1 = 𝑅1 𝑐𝑜𝑠(𝜑𝑚1 ) − 𝑅1 𝑐𝑜𝑠(𝜎), 𝜑𝑚1 = 𝜎 − (𝑑 − 𝑚1 )𝑠/𝑅1 . (ii)

the convex part

129

where 𝑥𝑚2 = 𝑅2 𝑠𝑖𝑛(𝜎) − 𝑅2 𝑠𝑖𝑛(𝜑𝑚2 ),𝑧𝑚2 = 𝑅2 𝑐𝑜𝑠(𝜎) − 𝑅2 𝑐𝑜𝑠(𝜑𝑚2 ), 𝜑𝑚2 = 𝜎 − (𝑚2 − 𝑑)𝑠/𝑅2 .

130

2.3.2 Beam steering and focusing

133

al

132

If the beam is steering and focusing, as displayed in Fig. 3 (b), taking the focusing distance is 𝑙0 , then the wave path 𝑙𝑚 can be computed. (i)

the concave part

urn

131

𝑙𝑚1 = √𝑅1 2 + 𝑂1 𝐹 2 − 2𝑅1 ∙ 𝑂1 𝐹 ∙ 𝑐𝑜𝑠((𝑑 − 𝑚1 )𝑠/𝑅1 − 𝜀)

where 𝑂1 𝐹 = √𝑅1 2 + 𝑙0 2 − 2𝑅1 ∙ 𝑙0 ∙ 𝑐𝑜𝑠(𝜋 − 𝜎 + 𝜃𝑠 ), 𝜀 can be calculated as

Jo

134

135

(7a)

(ii)

𝑅1 2 + 𝑂1 𝐹 2 − 𝑙0 2 ) , 𝜃𝑠 < 𝜎 2𝑅1 ∙ 𝑂1 𝐹 𝜀 = 0, 𝜃𝑠 = 𝜎 𝑅1 2 + 𝑂1 𝐹 2 − 𝑙0 2 −𝑎𝑐𝑜𝑠 ( ) , 𝜃𝑠 > 𝜎 2𝑅1 ∙ 𝑂1 𝐹 { 𝑎𝑐𝑜𝑠 (

the convex part 9

(7b)

Journal Pre-proof 𝑙𝑚2 = √𝑅2 2 + 𝑂2 𝐹 2 − 2𝑅2 ∙ 𝑂2 𝐹 ∙ 𝑐𝑜𝑠((𝑚2 − 𝑑)𝑠/𝑅2 + 𝛽) 136

(8a)

where 𝑂2 𝐹 = √𝑂1 𝐹 2 + (𝑅1 + 𝑅2 )2 − 2𝑂1 𝐹 ∙ (𝑅1 + 𝑅2 ) ∙ 𝑐𝑜𝑠(𝜀), 𝛽 can be given below

O

(8b)

of

𝑂1 𝐹𝑠𝑖𝑛(𝜀) 𝑎𝑠𝑖𝑛 ( ) , 𝑂1𝐹 ≪ (𝑅1 + 𝑅2 )⁡ 𝑂2 𝐹 (𝑅1 + 𝑅2 )𝑠𝑖𝑛(𝜀) 𝛽 = 𝜋 − 𝜀 − 𝑎𝑠𝑖𝑛 ( ) , 𝑂1 𝐹 > (𝑅1 + 𝑅2 )⁡𝑎𝑛𝑑⁡𝜀 > 0⁡ 𝑂2 𝐹 (𝑅1 + 𝑅2 )𝑠𝑖𝑛(𝜀) −𝜋 − 𝜀 − 𝑎𝑠𝑖𝑛 ( ) , 𝑂1 𝐹 > (𝑅1 + 𝑅2 )⁡𝑎𝑛𝑑⁡𝜀 ≤ 0 𝑂2 𝐹 {

p ro

1

Pr e-

α

0

φ

m

1

δ

n

m n θ θ

C

m

M

mn

θ

138

al

Fig. 4. Schematic diagram for calculating the pressure distribution of single concave components

urn

137

G (r,θ)

3. Acoustic pressure field for beam steering and focusing of single-layer components The flexible phased array transducer parameters will change due to the influences of curvature when

140

inspecting the curved components. It is necessary to investigate the acoustic pressure field for beam steering

141

and focusing to obtain the influent regularities of different factors on the transducer and the inspected

142

components. The radiated pressure field from a single element is composed of multiple line sources according

143

to the Huygens principle, and the pressure field of the entire phased array can be solved by superimposing

144

contributions of all elements with appropriate time delays. Therefore, we establish analytic models of the

Jo

139

10

Journal Pre-proof 145

sound field distribution for beam steering and focusing on the single-layer components with concave, convex

146

or concave/convex surfaces corresponding to the delay time calculation models in Section 2. The parameters

147

are shown in Table 2, and the other utilized parameters are listed in Table 1. Furthermore, the pressure

148

distribution for the double-layer structures with a concave or convex surface is computed in Appendix B. The flexible phased array is assumed to have 𝑀 elements, and each element is consists of a large number

150

(𝑁) of line sources located at equidistance,⁡𝑛 is an arbitrary segment of the element. Each element normalized

151

pressure distribution at an arbitrary point 𝐺(𝑟, 𝜃) is obtained by a superposition of multiple line sources,

152

which is given as follows 𝑁

p ro

of

149

𝑝(𝐺, 𝜔) 1 2𝑘𝑏 𝑒𝑥𝑝(𝑖𝑘𝑏𝑟̅̅̅̅̅) 𝑚𝑛 = √ ∑ 𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) 𝜌𝑐𝑣0 (𝜔) 𝑁 𝜋𝑖 𝑟𝑚𝑛 √̅̅̅̅̅ 𝑛=1

𝑠𝑖𝑛[𝑘𝑏𝑠𝑖𝑛𝜃𝑚𝑛 ⁄𝑁] 𝑘𝑏𝑠𝑖𝑛𝜃𝑚𝑛 ⁄𝑁

Pr e-

𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) =

(9a)

(9b)

𝑟𝑚𝑛 = 𝑟𝑚𝑛 ⁄𝑏 ̅̅̅̅̅

(9c)

𝑘 = 𝜔⁄𝑐

(9d)

In the practical detection of the flexible phased array probe, all the parameters in Eqs. (9) except for 𝑟𝑚𝑛

154

and 𝑠𝑖𝑛𝜃𝑚𝑛 are predetermined, so it is the key to calculate the values of 𝑟𝑚𝑛 and 𝑠𝑖𝑛𝜃𝑚𝑛 for deriving each

155

element normalized pressure field at any point 𝐺.

156

Table 2

157

Symbols of single-layer components geometric models for computing the sound field

urn

Basic parameters

al

153

Pressure

𝜌

Density of the specimen

𝑣0

Spatially uniform velocity of the element

𝑘 𝜔 𝑖

Jo

𝑝

Wave number Angular frequency Imaginary unit

11

Journal Pre-proof 𝑁

Number of segments of each element

𝑏

Half element width

𝑛

An arbitrary segment of the element Segment directivity

𝐺

Any point

𝑟

Distance from the origin to the point 𝐺

𝜃

Polar angle (counterclockwise is positive)

of

𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 )

Angle between the radial ray path and the actual ray path of the 𝑚th array element

𝑟𝑚

Wave path from the point 𝐺 to the center of the 𝑚th element

p ro

𝜃𝑚

Angle between the radial ray path and the actual ray path of the 𝑛th segment center of 𝜃𝑚𝑛

Pr e-

the 𝑚th array element

𝑟𝑚𝑛

Wave path from the 𝑛th segment center of the 𝑚th element to the point 𝐺

𝑟𝑚𝑛 ̅̅̅̅̅

Normalized distance

Concave surface 𝑂1 𝐺

Distance from the center 𝑂1 to the point 𝐺

𝛼0

Angle between the line 𝑂1 𝐺 and the line 𝑂1 𝐶

𝛼0

Distance from the center 𝑂2 to the focus 𝐺 Angle between the line 𝑂2 𝐺 and the line 𝑂2 𝐶

urn

𝑂2 𝐺

al

Convex surface

Concave/convex surface

159 160

Angle between the line 𝑂1 𝐺 and the centerline 𝑂1 𝑂2

𝛽0

Angle between the line 𝑂2 𝐺 and the centerline 𝑂1 𝑂2

Jo

158

𝜀0

3.1 The pressure distribution for the concave surface As seen in Fig. 4, setting the central position (𝐶) of the array as the origin, the centerline 𝑂1 𝐶 as the axis, a polar coordinate system is established. 𝑟𝑚𝑛 and 𝑠𝑖𝑛𝜃𝑚𝑛 can be expressed based on the cosine law 12

Journal Pre-proof 𝑟𝑚𝑛 = √𝑅1 2 + 𝑂1 𝐺 2 − 2𝑅1 ∙ 𝑂1 𝐺 ∙ cos⁡(𝜑𝑚 − 𝛼0 + 𝛿𝑛 ) 𝑠𝑖𝑛𝜃𝑚𝑛 =

𝑂1 𝐺 ∙ 𝑠𝑖𝑛(𝜑𝑚 − 𝛼0 + 𝛿𝑛 ) 𝑟𝑚𝑛

𝑂1 𝐺 = √𝑅1 2 + 𝑟 2 + 2𝑅1 ∙ 𝑟 ∙ 𝑐𝑜𝑠 𝜃 ,

162

𝑎𝑠𝑖𝑛 ( 𝑂

𝑟𝑠𝑖𝑛𝜃 1𝐺

𝜑𝑚 = (𝑚 −

𝑀+1 2

) 𝑠⁄𝑅1 ,

𝛿𝑛 = (𝑛 −

1+𝑁 2𝑏 2

)

𝑁

/𝑅1 , ⁡𝛼0 =

).

s

C

nm 1

M

θ

δ

φ

m

G (r,θ)

Pr e-

n

of

where

(11)

p ro

161

(10)

α

0

O

2

165 166

3.2 The pressure distribution for the convex surface

As observed in Fig. 5, setting the central position (𝐶) of the array as the origin, the line 𝑂2 𝐶 as the axis,

al

164

Fig. 5. Schematic diagram for calculating the pressure distribution of single convex components

a polar coordinate system is established. 𝑟𝑚𝑛 and 𝑠𝑖𝑛𝜃𝑚𝑛 can be expressed as

urn

163

𝑟𝑚𝑛 = √𝑅2 2 + 𝑂2 𝐺 2 − 2𝑅2 ∙ 𝑂2 𝐺 ∙ 𝑐𝑜𝑠(𝛼0 − 𝜑𝑚 − 𝛿𝑛 ) 𝑠𝑖𝑛𝜃𝑚𝑛 =

(12)

𝑂2 𝐺 ∙ 𝑠𝑖𝑛(𝛼0 − 𝜑𝑚 − 𝛿𝑛 ) 𝑟𝑚𝑛

where 𝑂2 𝐺 = √𝑅2 2 + 𝑟 2 − 2𝑅2 ∙ r ∙ 𝑐𝑜𝑠 𝜃 , 𝜑𝑚 = (𝑚 −

168

derived as follows

Jo

167

13

𝑀+1 2

) 𝑠⁄𝑅2 , 𝛿𝑛 = (𝑛 −

(13) 1+𝑁 2𝑏 2

)

𝑁

⁄𝑅2 , 𝛼0 can be

Journal Pre-proof 𝑟𝑠𝑖𝑛𝜃 𝑎𝑠𝑖𝑛 ( ) , 𝑟 ≤ 𝑅2 𝑂2 𝐺 𝑅2 𝑠𝑖𝑛𝜃 𝛼0 = 𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃, 𝑟 > 𝑅2 ⁡𝑎𝑛𝑑⁡𝜃 > 0⁡ 𝑂2 𝐺 𝑅2 𝑠𝑖𝑛𝜃 −𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃, 𝑟 > 𝑅2 ⁡𝑎𝑛𝑑⁡𝜃 ≤ 0 { 𝑂2 𝐺

(14)

σ

ε0 dE n1m1 θm1 θ θmn1

1

M

p ro

δn1

m2 n2 θm2 θmn2

of

O1

δn2

β0

Pr e-

G (r,θ)

Fig. 6. Schematic diagram for calculating the pressure distribution of single concave/convex components

169 170

O2

3.3 The pressure distribution for the concave/convex surface

If the specimen surface is concave/convex, the calculation of the ultrasonic field can be divided into

172

concave and convex parts and is schematized in Fig. 6. Assuming that the number of elements on the concave

173

part is 𝑑, we first figure out the pressure field of the concave part and then compute the convex part. Setting

174

the intersection 𝐸 as the origin, the vertical line connecting the origin 𝐸 as the axis, a polar coordinate

175

system is established. 𝑟𝑚𝑛 can be expressed based on cosine law and⁡𝑠𝑖𝑛𝜃𝑚𝑛 can be obtained by the law of

176

sines. (i)

urn

177

al

171

the concave part

Jo

𝑟𝑚𝑛1 = √𝑅1 2 + 𝑂1 𝐺 2 − 2𝑅1 ∙ 𝑂1 𝐺 ∙ 𝑐𝑜𝑠((𝑑 − 𝑚1 )𝑠/𝑅1 − 𝜀0 + 𝛿𝑛1 )

𝑠𝑖𝑛𝜃𝑚𝑛1 =

178

𝑂1 𝐺 ∙ 𝑠𝑖𝑛((𝑑 − 𝑚1 )𝑠/𝑅1 − 𝜀0 + 𝛿𝑛1 ) 𝑟𝑚𝑛1

where 𝑂1 𝐺 = √𝑅1 2 + 𝑟 2 − 2𝑅1 ∙ 𝑟 ∙ 𝑐𝑜𝑠(𝜋 − 𝜎 + 𝜃), 𝛿𝑛1 = (𝑛1 −

14

1+𝑁 2𝑏 2

)

𝑁

(15)

(16)

⁄𝑅1 , 𝜀0 can be calculated as

Journal Pre-proof

(ii)

(17)

𝑟𝑚𝑛2 = √𝑅2 2 + 𝑂2 𝐺 2 − 2𝑅2 ∙ 𝑂2 𝐺 ∙ 𝑐𝑜𝑠((𝑚2 − 𝑑)𝑠/𝑅2 + 𝛽0 + 𝛿𝑛2 )

(18)

the convex part

𝑂2 𝐺 ∙ 𝑠𝑖𝑛((𝑚2 − 𝑑)𝑠/𝑅2 + 𝛽0 + 𝛿𝑛2 ) 𝑟𝑚𝑛2

p ro

𝑠𝑖𝑛𝜃𝑚𝑛2 =

of

179

𝑅1 2 + 𝑂1 𝐺 2 − 𝑟 2 𝑎𝑐𝑜𝑠 ( ),𝜃 < 𝜎 2𝑅1 ∙ 𝑂1𝐺 𝜀0 = 0, 𝜃 = 𝜎 𝑅1 2 + 𝑂1 𝐺 2 − 𝑟 2 −𝑎𝑐𝑜𝑠 ( ),𝜃 > 𝜎 2𝑅1 ∙ 𝑂1 𝐺 {

180

where 𝑂2 𝐺 = √𝑂1 𝐺 2 + (𝑅1 + 𝑅2 )2 − 2𝑂1 𝐺(𝑅1 + 𝑅2 ) ∙ 𝑐𝑜𝑠(𝜀0 ) , 𝛿𝑛2 = (𝑛2 −

181

calculated in Eq. (20)

1+𝑁 2𝑏 2

)

𝑁

⁄𝑅2 , 𝛽0 can be

𝑂1 𝐺𝑠𝑖𝑛(𝜀0 ) 𝑎𝑠𝑖𝑛 ( ) , 𝑂1 𝐺 ≪ (𝑅1 + 𝑅2 )⁡⁡ 𝑂2 𝐺 (𝑅1 + 𝑅2 )𝑠𝑖𝑛(𝜀0 ) 𝛽0 = 𝜋 − 𝜀0 − 𝑎𝑠𝑖𝑛 ( ) , 𝑂1 𝐺 > (𝑅1 + 𝑅2 )⁡𝑎𝑛𝑑⁡𝜀0 > 0⁡ 𝑂2 𝐺 (𝑅1 + 𝑅2 )𝑠𝑖𝑛(𝜀0 ) −𝜋 − 𝜀0 − 𝑎𝑠𝑖𝑛 ( ) , 𝑂1 𝐺 > (𝑅1 + 𝑅2 )⁡𝑎𝑛𝑑⁡𝜀0 ≤ 0 𝑂2 𝐺 {

Pr e-

(19)

(20)

Therefore, an analytic model of the sound field distribution for the entire array beam steering and

183

focusing on the single-layer curved structure is established as shown in Eq. (21), which combines the time

184

delay laws of beam generation with each element normalized pressure field.

al

182

𝑀

𝑁

𝑚=1

𝑛=1

urn

𝑝(𝐆, 𝜔) 1 2𝑘𝑏 𝑒𝑥𝑝(𝑖𝑘𝑏𝑟̅̅̅̅̅) 𝑚𝑛 = ∑ exp(𝑖𝜔𝜏𝑚 ) [ √ ∑ 𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) ] 𝜌𝑐𝑣0 (𝜔) 𝑁 𝜋𝑖 𝑟𝑚𝑛 √̅̅̅̅̅

(21)

The proposed Eq. (21) is appropriate for elements of arbitrary width, as long as the suitable number of

186

segments is selected. We can set N = 1, 𝛿𝑛 = 0, 𝜃𝑚𝑛 = 𝜃𝑚 and 𝑟𝑚𝑛 = 𝑟𝑚 to obtain a simple expression

187

when analyzing the beam far-field (three near field lengths or greater) characteristics.

188

4. Simulation and discussion

Jo

185

189

The flexible phased array applied in simulation is a 16-element array probe, the center frequency 𝑓 is

190

2.5⁡MHz, the element width 2𝑏 is 0.9 𝑚𝑚 and the pitch 𝑠 is 1.17 𝑚𝑚. Assuming that there is only one

191

wave form and the wave speed 𝑐 is 5850 𝑚/𝑠, the concave radius 𝑅1 is 40 𝑚𝑚 and the convex radius 𝑅2 15

Journal Pre-proof 192

is 40 𝑚𝑚. When the surface is concave/convex, the element number of the concave part 𝑑⁡is 8.

193

4.1 Simulation Verification Simulation is performed to attain sound field distribution of the concave, convex and concave/convex

195

components without time delays and with time delays to verify our developed analytic models. The results

196

displayed in Fig. 7(a), Fig. 8(a) show that the sound energy is dispersed or concentrated spontaneously in

197

concave or convex components without time delays, respectively. And Fig. 9(a) shows that the ultrasonic

198

beam propagates along the steering angle of about 30° in the concave/convex components when there are no

199

time delays. But the ultrasonic beam can be dynamically steered and focused at the preset angle and focusing

200

distance with the delay time algorithms, no matter for the specimen with a concave (Fig. 7(b)), convex (Fig.

201

8(b)) or concave/convex (Fig. 9(b)) surface. The desired focus is beyond the maximum pressure point (the

202

actual focus) because of the beam diffraction effect. It can be seen from Fig. 7(c)-Fig. 9(c) that the steering

203

angle can reach ⁡45° without beam distortions, because the multiple line source model of pressure

204

distribution does not rely on the paraxial approximation, thus eliminating the size limitation of about 20

205

degrees [21]. It proves that the analytic expression of the entire array radiated wave field is feasible and valid

206

even with large steering angles.

Pr e-

p ro

of

194

The acoustic pressure distribution in Fig. 7(b)-Fig. 9(b) and Fig. 7(c)-Fig. 9(c) show the difference of

208

beam steering and focusing at different angles, but it is difficult to compare the pressure values quantitatively.

209

Therefore, extracting the axial ultrasonic pressure plots from these images to qualitatively evaluate the

210

focusing effect, as shown in Fig.10. Compared with the plane [2], the geometric focusing of the convex surface

211

enhances the focusing effect that of the plane, while the concave surface is just the opposite because of the

212

geometric divergence. As for the concave/convex components, the geometric focusing and divergent effects

213

are superimposed. The pressure amplitude is less than that of the plane when the steering angle is 0° but is

214

greater when the steering angle is 45°. Besides, with the increase of the steering angle in a certain range,

215

whether it is a planar array or a curved array, the deviation between the actual focus and the desired focus

216

position is larger, and the pressure amplitude decays.

Jo

urn

al

207

16

(a) No time delays

(b)⁡𝑙0 = 30⁡𝑚𝑚,⁡𝜃𝑠 = 0°

of

Journal Pre-proof

(c)⁡𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 45°

Fig. 7. Sound field distribution of the concave specimen without or with the time delays. The circles represent the desired

218

focal points.

(a) No time delays

Pr e-

p ro

217

(b) 𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 0°

(c) 𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 45°

Fig. 8. Sound field distribution of the convex specimen without or with the time delays. The circles represent the desired focal

220

points.

Jo

urn

al

219

(a) No time delays

(b) 𝑙0 = 30⁡𝑚𝑚,⁡𝜃𝑠 = 0°

(c) 𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 45°

221

Fig. 9. Sound field distribution of the concave/convex specimen without or with the time delays. The circles represent the

222

desired focal points. 17

of

Journal Pre-proof

(b) 𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 45°

p ro

(a) 𝑙0 = 30𝑚𝑚,⁡𝜃𝑠 = 0°

Fig. 10. Axial pressure normalized amplitude plots of the plane, concave, convex, and concave/convex specimen for beam

224

steering and focusing at different steering angles. The circles represent the actual focal points and the vertical dashed line

225

represents the desired focus position.

226

4.2 The inspection of the concave/convex components

Pr e-

223

The NDT of complex components with concave/convex surface is a typical case in the real industry

228

inspection, the process can be divided into three steps: (1) detection of the concave surface; (2) array scanning

229

of the concave/convex surface; (3) testing of the convex surface. Therefore, it is of great engineering

230

significance to study the detection of concave/convex components. To clearly describe and visually display

231

the detection process, simulation is applied to attain the wave propagation ray plots, time delays, and sound

232

field distribution for beam steering and focusing, as described in Figs. 11 -13.

Jo

urn

al

227

(a) 𝑙0 = 60𝑚𝑚, 𝜃𝑠 = −20° 233

(b)⁡𝑙0 = 30𝑚𝑚, 𝜃𝑠 = 0°

(c)⁡𝑙0 = 15𝑚𝑚, 𝜃𝑠 = 15°

Fig. 11. Ray paths of the concave specimen (a), concave/convex specimen (b) and convex specimen (c)

18

Journal Pre-proof

(c)⁡𝑙0 = 15𝑚𝑚, 𝜃𝑠 = 15°

Fig. 12. Time delays of the concave specimen (a), concave/convex specimen (b) and convex specimen (c)

(a)⁡𝑙0 = 60𝑚𝑚, 𝜃𝑠 = −20°

Pr e-

p ro

234

(b)⁡𝑙0 = 30𝑚𝑚, 𝜃𝑠 = 0°

of

(a)⁡𝑙0 = 60𝑚𝑚, 𝜃𝑠 = −20°

(b)⁡𝑙0 = 30𝑚𝑚, 𝜃𝑠 = 0°

(c)⁡𝑙0 = 15𝑚𝑚, 𝜃𝑠 = 15°

235

Fig. 13. Sound field distribution of the concave specimen (a), concave/convex specimen (b) and convex specimen (c). The

236

circles represent the actual focal points.

The propagation rays can be used to plan the scanning schemes reasonably, as observed in Fig. 11, to

238

ensure comprehensive coverage of the complex surface structures and then a reasonable steering angle and

239

focusing distance can be set. Each element time delay of the array transducer, as presented in Fig. 12, is utilized

240

to control each element signal transmission including signal excitation and reception. Based on the time delays,

241

acoustic pressure distribution in the curved components can be visually displayed in Fig. 13. The actual focal

242

points in Fig. 13(b) and Fig. 13(c) are near the desired focus, while the actual focus in figure Fig. 13(a) is far

243

away from the desired focus. This phenomenon can also be found in planar arrays when the focusing distance

244

is greater than the near-field length [2]. Therefore, it is necessary to consider the influence of the focusing

245

distance on the ultrasonic pressure field of curved components. The axial pressure normalized amplitude plots

246

at different focusing distances are shown in Fig.14. It can be summarized that with the increase of the focusing

247

distance, the deviation between the actual focus and the desired focus position is larger, and the pressure

Jo

urn

al

237

19

Journal Pre-proof amplitude decays obviously. The focusing effect converges to the pure steering when the focusing length

249

reaches infinity, which means that pure steering is adequate when the detecting position is far from the array

250

probe.

p ro

of

248

(a) Concave

(b) Concave/convex

(c) Convex

Fig. 14. Axial pressure normalized amplitude plots at different focusing distances of the concave specimen (a),

252

concave/convex specimen (b) and convex specimen (c) for beam steering and focusing. The circles represent the actual focal

253

points and the vertical dashed lines represent the desired focus positions. (𝜃𝑠 = 0°)

254

5. Conclusion

Pr e-

251

We derive the time delay laws and the sound field analytic expressions for beam steering and focusing

256

on the single-layer and double-layer complex curved components with a constant radius. The proposed time

257

delay laws are not only suitable for flexible array transducers, but also applicable to the curved rigid probes.

258

The pressure distribution calculation of the entire array is based on the multiple line source model and does

259

not rely on the paraxial approximation, thus eliminating the steering angle limitation of about 20 degrees.

260

From the numerical results of the sound field distribution and the axial pressure normalized amplitude plots,

261

we can conclude that the convex surface has a geometric focusing effect, while the concave surface has a

262

divergent effect. Besides, with the increase of the steering angle or the focusing distance, the deviation

263

between the actual focus and the desired focus position is larger, and the pressure amplitude decays.

urn

al

255

The key parameters of the inspection for complex components, such as the number of array elements,

265

element width, adjacent element distance, center frequency, the specimen curvature, the steering angle, and

266

the focusing distance, are interrelated and constrained. Their combined effects influence the radiated acoustic

267

field characteristics. Hence, based on the proposed theory of ultrasonic wave generation and propagation, the

268

parameters of the detection system can be optimized through simulation. This will effectively improve the

Jo

264

20

Journal Pre-proof 269

inspection quality and promote the development and application of the flexible phased array transducer in

270

NDT.

271

Conflicts of interest

273

The authors declare that they have no conflicts of interest. Acknowledgements

of

272

This work was supported by the National Natural Science Foundation of China [grant numbers

275

51905210]; the Jilin Provincial Department of Education under [grant number JJKH20170788KJ]; the

276

Fundamental Research Funds for the Central Universities; and the Jilin Provincial Department of Science &

277

Technology Fund Project [grant numbers 20180520072JH]. The authors would like to express their gratitude

278

to the reviewers for their helpful suggestions.

279

Appendix A. The time delay law for complex surface of double-layer components

Pr e-

p ro

274

We calculate the time delays for the concave or convex surface of double-layer structures based on Snell's

281

law and the principle of Fermat. The geometrical models are shown in Fig. A.1 and Fig. A.2, whose symbols

282

are listed in Table 3 and other applied symbols are demonstrated in Table 1. The time delay laws are not only

283

suitable for multi-layer structures with complex surfaces, but also effective for curved rigid probes with water

284

immersion method or special wedges.

al

280

As shown in Fig. A.1 and A.2, a ray path that travels from the center of 𝑚th element along the incident

286

angle 𝜃1𝑚 in the first medium passes through the intersection 𝐼m to the focus 𝐹 in the second medium at

287

the refracted angle 𝜃2𝑚 . The path must satisfy Snell's law 𝑠𝑖𝑛𝜃1𝑚 𝑠𝑖𝑛𝜃2𝑚 = 𝑐1 𝑐2

(A.1)

The time delay (𝜏𝑚 ) of 𝑚-th element is obtained on the basis of ray acoustics theory

Jo

288

urn

285

𝑙1𝑚 𝑙2𝑚 𝑙1𝑚 𝑙2𝑚 𝜏𝑚 = 𝑚𝑎𝑥 ( + )−( + ) 𝑐1 𝑐2 𝑐1 𝑐2

(A.2)

289

where 𝑙1m and 𝑙2𝑚 are the wave path in the first and second medium, 𝑐1 and 𝑐2 are longitudinal wave or

290

shear wave speeds in the two medium and 𝑐2 is greater than 𝑐1. 21

Journal Pre-proof To calculate the time delay, the task is to obtain the values of the wave path 𝑙1𝑚 and 𝑙2𝑚 , while the real

292

ray path, which depends on the position of the intersection 𝐼m , must be previously determined. Therefore, we

293

first set up the equations 𝜃1𝑚 and 𝜃2𝑚 for 𝛽𝑚 , and then apply these equations to solve the intersection

294

position 𝛽𝑚 according to Snell's law. Finally, we compute the values of the wave path 𝑙1𝑚 and 𝑙2𝑚 .

295

Table 3

296

Symbols of double-layer components geometric models for computing the time delays

of

291

𝑙0

Focal distance in the second medium

p ro

Basic parameters

Wave path from the center of 𝑚th element in the first medium

𝑙2𝑚

Wave path from the center of 𝑚th element in the second medium

𝜃1𝑚

Incident angle from the center of the 𝑚th element

𝜃2𝑚

Refracted angle from the center of the 𝑚th element

Pr e-

𝑙1𝑚

𝐷

Intersection of the line 𝑂1 𝐶 and the second medium surface

𝛽𝑚

Central angle between the intersection 𝐼m and the point 𝐷

Concave surface 𝑅11

Radius of the second medium

𝑅12

Radius of the first medium

al

(𝑅11 > 𝑅12)

Convex surface

Radius of the first medium

𝑅22

Radius of the second medium

(𝑅21 > 𝑅22 )

Jo

urn

𝑅21

22

Journal Pre-proof O

1

φ

m

of

α β

m

M

1 C

p ro

m

c

1

θ I θ

c

1m

2m

s

Pr e-

D θ

2

m

F (l ,θ ) 0

297 298 299

s

Fig. A.1. Schematic diagram for calculating the time delays of double-layer concave components

A.1 The time delay law for the concave surface

If the specimen is concave, as schematized in Fig. A.1, 𝜃1𝑚 and 𝜃2𝑚 can be obtained as follows

al

𝜃1𝑚 = 𝑎𝑡𝑎𝑛(𝑅12 𝑠𝑖𝑛(𝜑𝑚 − 𝛽𝑚 )⁄(𝑅11 − 𝑅12 𝑐𝑜𝑠(𝜑𝑚 − 𝛽𝑚 )))

(A.3)

𝜃2𝑚 = 𝑎𝑡𝑎𝑛(𝑂1 𝐹 𝑠𝑖𝑛(𝛽𝑚 − 𝛼)⁄(𝑂1 𝐹𝑐𝑜𝑠(𝛽𝑚 − 𝛼) − 𝑅11 ))

301

where 𝑂1 𝐹 = √𝑅11 2 + 𝑙0 2 + 2𝑅11 ∙ 𝑙0 ∙ 𝑐𝑜𝑠 𝜃𝑠 ,𝛼 = 𝑎𝑠𝑖𝑛(𝑙0 𝑠𝑖𝑛𝜃𝑠 ⁄𝑂1 𝐹 ),𝜑𝑚 = (𝑚 −

urn

300

(A.4) 𝑀+1 2

) 𝑠⁄𝑅12 .

Then according to Eq. (A.1), the following equation can be obtained 𝑓(𝛽𝑚 ) =

𝑠𝑖𝑛𝜃1𝑚 𝑠𝑖𝑛𝜃2𝑚 − =0 𝑐1 𝑐2

(A.5) 𝜋

𝜋

The equation 𝑓(𝛽𝑚 ) could be solved by iteration or function fzero in the interval [0, 2 ] or [− 2 , 0].

303

The wave path 𝑙1𝑚 and 𝑙2𝑚 can be expressed based on cosine law below

Jo

302

𝑙1𝑚 = √𝑅11 2 + 𝑅12 2 − 2𝑅11 𝑅12 𝑐𝑜𝑠(𝜑𝑚 − 𝛽𝑚 )

23

(A.6)

Journal Pre-proof 𝑙2𝑚 = √𝑅11 2 + 𝑂1 𝐹 2 − 2𝑅11 ∙ 𝑂1 𝐹 𝑐𝑜𝑠(𝛽𝑚 − 𝛼) 304

(A.7)

The time-of-flight 𝑡𝑚 can be obtained 𝑡𝑚 =

𝑙1𝑚 𝑙2𝑚 + 𝑐1 𝑐2

(A.8)

𝛽𝑚 could have multiple solutions based on Eqs. (A.3-A.5), but the real 𝛽𝑚 is the solution that

306

minimizes the time-of-flight 𝑡𝑚 based on Eqs. (A.6-A.8). Since the actual ray path is confirmed, then the

307

values of real wave path 𝑙1𝑚 and 𝑙2𝑚 can be computed based on Eqs. (A.6-A.7). Finally, we obtain the time

308

delay of each element by Eq. (A.2).

p ro

of

305

m

C

1

M

θ I θ

1m

c

m

1

Pr e-

D θ

s

2m

c

2

φ

m

β

F (l ,θ )

m

310 311 312

2

Fig. A.2. Schematic diagram for calculating the time delays of double-layer convex components

A.2 The time delay law for the convex surface

urn

309

s

al

O

0

If the specimen is convex, as shown in Fig. A.2, the calculation is the same as the concave case. 𝜃1𝑚 and 𝜃2𝑚 can be obtained as follows

Jo

𝜃1𝑚 = 𝑎𝑡𝑎𝑛(𝑅21 𝑠𝑖𝑛(𝛽𝑚 − 𝜑𝑚 )⁄(𝑅21 𝑐𝑜𝑠(𝛽𝑚 − 𝜑𝑚 ) − 𝑅22 )) 𝜃2𝑚 = 𝑎𝑡𝑎𝑛(𝑂2 𝐹 𝑠𝑖𝑛(𝛼 − 𝛽𝑚 )⁄(𝑅22 − 𝑂2 𝐹𝑐𝑜𝑠(𝛼 − 𝛽𝑚 )))

313

where 𝑂2 𝐹 = √𝑅22 2 + 𝑙0 2 − 2𝑅22 ∙ 𝑙0 ∙ 𝑐𝑜𝑠 𝜃𝑠 , 𝜑𝑚 = (𝑚 −

24

𝑀+1 2

) 𝑠⁄𝑅21 , 𝛼 can be expressed as

(A.9) (A.10a)

Journal Pre-proof 𝑙0 𝑠𝑖𝑛𝜃𝑠 𝑎𝑠𝑖𝑛 ( ) , 𝑙0 ≤ 𝑅22 𝑂2 𝐹 𝑅22 𝑠𝑖𝑛𝜃𝑠 𝛼 = 𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃𝑠 , 𝑙0 > 𝑅22 ⁡𝑎𝑛𝑑⁡𝜃𝑠 > 0⁡ 𝑂2 𝐹 𝑅22 𝑠𝑖𝑛𝜃𝑠 −𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃𝑠 , 𝑙0 > 𝑅22 ⁡𝑎𝑛𝑑⁡𝜃𝑠 ≤ 0 { 𝑂2 𝐹 The wave path 𝑙1𝑚 and 𝑙2𝑚 can be expressed based on cosine law below 𝑙1𝑚 = √𝑅21 2 + 𝑅22 2 − 2𝑅21 𝑅22 𝑐𝑜𝑠(β𝑚 − 𝜑𝑚 )

(A.11)

𝑙2𝑚 = √𝑅22 2 + 𝑂2 𝐹 2 − 2𝑅22 ∙ 𝑂2 𝐹 𝑐𝑜𝑠(𝛼 − β𝑚 )

(A.12)

p ro

of

314

(A.10b)

The only 𝛽𝑚 should be determined by Eqs. (A.9, A.10, A.5, A.11, A.12, A.8) based on Snell's law and

316

the principle of Fermat, then the values of real wave path 𝑙1𝑚 and 𝑙2𝑚 can be computed based on Eqs.

317

(A.10b, A.11, A.12), finally, we calculate the time delay of each element by Eq. (A.2).

318

Appendix B. Acoustic pressure field for beam steering and focusing of double-layer components

Pr e-

315

Based on the time delay laws of beam generation in Appendix A, analytic models of the sound field

320

distribution for beam steering and focusing on the concave, convex surface of double-layer components are

321

established. The parameters are shown in Table 4, and the other utilized parameters are listed in Table 1-3.

322

Each element normalized pressure distribution at an arbitrary point 𝐺(𝑟, 𝜃) is obtained by a superposition of

323

multiple line sources, which is given as follows

al

319

𝑁

urn

𝑇𝑝0 𝑒𝑥𝑝(𝑖𝑘1 𝑏𝑟̅̅̅̅̅̅ ̅̅̅̅̅̅) 𝑝(𝐆, 𝜔) 1 2𝑘1 𝑏 1𝑚𝑛 + 𝑖𝑘2 𝑏𝑟 2𝑚𝑛 = √ ∑ 𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) 𝜌1 𝑐1 𝑣0 (𝜔) 𝑁 𝜋𝑖 𝑛=1 𝑐2 𝑐𝑜𝑠 2 (𝜃1𝑚𝑛 ) √𝑟̅̅̅̅̅̅ 𝑟̅̅̅̅̅̅ 1𝑚𝑛 + 𝑐 2𝑚𝑛 2 1 𝑐𝑜𝑠 (𝜃2𝑚𝑛 ) 𝑠𝑖𝑛[𝑘1 𝑏𝑠𝑖𝑛𝜃𝑚𝑛 ⁄𝑁] 𝑘1 𝑏𝑠𝑖𝑛𝜃𝑚𝑛 ⁄𝑁

(B.1b)

𝑟1𝑚𝑛 = 𝑟1𝑚𝑛 ⁄𝑏 , ⁡𝑟̅̅̅̅̅̅ ̅̅̅̅̅̅ 2𝑚𝑛 = 𝑟2𝑚𝑛 ⁄𝑏

(B.1c)

𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) =

Jo

(B.1a)

𝑇𝑝0 =

2𝜌2 𝑐2 𝑐𝑜𝑠(𝜃1𝑚𝑛 ) 𝜌1 𝑐1 𝑐𝑜𝑠(𝜃2𝑚𝑛 ) + 𝜌2 𝑐2 𝑐𝑜𝑠(𝜃1𝑚𝑛 )

(B.1d)

324

Therefore, to calculate each element normalized pressure field at any point 𝐺, the task is to compute the

325

values of 𝑟1𝑚𝑛 , 𝑟2𝑚𝑛 and 𝑠𝑖𝑛𝜃𝑚𝑛 , the calculation is similar to the calculation of time delays in double-layer 25

Journal Pre-proof 326

components.

327

Table 4

328

Symbols of double-layer components geometric models for computing the sound field

Wave transmission coefficient

𝜌1

Density of the first medium

𝜌2

Density of the second medium

𝑣0

Spatially uniform velocity of the element

𝑘1

Wave number in the first medium

𝑘2

Wave number in the second medium

𝑟1𝑚𝑛

p ro

𝑇𝑝0

of

Basic parameters

Wave path from the 𝑛th segment center of the 𝑚th element in the first medium

medium

Pr e-

Wave path from the 𝑛th segment center of the 𝑚th element to the point 𝐺 in the second 𝑟2𝑚𝑛

Incident angle from the 𝑛th segment center of the 𝑚th element

𝜃2𝑚𝑛

Refracted angle from the 𝑛th segment center of the 𝑚th element

Jo

urn

al

𝜃1𝑚𝑛

26

Journal Pre-proof 1

δ

φ

n

m

of

O

α β 0

m

mn

C

mn θ

mn

mn

1

ρ ,c 2

2

2mn

Pr e-

θ

ρ ,c 1

θ I θ

1mn

D

M

p ro

β

1

G (r,θ)

Fig. B.1. Schematic diagram for calculating the pressure distribution of double-layer concave components

329 330

B.1 The pressure distribution for the concave surface

If the specimen is concave, as plotted in Fig. B.1, 𝜃1𝑚𝑛 and 𝜃2𝑚𝑛 can be employed as follows 𝜃1𝑚𝑛 = 𝑎𝑡𝑎𝑛(𝑅12 𝑠𝑖𝑛(𝜑𝑚 − β𝑚𝑛 + 𝛿𝑛 )⁄(𝑅11 − 𝑅12 𝑐𝑜𝑠(𝜑𝑚 − β𝑚𝑛 + 𝛿𝑛 )))

(B.2)

𝜃2𝑚𝑛 = 𝑎𝑡𝑎𝑛(𝑂1 𝐹 𝑠𝑖𝑛(β𝑚𝑛 − 𝛼0 )⁄(𝑂1 𝐺𝑐𝑜𝑠(β𝑚𝑛 − 𝛼0 ) − 𝑅11 ))

(B.3)

al

331

where 𝑂1 𝐺 = √𝑅11 2 + 𝑟 2 + 2𝑅11 ∙ 𝑟 ∙ 𝑐𝑜𝑠 𝜃 , 𝛼0 = asin(𝑟𝑠𝑖𝑛𝜃⁄𝑂1 𝐺 ) , 𝜑𝑚 = (𝑚 −

333

(𝑛 −

335 336

2

)

𝑁

/𝑅12

𝑀+1 2

) 𝑠⁄𝑅12 , 𝛿𝑛 =

Then according to Snell's law, the following equation can be obtained

Jo

334

1+𝑁 2𝑏

urn

332

𝑓(β𝑚𝑛 ) =

𝑠𝑖𝑛𝜃1𝑚𝑛 𝑠𝑖𝑛𝜃2𝑚𝑛 − =0 𝑐1 𝑐2

(B.4) 𝜋

𝜋

2

2

The equation 𝑓(β𝑚𝑛 ) could be solved by iteration or function fzero in the interval [0, ] or [− , 0]. Then the wave path 𝑟1𝑚𝑛 and 𝑟2𝑚𝑛 can be expressed based on cosine law

27

Journal Pre-proof

337

𝑟1𝑚𝑛 = √𝑅11 2 + 𝑅12 2 − 2𝑅11 𝑅12 𝑐𝑜𝑠(𝜑𝑚 − β𝑚𝑛 + 𝛿𝑛 )

(B.5)

𝑟2𝑚𝑛 = √𝑅11 2 + 𝑂1 𝐺 2 − 2𝑅11 ∙ 𝑂1 𝐺 𝑐𝑜𝑠(β𝑚𝑛 − 𝛼0 )

(B.6)

The time-of-flight 𝑡𝑚𝑛 can be obtained 𝑟1𝑚𝑛 𝑟2𝑚𝑛 + 𝑐1 𝑐2

(B.7)

of

𝑡𝑚𝑛 =

𝛽𝑚𝑛 could have multiple solutions based on Eqs. (B.2-B.4), but the real 𝛽𝑚𝑛 is the solution that

339

minimizes the time-of-flight 𝑡𝑚𝑛 based on Eqs. (B.5-B.7). Since the actual ray path is confirmed, then the

340

values of real wave path 𝑟1𝑚𝑛 and 𝑟2𝑚𝑛 can be computed based on Eqs. (B.5-B.6).

341

p ro

338

And sin𝜃𝑚𝑛 can be expressed based on sine law below

1

𝑅11 𝑠𝑖𝑛𝜃1𝑚𝑛 𝑅12

Pr e-

sin𝜃𝑚𝑛 =

mn θ

C

M

mn

D

θ I θ

1mn

θ

mn

2mn

φ

(B.8)

ρ ,c

1

1

ρ ,c 2

2

m

β β

343 344 345

G (r,θ)

mn

α

0

O

2

Fig. B.2. Schematic diagram for calculating the pressure distribution of double-layer convex components

B.2 The pressure distribution for the convex surface If the specimen is convex, as presented in Fig. B.2, the calculation is the same as the concave case. 𝜃1𝑚𝑛

Jo

342

urn

al

m

and 𝜃2𝑚𝑛 can be obtained as follows 𝜃1𝑚𝑛 = 𝑎𝑡𝑎𝑛(𝑅21 𝑠𝑖𝑛(𝛽𝑚𝑛 − 𝜑𝑚 − 𝛿𝑛 )⁄(𝑅21 𝑐𝑜𝑠(𝛽𝑚𝑛 − 𝜑𝑚 − 𝛿𝑛 ) − 𝑅22 ))

28

(B.9)

Journal Pre-proof 𝜃2𝑚𝑛 = 𝑎𝑡𝑎𝑛(𝑂2 𝐺 𝑠𝑖𝑛(𝛼0 − 𝛽𝑚𝑛 )⁄(𝑅22 − 𝑂2 𝐺𝑐𝑜𝑠(𝛼0 − 𝛽𝑚𝑛 ))) 346

where 𝑂2 𝐺 = √𝑅22 2 + 𝑟 2 − 2𝑅22 ∙ 𝑟 ∙ 𝑐𝑜𝑠 𝜃 , ⁡𝜑𝑚 = (𝑚 −

347

be depicted

𝑀+1 2

) 𝑠⁄𝑅21 , 𝛿𝑛 = (𝑛 −

(B.10a) 1+𝑁 2𝑏 2

)

𝑁

/𝑅21 , 𝛼0 can

of

𝑟𝑠𝑖𝑛𝜃 𝑎𝑠𝑖𝑛 ( ) , 𝑟 ≤ 𝑅22 𝑂2 𝐺 𝑅22 𝑠𝑖𝑛𝜃 𝛼0 = 𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃, 𝑟 > 𝑅22 ⁡𝑎𝑛𝑑⁡𝜃 > 0⁡ 𝑂2 𝐺 𝑅22 𝑠𝑖𝑛𝜃 −𝜋 − 𝑎𝑠𝑖𝑛 ( ) − 𝜃, 𝑟 > 𝑅22 ⁡𝑎𝑛𝑑⁡𝜃 ≤ 0 { 𝑂2 𝐺

p ro

Then the wave path 𝑟1𝑚𝑛 and 𝑟2𝑚𝑛 can be expressed based on cosine law

𝑟1𝑚𝑛 = √𝑅21 2 + 𝑅22 2 − 2𝑅21 𝑅22 𝑐𝑜𝑠(𝛽𝑚𝑛 − 𝜑𝑚 − 𝛿𝑛 )

(B.11)

𝑟2𝑚𝑛 = √𝑅22 2 + 𝑂2 𝐺 2 − 2𝑅22 ∙ 𝑂2 𝐺 𝑐𝑜𝑠(𝛼0 − 𝛽𝑚𝑛 )

(B.12)

Pr e-

348

(B.10b)

349

The only 𝛽𝑚𝑛 should be determined by Eqs. (B.9, B.10, B.4, B.11, B.12, B.7) based on Snell's law and

350

the principle of Fermat, then the values of real wave path 𝑟1𝑚𝑛 and 𝑟2𝑚𝑛 can be computed based on Eqs.

351

(B.10b, B.11, B.12).

352

And sin𝜃𝑚𝑛 can be expressed based on sine law below

𝑅22 𝑠𝑖𝑛𝜃1𝑚𝑛 𝑅21

(B.13)

al

𝑠𝑖𝑛𝜃𝑚𝑛 =

Therefore, based on the time delay laws of beam generation and each element normalized pressure field,

354

an analytic expression of the sound field distribution for the entire array beam steering and focusing on the

355

double-layer curved structures is established

urn

353

𝑀

𝑁

(B.14)

Jo

𝑇𝑝0 𝑒𝑥𝑝(𝑖𝑘1 𝑏𝑟̅̅̅̅̅̅ ̅̅̅̅̅̅) 𝑝(𝑮, 𝜔) 1 2𝑘1 𝑏 1𝑚𝑛 + 𝑖𝑘2 𝑏𝑟 2𝑚𝑛 = ∑ 𝑒𝑥𝑝(𝑖𝜔𝜏𝑚 ) √ ∑ 𝐷𝑏⁄𝑁 (𝜃𝑚𝑛 ) (𝜔) 𝜌1 𝑐1 𝑣0 𝑁 𝜋𝑖 𝑚=1 𝑛=1 𝑐2 𝑐𝑜𝑠 2 (𝜃1𝑚𝑛 ) √𝑟̅̅̅̅̅̅ 𝑟2𝑚𝑛 ̅̅̅̅̅̅ 1𝑚𝑛 + 𝑐 2 1 𝑐𝑜𝑠 (𝜃2𝑚𝑛 ) [ ] 356

References

357 358

[1] B.W. Drinkwater, P.D. Wilcox, Ultrasonic arrays for non-destructive evaluation: A review, NDT & E International, 39 (2006) 525-541. 29

Journal Pre-proof [2] L. Azar, Y. Shi, S.-C. Wooh, Beam focusing behavior of linear phased arrays, NDT & E International, 33 (2000) 189-198. [3] J. Camacho, J.F. Cruza, Auto-focused virtual source imaging with arbitrarily shaped interfaces, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 62 (2015) 1944-1956. [4] L.W. Schmerr Jr, Time Delay Laws (2-D), in: L.W. Schmerr Jr (Ed.) Fundamentals of Ultrasonic Phased Arrays, Springer International Publishing, Cham, 2015, pp. 99-111. [5] L.W. Schmerr Jr, Time Delay Laws (3-D), in: L.W. Schmerr Jr (Ed.) Fundamentals of Ultrasonic Phased Arrays, Springer International Publishing, Cham, 2015, pp. 169-177. [6] S. Mahaut, O. Roy, C. Beroni, B. Rotter, Development of phased array techniques to improve characterization of defect located in a component of complex geometry, Ultrasonics, 40 (2002) 165-169.

of

[7] S. Chatillon, G. Cattiaux, M. Serre, O. Roy, Ultrasonic non-destructive testing of pieces of complex geometry with a flexible phased array transducer, Ultrasonics, 38 (2000) 131-134.

[8] S. Robert, O. Casula, O. Roy, G. Neau, Real-time nondestructive testing of composite aeronautical structures with a self-

p ro

adaptive ultrasonic technique, Measurement Science and Technology, 24 (2013).

[9] F. Simonetti, M. Fox, Experimental methods for ultrasonic testing of complex-shaped parts encased in ice, NDT & E International, 103 (2019) 1-11.

[10] N. Xu, Z. Zhou, Numerical simulation and experiment for inspection of corner-shaped components using ultrasonic phased array, NDT & E International, 63 (2014) 28-34.

[11] M. Nagai, S. Lin, H. Fukutomi, Determination of shape profile by saft for application of phased array technique to complex geometry surface, AIP Conference Proceedings, 1430 (2012) 849-856.

Pr e-

[12] R. Long, P. Cawley, FURTHER DEVELOPMENT OF A CONFORMABLE PHASED ARRAY DEVICE FOR INSPECTION OVER IRREGULAR SURFACES, AIP Conference Proceedings, 975 (2008) 754-761.

[13] J. Russell, R. Long, P. Cawley, N. Habgood, INSPECTION OF COMPONENTS WITH IRREGULAR SURFACES USING A CONFORMABLE ULTRASONIC PHASED ARRAY, AIP Conference Proceedings, 1096 (2009) 792-799. [14] J.W. Mackersie, G. Harvey, A. Gachagan, DEVELOPMENT OF AN EFFICIENT CONFORMABLE ARRAY STRUCTURE, AIP Conference Proceedings, 1096 (2009) 785-791.

[15] G. Toullelan, O. Casula, E. Abittan, P. Dumas, APPLICATION OF A 3D SMART FLEXIBLE PHASED‐ARRAY TO PIPING INSPECTION, AIP Conference Proceedings, 975 (2008) 794-800.

[16] O. Casula, C. Poidevin, G. Cattiaux, P. Dumas, Control of complex components with Smart Flexible Phased Arrays, Ultrasonics, 44 (2006) e647-e651.

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[17] C.J.L. Lane, The inspection of curved components using flexible ultrasonic arrays and shape sensing fibres, Case Studies in Nondestructive Testing and Evaluation, 1 (2014) 13-18.

[18] K. Nakahata, S. Tokumasu, A. Sakai, Y. Iwata, K. Ohira, Y. Ogura, Ultrasonic imaging using signal post-processing for a

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flexible array transducer, NDT & E International, 82 (2016) 13-25. [19] H. Hu, X. Zhu, C. Wang, L. Zhang, X. Li, S. Lee, e. al, Stretchable ultrasonic transducer arrays for three-dimensional imaging on complex surfaces, Science Advances, 4 (2018) eaar3979. [20] E. Kühnicke, Three‐dimensional waves in layered media with nonparallel and curved interfaces: A theoretical approach, The Journal of the Acoustical Society of America, 100 (1996) 709-716. [21] R. Huang, L.W. Schmerr, A. Sedov, A New Multi‐Gaussian Beam Model for Phased Array Transducers, AIP Conference Proceedings, 894 (2007) 751-758.

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is currently pursuing her M.S. in the School of Mechanical and Aerospace Engineering, Jilin University, China. Her research interests include phased array ultrasonic testing theories and applications.

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Xinyu Zhao got his Ph.D. in the field of ultrasonic nondestructive testing from Harbin Institute of Technology, China in 2008. He is currently an associate professor in the Department of Welding Technology and Engineering at Dalian Jiaotong University. His research interests include ultrasonic nondestructive evaluation and industrial

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automation.

Sung-Jin Song received his Ph.D. in Engineering Mechanics from Iowa State University, Ames, Iowa, USA in 1991. He worked at Daewoo Heavy Industries, Ltd., Incheon, Korea for 5 years from 1983, where he was certified as ASNT Level III in RT, UT, MT and PT. He worked at Chosun University, Gwangju, Korea as Assistant Professor from 1993. Since 1998 he has been at Sungkyunkwan University, Suwon, Korea and is currently Professor of Mechanical Engineering.

Jianhai Zhang received his Ph.D. in Mechanical Engineering from Sungkyunkwan University, Korea in 2015. He is presently an assistant professor in the School of Mechanical and Aerospace Engineering, Jilin University, China. His current research interests include electromagnetic testing, nondestructive testing and evaluation theory and applications.

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Yanfang Zheng received her B.S. from Changchun University of Science and Technology, China in 2017. She

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