論文 / 著書情報
Article / Book Information
題目(和文)

高速な時間領域境界要素法及びその非線形超音波シミュレーションへの適用

Title(English)

Fast time-domain boundary element method and its application to
nonlinear ultrasonic simulation

著者(和文)

丸山泰蔵

Author(English)

Taizo Maruyama

出典(和文)

学位:博士(工学),学位授与機関:東京工業大学,
報告番号:甲第10240号,
授与年月日:2016年3月26日,
学位の種別:課程博士,
審査員:廣瀬 壮一,天谷 賢治,三上 貴正,アニール ワイジェ,中村 隆志,斎藤 隆泰

Citation(English)

Degree:Doctor (Engineering),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第10240号,
Conferred date:2016/3/26,
Degree Type:Course doctor,
Examiner:,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)FAST TIME - DOMAIN BOUNDARY ELEMENT METHOD AND ITS APPLICATION TONONLINEAR ULTRASONIC SIMULATIONbyTaizo Maruyama

Master of Engineering
Tokyo Institute of Technology, 2013
Bachelor of Engineering
Tokyo Institute of Technology, 2011————————————————————–
Submitted in Partial Fulfillment of theRequirements for the Degree of Doctor of Engineering in the
Department of Mechanical and Environmental Informatics
Graduate School of Information Science and Engineering
Tokyo Institute of Technology
2016

Academic Supervisor and Chief of Committee: Professor Sohichi Hirose
Committee Members:Professor Kenji Amaya
Associate Professor Anil C. Wijeyewickrema
Associate Professor Takahiro Saitoh
Associate Professor Takamasa Mikami
Lecturer Takashi Nakamura

Acknowledgments
I would like to express my gratitude to Professor Sohichi Hirose for his excellent support and providing me the precious study opportunity as a doctoral student in his laboratory. His shrewd advice
and enthusiastic direction always helped me to conduct this research. I especially would like toexpress my deepest appreciation to Associate Professor Takahiro Saitoh of Gunma University forhis guidance, encouragement, and discussion. He was an Assistant Professor of Tokyo Instituteof Technology until March 2012. I have been really helped by him since I was an undergraduate
student.
I would like to thank to Associate Professor Kazuyuki Nakahata of Ehime University for his
advice and many discussion. I would like to express my gratitude to Professor Terumi Touhei ofTokyo University of Science for his advice and many discussion. His advice was pertinent andhelped my research. I would like to thank to Professor Naoshi Nishimura of Kyoto University for
his advice and many discussion about a boundary element method and its acceleration. I would also
thank to Lecturer Hitoshi Yoshikawa of Kyoto University for many discussion about a time-domainboundary element method at conferences.
I am grateful to Professor Toshihiko Sugiura of Keio University for his useful advice about nonlinear ultrasonic generation phenomena. I wish to thank committee members of my doctoral research, Professor Kenji Amaya, Associate Professor Anil C. Wijeyewickrema, Associate Professor
Takamasa Mikami, and Lecturer Takashi Nakamura, for their enthusiastic discussion and usefuladvice.
I would also like to thank to members of Hirose laboratory for their encouragement and many
discussion. Especially, I could learn a lot from Assistant Professor Akira Furukawa’s sincere attitude
toward his education and research. Also, I thank to Tinh Q. Bui of Postdoctoral Fellowship for
Research Abroad for his advice and many discussion. Thank to him, my English skill became muchbetter than before.
I am also very grateful to Japan Society for the Promotion of Science for making my doctoral
study possible by the financial support.
Finally, I would like to express my gratitude to my parents for their understanding, support, andencouragement although I quitted the high school on account of personal reasons. They have always
supported my life.
Department of Mechanical and Environmental Informatics,
Graduate School of Information Science and Engineering,Tokyo Institute of Technology, Japan.

iii

Abstract
The nonlinear ultrasonic testing (NLUT) based on the contact acoustic nonlinearity (CAN) has attracted notice recently because it is expected to be a new technique for detection of closed cracks,
which cannot be detected by using the conventional linear ultrasonic testing. In the NLUT, the nonlinear ultrasonic waves which consist of higher- and sub-harmonic waves generated from the defects
are evaluated, and the location and size of the defect are predicted by analyzing the frequency characteristic of them.
The higher-harmonic generation due to the CAN was advocated over three decades ago. Thereafter, this phenomenon has been vigorously investigated, and its mechanism has been clearly understood from theoretical and experimental points of view. However, when the higher-harmonic
generation is caused owing to an interface crack, the contribution of material difference has notbeen investigated sufficiently. Moreover, the mechanism of sub-harmonic generation has not been
understood from a theoretical point of view yet although its generation has confirmed in the measurement. Therefore, further theoretical and/or numerical approaches are demanded in order toutilize this nonlinear phenomenon for the nondestructive evaluation (NDE).
In general, the higher-harmonic waves are generated by many causes such as transducer, couplant, and material nonlinearity. On the other hand, it is considered that the sub-harmonic waves are
generated only by crack faces with contact in this situation. Thus, if the sub-harmonic waves can beused for the NDE, the more precise detection and size measurement of closed cracks are expected.
At this stage, there are a few numerical approaches for sub-harmonic generation, and these simulations have been succeeded only in the case of very thin debonding plate. However, the subharmonic generation has also been observed for different situations in the realistic measurement.
Therefore, the author conducts numerical simulation of the sub-harmonic generation due to cracks
in an unbounded elastic solid and a surface breaking crack in an elastic half-space in order to investigate the causes.
The time-domain boundary element method (TD-BEM) is considered as an appropriate numerical method for this simulation because the (semi-)infinite domain is dealt with in this study, andthe nonlinear boundary conditions on crack faces can be treated exactly. In order to investigate stable sub-harmonic waves due to the CAN, a long-time transient analysis for the nonlinear boundary
problem has to be carried out. The computational cost of numerical simulation becomes large (especially for three-dimensional (3-D) problems).
In this dissertation, a convolution quadrature time-domain boundary element method (CQ-BEM),
which is known as the one of stable and accurate TD-BEMs, is accelerated using a fast multipolev

method (FMM) and a rapid convolution algorithm for solving large-scale wave problems. The author has applied the proposed fast CQ-BEM to the numerical simulation of NLUT due to the CAN.
First, the higher-harmonic generation due to an interface crack is investigated by using the developed 3-D TD-BEM. Second, many 2-D numerical simulations are conducted for the sub-harmonic
generation due to cracks in an unbounded infinite elastic solid and a surface breaking crack in anelastic half-space. Finally, some 3-D numerical simulations of the sub-harmonic generation are
carried out for cracks in an unbounded infinite elastic solid, and the author investigates the dimensional effects, i.e. difference from 2-D results. Through obtained numerical results, the causes andcharacteristics of the nonlinear ultrasonic generation phenomenon are discussed.

vi

Contents
Acknowledgmentsiii

Abstractv

List of figures

xix

List of Tables

xxi

List of Main Symbols

xxiii

1. Introduction
1.1. Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Previous researches of nonlinear ultrasonic testing . . . . . . . . . . . . . . . . . .
1.3. Time-domain boundary element method (TD-BEM) . . . . . . . . . . . . . . . . .
1.4. Organization of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
4
5

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)
2.1. Convolution quadrature method (CQM) . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Linear multistep-based CQM . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Implicit Runge-Kutta (IRK)-based CQM . . . . . . . . . . . . . . . . . .
2.1.3. Behavior of Laplace parameter . . . . . . . . . . . . . . . . . . . . . . . .
2.2. CQ-BEM for 3-D acoustic wave problems . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Formulation of boundary integral equation (BIE) . . . . . . . . . . . . . .
2.2.2. Discretization of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
7
7
9
12
13
13
14
15
18

3. Acceleration of CQ-BEM
3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM) . . . . . . . . . . . . . . . .
3.1.1. Time-marching process . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Fast multipole method (FMM) . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Some numerical techniques for efficient CQ-FMBEM . . . . . . . . . . .
3.1.4. Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.5. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
19
20
24
27
28

vii

Contents
3.1.6. CQ-FMBEM for 3-D elastic wave problems . . . . . . . . . . . . . . . .
3.1.7. CQ-FMBEM for 3-D coupled acoustic-elastic wave problems . . . . . . .
3.2. Acceleration for number of time steps . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. New time-marching process of CQ-FMBEM . . . . . . . . . . . . . . . .
3.2.2. Truncation condition for M2L using arrival time of influence waves . . . .
3.2.3. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
38
45
45
48
48
50

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bimaterial
53
4.1. Problem statement and BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2. Discretization of BIE and contact boundary conditions . . . . . . . . . . . . . . . 55
4.2.1. Nonlinear interface conditions . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.2. Modification of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3. Numerical procedure for nonlinear analysis . . . . . . . . . . . . . . . . . . . . . 58
4.4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1. Accuracy verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4.2. Nonlinear ultrasonic simulation . . . . . . . . . . . . . . . . . . . . . . . 61
4.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface
Breaking Cracks
77
5.1. Problem statement and BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1. Scattering problem by cracks in infinite domain . . . . . . . . . . . . . . 77
5.1.2. Scattering problem by surface breaking cracks in half-space . . . . . . . . 79
5.2. Regularization of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3. Discretization of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3.1. Scattering problem by cracks in infinite domain . . . . . . . . . . . . . . . 83
5.3.2. Scattering problem by surface breaking cracks in half-space . . . . . . . . 85
5.4. Nonlinear boundary conditions and numerical procedure . . . . . . . . . . . . . . 86
5.5. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.1. Linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.2. Higher-harmonic simulation . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.5.3. Simulation of sub-harmonic waves due to interior cracks . . . . . . . . . . 92
5.5.4. Simulation of sub-harmonic waves due to a surface breaking crack . . . . . 112
5.6. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
121
6.1. Problem statement and BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2. Regularization of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3. Discretization of BIE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

viii

Contents
6.4.
6.5.
6.6.
6.7.

Numerical integration over element surfaces . . . . . . . . . . . . . . . . . . . . .
Application of FMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear boundary conditions and numerical procedure . . . . . . . . . . . . . .
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.1. Linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.2. Higher-harmonic simulation . . . . . . . . . . . . . . . . . . . . . . . . .
6.7.3. Sub-harmonic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127
127
129
131
131
134
137
146

7. Final Remarks

149

A. Translation Operators and Recursions

153

B. Far-field Approximation of Scattered Waves
155
B.1. Inclusions and cracks in 2-D and 3-D . . . . . . . . . . . . . . . . . . . . . . . . . 155
B.2. Interface cracks between two semi-infinite elastic solids in 3-D . . . . . . . . . . . 157
B.3. Surface breaking cracks in 2-D elastic half-space . . . . . . . . . . . . . . . . . . 158
C. Free-field for two-layered elastic solid

161

Bibliography

164

ix

List of Figures
2.1. (a) Overall picture and (b) enlarged view around the origin of Laplace parameters s
required for CQMs with RKm and BDFk using computational parameters in Table
2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122.2. Analysis model for acoustic wave scattering by an obstacle in a 3-D infinite domain.

13

2.3. Acoustic wave scattering by a rigid sphere subjected to an incident plane wave. . .

16

2.4. Time variations of p/p0 obtained by using BDF1 at point X in Fig. 2.3. . . . . . .

16

2.5. Time variations of p/p0 obtained by using BDF2 at point X in Fig. 2.3. . . . . . .

16

2.6. Time variations of p/p0 obtained by using RK2 at point X in Fig. 2.3. . . . . . . .

17

2.7. Time variations of p/p0 obtained by using RK3 at point X in Fig. 2.3. . . . . . . .

17

3.1. Translation theorems (M2M, M2L, and L2L). . . . . . . . . . . . . . . . . . . . .

24

3.2. Scaling for M2M moment translated from Cϱ to C2ϱ using half diagonal length ofcube cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.3. Distance between two cells in M2L translation. . . . . . . . . . . . . . . . . . . .

27

3.4. Time variations of p/p0 obtained by CQ-BEM and CQ-FMBEM using RK3 at point
X in Fig. 2.3when cf ∆t/a = 0.069. . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.5. Absolute errors between p/p0 obtained by CQ-BEM and CQ-FMBEM, which are
shown in Fig. 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

3.6. Computational time comparison between CQ-BEM and CQ-FMBEM using RK3
for 3-D acoustic wave problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.7. Required memory comparison between CQ-BEM and CQ-FMBEM using RK3 for
3-D acoustic wave problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.8. Analysis model for elastic wave scattering by an obstacle in a 3-D infinite domain .

30

3.9. Elastic wave scattering by a spherical cavity subjected to an incident plane wave. .

34

3.10. Time variations of usc
1 /u0 obtained by using BDF2 at point X in Fig.

3.9when
Me = 9280. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.11. Time variations of usc
1 /u0 obtained by using RK2 at point X in Fig. 3.9when Me =9280. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.12. Time variations of usc
1 /u0 obtained by using RK3 at point X in Fig. 3.9when Me =9280. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.13. Time variations of usc
1 /u0 obtained by using RK3 at point X in Fig.

3.9whencT ∆t/a = 0.06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

xi

List of Figuresxii

3.14. Computational time comparison between CQ-BEM and CQ-FMBEM using BDF2
and RK3 for 3-D elastic wave problem. . . . . . . . . . . . . . . . . . . . . . . .

37

3.15. Required memory comparison between CQ-BEM and CQ-FMBEM using BDF2
and RK3 for 3-D elastic wave problem. . . . . . . . . . . . . . . . . . . . . . . .

37

3.16. Analysis model 3-D for coupled acoustic-elastic wave problems . . . . . . . . . .

39

3.17. Relative errors for calculating convolution integral (3.73) vs. time increments using
IRK-based CQM with RK2 and RK3. . . . . . . . . . . . . . . . . . . . . . . . .

42

3.18. Elastic wave scattering by a spherical obstacle subjected to an incident plane wave.

42

3.19. Time variations of far-field backscattered amplitude ΩL /(au0 ) for various obstacles
subjected to an incident plane P wave. . . . . . . . . . . . . . . . . . . . . . . . .

43

3.20. Time variations of far-field forward-scattered amplitude ΩL /(au0 ) for various obstacles subjected to an incident plane P wave. . . . . . . . . . . . . . . . . . . . .

43

3.21. Time variations of far-field backscattered amplitude ΩT V /(au0 ) for various obstacles subjected to an incident plane SV wave. . . . . . . . . . . . . . . . . . . . . .

44

3.22. Time variations of far-field forward-scattered amplitude ΩT V /(au0 ) for various obstacles subjected to an incident plane SV wave. . . . . . . . . . . . . . . . . . . .

44

3.23. (a) Coefficient matrix structure for time and (b) calculation order of sub-matrix inthe rapid convolution algorithm [31]. . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.24. Time variations of p/p0 obtained by conventional and new CQ-FMBEMs using
RK3 at point X in Fig. 2.3when cf ∆t/a = 0.069. . . . . . . . . . . . . . . . . . .

49

3.25. Absolute errors between p/p0 obtained by conventional and new CQ-FMBEMs,
which are plotted in Fig. 3.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.26. Computational time comparison between conventional and new CQ-FMBEMs using RK3 for 3-D acoustic wave problem varying the number of elements Me . . . .

50

3.27. Computational time comparison between conventional and new CQ-FMBEMs using RK3 for 3-D acoustic wave problem varying the number of time steps N . . . .

50

3.28. Ratio of M2L truncation for each sub-matrix in triangular Toeplitz system using (a)only conventional truncation condition (3.33) and (b) both conventional and newones when N = 128 and Me = 3176. . . . . . . . . . . . . . . . . . . . . . . . .

51

4.1. Analysis model for wave scattering by a debonding area on bi-material interface in
3-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

4.2. Contact phases on debonding area Sd . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.3. Flowchart of numerical procedure for nonlinear analysis. . . . . . . . . . . . . . .

58

4.4. Local coordinate system for stopping condition of relatively tangential movement. .

60

4.5. An incident plane wave scattering by a penny-shaped interface crack with radius a.

60

4.6. Vertical COD on Sd at each several time steps cT t/a when a penny-shaped crack issubjected to normal incidence of P wave. . . . . . . . . . . . . . . . . . . . . . . .

61

4.7. Time variations of vertical displacement at the center point of a penny-shaped crack
subjected to an incident plane P wave when kT a = 2.0 and σ st /σ0 = 0. . . . . . .

63

List of Figures
4.8. Time variations of vertical displacement at the center point of a penny-shaped crack
subjected to an incident plane P wave when kT a = 5.2 and σ st /σ0 = 0. . . . . . .

63

4.9. Normalized frequency spectrum of far-field amplitude of backscattered P wave ΩL /(au0 )when kT a = 2.0 and σ st /σ0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.10. Normalized frequency spectrum of far-field amplitude of backscattered P wave ΩL /(au0 )when kT a = 5.2 and σ st /σ0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.11. Time variations of vertical COD at the center point of a penny-shaped crack subjected to an incident plane P wave varying σ st /σ0 when kT a = 2.0. . . . . . . . .

65

4.12. Time variations of vertical COD at the center point of a penny-shaped crack subjected to an incident plane P wave varying σ st /σ0 when kT a = 5.2. . . . . . . . .

65

4.13. Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kT a = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.14. Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kT a = 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.15. Time variations of horizontal COD at the center of a penny-shaped crack subjected
to an incidence plane SV wave varying σ st /σ0 when kT a = 2.0. . . . . . . . . . .

67

4.16. Normalized Fourier amplitude corresponding to 3rd higher-harmonic component
A3ω /A03ω in far-field backscattered SV waves varying σ st /σ0 when kT a = 2.0. . .

67

4.17. Normalized frequency spectrum of far-field amplitude of backscattered SV wave
ΩT V /(au0 ) when kT a = 2.0 and σ st /σ0 = 0.6. . . . . . . . . . . . . . . . . . . .

67

4.18. Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying
θin when kT a = 2.0 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . . . . . .

68

4.19. Normalized frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 )varying θin when kT a = 2.0 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . 68
4.20. Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying
θin when kT a = 5.2 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . . . . . .

69

4.21. Normalized frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 )varying θin when kT a = 5.2 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . 69
4.22. Absolute values of reflection and transmission coefficients for displacement whena semi-infinite and flat interface between steel and aluminum is subjected to an
incident plane P wave with incident angle θin . . . . . . . . . . . . . . . . . . . . .

69

4.23. Time variations of vertical displacement on the center point of a penny-shaped interface crack subjected to an incident plane P wave when kTI a = 2.0 and σ st /σ0 = 0. 70
4.24. Time variations of vertical displacement on the center point of a penny-shaped interface crack subjected to an incident plane P wave when kTI a = 5.2 and σ st /σ0 = 0. 70
4.25. Normalized frequency spectrum of far-field amplitude of backscattered P wave ΩL /(au0 )when kTI a = 2.0 and σ st /σ0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 71

xiii

List of Figures
4.26. Normalized frequency spectrum of far-field amplitude of backscattered P wave ΩL /(au0 )when kTI a = 5.2 and σ st /σ0 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.27. Time variations of vertical COD at the center point of a penny-shaped interface
crack subjected to an incident plane P wave varying σ st /σ0 when kTI a = 2.0. . . .

72

4.28. Time variations of vertical COD at the center point of a penny-shaped interface
crack subjected to an incident plane P wave varying σ st /σ0 when kTI a = 5.2. . . .

72

4.29. Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kTI a = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.30. Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kTI a = 5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

4.31. Time variations of horizontal COD at the center of a penny-shaped interface crack
subjected to an incidence plane SV wave varying σ st /σ0 when kTI a = 2.0. . . . . .

74

4.32. Normalized Fourier amplitude corresponding to 3rd higher-harmonic component
A3ω /A03ω in far-field backscattered SV waves varying σ st /σ0 when kTI a = 2.0. . .

74

4.33. Normalized frequency spectrum of far-field amplitude of backscattered SV wave
ΩT V /(au0 ) when kTI a = 2.0 and σ st /σ0 = 0.4. . . . . . . . . . . . . . . . . . . .

74

4.34. Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying
θin when kTI a = 2.0 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . . . . . .

75

4.35. Normalized frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 )varying θin when kTI a = 2.0 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . 75
4.36. Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying
θin when kTI a = 5.2 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . . . . . .

76

4.37. Normalized frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 )varying θin when kTI a = 5.2 and σ st /σ0 = 0.0. . . . . . . . . . . . . . . . . . . . 76
5.1. Analysis model for elastic wave scattering by a crack in a 2-D infinite domain. . . .

78

5.2. Analysis model for wave scattering by a surface breaking crack in a 2-D elastic
half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

5.3. Spatial shape functions with respect to displacement and traction for a crack in an
infinite domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

5.4. (a) Double node model and (b) triple node model for connection point between
surface breaking crack and free-surface. . . . . . . . . . . . . . . . . . . . . . . .

85

5.5. A straight-line crack subjected to an incident plane wave. . . . . . . . . . . . . . .

87Time variations of dynamic SIF of mode I when θin = 30◦ . . . . . . . . . . . . . .

88

5.7. Time variations of (a) COD at the center point of a crack and (b) far-field amplitudeof backscattered P wave when a straight-line crack is subjected to an incident plane
P wave. (c) is normalized frequency spectrum of far-field amplitude. . . . . . . . .

90

5.6.

xiv

List of Figures
5.8. Time variations of (a) tangential traction at the center point of a crack and (b) farfield amplitude of backscattered S wave when a straight-line crack is subjected to
an incident plane S wave, and σ st /σ0 = 1.0. (c) is normalized frequency spectrumof far-field amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

5.9. Fourier amplitude corresponding to 2nd higher-harmonic wave normalized by theone when ug /u0 = σ st /σ0 = 0.0 for (a) various kT a and ug /u0 , and (b) various
kT a and σ st /σ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.10. Time variations of far-field amplitude of backscattered P wave when kT a = 5.0. . .

92

5.11. Elastic wave scattering by (a) a half-sine-shaped crack and (b) two straight-line
cracks arranged in parallel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.12. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena half-sine-shaped crack is subjected to an incident plane P wave with nin = 30
(h/a = 0.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

5.13. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena half-sine-shaped crack is subjected to an incident plane P wave with nin = 30
(h/a = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.14. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena half-sine-shaped crack is subjected to an incident plane P wave with nin = 30
(h/a = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

5.15. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena half-sine-shaped crack is subjected to an incident plane P wave with nin = 30
(h/a = 1.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.16. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena half-sine-shaped crack is subjected to an incident plane P wave with nin = 30
(h/a = 2.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

5.17. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a curved crack is subjected to an incident plane P wave
(kT a = 1.2, ug /u0 = 2.0, h/a = 2.0, and θin = 60◦ ). . . . . . . . . . . . . . . . .

99

5.18. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a curved crack is subjected to an incident plane P wave
(kT a = 2.5, ug /u0 = 0.0, h/a = 2.0, and θin = 60◦ ). . . . . . . . . . . . . . . . . 100
5.19. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 1.2, h/a = 2.0, and θin = 60◦ . . . . 100
5.20. Displacement fields around a curved crack from f in t = 20.0 to 22.0 when h/a =2.0, θin = 60◦ , kT a = 1.2, and ug /u0 = 2.0. . . . . . . . . . . . . . . . . . . . . 101

xv

List of Figures
5.21. Histories of displacement at points (a) C1 and C2, and (b) C3 and C4 in the lower
right of Fig. 5.20from f in t = 20.0 to 22.0 when h/a = 2.0, θin = 60◦ , kT a = 1.2,
and ug /u0 = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.22. Histories of displacement at points C1–C4 in the lower right of Fig. 5.20from
f in t = 20.0 to 21.0 when h/a = 2.0, θin = 60◦ , kT a = 1.2, and (a) ug /u0 = 0.5
and (b) ug /u0 = 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.23. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when
two straight-line cracks arranged in parallel are subjected to an incident plane P
wave with nin = 30 (h/a = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.24. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when
two straight-line cracks arranged in parallel are subjected to an incident plane P
wave with nin = 30 (h/a = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.25. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when
two straight-line cracks arranged in parallel are subjected to an incident plane P
wave with nin = 30 (h/a = 1.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.26. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when two straight-line cracks arranged in parallel are subjected to an incident plane P wave (kT a = 2.5, ug /u0 = 2.0, h/a = 0.5, andθin = 0◦ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.27. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 2.5, h/a = 0.5, and θin = 0◦ . . . . . 107
5.28. Displacement fields around two cracks arranged in parallel from f in t = 20.0 to
22.0 when h/a = 0.5, θin = 0◦ , kT a = 2.5, and ug /u0 = 2.0. . . . . . . . . . . . 108
5.29. Histories of displacement at points (a) D1 and D2, and (b) D3 and D4 in the lower
right of Fig. 5.28from f in t = 20.0 to 22.0 when h/a = 0.5, θin = 0◦ , kT a = 2.5,
and ug /u0 = 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.30. Histories of displacement at points D1–D4 in the lower right of Fig. 5.28fromf in t = 20.0 to 21.0 when h/a = 0.5, θin = 0◦ , kT a = 2.5, and (a) ug /u0 = 1.0
and (b) ug /u0 = 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.31. Frequency responses for the model in Fig. 5.11(a) using the frequency-domain
BEM and assuming that crack faces are always separated. The norm ||[u]||/u0 at
point A in Fig. 5.11(a) for various θin and kT a is plotted when (a) h/a = 0.0, (b)0.5, (c) 1.0, (d) 1.5, and (e) 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.32. Frequency responses for the model in Fig. 5.11(b) using the frequency-domain
BEM and assuming that crack faces are always separated. The norm ||[u]||/u0 at
point B in Fig. 5.11(b) for various θin and kT a is plotted when (a) h/a = 0.5, (b)1.0, and (c) 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xvi

List of Figures
5.33. Elastic wave scattering by a surface breaking crack in an elastic half-space. . . . . 112
5.34. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena surface breaking crack is subjected to an incident plane P wave with nin = 15(θc = 0◦ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.35. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena surface breaking crack is subjected to an incident plane P wave with nin = 15(θc = 15◦ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.36. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena surface breaking crack is subjected to an incident plane P wave with nin = 15(θc = 30◦ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.37. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin whena surface breaking crack is subjected to an incident plane P wave with nin = 15(θc = 45◦ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.38. (a) Time variation and (b) normalized frequency spectrum of far-field amplitudeof backscattered P wave when a surface breaking crack is subjected to an incident
plane P wave (kT a = 3.7, ug /u0 = 0.0, θc = 45◦ , and θin = 75◦ ). . . . . . . . . . 117
5.39. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 3.7, θc = 45◦ , and θin = 75◦ . . . . . 117
5.40. Displacement fields around a surface breaking crack from f in t = 10.0 to 12.0 whenθc = 45◦ , θin = 75◦ , kT a = 3.7, and ug /u0 = 0.0. . . . . . . . . . . . . . . . . . 118
5.41. History of displacement at point E in the lower right of Fig. 5.40from f in t = 10.0
to 12.0 when θc = 45◦ , θin = 75◦ , kT a = 3.7, and ug /u0 = 0.0. . . . . . . . . . . 119
5.42. Frequency responses for the model in Fig. 5.33using the frequency-domain BEM
and assuming that crack faces are always separated. The norm ||[u||/u0 at the origin
O in Fig. 5.33for various θin and kT a is plotted when (a) θc = 0◦ , (b) 15◦ , (c) 30◦ ,and (d) 45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1. Analysis model for elastic wave scattering by a crack in a 3-D infinite domain. . . . 122
6.2. Two types of triangular tip element for crack front. . . . . . . . . . . . . . . . . . 124
6.3. Schematic of element subdivision and coordinate transformation for case 1 tip element in Fig. 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4. Schematic of potential calculation to ensure (a) Eq. (6.27) and (b) Eq. (6.28). . . . 130
6.5. A penny-shaped crack with radius a subjected to an incident plane P wave. . . . . . 131
6.6. Mesh structure of a penny-shaped crack when Me = 656 and Md = 361. . . . . . 131

xvii

List of Figures
6.7. (a) Vertical COD at each several time-steps and (b) time variations of normalized
dynamic SIF when a penny-shaped crack is subjected to normal incidence of plane
P wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.8. Computational time of CQ-FMBEM using RK3 for 3-D crack problems varying thenumber of elements Me when N = 128. . . . . . . . . . . . . . . . . . . . . . . . 133
6.9. Computational time of CQ-FMBEM using RK3 for 3-D crack problems varying thenumber of time steps N when Me = 1464. . . . . . . . . . . . . . . . . . . . . . 133
6.10. Time variations of (a) COD at the center point of a crack and (b) far-field amplitudeof backscattered P wave when a penny-shaped crack is subjected to an incident
plane P wave. (c) is normalized frequency spectrum of far-field amplitude. . . . . . 135
6.11. Time variations of (a) tangential traction at the center point of a crack and (b) farfield amplitude of backscattered SV wave when a penny-shaped crack is subjected
to an incident plane SV wave, and σ st /σ0 = 1.0. (c) is normalized frequency
spectrum of far-field amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.12. Elastic wave scattering by (a) a saddle-shaped crack and (b) two penny-shaped
cracks arranged in parallel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.13. Frequency responses for the model in Fig. 6.12(a) using the frequency-domain
BEM and assuming that crack faces are always separated. The norm ||[u]||/u0 at
point F in Fig. 6.12(a) for various θin and kT a is plotted when (a) w/a = 1.0 and(b) 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.14. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when a saddle-shaped crack is subjected to an incident plane P wave (w/a = 1.0, θin = 60◦ , and nin = 15). . . . . . . . . . . . . . 139
6.15. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when a saddle-shaped crack is subjected to an incident plane P wave (kT a = 2.8, w/a = 1.0, θin = 60◦ , and nin = 15). . . . . . . 140
6.16. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane
P wave (kT a = 2.8, ug /u0 = 3.4, w/a = 1.0, θin = 60◦ , and nin = 15). . . . . . . 140
6.17. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when a saddle-shaped crack is subjected to an incident plane P wave (w/a = 2.0, θin = 60◦ , and nin = 15). . . . . . . . . . . . . . 141
6.18. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane
P wave (kT a = 1.4, ug /u0 = 2.6, w/a = 2.0, θin = 60◦ , and nin = 15). . . . . . . 142
6.19. (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane
P wave (kT a = 1.4, ug /u0 = 2.6, w/a = 2.0, θin = 60◦ , and nin = 30). . . . . . . 143

xviii

List of Figures
6.20. Frequency responses for the model in Fig. 6.12(b) using the frequency-domain
BEM and assuming that crack faces are always separated. The norm ||[u]||/u0 at
point G in Fig. 6.12(b) for various h/a and kT a is plotted when θin = 0◦ . . . . . .
6.21. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when two penny-shaped cracks arranged in parallel are subjected to an incident plane P wave (h/a = 0.5, θin = 0◦ , and nin = 15). .
6.22. (a) Time variation and (b) normalized frequency spectrum of far-field amplitudeof backscattered P wave when two penny-shaped cracks arranged in parallel are
subjected to an incident plane P wave (kT a = 3.0, ug /u0 = 3.2, h/a = 0.5,
θin = 0◦ , and nin = 15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.23. (a) Time variation and (b) normalized frequency spectrum of far-field amplitudeof backscattered P wave when two penny-shaped cracks arranged in parallel are
subjected to an incident plane P wave (kT a = 3.0, ug /u0 = 3.2, h/a = 0.5,
θin = 0◦ , and nin = 30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.24. Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when two penny-shaped cracks arranged in parallel are subjected to an incident plane P wave (h/a = 1.0, θin = 0◦ , and nin = 15). .

143

144

145

146

147

B.1. Interface crack of two-layered elastic solid. . . . . . . . . . . . . . . . . . . . . . 157
B.2. Surface breaking crack in elastic half-space. . . . . . . . . . . . . . . . . . . . . . 158
C.1. Reflection and transmission of the incident plane wave on the interface between twoelastic solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
C.2. (a) Real parts and (b) imaginary parts of reflection and transmission coefficients of
P wave incidence for various incident angles θin . . . . . . . . . . . . . . . . . . . 163
C.3. (a) Real parts and (b) imaginary parts of reflection and transmission coefficients of
SV wave incidence for various incident angles θin . . . . . . . . . . . . . . . . . . 163

xix

List of Tables
2.1. Computational parameters used for calculating Laplace parameters s in Fig. 2.1. . .

12

3.1. Computational time and required memory of CQ-FMBEM when N = 128 and
Me = 5048. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Material constants for base material and inclusions. . . . . . . . . . . . . . . . . .
3.3. Computational time (sec). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38
43
45

4.1. Material constants for steel and aluminum. . . . . . . . . . . . . . . . . . . . . . .
4.2. Friction coefficients for steel-steel and steel-aluminum interfaces. . . . . . . . . .

62
62

6.1. Relation between coordinate systems (d1 , d2 ) and (d∗1 , d∗2 ) for sub-elements E1 , E2 ,and E3 in Fig. 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xxi

List of Main Symbols
Material propaties:
C: Elastic modulas tensor
λ: First Lamé constantµ: Second Lamé constant
ρ: Density
cf : wave velocity in fluidcL : P wave velocity
cT : S wave velocityµs : Static friction coefficientµd : Dynamic friction coefficient

Geometry:
x: Field pointy: Source point
r: Distance between x and y
θ: Zenith angle
ϕ: Azimuthal angle
n: Unit normal vector
s: Unit tangential vector
x0 , x1 : Local expansion point
y0 , y1 : Multipole point

Wave problems:
t: Timef : Frequency
ω: Circular frequency
kL : Wavenumber of P wave
kT : Wavenumber of S wave
θin : Incident angle

Fundamental solutions:
G(x, y, t): Fundamental solution of pressure for acoustic wave problem
H(x, y, t): Normal derivative of G(x, y, t) with respect to y
U (x, y, t): Fundamental solution of displacement for elastic wave problem
T (x, y, t): Fundamental solution of traction for elastic wave problem

xxiiiList of Tables
W (x, y, t): Hypersingular kernel for elastic wave problem(ˆ): Laplace-domain function

Numerical simulations:∆t: Time increments: Laplace parameter
N : Number of time steps
Me : Number of elements
a: Representative length
ug : Pre-opening displacement of crack
σ st : Compressive static stress over crack

Physical quantities:
p: Pressure
u: Displacement[u]: Crack opening displacementt: Traction force
σ: Stress

Functions:δ(x): Dirac delta distribution
H(x): Heaviside function
Ynm (x̂): Spherical harmonics
Pnm (x): Associated Legendre functionin (z): Modified spherical Bessel function of the first kind
kn (z): Modified spherical Bessel function of the second kindSpecial items:
δij : Kronecker delta
eijk : Component of 3-D permutation tensor
eij : Component of 2-D permutation tensor

xxivChapter 1

Introduction
1.1. Background of study
The nondestructive evaluation (NDE) of existing structures such as nuclear plants, bridges, and tunnels is important to detect flaws at an early stage and to prevent accidents. The inspection of welded
joints is also important to raise its quality in manufacturing. Therefore, it is considered that cost
reduction and safe management can be attained by effective utilization of the nondestructive testing(NDT). Among many techniques of NDE, the ultrasonic testing (UT) and radiographic testing (RT)are often used for the flaw detection on the inside of material. The UT has some advantages in its
portability and safety compared with the RT, and it is therefore considered as appropriate for theinspection of infrastructures.
However, it is difficult to detect closed cracks, which are flaws at an early stage, using the conventional and linear UT (LUT) that is based on mismatch of acoustic impedance between base material
and flaws. This is because an incident wave is hardly scattered on contacting crack faces, and adequate scattered waves cannot be received in the LUT. On the other hand, a nonlinear UT (NLUT)
based on the contact acoustic nonlinearity (CAN) has attracted notice as a new technique for detection of closed cracks recently [89]. This method is based on the nonlinear ultrasonic waves which
are very sensitive to degradation of material properties at the early stage of damage. The nonlinear
ultrasonic waves, which consist of higher- and sub-harmonic waves, are considered as generated
by the CAN. The higher- and sub-harmonic wave components correspond to the integer times andinteger division of the center frequency of incident wave, respectively.
There are two different mechanisms of higher-harmonic generation, and they have been mostly
understood from theoretical and experimental points of view. The first is a non-symmetrical stressstrain diagram under perpendicular vibration to a discontinuous interface such as a closed crack.
Thereby, the stiffness crossing the discontinuous interface for compression is harder than that fortension. The clapping motion of crack faces is then caused, and it excites higher-harmonic waves
of both odd and even orders. The second is energy dissipation because of the interface roughness.
The friction force is generated under some compressive stress, and an incident wave is distorted in asymmetrical way. It is known that odd order higher-harmonic waves are generated by this dynamic
friction phenomenon.

11. Introduction
In the NLUT, an incident wave with large amplitude is used for the inspection of closed cracks.
Then, the incident wave makes crack faces vibrate with contacting each other, and this vibration
causes the clapping motion and dynamic friction on the crack faces. Consequently, the transmitted
wave through closed cracks contains the nonlinear ultrasonic wave components. Therefore, it isconsidered as possible to detect the location of the closed crack and measure its size via the spectrum
analysis of received waves.
However, at this stage, the generation mechanism of sub-harmonic wave in the NLUT has notbeen understood clearly unlike the one of higher-harmonic wave. On the other hand, the excitation of sub-harmonic waves have been observed in the measurement [86]. In order to reveal thegeneration mechanism of sub-harmonic waves, further theoretical and/or numerical approaches are
implemented in this study. In general, the higher-harmonic waves are generated by many causes
except flaws such as transducer, and couplant. On the other hand, it is considered that the subharmonic waves are generated only by flaws in the NLUT. Thus, if the sub-harmonic waves can beused for the NDE, the more precise detection and size measurement of closed cracks are expected.
Taking into account the numerical approaches for the NLUT, there are two requirements. The
first is to accurately calculate physical quantities on the crack faces such as displacement and stressbecause they significantly effect on the states of crack faces such as contact and non-contact. In addition, appropriate treatment of boundary conditions on crack faces is also important for accuracy.
On the other hands, nonlinear boundary value problems have to be solved, and the generation mechanism of sub-harmonic waves is unknown. Hence, we need to conduct many numerical simulations
for various parameters. From the above, the second requirement is a fast solver of time-dependent
partial differential equations (PDEs), especially for three-dimensional (3-D) problems.
Considering the first requirement, the boundary element method (BEM) is desirable because it
can exactly deal with the nonlinear boundary conditions on crack faces, and accuracy of physicalquantities on boundaries is much better than the finite element method (FEM) and finite difference
method (FDM). To numerically solve the time-dependent problems with nonlinear boundary conditions, a time-domain BEM (TD-BEM) is utilized in this study. The acceleration of TD-BEM must
be considered for 3-D problems owing to the second requirement. Therefore, we propose a new
fast TD-BEM for the nonlinear ultrasonic simulation and implement many numerical simulations inorder to investigate the phenomena of nonlinear ultrasonic generation due to closed cracks.

1.2. Previous researches of nonlinear ultrasonic testing
The generation of higher-harmonic waves due to CAN was advocated over thirty years ago [15], andmany experimental and theoretical approaches have been carried out to investigate its mechanism
and characteristic. For instances of experimental approach, 2nd and 3rd higher-harmonic wave
components in the reflected P waves from a contact surface between two solid blocks have been
investigated by Severin and Solodov [81]. The effects of CAN on surface Rayleigh waves have also
been investigated by Solodov et al. [85]. Moreover, some flaw detection techniques based on the
NLUT have been developed recently. For instance, 2nd higher-harmonic waves have been used for2

1.2. Previous researches of nonlinear ultrasonic testing
the synthetic aperture focusing technique (SAFT) [66].
On the other hands, the generation of sub-harmonic waves due to CAN has been first observedby Solodov and Vu [86]. At that time, it was considered that the sub-harmonic wave is generated
just before the chaotic vibration and an unstable phenomenon. However, comparatively stable subharmonic waves have been observed by Yamanaka et al. [98] in the ultrasonic measurement for asurface breaking crack, and the depth measurement technique of a surface breaking crack has beendeveloped by Ohara et al. [65]. Moreover, Hayashi and Biwa [34] have shown that stable subharmonic waves are generated in the case that a thin layer exists between two solid blocks throughexperiment and 1-D numerical simulation. However, the generation mechanism of sub-harmonic
waves has not been understood clearly in all cases of above. It is difficult to intuitively comprehend
the generation mechanism of sub-harmonic wave unlike that of higher-harmonic wave.
As previous theoretical and numerical approaches, the 1-D numerical simulation modeling theclapping motion on the discontinuous interface has been first implemented by Richardson [70].
Thereafter, some numerical simulations have been implemented in 2-D and 3-D modeling the clapping motion and dynamic friction on the crack faces. For instance, 2-D numerical simulations havebeen conducted by the TD-BEM [56, 39]. They have used the Coulomb friction law with constantstatic and dynamic friction coefficients. Their methods have been extended to the debonding area on
the interface of bi-material in 2-D [77]. Also, the 3-D axisymmetric problem, i.e. a penny-shaped
crack subjected to normal incidence of P wave, has been solved numerically [38]. However, thesenumerical approaches using TD-BEM have not succeeded in the simulation of sub-harmonic generation yet. On the other hand, the 2-D antiplane simulation of FEM for the discontinuous interface
with dynamic friction conditions has been implemented by Meziane et al. [59]. The 2-D FDM
simulation for cracks only with clapping motion has also been conducted by Kimoto and Ichikawa[42]. Some 3-D sub-harmonic simulations have been conducted for delamination of thin materials
using the FDM [79] and FEM [20]. However, they have presented the sub-harmonic generation onlyin the case that the thickness of delaminated layer is much smaller than its diameter and specificallyless than 1/20.
From the above, it is necessary to investigate the physical phenomenon of sub-harmonic generation due to cracks. There also remains implementation of the 3-D nonlinear ultrasonic simulation
for a debonding area on the interface of bi-material. Therefore, we investigate the characteristic ofhigher-harmonic generation in the case that a debonding area is subjected to an incident wave bymeans of the 3-D TD-BEM. Subsequently, 2-D simulations of sub-harmonic generation by cracks
in an infinite domain and a surface breaking crack in a half-space are implemented in order to investigate its causes. Finally, sub-harmonic simulations due to cracks in a 3-D infinite domain are
carried out, and obtained results are compared between 2-D and 3-D.

3

1. Introduction

1.3. Time-domain boundary element method (TD-BEM)
In this section, we explain the TD-BEM, which is utilized for transient elastic wave scattering
problems for the nonlinear ultrasonic simulation in this study. In general, the BEM is widely used
for transient analyses of wave propagation in fluid and solid media [13, 25]. The transient elastic
wave problems have been investigated by many authors based on different methods, for instance,
the frequency- [63] or Laplace-domain BEM [49] with the construction of time-domain results byinverse Fourier or Laplace transform, respectively. However, the indirect methods cannot solve
nonlinear boundary value problems such as the nonlinear ultrasonic simulation. Therefore, for suchproblems, the direct TD-BEM is clearly preferred [35, 36].
In general, it is well known that the conventional TD-BEM in which a collocation method is used
for the time discretization, causes numerical instability for small time step size in time-marchingprocess. This issue has been overcome by the Galerkin discretization of both space and time integrals in boundary integral equations (BIEs) [29, 30]. However, it is difficult to solve wave propagation problems for dispersive media such as viscoelastic and poroelastic ones because explicit
time-domain fundamental solutions for those complex problems are not available. On the other
hand, the convolution quadrature method (CQM) which was pioneered by Lubich [46, 47], hasbeen developed as a technique to approximate convolution integrals in the past two decades. The
application of the CQM to a TD-BEM (CQ-BEM) is suitable for transient analysis of wave propagation problems [82]. By using the CQM for discretizing the convolution integrals in boundary
integral equations, the accuracy and numerical stability of the time-marching process are substantially improved. The CQ-BEM requires the Laplace-domain fundamental solutions. Therefore, asa result, the dispersion issues encountered in viscoelastic and poroelastic problems can be overcome straightforwardly [75, 78, 83]. Recently, several studies have shown that the CQ-BEM using
an implicit Runge-Kutta (IRK) scheme performs better accuracy than that using a linear multistep
method [48, 7]. In addition, the stability and error analysis of the IRK-based CQM were conductedby Calvo et al. [16] and Banjai et al. [5, 6] based on analytical and numerical approaches. The error
and behavior of influence functions for heat and wave problems using the IRK-based CQM werealso analyzed by Monegato et al. [60].
Although the CQ-BEM is powerful, there remains limitations in solving large-scale problems
because of low computational efficiency for matrix-vector products of the retarded potentials. The
computation of influence functions in the CQ-BEM framework generally requires a great deal ofcomputational time and memory resource compared to that of the conventional TD-BEM. The IRKbased CQ-BEM also requires much computational cost. To overcome these disadvantages and tosolve large-scale problems, the CQ-BEM accelerated by the fast multipole method (FMM) [72, 26](CQ-FMBEM) has been developed [75, 76]. The FMM has been developed as a technique to reducethe computational time and memory usage for solving large-scale problems. The BEM based on
FMM (FMBEM) has been developed for wave propagation problems in frequency-domain [62].
The conventional TD-BEM has also been accelerated by Takahashi et al. [91, 92] based on theplane wave time-domain (PWTD) algorithm [24]. In addition, the H-matrix and adaptive cross41.4. Organization of dissertation
approximation (ACA) [9, 10], which is an efficient method for matrix-vector product calculation,
has been employed for the acceleration of the CQ-BEM [58].
However, the computational cost for a number of time steps is expensive even if the FMM or
H-matrix is applied to matrix-vector products with respect to space. Banjai and Kachanovska haveapplied the rapid algorithm of convolution integral [31] and H-matrix to the CQ-BEM for the 3-D
wave equation and shown computational efficiency [8]. However, their method requires comparatively much memory although it is order of O(N Me log Me ), where N and Me are the numbersof time steps and elements, respectively. Moreover, no 3-D elastodynamic formulation for CQFMBEM has been proposed yet [18]. Therefore, we propose a new IRK-based CQ-FMBEM for
3-D elastodynamics which is accelerated for number of time steps using the rapid convolution algorithm with the fast Fourier transform (FFT), in order to utilize it for the nonlinear ultrasonicsimulation.

1.4. Organization of dissertation
This dissertation consists of seven chapters, and these outlines except this one are shown as follows:
Chapter 2 The formulations of linear multistep- and IRK-based CQMs are briefly presented.
Then, we discuss the expected accuracy in the TD-BEM using each CQMs by comparing
the arguments of Laplace-domain fundamental solutions. We also present the application of
IRK-based CQM to TD-BEM for 3-D acoustic wave problems and implement some numerical simulations. The accuracy is discussed through numerical results obtained by CQ-BEMs.
Chapter 3 We explain the acceleration techniques of the CQ-BEM for 3-D acoustic and elastic
wave problems. First, the IRK-based CQ-BEM for 3-D acoustic wave problems is accelerated
by the FMM with respect to spatial components. Second, the CQ-FMBEM is extended to 3D elastic wave and coupled acoustic-elastic wave problems, and some numerical simulations
are implemented. Finally, the CQ-FMBEM is accelerated with respect to the number of time
steps using the rapid convolution algorithm, and we demonstrate the computational efficiencyof the new CQ-FMBEM from numerical results.
Chapter 4 We implement the 3-D numerical simulation of higher-harmonic generation due to aninterface crack between two semi-infinite elastic solids. In this simulation, the collocation
method and piecewise constant approximation are used for the spatial discretization. The
higher-harmonic generation by a penny-shaped interface crack subjected to an incident plane
wave is simulated varying the pre-compressive stress over the crack faces. Then, the scatteredwaves are evaluated by means of the far-field approximation, and their frequency spectra areinvestigated.
Chapter 5 The 2-D simulations of sub-harmonic generation due to cracks in an infinite domain
and a surface breaking crack in an elastic half-space are carried out. We use the Galerkinmethod for the spatial discretization in the sub-harmonic simulation because we deal with5

1. Introduction
cracks of complex shape or arrangement. No acceleration technique is utilized for the 2-D
CQ-BEM because the computational cost of the simulations presented in this chapter is notvery expensive.
Chapter 6 We extend the 2-D numerical method for cracks with nonlinear boundary conditionsin an infinite elastic solid presented in Chapter 5 to the 3-D and implement some numericalsimulations. For the 3-D sub-harmonic simulation, we use the new CQ-FMBEM in order
to conduct many simulations with a long analysis time. We conduct some sub-harmonic
simulations with similar models to the 2-D ones and compare the obtained results between
2-D and 3-D.
Chapter 7 In this chapter, we state conclusions.

6

Chapter 2

Convolution Quadrature Time-domain
Boundary Element Method (CQ-BEM)
In this chapter, we present the CQM formulation and its application to a TD-BEM. Then, 3-D acoustic wave problems are numerically solved for comparison of accuracy between the linear multistepand IRK-based CQ-BEMs.

2.1. Convolution quadrature method (CQM)
We briefly explain the formulations of the linear multistep- and IRK-based CQMs, which are discretization methods of convolution integrals. The linear multistep-based CQM was pioneered by
Lubich [46, 47], and then, the IRK-based one has been developed by Lubich and Ostermann [48].
The Laplace transform of a kernel function is used in both CQMs.

2.1.1. Linear multistep-based CQMFirst, the formulation of linear multistep-based CQM is explained. The convolution integral withrespect to time t,u(t) = (k ∗ g) (t) ≡∫ tk(t − τ )g(τ )dτ

0 ≤ t,(2.1)

0

is approximated by the linear multistep-based CQM. In the CQM, k̂(s), which is the Laplace transform of kernel function k(t), is used for the computation of quadrature weight, where s is the
Laplace parameter. We define the notations with respect to the Laplace transform and its inverse as
follows:∫ ∞k̂(s) = L [k(t)] ≡k(t)e−st dt,(2.2)
0∫ h+iR[]1−1k(t) = Lk̂(s) ≡ limk̂(s)est ds,(2.3)
R→∞ 2πi h−iR7

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)
where i is the imaginary unit. In Eq. (2.3), h is a real number and h > 0. Substituting Eq. (2.3) into
Eq. (2.1) yields
1u(t) = lim
R→∞ 2πi∫ h+iRk̂(s)ys (t)ds,

(2.4)

h−iR

∫ t s(t−τ )g(τ )dτ and the solution of the following initial value prob0 e

where ys is given by ys (t) =
lem:

∂y(t)
= sy(t) + g(t),∂ty(0) = 0.

(2.5)

Note that the difference between the linear multistep-based CQM and IRK-based one is each approximation scheme of Eq. (2.5). The second order backward differentiation formula (BDF2),
which is one of the A-stable linear multistep methods [32], is often used for this approximationin the linear multistep-based CQM. The A-stable scheme should be taken to compute convolution
integrals stably, and there is no A-stable linear multistep method with k-th order (k > 2) according
to the second Dahlquist barrier [32].
According to [46, 47], Eq. (2.4) can be approximated as follows:
un = ∆tn∑ωn−j gj(n = 0, ..., N − 1),

(2.6)

j=0

where un ≃ u(n∆t), gn ≃ g(n∆t), and ∆t is the time increment. In addition, N is the number of
total time steps. Taking into account generating polynomials with variable ζ, the quadrature weightωn in Eq. (2.6) can be written as follows:
∆t

∞
∑

[
ωn ζ n = L−1 k̂(s)

(

n=0

)−1 ]
γ(ζ)
−s
,
∆t

(2.7)

where γ is the quotient of the generating polynomial. If we use the k-th order BDF, γ is given as
follows:γ(ζ) =

k∑(1 − ζ)i
i=1

i

.

(2.8)

The inverse Laplace transform in Eq. (2.7) is evaluated by the residue theorem. Then, the quadrature
weight ωn can be described in the Cauchy integral form and approximated by using a trapezoidal
rule with the number of steps L as follows:∆tωn =

1
2πi

[ (
)]
[ (
)]
γ(ζl )
γ(ζl )−1k̂
ζ −n−1 dζ ≃ F̃ln
k̂
,
∆t
∆t
|ζ|=R

∫

(2.9)

−1
where the operator F̃ln
indicates the inverse discrete Fourier transform for the CQM. The corre-

8

2.1. Convolution quadrature method (CQM)
sponding forward transform F̃nl and it are defined byF̃nl [hn ] ≡ Rl

N
−1
∑

2πinl

hn e N ,(2.10)

n=0−1
F̃ln[hl ] ≡

L−1
R−n ∑

L

hl e− L .
2πinl

(2.11)

l=0

In Eq. (2.9), ζl = Re2πil/L , and R is the radius of contour integral. If we select L as L = N , Eqs.
(2.10) and (2.11) can be evaluated by using the FFT with O(L log L) computational complexity. A√
specific integral path R has to be selected for Eq. (2.9) and is calculated by RL = ϵ with an error√of magnitude O( ϵ), where ϵ is the error parameter of the numerical calculation of Eq. (2.9).

2.1.2. Implicit Runge-Kutta (IRK)-based CQMSecond, the formulation of the IRK-based CQM is shown. As mentioned in later, note that the IRKbased CQM involves matrix form of the quadrature weight instead of ωn . A detailed explanation ofboth explicit and implicit RK methods can be found in [33].
The RK method is applied to the following initial value problem:∂y(t)= f (t, y),∂ty(0) = 0.

(2.12)

Eq. (2.12) is approximated by the recursions with yn ≃ y(n∆t) as follows:
Yni = yn + ∆tyn+1 = yn + ∆t

m
∑
j=1
m
∑

aij f ((n + cj )∆t, Ynj )(i = 1, ..., m),

bj f ((n + cj )∆t, Ynj ) ,(2.13)(2.14)

j=1

where aij , bi , and ci are Butcher’s parameters and frequently described in the Butcher table with thestage number of RK method m as follows:
c

A
bT

,



 A = (aij ) (1 ≤ i, j ≤ m),b = (b1 , ..., bm )T ,c = (c1 , ..., cm )T .

(2.15)

The approximations in Eqs. (2.13) and (2.14) are implicit when all of the upper right components
of matrix A are not equal to zero.

9

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)
In this study, the Radau IIA method [33], which is one of the A-stable IRK methods with (2m −1)-th order, is used for the CQM formulation. The coefficient parameters aij , bi , and ci satisfy thefollowing conditions:0 < c1 < ... < cm = 1,

bi = ami(i = 1, ..., m).

(2.16)

Some other specific coefficients pertaining to the Radau IIA method can also be found in [33].
In order to discretize the convolution integral in Eq. (2.1), the following vectors with m components are defined as
Un = (u((n + cj )∆t))mj=1 ,Gn = (g((n + cj )∆t))mj=1 ,and the generating functions U (ζ) and G(ζ) are also defined as follows:
U (ζ) =∞∑nUn ζ ,

ζ=0

G(ζ) =∞∑

Gn ζ n .

ζ=0Considering the approximation of Eq. (2.5) using the Radau IIA method and using these generating
functions, Eq. (2.1) can be approximated as[]
U (ζ) = L−1 k̂(s) (D(ζ, s))−1 G(ζ).

(2.17)In Eq. (2.17), m × m matrix D(ζ, s) is defined by
Υ(ζ)
− sI,
D(ζ, s) =
∆t
(
)−1
T ζ
Υ(ζ) = 1b+A,
1−ζ(2.18)(2.19)

where I is the identity matrix, and 1 = (1, ..., 1)T . From Eq. (2.18), D(ζ, s) corresponds tothe characteristic equation of Υ(ζ)/∆t. If this characteristic equation does not have any multiple
roots, (D(ζ, s))−1 has first-order poles at s = λβ (β = 1, ..., m), where λβ are the eigenvalues ofΥ(ζ)/∆t. Therefore, the inverse Laplace transform in Eq. (2.17) can be evaluated by means of theresidue theorem as follows:m∑
U (ζ) = k̂(λβ )Eβ (ζ) G(ζ),(2.20)
β=1

adj (D(ζ, λβ ))Eβ (ζ) = lim (λβ − s) (D(ζ, s))−1 = ∏m,s→λβ
j=1,j̸=β (λj − λβ )(2.21)

where adj(·) indicates the adjugate matrix. The modification in Eq. (2.21) can be attained by∏considering that |D(ζ, s)| = m
j=1 (λj − s) owing to the matrix diagonalization. Defining the n-th

10

2.1. Convolution quadrature method (CQM)
step quadrature weight matrix Wn as∞∑n

Wn ζ =n=0m∑

k̂(λβ )Eβ (ζ),(2.22)

β=1

the following discretized convolution formula can be obtained from Eq. (2.20) by means of the
Cauchy product.
Un ≃

n
∑

Wn−k Gk .

(2.23)

k=0

The quadrature weight matrix Wn can be expressed by the Cauchy integral form and approximatedby using a trapezoidal rule with the number of steps L as follows:

Wn =

1
2πi

∫
|ζ|=R




m
∑





−1 
k̂(λβ )Eβ (ζ) ζ −n−1 dζ ≃ F̃ln

β=1

m
∑


k̂(λlβ )Eβ (ζl ) ,(2.24)

β=1

where λlβ are the eigenvalues of Υ(ζl )/∆t. A specific integral path R must be selected for Eq.
√√(2.24) and is calculated by RL = ϵ with an error of magnitude O( ϵ), where ϵ is the error
parameter of the numerical calculation of Eq. (2.24). As presented in [4], R must also be selected
to avoid the issue that the characteristic equation of Υ(ζl )/∆t has multiple roots.
The computational manner of λlβ is presented here. For m = 2, it is easy to exactly determine λlβusing the quadratic formula as follows:
λl1 =

1
∆t

)
(
√
2 + ζl + ζl2 + 10ζl − 2 ,

λl2 =

1
∆t

)
(
√
2 + ζl − ζl2 + 10ζl − 2 .

Therefore, R must be selected to avoid ζl2 + 10ζl − 2 = 0 (multiple roots) for m = 2 as presented
in [4]. Similarly, the determination of λlβ for m = 3 and 4 can be implemented by using the corresponding formulas. However, from a numerical point of view, some rounding errors are introduced
when m ≥ 3 (especially for large m). Therefore, an iterative decomposition must be used to calculate λlβ in order to avoid this issue. In this study, the QR algorithm is used for the evaluation of λlβbecause Υ(ζl )/∆t usually becomes an asymmetric complex matrix.
The derivation of IRK-based CQM formulation presented in this paper is a little different from [4,7], and they have decomposed the quadrature weight matrix after evaluation of the inverse Laplacetransform. In this study, we evaluate the inverse Laplace transform after the decomposition of thematrix D(ζ, s). The presented derivation is valid if k̂(s) is an analytic function for ℜ[s] > 0 andthe characteristic equation of Υ(ζ)/∆t does not have any multiple roots, where ℜ[z] denotes thereal part of complex value z. The resulting quadrature weight Wn is identical to the one in [4, 7].

112. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)
1.5

3

(a)

(b)

RK3
RK2
BDF2
BDF1

2

RK3
RK2
BDF2
BDF1

1.0

1

0.5

0

0.0

-1

-0.5

-2

-1.0

-1.5

-3
0

1

2

3

4

0.0

0.1

0.2

0.3

0.4

Figure 2.1.: (a) Overall picture and (b) enlarged view around the origin of Laplace parameters s
required for CQMs with RKm and BDFk using computational parameters in Table 2.1.

Table 2.1.: Computational parameters used for calculating Laplace parameters s in Fig. 2.1.
L
∆t
R

RK3
100
3.0
0.871

RK2
150
2.0
0.933

BDF2
300
1.0
0.955

BDF1
300
1.0
0.955

2.1.3. Behavior of Laplace parameter
The arguments of Laplace-domain kernel γ(ζl )/∆t and λlβ correspond to the Laplace parameter s
in the linear multistep- and IRK-based CQMs, respectively. Their behavior significantly effects on
the accuracy because they correspond to the circular frequencies in frequency-domain. In addition,
considering the conjugate features of γ(ζl )/∆t and λlβ , we can reduce the computation number ofquadrature weights in the CQM formulation to about half. Hereinafter, BDFk and RKm indicate
the kth order BDF- and the m-stage IRK-based CQM or CQ-BEM, respectively.
Laplace parameters s which are required for the various CQMs are shown in Fig. 2.1 whenϵ = 10−12 and computational parameters in Table 2.1 are used. ℜ[z] and ℑ[z] indicate the real andimaginary parts of z, respectively. As we can see from Fig. 2.1 (a), some s are complex conjugate
of others. Specifically, γ(ζL−l ) = γ(ζl ) and λL−l
= λlβ for l = 1, ..., L/2 − 1, where ( ) indicatesβcomplex conjugate. Therefore, we need to compute γ(ζl )/∆t or λlβ only for l = 0, ..., L/2.
The behavior of s around the imaginary axis is important for the construction of time-domain
original kernel k(t) with high frequency components because ℑ[s] and ℜ[s] correspond to the circular frequency and attenuation, respectively. As shown in Fig. 2.1 (b), higher frequency components
near the imaginary axis are contained in RK3 and RK2 than in BDF2 and BDF1. This comparison is conducted under the same division number 300 for the same time range [0, 300] as shown in
Table 2.1. Therefore, it is expected that the performance of IRK-based CQM is better in accuracy

12

2.2. CQ-BEM for 3-D acoustic wave problems

Figure 2.2.: Analysis model for acoustic wave scattering by an obstacle in a 3-D infinite domain.

for high frequency components than that of linear multistep-based one. Hereinafter, ϵ is taken to beϵ = 10−12 for all numerical simulations.

2.2. CQ-BEM for 3-D acoustic wave problems
The application of the IRK-based CQM to the TD-BEM for 3-D acoustic wave problems is presented. The IRK-based CQ-BEM has been developed by Maruyama et al. [51] and Banjai et al.
[7]. The detailed formulation of the linear multistep-based CQ-BEM can be found in the reference
papers [75, 58], and it is omitted here.

2.2.1. Formulation of boundary integral equation (BIE)
Let S be a closed boundary with unit normal vector n in R3 as depicted in Fig. 2.2. D is anexterior domain, which can be considered as an incompressible and non-viscous fluid. The pressurep satisfies the following wave equation:c2f ∇2 p(x, t) − p̈(x, t) = 0

x ∈ D,(2.25)

where cf is the velocity of wave propagation in D, and ( ˙ ) indicates the partial differentiation withrespect to time t. We assume the following initial condition:
psc (x, 0) = ṗsc (x, 0)

x ∈ D,(2.26)

where the superscript ”sc” indicates the scattered wave. Considering the Sommerfeld radiation
condition for scattered waves and initial condition (2.26), the time-domain BIE for pressure p isformulated by means of the usual manner as follows:
∫ t∫

in

G(x, y, t − τ )q(y, τ )dSy dτ∫ t∫−p.v. H(x, y, t − τ )p(y, τ )dSy dτ

C(x)p(x, t) = p (x, t) +

0

0

S

x ∈ S,(2.27)

S

where the superscript ”in” indicates the incident wave, C(x) is the free term [14] depending on theboundary shape at x, and q is the normal derivative of p. G is the fundamental solution of pressure13

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)
for 3-D acoustic wave problems in time-domain, and H is its normal derivative. In addition, thesymbol p.v. indicates the Cauchy principal integral.

2.2.2. Discretization of BIE
The discretization of BIE (2.27) using the IRK-based CQM is presented. The collocation method
and piecewise constant approximation are utilized for spatial discretization. If the m-stage Radau
IIA method is used and the boundary S is divided into Me boundary elements, the discretized BIE
at n-th and i-th sub-step in time is written as follows:
[
]
Me ∑
n ∑
m
∑
1 i;n
ij;n−k j;kij;n−k j;kp = pin;i;n+Gγαqα − Hγαpα
γ2 γ
k=0 α=1 j=1(i = 1, ..., m), (n = 0, ..., N − 1),(2.28)

whereui;nγ = u(xγ , (n + ci )∆t)(u = p, q, or pin ),α and γ indicate the indices of a boundary element, and xγ denotes the collocation point of γ-th
ij;κ
ij;κ
element. The collocation point is selected as the barycenter of element. In addition, Gγαand Hγα
are the influence functions defined by
−1
ij;κ
Gγα
= F̃lκ[ m) ∫∑(Eβ (ζl )
β=1

−1
ij;κ
Hγα
= F̃lκ

[ m
∑(

ij

]
Ĝ(xγ , y, λlβ )dSy∫)
Eβ (ζl ) p.v.

β=1

ij

,

Sα

(2.29)]

Ĥ(xγ , y, λlβ )dSy,

(2.30)

Sα

where Sα is the surface of α-th element. In Eqs. (2.29) and (2.30), Ĝ and Ĥ are the Laplace-domain
fundamental solutions for pressure and its normal derivative, respectively, and given as follows:e−sf r,4πr
Ĥ(x, y, s) = n(y) · ∇y Ĝ(x, y, s),Ĝ(x, y, s) =(2.31)(2.32)

where r = |y − x|, ∇y = (∂/∂y1 , ∂/∂y2 , ∂/∂y3 ), and sf = s/cf . Note that the integral kernels Ĝ
and Ĥ in Eqs. (2.29) and (2.30) do not depend on time t. Rearranging Eq. (2.28) according to thetime step, the following expression is obtained:14

2.3. Numerical results

Me ∑
m [(
∑
1
α=1 j=12)ij;0δγα δij + Hγα]ij;0 j;npj;nα − Gγα qα= pin;i;n
−
γ

Me ∑
n−1
m
∑∑

[

]
ij;n−k j;kij;n−k j;kHγαpα − Gγαqα ,(2.33)

k=0 α=1 j=1

where δij is the Kronecker delta.
For the n-th time step, all of the quantities on the right-hand side of Eq. (2.33) are known.
Therefore, the unknown values p and q at the n-th time step can be obtained by solving Eq. (2.33).
In Eq. (2.33), the total number of unknowns is mMe N . Therefore, the computational time is
proportional to O(m2 N 2 Me2 ) order without any acceleration techniques even if we use an iterativesolver, such as the Krylov subspace method [95].

2.3. Numerical results
For numerical examples, a scattering problem as depicted in Fig. 2.3 is numerically solved tocompare the accuracy between the linear multistep- and IRK-based CQ-BEMs. An incident plane
wave is scattered by a rigid sphere with radius a, and the pressure fields at the point X in Fig. 2.3
are investigated using different time increments. The boundary condition on S is given by q = 0
here. The following plane wave with amplitude p0 propagating in x1 -direction is used as an incident
plane wave.
in
(2.34a)p (x, t) = p0 (1 − cos 2πΛf ) H(Λf )H(1 − Λf ),cf t − (x1 + a)
Λ =,(2.34b) f
λinfwhere λinf is the wavelength corresponding to the center frequency of the incident wave, and H(·)
is the Heaviside function. The surface of a sphere is discretized into Me = 3176 flatly triangularinboundary elements, and λin
f is given by λf /a = 1.0. This problem has been analytically solvedin the frequency-domain by Pao and Mow [67]. Therefore, the transient solutions, which are hereinafter called the ”reference solutions”, can be constructed by using the inverse Fourier transformof the frequency-domain analytical solutions.
The time variations of pressure p/p0 at point X in Fig. 2.3 obtained by BDF1, BDF2, RK2, andRK3 are shown for different time increments in Figs. 2.4, 2.5, 2.6, and 2.7, respectively. As seenin Fig. 2.4, the BDF1 cannot yield accurate results even if the smallest time increment cf ∆t/a =0.0087 is used. In particular, the amplitude of p/p0 at cf t/a = 1 to 2 is different from the reference
solutions, and large oscillation at cf t/a = 4 to 8 is observed. In additions, the oscillation at cf t/a =4 to 8 is hardly improved with decreasing time increments. Consequently, it is considered as difficult
to compute p/p0 accurately using the BDF1.
As seen in Fig. 2.5, the BDF2 with the smallest time increment cf ∆t/a = 0.0087, can yield good15

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)

Figure 2.3.: Acoustic wave scattering by a rigid sphere subjected to an incident plane wave.
0.5

Reference

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 2.4.: Time variations of p/p0 obtained by using BDF1 at point X in Fig. 2.3.
0.5

Reference

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 2.5.: Time variations of p/p0 obtained by using BDF2 at point X in Fig. 2.3.

results at the time up to cf t/a = 5. However, p/p0 is not accurate compared with the reference
solutions after cf t/a = 5, and fluctuations can be seen clearly. In particular, larger time increments,

162.3. Numerical results
0.5

Reference

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 2.6.: Time variations of p/p0 obtained by using RK2 at point X in Fig. 2.3.
0.5

Reference

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 2.7.: Time variations of p/p0 obtained by using RK3 at point X in Fig. 2.3.

e.g., cT ∆t/a = 0.035 or 0.017, give less accurate results. As a result, a small time increment isrequired for accurate computation of p/p0 for the time range between cf t/a = 5 to 8.
As shown in Figs. 2.6 and 2.7, much more accurate results can be obtained by the RK2 andRK3 for the cases of cf ∆t/a = 0.035 than the BDF2. The fluctuations of p/p0 increase withincreasing time increments. p/p0 calculated by the RK2 and RK3 using cf ∆t/a = 0.035 seemto be sufficiently accurate. From the above, the selection of time increments is important, and it
has a strong influence on the scattered waves obtained by the CQ-BEM. Therefore, an adequately
small time increment must be chosen to obtain sufficiently accurate results for practical engineering
applications.

17

2. Convolution Quadrature Time-domain Boundary Element Method (CQ-BEM)

2.4. Concluding remarks• In Chapter 2, we have explained both linear multistep- and IRK-based CQMs and successfully
applied them to the TD-BEM for 3-D acoustic wave problems. To simplify and focus on thetime discretization, the collocation method and piecewise constant approximation have beenused for the spatial discretization.
• It was confirmed that there is large difference in the behavior of Laplace parameters s among
schemes which are used for CQM formulations. This difference, especially behavior near theimaginary axis, is greatly concerned with the available range of circular frequencies. It seems
that the single-step methods, such as RK methods have some advantages in the calculationof high frequency components. This is because the influence functions for wave problems
contain very rapid fluctuation.
• The BDF1 cannot yield accurate solutions even if the small time increment is used, and better
numerical solutions have been obtained by the BDF2 with the small time increment. However, the performance of IRK-based CQ-BEM is much better in accuracy than that of linear
multistep-based one when the same number of discretization points are used for the sametime range.
• The computational complexity and required memory of IRK-based CQ-BEM without any
acceleration techniques are O(m2 N 2 Me2 ) even if an iterative solver is used.

18

Chapter 3

Acceleration of CQ-BEM
It is difficult to solve large-scale problems using the CQ-BEM without any acceleration techniques
because its computational and memory complexities are O(m2 Me2 N 2 ) as shown in the previous
chapter. Therefore, we present some acceleration techniques for the CQ-BEM in this chapter.
First, application of the FMM [72, 26] to the CQ-BEM is explained. Then, the formulations of
CQ-FMBEM for 3-D acoustic wave, elastic wave, and their coupling problems, which have beendeveloped by Maruyama et al. [52, 53], are shown. Finally, we apply the rapid convolution algorithm using FFT [31] to the CQ-FMBEM for 3-D acoustic wave problems and demonstrate thecomputational complexity of our proposed method through some numerical results.

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)
The FMM is applied to the IRK-based CQ-BEM to achieve stable, accurate, and efficient calculation
for large-scale wave propagation problems. In what follows, we present the time-marching processof the IRK-based CQ-FMBEM, the multipole and local expansions, and translation theorems. In
addition, scaling and truncation techniques for the IRK-based CQ-FMBEM are introduced in orderto enhance the stability and efficiency of the proposed method.

3.1.1. Time-marching process
Application of the FMM to the IRK-based CQ-BEM complicates the time-marching process because the influence functions defined in Eqs. (2.29) and (2.30) include the inverse discrete Fourier
ij;κ
transforms in Eq. (2.11). The time-marching process for the terms including Gγα
in the right-handside of Eq. (2.33) is explained as an example. The retarded potential vector with mMe components
Rn is defined byn(R )m(γ−1)+i =

Me ∑
n−1
m
∑∑
k=0 α=1 j=1

ij;n−k j;k
Gγα
qα =

(n−1
∑
k=0

)
G

n−k k

.

q

(3.1)

m(γ−1)+i

The multiplication of mMe × mMe matrix and mMe dimensional vector is written by Gn−k tk here.
To evaluate Rnγ using the FMM, the inverse discrete Fourier transform and the summations in Eqs.

19

3. Acceleration of CQ-BEM(2.29) and (2.33) are rearranged as follows:
(
)
Gκ qk

m(γ−1)+i

=

Me ∑
m
∑



m (
) ∫
∑
−1 
F̃lκ
Eβ (ζl )α=1 j=1−1 = F̃lκ

ij

β=1

{M
m
e
∑
∑

β=1


Ĝ(xγ , y, λlβ )dSy  qαj;kSα}i;k;β;l 
Ĝβ;l
,
γα qα

(3.2)

α=1

where
∫
Ĝβ;l
γα =

Ĝ(xγ , y, λlβ )dSy ,qi;k;β;l
=
α

Sα

m (
)
∑
Eβ (ζl ) qαj;k .
ij

j=1

The term inside braces { } in Eq. (3.2) is first calculated for all of γ, β, i, and l via the FMM. Then,
taking the summation with respect to β and evaluating the inverse discrete Fourier transform, Gκ qk
for all κ (κ = 0, ..., N − 1) are obtained. Finally, the corresponding retarded potential vector Rk+κis updated. This calculation can be expressed in the following matrix form:










R1
R2
R3
..
.













G1
G2
G3
..
.



G1


G2 G1
=

















 RN −1 
GN −1
···



0


q








1


q





2
q
.




.. 
..



.
. 








1
N
q −2 
G

0(3.3)

Once a set of the boundary values qk at k-th step is obtained by solving Eq. (2.33), Gκ qk for allκ can be calculated. Then, the resulting vectors are added into Rn for the rest time steps (n =k + 1, ..., N − 1) as Rn + Gn−k qk ⇒ Rn . If this procedure is conducted at each time step, Rn is
updated for the next time step calculation. Note that Cauchy data qαj;k are treated in time-domain
although the FMM is implemented in Laplace-domain. In this algorithm, the FMM is implemented
for l = 0, ..., L/2 at each time step owing to the following relation:
m
∑

{M
e
∑

β=1

α=1

}
Ĝβ;L−lqαi;k;β;L−l
γα

=

m
∑

{M
e
∑

β=1

α=1

}
i;k;β;l
Ĝβ;l
γα qα

(l = 1, ..., L/2 − 1).

ij;κ
ij;κ
The terms including Hγαin Eq. (2.33) can be evaluated as well as Gγα.

3.1.2. Fast multipole method (FMM)
In Eq. (3.2), the FMM is implemented for the Laplace-domain fundamental solutions. The multipole and local expansions for Laplace-domain fundamental solutions can be derived as well as thosein frequency-domain [23]. The FMM formulation based on Wigner 3-j symbol [57] with O(Nt3 )arithmetic complexity [28] is used in this study, where Nt is the truncation number of infinite sum-20

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)
mations in multipole and local expansions.

Multipole and local expansions
The expansions of fundamental solution of modified Helmholtz equation corresponding to somewave velocities are required for the derivation of multipole and local expansions for elastic wave
problems in Section 3.1.6. Therefore, the expansions of Ĝφ , which is the fundamental solution of
the modified Helmholtz equation with wave velocity cφ , are shown here. Note that Ĝ in Eq. (2.31)
is Ĝ = Ĝf . According to [23], the expansions of Ĝφ are given byĜφ (x, y, s) =∞ ∑a∑sφ (2a + 1)
a=0 b=−a

{
×

4π

φ
φ
F̄a,b
(y − y0 )Oa,b(x − y0 )for |y − y0 | ≤ |x − y0 |,(3.4a)φφŌa,b(y − x0 )Fa,b(x − x0 )for |y − x0 | > |x − x0 |,(3.4b)φφ
φ
φwhere sφ = s/cφ , and Fa,b(x), F̄a,b(x), Oa,b(x), and Ōa,b(x) are defined by
φFa,b(x) = ia (sφ |x|)Yab (x̂),φF̄a,b(x) = ia (sφ |x|)Yab (x̂),

(3.5)

φ
Oa,b
(x) = ka (sφ |x|)Yab (x̂),φŌa,b(x) = ka (sφ |x|)Yab (x̂).

(3.6)

In Eqs. (3.5) and (3.6), x̂ = x/|x|, and ia and ka are the modified spherical Bessel functions of thefirst and second kinds, respectively. Yab are the spherical harmonics which are defined by√
Yab (x̂) =(a − b)! b
P (cos θ)eibϕ ,(a + b)! a(3.7)

where θ and ϕ are the zenith and azimuthal angles of x̂ in the spherical coordinate system, respectively, and Pab is the associated Legendre function.
In order to derive the multipole and local expansions, Gi;k;β;land Hi;k;β;lare defined by
γα
γα
[∫

]

i;k;β;lGγα=qi;k;β;l
,
α
Sα
[∫
]
i;k;β;lφlHγα =Ĥ (xγ , y, λβ )dSy pi;k;β;l
,
α
Ĝ

φ

(xγ , y, λlβ )dSy

(3.8)
(3.9)

Sα

where Ĥ φ is the normal derivative of Ĝφ , andpi;k;β;l
=
α

m (
)
∑
Eβ (ζl ) pj;k
α .
j=1

ij

In Eq. (3.9), the symbol p.v. is omitted because γ (field point) is never equal to α (source point) inan FMM far-field calculation. Substituting Eq. (3.4a) into Eqs. (3.8) and (3.9) yields the following

21

3. Acceleration of CQ-BEM
multipole expansions:
i;k;β;l
Gγα
=

∞ ∑
a
∑
sφ (2a + 1)
a=0 b=−a=Hi;k;β;lγα]

φ
G;i;k;β;lOa,b(xγ − y0 )Ma,b;α
(y0 )

4π

∞ ∑
a
∑
sφ (2a + 1)
a=0 b=−a[,(3.10),(3.11)

s=λlβ

[

]
φ
H;i;k;β;lOa,b(xγ − y0 )Ma,b;α
(y0 )

4π

s=λlβ

G;i;k;β;l
H;i;k;β;lwhere Ma,b;αand Ma,b;α
are the multipole moments with respect to Ĝφ and Ĥ φ , respectively.
The infinite summations in Eqs. (3.10) and (3.11) are truncated by Nt terms in the numericalsimulation. The explicit expressions of the multipole moments in Eqs. (3.10) and (3.11) are written
as follows:∫
G;i;k;β;lφi;k;β;lMa,b;α (y0 ) = qαF̄a,b(y − y0 )dSy ,(3.12)∫Sα
H;i;k;β;lφMa,b;α(y0 ) = pi;k;β;ln(y) · ∇y F̄a,b(y − y0 )dSy .
(3.13)αSαNote that all multipole moments can be evaluated independently on a field point xγ .
i;k;β;lThe local expansions for Gγαand Hi;k;β;lare obtained by substituting Eq. (3.4b) into Eqs.
γα(3.8) and (3.9) as follows:
Gi;k;β;l=γαa=0 b=−ai;k;β;l
=
Hγα

[

∞ ∑
a
∑
sφ (2a + 1)

4π

∞ ∑
a
∑
sφ (2a + 1)
a=0 b=−a

4π

]

φ
Fa,b
(xγ − x0 )LG;i;k;β;l
(x0 )
a,b;α

[

,

(3.14),(3.15)

s=λlβ

]

φ
Fa,b
(xγ − x0 )LH;i;k;β;l
(x0 )
a,b;α
s=λlβ

H;i;k;β;lwhere LG;i;k;β;land La,b;α
are the coefficients of local expansion. The coefficients of local
a,b;α
φ
φ
expansion can be obtained by replacing F̄a,b(y − y0 ) by Ōa,b(y − x0 ) in Eqs. (3.12) and (3.13) as
follows:∫
G;i;k;β;lφi;k;β;lLa,b;α (x0 ) = qαŌa,b(y − x0 )dSy ,(3.16)∫Sαφ
LH;i;k;β;l(x0 ) = pi;k;β;ln(y) · ∇y Ōa,b(y − x0 )dSy .
(3.17)
αa,b;αSαTranslation theorems (M2M, M2L, and L2L)
In the FMM algorithm, multipole expansion to multipole expansion (M2M), multipole expansion to
local expansion (M2L), and local expansion to local expansion (L2L), are required [62]. These threetranslation theorems for Ĝφ are described by the following equations for the schematic dipected inFig. 3.1:

22

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

G;i;k;β;lM2M : Ma,b;α(y1 ) =

∞ ∑c∑

G;i;k;β;l(M|M)b,d(y0 ),a,c (y0 → y1 ; sφ )Mc,d;α(3.18)

G;i;k;β;l(M|L)b,d(y1 ),a,c (y1 → x1 ; sφ )Mc,d;α(3.19)

G;i;k;β;l(L|L)b,d(x1 ).
a,c (x1 → x0 ; sφ )Lc,d;α(3.20)

c=0 d=−c

M2L : LG;i;k;β;l(x1 ) =a,b;α
L2L : LG;i;k;β;l(x0 ) =
a,b;α

∞ ∑
c
∑
c=0 d=−c∞ ∑c∑c=0 d=−c

G;i;k;β;l
H;i;k;β;lIf Ma,b;αand LG;i;k;β;lin Eqs. (3.18)–(3.20) are replaced with Ma,b;αand LH;i;k;β;l, respeca,b;α
a,b;α
b,d
φ
tively, the translation formulas for Ĥ are derived. In Eqs. (3.18)–(3.20), (E|F)a,c (E, F = M orL) are the translation operators including Wigner 3-j symbol. The operator (E|F)b,da,c can be calcu4
lated recursively in O(Nt ) arithmetic complexity [28]. Eqs. (3.18)–(3.20) can be recast using theorthogonalization of spherical harmonics in O(Nt3 ) arithmetic complexity:


a
∑


G;i;k;β;lG;i;k;β;l
a

M̃
(y
)
=
Ma,b;α(y0 )Ddb(RT ),

0
a,d;α



b=−a




∞
 G;i;k;β;l∑
G;i;k;β;lM̃c,d;α (y1 ) =(M|M)dc,a (y0 → y1 ; sφ )M̃a,d;α(y0 ),


a=|d|



c

∑

 G;i;k;β;l
G;i;k;β;lcM̃c,d;α(y1 )Dbd
(R) ,
M
(y
)
=
1

 c,b;α(3.21a)(3.21b)(3.21c)

d=−c


a
∑


G;i;k;β;lG;i;k;β;laM̃a,d;α (y1 ) =Ma,b;α(y1 )Ddb(RT ),




b=−a




∞

∑
G;i;k;β;lL̃G;i;k;β;l(x)=(M|L)dc,a (y1 → x1 ; sφ )M̃a,d;α(y1 ),
1
c,d;α


a=|d|



c

∑


G;i;k;β;lcL̃G;i;k;β;l(x1 )Dbd(R) ,L(x1 ) =
c,d;α
 c,b;α(3.22a)(3.22b)(3.22c)

d=−c


a
∑


G;i;k;β;laL̃a,d;α (x1 ) =LG;i;k;β;l(x1 )Ddb(RT ),

a,b;α



b=−a




∞
 G;i;k;β;l∑L̃c,d;α (x0 ) =(L|L)dc,a (x1 → x0 ; sφ )L̃G;i;k;β;l(x1 ),
a,d;α


a=|d|



c

∑


G;i;k;β;lcL(x0 ) =
L̃G;i;k;β;l(x0 )Dbd
(R) ,

c,d;α
 c,b;α(3.23a)(3.23b)(3.23c)d=−ca (R) are the rotation matrices of spherical harmonics and can be calculated recursively
where Dbd
in O(Nt3 ) computational complexity [19]. In Eqs. (3.21)–(3.23), (E|F)ba,c = (E|F)b,b
a,c , and the23

3. Acceleration of CQ-BEM
Boundary element

L2L
M2M

M2L

Field point
Source point
Local expansion point
Multipole point

Figure 3.1.: Translation theorems (M2M, M2L, and L2L).

rotation matrices of vector R are defined by
 (y0 − y1 ) · R = |y0 − y1 |e3 in Eq. (3.21),(x1 − y1 ) · R = |x1 − y1 |e3 in Eq. (3.22),(x0 − x1 ) · R = |x0 − x1 |e3 in Eq. (3.23),
where e3 is given by e3 = (0, 0, 1). All of the calculations in Eqs. (3.21)–(3.23) and the evaluationsof (E|F)ba,c are conducted in O(Nt3 ) computation. The detailed expressions of (E|F)ba,c and therecursive computation are presented in Appendix A.

3.1.3. Some numerical techniques for efficient CQ-FMBEM
Two typical techniques that enhance the stability and efficiency of the CQ-FMBEM are presented.
The first is scaling of the multipole moments (3.12) and (3.13) with the modified spherical Bessel
functions ia and ka . ia (z) and ka (z) exponentially tend to infinity and zero for a large argument z,
respectively. This fact sometimes causes instability of the translation formulas (3.18)–(3.20) whencell size is large. The second is the truncation of M2L translations. Some M2L translations for theretarded potential calculation can be neglected because the fundamental solution in Laplace-domain
rapidly approaches to zero if the distance between field and source points r = |x − y| is large.

Scaling of modified spherical Bessel function
When the absolute value of the argument |z| is large, the asymptotic expressions of ia (z) and ka (z)are given byia (z) ∼

24

ez
,
2z

ka (z) ∼e−z.
z(3.24)

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Figure 3.2.: Scaling for M2M moment translated from Cϱ to C2ϱ using half diagonal length of cube
cell.

Therefore, the scaled modified spherical Bessel functions ĭa and k̆a are defined byĭa (z) = e−z ia (z),k̆a (z) = ez ka (z).

(3.25)

Scaling of multipole and local expansions
To utilize the scaled modified spherical Bessel functions in Eq. (3.25), the multipole and local
expansions and the translation operators also have to be modified. Taking into account ϱ, whichis the half diagonal length of a cube cell Cϱ as shown in Fig. 3.2, the scaled multipole moment
G;i;k;β;lφ, the scaled local expansion coefficient ϱ LG;i;k;β;l, and the scaled functions ϱ Fa,band
ϱ Ma,b;αa,b;αφϱ Oa,b are defined as follows:
G;i;k;β;lG;i;k;β;l= e−sφ ϱ Ma,b;α,ϱ Ma,b;αG;i;k;β;l= esφ ϱ LG;i;k;β;l,ϱ La,b;αa,b;α(3.26)

φ−sφ ϱ φFa,b ,
ϱ Fa,b = eφsφ ϱ φOa,b .
ϱ Oa,b = e(3.27)

Consequently, the scaled multipole and local expansions for Gi;k;β;l
are recast as follows:
γαGi;k;β;l
=
γα

∞ ∑
a
∑
sφ (2a + 1)
a=0 b=−aGi;k;β;l
=
γα

4π

∞ ∑
a
∑
sφ (2a + 1)
a=0 b=−a

4π

[

]

φ
G;i;k;β;l(y0 )ϱ Oa,b (xγ − y0 )ϱ Ma,b;α

[

,

]

(3.28)

s=λlβ

φ
G;i;k;β;l(x0 )ϱ Fa,b (xγ − x0 )ϱ La,b;α.

(3.29)s=λlβ

The scaled multipole and local expansions for Hi;k;β;lcan be obtained in the same manner as the
γαi;k;β;lones for Gγα .

25

3. Acceleration of CQ-BEM
Scaling of M2M, M2L, and L2L
Considering the sizes of cells before and after translations, the scaling of translation formulas(M2M, M2L, and L2L) for Eqs. (3.28) and (3.29) can be obtained. Taking into account the variationof cell size from ϱ to 2ϱ as shown in Fig. 3.2, the scaling of M2M formulas (3.21) is expressed asfollows:a∑G;i;k;β;lG;i;k;β;l
a

M̃
(y
)
=
(y0 )Ddb(RT ),(3.30a)ϱ0ϱ Ma,b;α
a,d;α



b=−a




∞

∑
G;i;k;β;lM̃
(y)=(M|M)dc,a (y0 → y1 ; sφ )ϱ M̃G;i;k;β;l(y0 ),(3.30b)
2ϱ
1
c,d;α
a,d;α


a=|d|



c

∑


G;i;k;β;lG;i;k;β;l
c


M
(y
)
=
(y1 )Dbd(R) ,(3.30c)1
2ϱ M̃c,d;α 2ϱ c,b;αd=−cwhere (E|F)ba,c (E, F = M or L) is the scaled translation operator, and the detail is given in
Appendix A. The scaled M2L and L2L formulas can be obtained by analogous modifications ofEqs. (3.22) and (3.23) as follows:

a
∑


G;i;k;β;lG;i;k;β;l
a

M̃
(y
)
=
(y1 )Ddb(RT ),
ϱ
1
ϱ Ma,b;α
a,d;α



b=−a




∞
 G;i;k;β;l∑(x1 ) =e−sφ (dM2L −2ϱ) (M|L)dc,a (y1 → x1 ; sφ )ϱ M̃G;i;k;β;l(y1 ),
ϱ L̃c,d;α
a,d;α


a=|d|



c

∑

 G;i;k;β;l
G;i;k;β;l
c


L
(x
)
=
(x1 )Dbd(R) ,1
ϱ L̃c,d;α ϱ c,b;α(3.31a)(3.31b)(3.31c)

d=−c


a
∑


G;i;k;β;lG;i;k;β;la(x1 ) =
(x1 )Ddb(RT ),
2ϱ L̃a,d;α2ϱ La,b;α



b=−a




∞

∑
G;i;k;β;l(x0 ) =
(L|L)dc,a (x1 → x0 ; sφ )2ϱ L̃G;i;k;β;l(x1 ),
ϱ L̃c,d;α
a,d;α


a=|d|



c

∑


G;i;k;β;lG;i;k;β;lcL(x0 ) =(x0 )Dbd(R) .
ϱ L̃c,d;α ϱ c,b;α(3.32a)(3.32b)(3.32c)d=−c

In Eq. (3.31), dM2L denotes the distance between the multipole point y1 and local expansion point
x1 as shown in Fig. 3.1.
Truncation of M2L
From Eq. (3.31b), the M2L moment tends to zero for a large argument sφ (dM2L − 2ϱ), where
dM2L − 2ϱ corresponds to the minimum distance between two cells rmin , and it holds that dM2L −2ϱ ≤ rmin as shown in Fig. 3.3. Negligibly small values of ϱ L̃G;i;k;β;l(x1 ) in M2L (3.31) are
c,d;α

26

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Figure 3.3.: Distance between two cells in M2L translation.

excluded from the translation process in order to accelerate the FMM calculation. To this end,
the asymptotic behavior of M2L including exp[−sφ (dM2L − 2ϱ)] is investigated. The truncation
condition is expressed as follows:ℜ [sφ (dM2L − 2ϱ)] > b log 10,(3.33)

where b is the number of significant figures required in M2L computation. The truncation procedureis briefly summarized as follows:
1. sφ (dM2L − 2ϱ) is evaluated before M2L.
2. The truncation condition defined in Eq. (3.33) is calculated.
3. If the truncation condition defined in Eq. (3.33) is satisfied, M2L is neglected. However,when Eq. (3.33) is not satisfied, M2L must be conducted as usual.
This truncation technique can be applied to the large number of M2L translations to save computational time. The truncation parameter b is given by b = 20 in this study.

3.1.4. Computational complexity
We briefly conclude the computational complexity of our proposed method for N and Me here.
The computational complexity of one matrix-vector product for spatial components is consideredas O(Me ) or O(Me log Me ) owing to use of the FMM [62]. The computational complexities of
FMM and FFT are dominant in the time-marching process presented in Section 3.1.1. The numberof FMM implementation is O(N 2 ), and the number of FFT calculations is O(N Me ). Therefore,
the computational complexity of FMM is O(N 2 Me ) or O(N 2 Me log Me ), while the one for FFT
is O(N 2 Me log N ). In practical numerical computations, the computational time of FMM usually
becomes much larger than that of FFT, and O(N 2 Me ) or O(N 2 Me log Me ) is considered as the total
computational complexity of the proposed method. On the other hand, the memory complexity is
O(N Me ) owing to the FMM [62].

273. Acceleration of CQ-BEM
0.5

Reference
CQ - BEM
CQ - FMBEM

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 3.4.: Time variations of p/p0 obtained by CQ-BEM and CQ-FMBEM using RK3 at point X
in Fig. 2.3 when cf ∆t/a = 0.069.
4x10-3Absolute error

3x10-32x10-3

1x10-3

0
0

1

2

3

4

5

6

7

8

Figure 3.5.: Absolute errors between p/p0 obtained by CQ-BEM and CQ-FMBEM, which are
shown in Fig. 3.4.

3.1.5. Numerical results
To verify the accuracy and to demonstrate the computational efficiency of the proposed CQ-FMBEM,
some numerical examples are presented in this section. The same analysis model as Section 2.3 depicted in Fig. 2.3 is used for this numerical simulation. The truncation number for the summationof multipole and local expansions, such as Eqs. (3.10) and (3.11), is chosen as Nt = 10 for all
simulations in this study.
First, the comparison of numerical solutions obtained by the CQ-BEMs with and without the
FMM for accuracy verification. Fig. 3.4 shows numerical solutions for the same simulation as Fig.
2.7 when cf ∆t/a = 0.069, and the dashed line is obtained by the CQ-FMBEM with RK3. As we28

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Computational time (sec)106

CQ - BEM
CQ - FMBEM
105

104

103

102
102

103

104

105

Number of elements

Figure 3.6.: Computational time comparison between CQ-BEM and CQ-FMBEM using RK3 for
3-D acoustic wave problem.
106

Required memory (MB)

105

10

CQ - BEM
CQ - FMBEM

4

103
102
101
100
102

103

104

105

Number of elements

Figure 3.7.: Required memory comparison between CQ-BEM and CQ-FMBEM using RK3 for 3-D
acoustic wave problem.

can see from Fig. 3.4, the waveforms obtained by the CQ-BEM and CQ-FMBEM are very similar.
On the other hand, the absolute errors between p/p0 obtained by CQ-BEM and CQ-FMBEM areplotted in Fig. 3.5. These errors are caused by the approximation in the FMM and considered as
sufficiently small from Fig. 3.5.
Second, the computational efficiency of CQ-FMBEM is confirmed. All of the computations here
are conducted by a single processor. Discretized BIE (2.33) is numerically solved by both CQ-BEM

29

3. Acceleration of CQ-BEM

Figure 3.8.: Analysis model for elastic wave scattering by an obstacle in a 3-D infinite domain

and CQ-FMBEM, and the non-preconditioned GMRES [74], which is one of the Krylov subspacesolvers, is used. In the CQ-BEM, all of the influence functions (2.30) are stored after the calculations
for all elements and time steps.
The computational time and required memory for both CQ-BEM and CQ-FMBEM with RK3 areshown in Figs. 3.6 and 3.7, respectively, when cf ∆t/a = 0.069 and N = 128. As shown in Fig.
3.6, the CQ-FMBEM is faster than the CQ-BEM when the number of elements Me is Me = 5000 or
more. Moreover, the memory required by the CQ-FMBEM is much less than that by the CQ-BEM
as shown in Fig. 3.7. Consequently, the CQ-FMBEM is more efficient for large-scale problems thanthe CQ-BEM.

3.1.6. CQ-FMBEM for 3-D elastic wave problems
The CQ-FMBEM is applied to 3-D elastic wave problems in this section. The CQ-FMBEM for thisproblem has been developed by Maruyama et al. [55]. Only the formulation of BIE and multipole
and local expansions are presented because the calculation algorithm is same as the one for 3-D
acoustic problems.
Let S be a closed boundary with unit normal vector n in R3 as depicted in Fig. 3.8. D is anexterior domain, which is a homogeneous, isotropic, and linearly elastic solid. Disregarding the
body force, the displacement u satisfies the following governing equation:()
c2T ∇2 u(x, t) + c2L + c2T ∇∇ · u(x, t) = ü(x, t)

x ∈ D,(3.34)

where cL and cT are the velocities of P and S waves, respectively, and expressed using the density
ρ and the Lamé constants λ and µ as follows:√cL =λ + 2µ
,
ρ

√
cT =

µ
.
ρ

We assume that the solution u of Eq. (3.34) satisfies the following initial condition:usc (x, 0) = u̇sc (x, 0) = 0

x ∈ D.

(3.35)

Considering the Sommerfeld radiation condition for scattered waves and initial condition (3.35), the30

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)
time-domain BIE for displacement u is formulated by means of the usual manner as follows:∫ t∫in

C(x)u(x, t) = u (x, t) +
0

−

∫ t

U (x, y, t − τ ) · t(y, τ )dSy dτ
∫p.v. T (x, y, t − τ ) · u(y, τ )dSy dτ,S0(3.36)

S

where C(x) is also the free term [14] depending on the boundary shape at x, and t is the traction
force. U and T are the fundamental displacement and traction solutions for 3-D elastodynamic
problems in time-domain, respectively [1]. The displacement formulation defined in Eq. (3.36) isused for the boundary element analysis of elastic wave scattering in this chapter.

CQ-FMBEM formulation
In numerically solving BIE (3.36), the convolution integrals are evaluated by means of the IRKbased CQM, and the surface integrals over the boundary S are discretized by the piecewise constant
approximation. The discretized BIE at n-th and i-th sub-step in time is written as follows:
[
]
Me ∑
n ∑
m
∑
1 i;n
ij;n−kin;i;nj;kij;n−kj;kUγαu = uγ+· tα − Tγα· uα
2 γ
k=0 α=1 j=1(i = 1, ..., m), (n = 0, ..., N − 1),(3.37)

where
in
pi;n
γ = p(xγ , (n + ci )∆t), (p = u, t, or u ).
ij;κ
ij;κ
In Eq. (3.37), Uγαand Tγα
are the influence functions defined by

−1
ij;κ
Uγα
= F̃lκ[ m) ∫∑(Eβ (ζl )
β=1

−1
ij;κ
Tγα
= F̃lκ

ij

[ m
∑(
β=1

]
Û (xγ , y, λlβ )dSy

,

Sα

]

∫
)
Eβ (ζl ) p.v.
ij

(3.38)T̂ (xγ , y, λlβ )dSy,

(3.39)

Sα

where Û and T̂ are the Laplace-domain fundamental solutions for displacement and traction, respectively. Rearranging Eq. (3.37) according to the time step, the following expression is obtained.
M ∑
m [(
∑
1
α=1 j=12)ij;0Iδγα δij + Tγα= uin;i;n
−
γ

]
ij;0
j;n
· uj;nα − Uγα · tα
n−1
M ∑
m
∑∑

[
ij;n−kij;n−kTγα· uj;k· tj;kα − Uγαα].

(3.40)

k=0 α=1 j=1

31

3. Acceleration of CQ-BEM
The matrix-vector calculation of retarded potential terms in the right-hand side of Eq. (3.40) is
rapidly evaluated by the FMM in order to reduce the computational cost. Only the multipole andlocal expansions are presented here because the time-stepping procedure and translation theorems
are same as the CQ-FMBEM for 3-D acoustic wave problems.
i;k;β;lIn order to derive the multipole and local expansions, Ui;k;β;lI;γα and TI;γα are defined by[∫
Ui;k;β;lI;γα =]ÛIK (xγ , y, λlβ )dSy[∫

Ti;k;β;lI;γα =Sα

Sαti;k;β;lK;α ,(3.41)]T̂IK (xγ , y, λlβ )dSy ui;k;β;lK;α ,(3.42)

whereui;k;β;l
K;α =

m (
)
∑
Eβ (ζl ) uj;k
K;α ,
ij

j=1

ti;k;β;l
K;α =

m (
∑
j=1

)
Eβ (ζl ) tj;k
K;α .
ij

Û and T̂ in Eqs. (3.41) and (3.42) are expressed using the fundamental solution of modified
Helmholtz equation Ĝφ as follows:
[
(
)]
1
1
∂2
T
T
L
ÛIK (x, y, s) =Ĝ (x, y, s)δIK + 2
Ĝ (x, y, s) − Ĝ (x, y, s) ,µsT ∂xI ∂yK∂
T̂IK (x, y, s) = nJ (y)CKJpq
ÛIp (x, y, s),∂yq(3.43)(3.44)

where Cijkl is a component of the elastic modulus tensor. The multipole and local expansions for
Laplace-domain fundamental solutions can be derived in a similar way to the ones in frequencydomain [99, 17].
Taking into account the scaling of modified Bessel functions, substituting Eqs. (3.4a), (3.43), and(3.44) into Eqs. (3.41) and (3.42) yields the following multipole expansions:
[

{

}
∂
UL ;i;k;β;l
L
Aa
(y0 )
ϱ O (xγ − y0 ) ϱ Ma,b;α∂xI a,b
a=0 b=−a]}{∂
UT ;i;k;β;l
T
,
(y0 )
+ Ba etIJ
ϱ O (xγ − y0 ) ϱ Mt,a,b;α∂xJ a,b
s=λlβ
[ {
}
∞ ∑
a
∑
∂
i;k;β;lTL ;i;k;β;lL
TI;γα =Aa(y0 )
ϱ O (xγ − y0 ) ϱ Ma,b;α∂xI a,b
a=0 b=−a]}{∂T;i;k;β;lT
T,+ Ba etIJ(y0 )ϱ O (xγ − y0 ) ϱ Mt,a,b;α∂xJ a,bl

Ui;k;β;lI;γα =∞ ∑a∑(3.45)(3.46)s=λβ

L ;i;k;β;l
T ;i;k;β;lwhere ϱ MΠand ϱ MΠare the scaled multipole moments with respect to P and Sa,b;αt,a,b;αwaves, respectively, and the superscript Π is Π = U or T . In addition, eijk indicates a component

32

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)
of the 3-D permutation tensor, and Aa and Ba are defined by
Aa = −sL (2a + 1),
4πµs2TBa =2a + 1.
4πµsTi;k;β;lIn Eqs. (3.45) and (3.46), each of Ui;k;β;lI;γα and TI;γα has four multipole moments: one for P waves
ΠT ;i;k;β;l
L ;i;k;β;l(ϱ MΠ) and the others for S waves (ϱ Mt,a,b;α, t = 1, 2, 3). The explicit expressions of thea,b;αmultipole moments in Eqs. (3.45) and (3.46) are shown as follows:

]
∂
L
F̄
(y
−
y
)
dSy ,
ϱ a,b
0
Sα ∂yK[]∫∂
UT ;i;k;β;li;k;β;lT(y0 ) = tK;αetJK
ϱ Mt,a,b;αϱ F̄ (y − y0 ) dSy ,∂yJ a,bSα[]∫∂
TL ;i;k;β;li;k;β;lL(y0 ) = uK;αCKJpq nJ (y)ϱ Ma,b;αϱ F̄ (y − y0 ) dSy ,∂yp ∂yq a,b
Sα
]
[
∫
∂
TT ;i;k;β;li;k;β;lT(y0 ) = uK;αetup CKJpq nJ (y)ϱ F̄ (y − y0 ) dSy .
ϱ Mt,a,b;α∂yq ∂yu a,bSα∫[UL ;i;k;β;l(y0 ) = ti;k;β;lϱ Ma,b;αK;α(3.47)(3.48)(3.49)(3.50)

i;k;β;lThe local expansions for Ui;k;β;lI;γα and TI;γα are obtained by substituting Eq. (3.4b) into Eqs. (3.41)
and (3.42) as follows:[}∂
UL ;i;k;β;l
L
F
(x
−
x
)
(x0 )
ϱ
γ
0
ϱ La,b;α∂xI a,b
a=0 b=−a]}{∂
UT ;i;k;β;l
T
(x0 )
,
+ Ba etIJ
ϱ F (xγ − x0 ) ϱ Lt,a,b;α∂xJ a,b
s=λlβ
[ {
}
∞ ∑
a
∑
∂
i;k;β;lTL ;i;k;β;lL
Aa
TI;γα =(x0 )
ϱ F (xγ − x0 ) ϱ La,b;α∂xI a,b
a=0 b=−a]{}∂T;i;k;β;lT
T+ Ba etIJ(x0 ),ϱ F (xγ − x0 ) ϱ Lt,a,b;α∂xJ a,bl
Ui;k;β;lI;γα =

∞ ∑
a
∑

{

Aa

(3.51)(3.52)s=λβ

L ;i;k;β;l
T ;i;k;β;lwhere ϱ LΠand ϱ LΠare the coefficients of local expansion with respect to P and Sa,b;αt,a,b;αφwaves, respectively. The coefficients of local expansion can be written by replacing ϱ F̄a,b(y − y0 )φby ϱ Ōa,b (y − x0 ) in Eqs. (3.47)–(3.50) as follows:

]
∂
L
Ō
(y
−
x
)
dSy ,
ϱ a,b
0
Sα ∂yK][∫∂
UT ;i;k;β;li;k;β;lT(x0 ) = tK;αetJK
ϱ Ō (y − x0 ) dSy ,ϱ Lt,a,b;α∂yJ a,bSα][∫∂
TL ;i;k;β;li;k;β;lL(x0 ) = uK;αCKJpq nJ (y)ϱ Ō (y − x0 ) dSy ,ϱ La,b;α∂yp ∂yq a,bSα∫UL ;i;k;β;l(x0 ) = ti;k;β;lϱ La,b;αK;α[(3.53)(3.54)(3.55)

33

3. Acceleration of CQ-BEM

Figure 3.9.: Elastic wave scattering by a spherical cavity subjected to an incident plane wave.
]∂
Tetup CKJpq nJ (y)ϱ Ō (y − x0 ) dSy .
∂yq ∂yu a,bSα∫TT ;i;k;β;li;k;β;l(x0 ) = uK;αϱ Lt,a,b;α[(3.56)φ
φ
In Eqs. (3.47)–(3.50) and Eqs. (3.53)–(3.56), ϱ Ōa,band ϱ F̄a,b
are defined byφ−sφ ϱ φF̄a,b ,
ϱ F̄a,b = eφsφ ϱ φŌa,b .
ϱ Ōa,b = e(3.57)

L ;i;k;β;l
L ;i;k;β;lIf ϱ MG;i;k;β;l, ϱ LG;i;k;β;l, and sφ in Eqs. (3.30)–(3.32) are replaced with ϱ MΠ
, ϱ LΠ
,
a,b;α
a,b;α
a,b;α
a,b;α
and sL , respectively, the translation formulas for the P wave moments are derived. Similarly, reG;i;k;β;l
T ;i;k;β;l
T ;i;k;β;lplacing ϱ Ma,b;α, ϱ LG;i;k;β;l, and sφ in Eqs. (3.30)–(3.32) by ϱ MΠ
, ϱ LΠ
, and sT ,a,b;αt,a,b;αt,a,b;αrespectively, the translation formulas for the S wave moments are also obtained.

Numerical results
The scattering of an incident plane wave by a spherical cavity is numerically solved to show theaccuracy and efficiency of the CQ-FMBEM for 3-D elastodynamics. The computational time andrequired memory are confirmed by comparing the results from CQ-BEMs with and without FMM.
The boundary condition on S is assumed to be t = 0, and Poisson’s ratio ν is given by ν = 0.25.
An incident plane P wave scattering by a spherical cavity with radius a as depicted in Fig. 3.9,
is numerically solved to verify the accuracy of the CQ-FMBEM for 3-D elastodynamics. Displacement fields at the point X in Fig. 3.9 are investigated for CQ-FMBEMs based on different schemes
and various time increments. The following Ricker wave [71] with amplitude u0 is used as anincident plane wave.
√

π

in

u0 dφ (α − 0.5) e−α , u (x, t) =
2
[
(
)]

x +a 2 α = πf in t − ts − 1,
cL(3.58a)(3.58b)where dφ is the displacement vector of the incident wave, and φ = L, T V , or T H and indicatesa P, SV, or SH wave, respectively. In this section, we present only the results of P wave incidence,and therefore, φ = L and dL = (1, 0, 0) are given. In addition, f in is the center frequency of34

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)
0.10
0.05
0.00
-0.05

Reference

-0.10
-0.15
-0.20
0

1

2

3

4

5

6

7

Figure 3.10.: Time variations of usc
1 /u0 obtained by using BDF2 at point X in Fig. 3.9 when Me =
9280.

0.10
0.05
0.00
-0.05

Reference

-0.10
-0.15
-0.20
0

1

2

3

4

5

6

7

Figure 3.11.: Time variations of usc
1 /u0 obtained by using RK2 at point X in Fig. 3.9 when Me =9280.

the incident wave, and ts is the peak time parameter. In this analysis, ts = 1/f in and f in =inin and given by λin /a =acL /λin
L are considered, where λL is the P wavelength with respect to f
L√
3. The boundary surface of the cavity is discretized into Me = 9280 flatly triangular boundary
elements. The reference transient solutions are constructed by the inverse Fourier transform because
this problem has also been analytically solved in the frequency-domain by Pao and Mow [67].
The time variations of usc
1 /u0 at point X in Fig. 3.9 obtained by BDF2, RK2, and RK3 are shownfor different time increments in Figs. 3.10, 3.11, and 3.12, respectively. As seen from Fig. 3.10,
the BDF2 with the smallest time increment cT ∆t/a = 0.015, can yield very good results at the35

3. Acceleration of CQ-BEM

0.10
0.05
0.00
-0.05

Reference

-0.10
-0.15
-0.20
0

1

2

3

4

5

6

7

Figure 3.12.: Time variations of usc
1 /u0 obtained by using RK3 at point X in Fig. 3.9 when Me =
9280.

0.10
0.05
0.00
-0.05
-0.10

Reference

-0.15
-0.20
0

1

2

3

4

5

6

7

Figure 3.13.: Time variations of usc
1 /u0 obtained by using RK3 at point X in Fig. 3.9 whencT ∆t/a = 0.06.

time up to cT t/a = 3. However, usc
1 /u0 is not accurate compared with the reference solutions
after cT t/a = 3, and fluctuations can be seen clearly. In particular, larger time increments, e.g.,cT ∆t/a = 0.06 or 0.03, give less accurate results. Consequently, a small time increment is required
for accurate computation of usc
1 /u0 for the time range between cT t/a = 4 to 7.
As shown in Figs. 3.11 and 3.12, better accuracy can be obtained by the RK2 and RK3 for thecases of cT ∆t/a = 0.06 and 0.03 than that by the BDF2. The fluctuations of usc
1 /u0 increase with36

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Computational time (sec)

106

105

104

CQ - BEM with BDF2
CQ - FMBEM with BDF2
CQ - BEM with RK3
CQ - FMBEM with RK3

103

102
102

103

104

105

Number of elements

Figure 3.14.: Computational time comparison between CQ-BEM and CQ-FMBEM using BDF2 andRK3 for 3-D elastic wave problem.
106

Required memory (MB)10

CQ - BEM with BDF2
CQ - FMBEM with BDF2
CQ - BEM with RK3
CQ - FMBEM with RK3

5

104
103
102
101
100
102

103

104

105

Number of elements

Figure 3.15.: Required memory comparison between CQ-BEM and CQ-FMBEM using BDF2 andRK3 for 3-D elastic wave problem.

increasing time increments. usc
1 /u0 calculated by the RK2 and RK3 using cT ∆t/a = 0.03 and 0.06
seem to be sufficiently accurate with the exception of a slight oscillation at about cT t/a = 5.
The effect of spatial discretization of the BIE on the numerical solutions is next investigated. Fig.
3.13 shows usc
1 /u0 as a function of time at point X in Fig. 3.9 in the cases of Me = 9280 and 50230.
These results are calculated by the RK3 using cT ∆t/a = 0.06. As expected, a large number ofboundary elements, Me = 50230, give more accurate results than Me = 9280. The accuracy of the37

3. Acceleration of CQ-BEM

Table 3.1.: Computational time and required memory of CQ-FMBEM when N = 128 and Me =5048.
Computational time (sec)
Required memory (MB)BDF2 (m = 1)
45291
91

RK2
115860174

RK3161140272

numerical results can be improved either by using high order elements for the spatial discretization
or by increasing the number of boundary elements.
The computational time and memory needed by the CQ-FMBEM with BDF2 and RK3 are shownin Figs. 3.14 and 3.15, respectively. The time increment cT ∆t/a = 0.08 and the total time stepN = 128 are used here, and all computations are performed by a single processor. As shown in
Fig. 3.14, the CQ-FMBEM with BDF2 is faster than the CQ-BEM with BDF2 when the numberof elements Me is Me = 5000 or more. Moreover, the memory required by the CQ-FMBEM
is much less than that by the CQ-BEM as shown in Fig. 3.15. The CQ-BEM with BDF2 could
not be performed when the number of elements Me is Me = 4000 or more because of memory
restriction. The similar behavior is seen for RK3. Consequently, the CQ-FMBEM is more efficient
for large-scale problems than the CQ-BEM.
Next, the relation between the stage number of IRK method m and the computational efficiencyof the proposed method is considered. The sub-matrix of influence functions becomes 3m × 3m
size as shown in Eqs. (3.38) and (3.39). However, the argument λlβ used in Eqs. (3.38) and (3.39),
which corresponds to the Laplace parameter, has m components per l. Only m kinds of translation
operators for each of M2M, M2L, and L2L are required in the FMM although the multiplications in
the translation theorem and near-field contribution require O(m2 ) calculation. Therefore, for smallm and large Me , it is expected that the combination of the FMM and IRK-based CQ-BEM can
reduce the computational cost as the stage number of the IRK method increases. Table 3.1 presents
the computational time and memory for BDF2, RK2, and RK3 when N = 128 and Me = 5048. As
shown in Table 3.1, the computational time seems to be not O(m2 ) but faster than it. However, thebehavior of computational time with respect to m is not stable because the M2L truncation depends
on λlβ . On the other hand, the memory complexity for m is O(m), and this fact is obviously
confirmed from Table 3.1.

3.1.7. CQ-FMBEM for 3-D coupled acoustic-elastic wave problems
In this section, the formulation and numerical results of the CQ-FMBEM for the 3-D coupledacoustic-elastic wave problems are presented. The CQ-FMBEM for this problem has been developed by Maruyama et al. [53].
Let S be a closed interface between a fluid domain Df and solid domain Ds as depicted in Fig.
3.16. The superscripts of normal vector n, f and s, indicate respective domains, and s denotes theunit tangential vector to S. Disregarding the body force, the pressure p in Df and displacement u

38

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Figure 3.16.: Analysis model 3-D for coupled acoustic-elastic wave problems

in Ds satisfy the following governing equations:c2f ∇2 p(x, t) = p̈(x, t)

x ∈ Df ,(3.59)

c2T ∇2 u(x, t) + (c2L − c2T )∇∇ · u(x, t) = ü(x, t)

x ∈ Ds .

(3.60)Taking into account the continuity of displacement and traction force on the interface S, the interface
conditions are given as follows:− ρf nf (x) · ü(x, t) = q(x, t)n (x) · t(x, t) = −p(x, t)s(x) · t(x, t) = 0

x ∈ S,x ∈ S,sx ∈ S,(3.61)(3.62)(3.63)

where ρf is the density of fluid. Assuming that the incident wave exists in Ds and that Df is closed
inner domain as depicted in Fig. 3.16, the initial conditions for wave scattering problems are given
as follows:x ∈ Ds ,(3.64)u(x, 0) = u̇(x, 0) = 0

x ∈ S,(3.65)

p(x, 0) = ṗ(x, 0) = 0

x ∈ Df , S.

(3.66)u(x, 0) = uin (x, 0)Using boundary conditions (3.61)–(3.63), the time-domain BIEs for pressure (2.27) and displacement (3.36) can be solved simultaneously.
Substituting Eqs. (3.61)–(3.63) into BIEs (2.27) and (3.36) and assuming that pin = 0 yield thefollowing BIEs:∫ t∫
C(x)η(x, t) = ρf0

G(x, y, t − τ )nf (y) · ü(y, τ )dSy dτ
S
∫ t
∫
−
p.v. H(x, y, t − τ )η(y, τ )dSy dτ,0(3.67)S

39

3. Acceleration of CQ-BEM∫ t∫

C(x) · u(x, t) = u (x, t) +U (x, y, t − τ ) · ns (y)η(y, τ )dSy dτ
0
S
∫ t
∫
−
p.v. T (x, y, t − τ ) · u(y, τ )dSy dτ,in0(3.68)

S

where η is defined by η = ns · t. The second derivative of displacement with respect to time ü
is contained in the right-hand side of Eq. (3.67). This second derivative has to be numericallytreated by some techniques in order to simultaneously solve Eqs. (3.67) and (3.68). Using the CQM
formulation, it is easy to treat time differential terms.
Taking into account the identity of convolution integral()
d(k ∗ g) (t) = k̇ ∗ g (t) + k(0)g(t)dt
and initial condition (3.65), the second term in the right-hand side of Eq. (3.67) can be modified asfollows:∫ t∫
G(x, y, t − τ )nf (y) · ü(y, τ )dSy dτ
ρf
0
S
∫ t∫
= ρf
G̈(x, y, t − τ )nf (y) · u(y, τ )dSy dτ.
(3.69)0S

As shown in Eq. (3.69), derivatives of quantity are no longer included. Next, the discretizationof Eqs. (3.67) and (3.68) and the numerical treatment of differentiated fundamental solution G̈ in
Eq.(3.69) using the CQM are presented.
CQ-BEM formulation
In numerically solving the BIEs (3.67) and (3.68), the convolution integrals are evaluated by meansof the IRK-based CQM, and the surface integrals over the interface S are discretized by the piecewise constant approximation. The discretized BIEs at n-th and i-th sub-step in time are written as
follows:]
1 i;n ∑ ∑ ∑ [ij;n−k f
ij;n−k j;kηγ =ρf Dγαnα · uj;k
−
H
η
,
α
γα
α
2

(3.70)

]
∑∑∑[
1 i;n
s j;k
ij;n−kj;kij;n−k
·
n
η
−
T
·
u
uγ = uin;i;n
+
U
α α
γα
α ,
γ
γα
2

(3.71)nMm

k=0 α=1 j=1

n

M

m

k=0 α=1 j=1

ij;κ
where Dγαis the influence function with respect to the differentiated fundamental solution G̈ andgiven as follows:403.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

−1 
ij;κ
Dγα
= F̃lκ

m
∑


(
) ∫
(λlβ )2 Eβ (ζl )Ĝ(xγ , y, λlβ )dSy  .
ij

β=1(3.72)

Sα

ij;κ
ij;κ
(λlβ )2 in Eq. (3.72) is difference between Dγαand Gγαin Eq. (2.29). This term is caused by
the differentiation with respect to time because Laplace-domain kernel functions are used here. The
Laplace transform of time differentiated function is given as follows:[]
L k̇(t) = sk̂(s).
From the above, BIEs (3.67) and (3.68) can be solved simultaneously with the easy treatment of thesecond derivative of displacement.
We confirm the numerical errors of calculating the second differentiation of fundamental solution in the CQM through numerical experiment. Therefore, the following convolution integral iscomputed in two ways, and their results are compared.
∫ t u(t) =G(x, y, t − τ )ψ̈(τ )dτ

0 < t,

03 t

ψ(t) = t e H(t),(3.73a)(3.73b)where G is specifically described as follows:()
1r
G(x, y, t) =
δ t−
,
4πr
cf

(3.74)

where δ(·) is the Dirac delta distribution. The exact solution of u in Eq. (3.73) can be obtained asfollows:()
r
1
ψ̈ t −.
(3.75)
u(t) =
4πrcf
We investigate the relative errors caused by numerical computation of Eq. (3.73) in the following
two ways.
case 1 The CQM is applied to Eq. (3.73a) after differentiation of ψ. In other words, the following
equation is evaluated by means of the CQM.
∫ tū(t) =G(x, y, t − τ )(τ 3 + 6τ 2 + 6τ )eτ dτ.

(3.76)

0case 2 As well as the proposed method, the convolution integral including the second differentiation of fundamental solution is computed by the CQM. Namely, the following equation iscomputed by the CQM.
∫ tū(t) =G̈(x, y, t − τ )τ 3 eτ dτ.

(3.77)

041

3. Acceleration of CQ-BEM
100
10-1
10-2
10-3
10-4

Relative error

10-5
10-6
10-7
10-8
case1 RK2
case2 RK2
case1 RK3
case2 RK3

10-9
10-10
10-11
10-12
10-2

10-1

100

Figure 3.17.: Relative errors for calculating convolution integral (3.73) vs. time increments using
IRK-based CQM with RK2 and RK3.

Forward-scattered wave

Backscattered wave

Figure 3.18.: Elastic wave scattering by a spherical obstacle subjected to an incident plane wave.

The total analysis time is cf t/r = 2, and we investigate computational errors varying the timeincrement cf ∆t/r. In addition, the RK2 and RK3 are used for the CQM. The relative errors |(u −ū)/u| caused by both cases are plotted in Fig. 3.17. As we can see from Fig. 3.17, behavior of errors
is unstable when cf ∆t/r is comparatively large. However, the slopes of relative errors caused bycase 1 and case 2 are identical, and their behavior is stable when cf ∆t/r is sufficiently small. An
increase of computational error from case 1 to case2 is considered as caused by the rapid fluctuationof differentiated fundamental solution. However, we can use the numerical approach of case 2 if
cf ∆t/r is sufficiently small.
Numerical results
The numerical examples deal with the scattering problems of an incident plane wave by a sphericalcavity, elastic inclusion, and fluid-filled inclusion as depicted in Fig. 3.18. In this section, thefar-field approximations of scattered wave usc;far are evaluated, and their details can be found in42

3.1. Fast multipole accelerated CQ-BEM (CQ-FMBEM)

Table 3.2.: Material constants for base material and inclusions.
Base material
Elastic inclusion
Fluid-filled inclusion

P wave velocity
cL√
cL /√3cL / 3

S wave velocity
cT√
cT / 3–

Density
ρ
ρ/2
ρ/2

4

2

0

CavityElastic inclusion
Fluid-filled inclusion

-2

-4
0

5

10

15

20

Figure 3.19.: Time variations of far-field backscattered amplitude ΩL /(au0 ) for various obstacles
subjected to an incident plane P wave.
15
10
5
0
-5

CavityElastic inclusion
Fluid-filled inclusion

-10
-15
0

5

10

15

20

Figure 3.20.: Time variations of far-field forward-scattered amplitude ΩL /(au0 ) for various obstacles subjected to an incident plane P wave.

Appendix B. The Ricker wave described in Eq. (3.58) is used as an incident plane wave. We
present numerical results of P and SV wave incidence, and dφ is given by dL = (1, 0, 0) and√dT V = (0, 1, 0). Time parameters are set as λin
3, ts = 1/f in , cT ∆t/a = 0.12, andL /a =

43

3. Acceleration of CQ-BEM
4

2

0

-2

CavityElastic inclusion
Fluid-filled inclusion

-4

-6
0

2

4

6

8

10

12

Figure 3.21.: Time variations of far-field backscattered amplitude ΩT V /(au0 ) for various obstacles
subjected to an incident plane SV wave.
30
20
10
0
-10

CavityElastic inclusion
Fluid-filled inclusion

-20
-30
-40
0

2

4

6

8

10

12

Figure 3.22.: Time variations of far-field forward-scattered amplitude ΩT V /(au0 ) for various obstacles subjected to an incident plane SV wave.

N = 128. In addition, the RK3 is used for the CQM. Material constants for base material, elastic
inclusion, and fluid-filled inclusion are shown in Table 3.2, and Poisson’s ratio of base material ν isgiven by ν = 0.25. As shown in Table 3.2, the acoustic impedance of elastic inclusion for P wave
is identical with the one of fluid-filled inclusion. For all numerical simulations in this section, thesurface of obstacle is divided into Me = 9280 flatly triangular boundary elements.
Figs. 3.19 and 3.20 show the time histories of ΩL /(au0 ) corresponding to backscattered andforward-scattered waves, respectively, when an incident plane P wave is scattered by a various
spherical obstacle. Ωφ is the far-field scattered amplitude and found in Appendix B. As shown in
Fig. 3.19, the amplitudes of the first wave at about (cL t − x)/a = 0.7 generated by elastic andfluid-filled inclusions are almost same because their acoustic impedances for P wave are identical.

44

3.2. Acceleration for number of time steps

Table 3.3.: Computational time (sec).
P wave scattering
S wave scattering

Cavity41959
41999

Elastic inclusion
220103223366

Fluid-filled inclusion
114366114952

In addition, the phase of waveform at about (cL t − x)/a = 0.7 is the inverse of incident wave,
and then, large amplitude waves arrive in the case of elastic inclusion. These phenomena have also
been observed in the previous 2-D simulation [64]. On the other hand, there is only slight difference
among far-field forward-scattered waves by various obstacles as shown in Fig. 3.20.
Figs. 3.21 and 3.22 show the time histories of ΩT V /(au0 ) corresponding to backscattered andforward-scattered waves, respectively, when an incident plane SV wave is scattered by various
spherical obstacles. In this simulation, v̂ in Appendix B is given by v̂ = (0, 1, 0). As shown in
Fig. 3.21, the amplitudes of the first wave in the cases of cavity and fluid-filled inclusion are almost
same, and the one in the case of elastic inclusion is comparatively small. This is because S waves
are completely reflected on the cavity and fluid-solid interfaces. In addition, the secondary waves
at about (cT t − x)/a = 6.0 are greatly dissimilar among different obstacles. There is only slightdifference among far-field forward-scattered waves as shown in Fig. 3.22 similarly to the case of P
wave scattering.
The computational time required for solving the boundary value problems is shown in Table 3.3.
The numerical simulations were conducted using the OpenMP parallelization with 24 threads. As
shown in Table 3.3, the most expensive case costs about 3 days.

3.2. Acceleration for number of time steps
The acceleration of CQ-BEM for the number of elements using the FMM had been discussed until
the previous section. However, there remains the issue of computational cost with respect to thenumber of time steps N , and the computational complexity of CQ-FMBEM with respect to N isstill O(N 2 ). This computational complexity prevents the application of CQ-FMBEM to a longtime analysis. Therefore, the time-marching process of CQ-FMBEM is improved using the rapid
convolution algorithm [31] without error increasing in this section. A new M2L truncation condition
can be introduced through the rapid convolution algorithm. Consequently, the total computational
complexity of a new CQ-FMBEM becomes about O(N Me log N log Me ), and that is demonstratedby showing some numerical results.

3.2.1. New time-marching process of CQ-FMBEM
In this section, the calculation process of matrix-vector products with respect to time for Eq.(3.3)
is improved by using the rapid convolution algorithm. The right-hand side of Eq. (3.3) consists of
the Toeplitz matrix depicted in Fig. 3.23 (a) and the boundary value vector. Moreover, we can use

45

3. Acceleration of CQ-BEM

(a)

(b)

Figure 3.23.: (a) Coefficient matrix structure for time and (b) calculation order of sub-matrix in the
rapid convolution algorithm [31].

Algorithm 1 Select h at n-th step.
1: i ⇐ n
2: for j = 0, ... do3:k ⇐ 2j4:
if k = i then
5:
h⇐k
6:
exit
7:
else if k > i then8:i ⇐ i − 2j−1
9:goto 210:
end if
11: end for
only past boundary values, i.e. q0 , ..., qn−1 at n-th step. This type of matrix-vector products canbe evaluated by the rapid algorithm using FFT with O(N log2 N ) computational complexity [31].
In this algorithm, the product of 2n × 2n sub-matrix and 2n dimensional sub-vector is calculated atevery time step because of using the FFT.
According to [31], the multiplication of sub-matrix and sub-vector is conducted in the ordershown in Fig. 3.23 (b). Specifically, we calculate the following matrix-vector product at n-th step.
 

n
Gh
Gh−1 · · ·
S









 Sn+1   Gh+1
Gh
···

=
.
.
.

..
..
..




 



 n+h−1 G2h−1 G2h−2 · · ·
S



n−h
q
G1 






2 
n−h+1


q
G 

.
..
..  


.
. 






qn−1
Gh

(3.78)

Then, the resulting vector Sκ is used to update Rκ as follows:Rκ + Sκ ⇒ Rκfor κ = n, ..., n + h − 1.

An appropriate h has to be chosen for efficient computation using the FFT, and h at n-th step canbe determined by Algorithm 1. As shown in Algorithm 1, h is always an exponent of two.

46

3.2. Acceleration for number of time steps
Now, we explain the computational procedure of Eq. (3.78) using the rapid convolution algoij;lrithm. The Fourier-domain influence function G̃γαis defined by
ij;l
G̃γα
= F̃κlm () ∫[ ij;κ ] ∑Gγα =Eβ (ζl )ij

β=1

Ĝ(xγ , y, λlβ )dSy ,(3.79)

Sαland the corresponding component of matrix in Eq. (3.78) is described as G̃ . The computational
procedure of the rapid convolution algorithm at n-th step is summarized as follows:
1. The vector consisting of 2h components {qn−h , ..., qn−1 , 0, ..., 0}T is transformed by discrete
Fourier transform (2.10) as follows:[]{q̃0 , ..., q̃2h−1 }T = F̃ {qn−h , ..., qn−1 , 0, ..., 0}T .

(3.80)

l

2. The components of Fourier-domain matrix G̃ and the vector in the left-hand side of Eq. (3.80)
are multiplied, and the resulting vector is transformed by inverse discrete Fourier transform(2.11) as follows:[ 0]2h−1 2h−1 T{P0 , ..., P2h−1 }T = F̃ −1 {G̃ q̃0 , ..., G̃q̃} .

(3.81)l

In the right-hand side of Eq. (3.81), G̃ q̃l for each l includes the matrix-vector productswith respect to space. Therefore, the spatial matrix-vector products are rapidly evaluated by
the FMM. In addition, the following identity is also used for the efficient calculation of theright-hand side of Eq. (3.81).
G̃2h−l 2h−lq̃l= G̃ q̃l(l = 1, ..., h − 1).

(3.82)As a result of Eq. (3.82), h + 1 spatial matrix-vector products are required in the right-handside of Eq. (3.81).
3. Considering the relation between Sκ in the left-hand side of Eq. (3.78) and Pκ in the left-handside of Eq. (3.81),
{
}T {
}T
Sn , ..., Sn+h−1= Ph , ..., P2h−1 ,(3.83)

Sκ can be determined, and Rκ can be updated.
In order to use the FFT for the forward and inverse discrete Fourier transforms in the above, L isgiven by L = 2h. In the total calculation process of retarded potential term, the computational complexities of FFT and FMM are about O(N Me log2 N ) and O(N Me log N log Me ), respectively.
The computational time of FFT is usually much smaller than that of FMM, and therefore, the total
computational complexity is considered as about O(N Me log N log Me ). In addition, the memory
requirement of the proposed CQ-FMBEM is in O(N Me ) order.

47

3. Acceleration of CQ-BEM

3.2.2. Truncation condition for M2L using arrival time of influence waves
A new condition is introduced to the truncation of M2L in this section. The attenuative featureof Laplace-domain fundamental solution has been used for the conventional M2L truncation in
Section 3.1.3. Moreover, arrival time of influence waves is considered for the additional truncation
condition.
As shown in Eq. (3.78), the influence functions from first to (2h − 1)-th step are used for theevaluation of influence waves at n-th step. Therefore, the M2L calculation between two cells which
satisfy the following condition can be neglected.
cφ ∆t(2h − 1 + cm ) < dM2L − 2ϱ,(3.84)

where dM2L − 2ϱ is used instead of the minimum distance between two cells rmin , similarly to
Section 3.1.3. The notations, dM2L , ϱ, and rmin , are found in Fig. 3.3. This new truncation iseffective in sub-matrices near the diagonal of Toeplitz matrix with respect to time because it isexpected that many components before arrival of influence waves are included in them.

3.2.3. Numerical results
To verify accuracy and to demonstrate the computational efficiency of the new CQ-FMBEM, somenumerical results are presented here. The same analysis model as Section 2.3 depicted in Fig. 2.3
is used for this numerical simulation. All numerical simulations in this section are conducted using
the RK3.
First, we compare numerical solutions obtained by the conventional CQ-FMBEM presented in
Section 3.1 with the ones obtained by the new CQ-FMBEM proposed in this section. Fig. 3.24
shows the time histories of p/p0 at point X in Fig. 2.3 when cf ∆t/a = 0.069. As we can see from
Fig. 3.24, the waveforms of p/p0 obtained by the conventional and new CQ-FMBEMs are very
similar. On the other hand, absolute errors between p/p0 obtained by the conventional and new
CQ-FMBEMs are plotted in Fig. 3.25. These errors are caused by the rapid convolution algorithm
and new M2L truncation. Comparing between Figs. 3.5 and 3.25, the magnitude of errors in Fig.
3.25 is very small compared with that caused by the FMM.
Second, the computational efficiency of the new CQ-FMBEM is presented when computations
are conducted by a single processor. The computational time varying the number of elements Me
for both conventional and new CQ-FMBEMs is shown in Fig. 3.26. As shown in Fig. 3.26, the
new CQ-FMBEM is much faster than the conventional one. The slopes of regression lines for theconventional and new CQ-FMBEMs are larger than O(Me ) and a little smaller than O(Me log Me ).
On the other hand, the computational time varying the number of time steps N for both conventionaland new CQ-FMBEMs are shown in Fig. 3.27. As shown in Fig. 3.27, the slopes of regression
lines for the conventional and new CQ-FMBEMs are greatly different. The slope of regression
line for the new CQ-FMBEM is very similar to O(N log N ) although the one for the conventional
CQ-FMBEM is about O(N 2 ). From the above, the total computational complexity of the new48

3.2. Acceleration for number of time steps
0.5

Reference
Conv. CQ - FMBEM
New CQ - FMBEM

0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
0

1

2

3

4

5

6

7

8

Figure 3.24.: Time variations of p/p0 obtained by conventional and new CQ-FMBEMs using RK3
at point X in Fig. 2.3 when cf ∆t/a = 0.069.

Absolute error

4x10-53x10-52x10-5

1x10-5

0
0

1

2

3

4

5

6

7

8

Figure 3.25.: Absolute errors between p/p0 obtained by conventional and new CQ-FMBEMs, which
are plotted in Fig. 3.24.

CQ-FMBEM is considered as O(N Me log N log Me ), which is much faster than the conventionalO(N 2 Me log Me ).
Finally, the situation of M2L truncation is presented in Fig. 3.28 when N = 128 and M = 3176.
The ratio of the truncated number to the total number of M2L moments is plotted in Fig. 3.28.
The case of using only conventional condition (3.33) is presented in Fig. 3.28 (a). As shown in
Fig. 3.28 (a), the ratio of M2L truncation is about 0.2 on every sub-matrix. On the other hand, thecase of using both conventional and new conditions is presented in Fig. 3.28 (b). The ratio of M2Ltruncation becomes large as the sub-matrix approaches the diagonal. This is because the new M2Ltruncation condition (3.84) is effective in the sub-matrices of small size.

49

3. Acceleration of CQ-BEM

Computational time (sec)106

Conv . CQ - FMBEM
New CQ - FMBEM

105

104

103

102
102

103

104

105

Number of elements

Figure 3.26.: Computational time comparison between conventional and new CQ-FMBEMs using
RK3 for 3-D acoustic wave problem varying the number of elements Me .

Computational time (sec)

105

104

103

Conv . CQ - FMBEM
New CQ - FMBEM
102

101
101

102

103

104

Number of time steps

Figure 3.27.: Computational time comparison between conventional and new CQ-FMBEMs using
RK3 for 3-D acoustic wave problem varying the number of time steps N .

3.3. Concluding remarks• In Chapter 3, we have successfully applied the FMM to the IRK-based CQ-BEM for 3-D
acoustic and elastic wave problems and their coupled one. Moreover, the application of therapid convolution algorithm to the time-marching process in CQ-FMBEM has been shown,and it has significantly improved the computational efficiency of the CQ-FMBEM with re-50

3.3. Concluding remarks
0.179
0.216
0.222
0.234

1.000
0.942
0.693
0.296

0.239

0.239

0.236

0.236

1.0

0 . 75

0.5
0.235

0.235

0 . 25

0.0
(a)

(b)

Figure 3.28.: Ratio of M2L truncation for each sub-matrix in triangular Toeplitz system using (a)only conventional truncation condition (3.33) and (b) both conventional and new ones
when N = 128 and Me = 3176.

spect to the number of time steps.
• In the CQ-FMBEM, many M2L moments can be truncated using condition (3.33) because theFMM calculation is carried out in Laplace-domain. Using the rapid convolution algorithm[31] enables the additional truncation condition (3.84) for M2L calculation. The additional
condition is effective in the small sub-matrices near the diagonal of the coefficient matrix withrespect to time.
• The other fast CQ-BEM proposed by Banjai and Kachanovska [8], which is based on the
H-matrix, requires O(N Me log Me ) memory. Their method [8] can be more widely applied
to various problems than our proposed CQ-FMBEM because the applicability of our method
depends on the multipole expansion of fundamental solutions. However, our proposed CQFMBEM requires only O(N Me ) memory and less than their method.
• In Chapters 2 and 3, only the combination of collocation method and piecewise constant
approximation has been used for the spatial discretization. There remains some problems inaccuracy with respect to spatial discretization as shown in Section 3.1.6. However, it is notvery difficult to use the higher order element or to apply the Galerkin discretization to the
proposed CQ-FMBEM. It is emphasized that the computational and memory complexities
are not changed if these improvements of spatial discretization are considered.

51

Chapter 4

3-D Simulation of Higher-harmonic
Waves Due to an Interface Crack of
Bi-material
The boundary element formulation and numerical implementation for wave scattering by an interface crack with contact boundary conditions between two semi-infinite elastic solids are presented inthis chapter. For this problem, the numerical method has been developed and some higher-harmonic
simulations have been implemented by Maruyama et al. [54]. First, the BIE is formulated for wave
scattering by an interface crack. Second, the contact boundary conditions on the interface crack aredescribed, and the procedure of numerical implementation is explained using the flow chart. Finally,
we show some numerical results and discuss the characteristic of the higher-harmonic generation.

4.1. Problem statement and BIE
A 3-D boundary element analysis model for nonlinear ultrasonic simulation is considered for twolayered semi-infinite media including debonding areas as shown in Fig. 4.1. This model consistsof two semi-infinite domains DI and DII , which are homogeneous, isotropic, and linearly elastic
solids, and the interfaces between them, Sh and Sd , denote bonding and debonding areas, respectively. In addition, nς is defined as an unit outer normal vector to Dς (ς = I or II), where thesuperscript denotes the respective domain. Disregarding the body force, displacement u satisfies
the following governing equation in each domain.
[( )( ς )2 2 ς( )2 ]
2cT ∇ u (x, t) + cςL − cςT∇∇ · uς (x, t) = üς (x, t)

x ∈ Dς .

(4.1)

In this study, a plane wave is used as the incident wave in order to investigate fundamental motionsof an interface crack with contact boundary conditions. For the layered media subjected to an
incident plane wave, the free field formulation [37] is usually used in the BEM in order to enhance
the computational efficiency and accuracy. We assume that the interface Sh is flat and that the53

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

Figure 4.1.: Analysis model for wave scattering by a debonding area on bi-material interface in 3-D.

boundary conditions on Sh are given by the following continuity of displacement and traction:tI = −tII .

uI = uII ,(4.2)

We also assume that uin exists in DI . Then, the free field ufree , which consists of incident wave
uin , reflected wave uref , and transmitted wave utrans , can be calculated analytically as presented inAppendix C, and the following relations hold.
ufree;I = uin;I + uref;I ,ufree;II = utrans;II .

(4.3)

Moreover, the superposition principle holds in DI and DII because they are linearly elastic solids.
Therefore, defining the scattered wave usc as the disturbance of ufree by the debonding area Sd , thetotal displacement field u is described as follows:u = ufree + usc .

(4.4)

Considering the initial condition
usc;ς (x, 0) = u̇sc;ς (x, 0) = 0

x ∈ Dς

(4.5)

and that usc satisfies the Sommerfeld radiation condition, the BIE can be formulated for usc as
follows:∫ t∫
1 sc;ς
U ς (x, y, t − τ ) · tsc;ς (y, τ )dSy dτu (x, t) =20
Sh +Sd∫ t∫−p.v.
T ς (x, y, t − τ ) · usc;ς (y, τ )dSy dτ x ∈ Sh , Sd .
(4.6)
0

Sh +SdSubstituting Eq. (4.4) into Eq. (4.6), the BIE can be rewritten for the total wave field u as follows:∫ t∫

U (x, y, t − τ ) · t (y, τ )dSy dτ −ς0

Sh +Sd∫ tς∫T ς (x, y, t − τ ) · uς (y, τ )dSy dτ

p.v.
0

Sh +Sd1− uς (x, t) = g free;ς (x, t)
254

x ∈ Sh , S d ,(4.7)

4.2. Discretization of BIE and contact boundary conditions
where g free is equivalent to the left-hand side of Eq. (4.7) in which u and t are replaced with ufree
and tfree , respectively. Therefore, g free is the known quantity in the boundary element analysis.

4.2. Discretization of BIE and contact boundary conditions
The asymptotic behavior of stress field in the vicinity of interface crack front between dissimilarmaterials shows an oscillating singularity [96, 69]. This fact is also known for the anisotropic bimaterial [90]. However, it is cumbersome to use the special element which can describe the proper
behavior of displacement and stress in the vicinity of crack front. The usual element (specifically,piecewise linear element) has been used for the vicinity of crack front by Wünsche et al. [97], andThey obtained accurate solutions. Moreover, in this study, we evaluate displacement at far-field as
the received wave. Therefore, it is expected that sufficiently accurate solutions are obtained if theusual elements are used for the interpolation of quantities in the vicinity of the interface crack front.
In numerically solving BIE (4.7), the convolution integrals are evaluated by means of the IRK
based CQM, and the surface integrals over Sh and Sd are discretized by the piecewise constantboundary elements. If the interface including the debonding area is divided into Me boundary
elements, the discretized BIE at n-th step and i-th sub-step in time are written as follows:
Me ∑
m [
∑

ς;ij;0Uγα· tς;j;n−αα=1 j=1

−

Me ∑
n−1
m
∑∑

}
]
{
1
ς;j;n
ς;ij;0= gγfree;ς;i;nTγα + Iδγα δij · uα
2

[

]
{
}
{
}
ς;ij;n−k
ς;ij;n−kUγα· tς;j;k− tfree;ς;j;k− Tγα· uς;j;k− ufree;ς;j;k
,
α
α
α
α

(4.8)

k=0 α=1 j=1

where gγfree;ς;i;n is equivalent to the left-hand side of Eq. (4.8) in which uς;j;n
and tς;j;n
are replacedααfree;ς;j;n
free;ς;j;nwith uαand tα, respectively. For n-th time step, all of the quantities on the right-handside of Eq. (4.8) are known. Therefore, considering the boundary conditions on Sh and Sd , theunknown values uς;j;n
and tς;j;n
can be obtained by solving Eq. (4.8). In additions, the underlinedααportion in Eq. (4.8) is rapidly evaluated by the FMM as presented in Section 3.1.6.

4.2.1. Nonlinear interface conditions
In order to simultaneously solve BIEs (4.8) with ς = I and II, the appropriate boundary conditions
should be selected at each time step. Eq. (4.2) holds on the bonding interface Sh . For the debondingarea Sd , we consider three types of interface state in a similar way to the previous researches [38,39, 77]. The interface phases consist of ”separation”, ”stick”, and ”slip” as shown in Fig. 4.2, andthey are summarized as follows:55

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
Contact condition
Stick

Debonding

Separation

Slip

Figure 4.2.: Contact phases on debonding area Sd .

Separation
The separation phase means that two surfaces of upper and lower materials are separated with notraction as depicted in the lower left of Fig. 4.2. Thus, the following traction-free condition is given
for this phase.
tI = tII = 0.

(4.9)

Stick contact
For the stick phase, the surfaces of two materials are contacting and not relatively moving due to thestatic friction force as depicted in the upper right of Fig. 4.2. However, the discontinuity of relative
displacement in tangential direction is allowed unlike the perfect bonding on Sh . From the above,
the interface conditions are given as follows:[u3 ] = −ug ,[u˙s ] = 0,tI = −tII ,(4.10)

where [u] is the crack opening displacement (COD) and defined by [u] = uII −uI , and us indicatesthe displacement vector in the plane of a crack surface. ug denotes the pre-opening displacement of
the debonding surface.

Slip contact
Although the slip phase is also contact state similarly to the stick phase, it allows the relatively
tangential movement with dynamic friction force. The traction force is continuous in any direction
because the surfaces of two materials are contacting under this phase. The friction force is evaluatedby using the Coulomb friction law in this study, and therefore, the following interface conditions56

4.2. Discretization of BIE and contact boundary conditions
are given for this phase.
[u3 ] = −ug ,tI3 = −tII3,

tIs = −tIIs =()[u˙s ]µd −tI3 ,[u˙s ](4.11)

where µd is the dynamic friction coefficient, and ts is the traction vector in the plane of a crack
surface. The moving direction defined by [u˙s ]/ [u˙s ] is significant to the determination of friction
force vector in the dynamic problem [39, 56, 59].

4.2.2. Modification of BIE
The interface conditions for the stick and slip phases (4.10) and (4.11) contain the tangential velocities of COD, and it is difficult to accurately compute them using ordinary numerical differentiation,such as backward differentiation. Moreover, the tangential velocity of COD consists of x1 - and x2 components in 3-D problems although it is enough to evaluate sgn([u˙1 ]) in the 2-D case [39, 77].
Therefore, the BIE is modified to the appropriate form in order to accurately evaluate the tangential
velocity of COD.
The vector w is defined by w = (u̇1 , u̇2 , u3 ), and discretized BIE (4.8) is modified to the one for
w. Taking into account initial condition (4.5), the BIE for scattered wave (4.6) can be modified to
the following expression in a similar way to the modification in Section 3.1.7.
1 sc;ς
w (x, t) =
2
−

∫ t∫
0

∫ t

X ς (x, y, t − τ ) · tsc;ς (y, τ )dSy dτ

Sh +Sd∫Y ς (x, y, t − τ ) · wsc;ς (y, τ )dSy dτp.v.
0x ∈ Sh , S d ,(4.12)

Sh +Sdwhere X and Y are defined by
T11 T12 Ṫ13
Y =  T21 T22 Ṫ23  ,Γ31 Γ32 T33

U̇11 U̇12 U̇13

X =  U̇21 U̇22 U̇23  ,U31 U32 U33



(4.13)

respectively, and Γ is∫ t
Γ(x, y, t) =T (x, y, τ )dτ.
0

The discretized BIE for w can be obtained in a similar manner to derivation of Eq. (4.8) as follows:
{
}
]
Me ∑
m [
∑
1
ς;ij;0ς;j;nς;ij;0ς;j;nXγα · tα − Yγα + Iδγα δij · wα
= hfree;ς;i;nγ
2α=1 j=1
[
]
Me ∑
n−1
m
{
}
{
}
∑∑
ς;ij;n−kς;j;kfree;ς;j;k
ς;ij;n−kς;j;kfree;ς;j;k−Xγα· tα − tα− Yγα· wα − wα,(4.14)

k=0 α=1 j=1

57

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

Start
No
Read input data

“separation”→ “stick”

No“stick” → “slip”

No

Yes

Yes
No

Calculate free fields for displacement,
velocity, and traction on the surface

Yes

Yes

No
Yes
Calculate the known vector

No“slip” → “stick”“stick” or “slip”→ “separation”

Yes

Assemble the unknown vector,
and solve the BIEs simultaneously: number of elements on: static friction coefficient: total number of time steps

No

Evaluate the scattered
far-fields

Yes
No

Write
output data
End

Yes

Figure 4.3.: Flowchart of numerical procedure for nonlinear analysis.

where hfree;ς;i;n
is equivalent to the left-hand side of Eq. (4.14) in which wας;j;n and tς;j;nare
γ
αfree;ς;j;n
free;ς;j;n
φ;ij;κ
φ;ij;κ
replaced with wαand tα, respectively. The influence functions Xγα
and Yγα
correspond to X and Y , respectively, and contain time differentiation and integration. However,these can be easily evaluated by considering the following identities with respect to the Laplacetransform:
[∫ t
]
[
]
1
L k̇(t) = sk̂(s), Lk(τ )dτ = k̂(s).
s0Using discretized BIE (4.14) and dealing with the tangential velocities of COD as variables, thedirection of relative movement and tangential traction on the debonding area in the slip phase canbe evaluated precisely.

4.3. Numerical procedure for nonlinear analysis
The transition and constraint conditions for the iterative calculation of nonlinear analysis are presented in this section. The overall procedure of numerical simulation is shown in Fig. 4.3. At atime step in the CQ-BEM, discretized BIE (4.14) is solved assuming that the interface conditions
on each element are the same as those at the previous time step. If the additional constraint conditions enclosed by the double rhombuses in Fig. 4.3 are not satisfied by the obtained solutions, theinterface condition on the element, which is one of separation, stick, and slip, is changed to anothercondition, and then, the system of equations is assembled and solved again. After conducting the58

4.4. Numerical results
iterative calculations, if both the boundary and constraint conditions on all elements are satisfied,
the time step proceeds to the next one.
Some remarks concerning the numerical calculations are summarized as follows:• At the initial time step, the interface condition of separation is given for all elements on thedebonding area Sd if the pre-opening displacement g > 0. On the other hand, all elements
are taken to be the stick phase if ug = 0.
• There are two options for the phase shift from separation to one of two contact conditions,i.e. slip and stick. In this study, the priority is given to the change from separation to stick if[u3 ] > −ug for the separation condition is violated on the element. In practice, it is consideredthat the slip is not started unless the tangential force exceeds the static friction force which is
generated by the contact of debonding surfaces.
• The stick phase is changed to the slip one if the tangential traction force exceeds the static
friction force. Specifically, this transition occurs if the following condition is violated.
()
tIs < µs −tI3 ,(4.15)

where µs is the static friction coefficient. On the other hand, the transition from contact state
to separation is caused if the following condition is violated.
tI3 ≤ 0.

(4.16)• In numerical calculations, it is difficult to achieve the condition |[u˙s ]| = 0 exactly in thetransition from stick to slip. Therefore, we set |[u˙s ]| = 0 if the following condition is notsatisfied.
prev( stop )[u˙s ] · [u˙s ],prev = ξ < cos θ[u˙s ] [u˙s ](4.17)

where the superscript ”prev” indicates the quantity at the previous time step. Violating Eq.
(4.17) means that the direction of relatively tangential movement is changed more than θstop .
Considering the 2-D local coordinate system (s1 , s2 ) on the debonding surface in which s1 is
prev
prev
parallel to [u˙s ]/|[u˙s ]| as depicted in Fig. 4.4, a sign of s1 -component of [u˙s ]/|[u˙s ]| is
prev
prev
different from the one of [u˙s ]/|[u˙s ]| if the direction of relatively tangenrial movement◦stop
is changed more than 90 . Therefore, θis given by 90◦ in this study.

4.4. Numerical results
In this section, we present some numerical results for two types of material combination using theanalysis model depicted in Fig. 4.5. The first is a penny-shaped debonding area on the steel-steel

594. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

Figure 4.4.: Local coordinate system for stopping condition of relatively tangential movement.

Penny-shaped debonding

Bi-material interface

Figure 4.5.: An incident plane wave scattering by a penny-shaped interface crack with radius a.

interface subjected to an incident plane wave, and this model is equivalent to a crack in an infinite
elastic solid. The second is a penny-shaped debonding area on the interface of the steel-aluminum
coupled medium subjected to an incident plane wave. Numerical results of scattering problems due
to normal incidence of P and S waves and oblique incidence of P wave are shown for both cases. In
advance of the nonlinear ultrasonic simulation, we show numerical results for the linear problem toverify the numerical method. The CQ-FMBEM with RK2 is used for all numerical simulations inthis section.

4.4.1. Accuracy verification
To verify accuracy of the proposed CQ-FMBEM, we solve the scattering problem by an interface
crack when the debonding area is always separated. The material constants of DI and DII are
given by identical values, and therefore, this problem is equivalent to the scattering problem by apenny-shaped crack in an infinite domain. If the material constants of DI and DII are identical, weomit the superscripts I and II. We use an incident plane P wave normal to the crack face for thisverification, and it is defined by the monotonically increasing function as follows:uin
i (x, t) = −u0 δi360cT cL t − x3
H (cL t − x3 ) .
cLa(4.18)

4.4. Numerical results
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

Figure 4.6.: Vertical COD on Sd at each several time steps cT t/a when a penny-shaped crack issubjected to normal incidence of P wave.

Time increment ∆t is taken as cT ∆/a = 0.1, and Poisson’s ratio ν is given by ν = 0.3. Vertical
CODs at each several time steps are shown in Fig. 4.6. As shown in Fig. 4.6, the CODs near thecenter point of the penny-shaped crack becomes about 2cT t/a from cT t/a = 0.1 to 0.5 because theincident wave is completely reflected on Sd . The diffracted P wave reaches the center point of crack
at cT t/a ≃ 0.53. Therefore, the CODs are larger than 2cT t/a due to the influence of diffracted
waves after cT t/a = 0.6. From the above, it seems that the proposed CQ-FMBEM is verified.

4.4.2. Nonlinear ultrasonic simulation
Next, we present numerical results of the nonlinear ultrasonic simulation considering the nonlinear
interface conditions. The following sinusoidal burst wave with nin cycles is used as an incident
wave in this simulation.
() ( I ) ( in
)
in
I
I

(4.19a) u (x, t) = u0 dφ sin 2πΛφ H Λφ H n − Λφ ,()
p · (x − χ)
I
in

t−
,
(4.19b) Λφ = fcIφwhere φ = L, T V , or T H and indicates a P, SV, or SH wave, respectively. p is the propagation
vector of the incident palne wave and given by p = (sin θin , 0, cos θin ) using the incident angle
θin . The displacement vector is given by dL = p for the incident P wave, and we assume thatdT H = (0, 1, 0) and dT V = dT H × p in this study. χ is a point on the wavefront of the incident
wave at t = 0 and given by χ = (−a, 0, 0). nin is given by nin = 3 in this section. The free-fields
are calculated from Eq. (4.19) as presented in Appendix C. The far-field approximation of scatteredwave (see Appendix B) is used for the frequency spectrum analysis. The time increment is taken to
be f in ∆t = 0.08, and material constants for steel and aluminum media are shown in Table 4.1. The
static and dynamic friction coefficients for steel-steel and steel-aluminum interfaces are also shownin Table 4.2 [22].

614. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

Table 4.1.: Material constants for steel and aluminum.
Steel
AluminumcL [m/s]
5800
6400

cT [m/s]
3100
3040

ρ [kg/m3 ]
7690
2700

Table 4.2.: Friction coefficients for steel-steel and steel-aluminum interfaces.
Steel-steel
Steel-aluminum

µs
0.74
0.61

µd
0.57
0.47

In some numerical simulations, we consider the static compressive stress normal to the debondingarea σ st . Adding σ st into the free-field of traction, σ st can be considered in the case that g = 0
because σ st does not cause COD at t ≤ 0 due to the stick state. In this study, σ st is given over thedebonding area uniformly.
We explain some symbols used for the frequency spectrum analysis of nonlinear ultrasonic waves
here. Aω , A2ω , and A3ω indicate the Fourier amplitudes corresponding to fundamental frequency,
2nd higher-harmonic component, and 3rd higher-harmonic component, respectively. This type ofthe symbol with superscript 0 such as A0ω , A02ω , or A03ω , indicates the quantity in the case thatug /u0 = σ st /σ0 = 0.0. The nondimensionalization parameter σ0 is the maximum stress of theincident wave and given by
σ0 = 2πu0 f in ρcφ ,(4.20)

where φ is taken to be same as the incident wave.
Debonding area on steel-steel interface
The scattering problems by a penny-shaped crack in an infinite elastic solid are numerically solvedby the proposed CQ-FMBEM. Therefore, the material constants of steel are used for DI and DII ,and we omit the superscripts I and II.
Numerical resutls using an incident plane P wave with θin = 0◦ are shown. Time histories ofvertical displacement on the top and bottom surfaces at the center point of a crack are shown in Figs.
4.7 and 4.8 when the center wavenumbers of the incident wave are given by kT a = 2πaf in /cT =2.0 and 5.2, respectively. Opening and closing of crack faces, i.e. clapping motions, are repeatedperiodically as shown in Fig. 4.7. The crack faces are rapidly closed by the diffracted wave after the
opening of crack faces. On the other hand, the waveforms of displacement in the case kT a = 5.2
are not clearly periodic as we can see in Fig. 4.8. If the frequency of incident wave is higher thanthe eigenfrequency of mode I for the linear system, it takes time that the vibration of crack faces

62

4.4. Numerical results
2

Top
Bottom

1
0
-1
-2
-3
-4
0

5

10

15

20

25

Figure 4.7.: Time variations of vertical displacement at the center point of a penny-shaped crack
subjected to an incident plane P wave when kT a = 2.0 and σ st /σ0 = 0.
2

Top
Bottom

1
0
-1
-2
-3
-4
0

2

4

6

8

10

Figure 4.8.: Time variations of vertical displacement at the center point of a penny-shaped crack
subjected to an incident plane P wave when kT a = 5.2 and σ st /σ0 = 0.

becomes periodic. This is because the arrival time of diffracted waves is large compared with theperiod of incident wave.
The frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 ) are shown in
Figs. 4.9 and 4.10 when kT a = 2.0 and 5.2, respectively. They are normalized by the maximum
value around the fundamental frequency in each spectrum, and the time history of ΩL /(au0 ) isshown in each small window in Figs. 4.9 and 4.10. From Fig. 4.9, the similar waveform to [38] is
obtained, and some rapid fluctuations can be seen. It is confirmed that this sharp waveform includes
higher-harmonic components of 2nd, 3rd, 4th, etc. orders. Similarly, the sharp waveform is obtainedin the case that kT a = 5.2 as shown in Fig. 4.10. As described in Eqs. (B.2) and (B.6), the farfield scattered wave depends on the velocity of the COD [35]. Therefore, the sharp waveforms areconstructed from the rapid fluctuation of the velocity of COD due to the closing motion of crack
faces, and higher-harmonic waves are generated strongly.
Time histories of vertical COD at the center point of a crack for various static compressive stressσ st /σ0 are shown in Figs. 4.11 and 4.12 when kT a = 2.0 and 5.2, respectively. The normalized63Normalized Fourier amplitude

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Normalized Fourier amplitude

Figure 4.9.: Normalized frequency spectrum of far-field amplitude of backscattered P wave
ΩL /(au0 ) when kT a = 2.0 and σ st /σ0 = 0.
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0.0

5.2

10 . 415 . 6

20 . 826 . 0Figure 4.10.: Normalized frequency spectrum of far-field amplitude of backscattered P wave
ΩL /(au0 ) when kT a = 5.2 and σ st /σ0 = 0.

Fourier amplitudes corresponding to the 2nd and 3rd higher-harmonic components, A2ω /A02ω andA3ω /A03ω , are plotted in Figs. 4.13 and 4.14. From Fig. 4.11, amplitude of vertical COD decreases
monotonously in the case that kT a = 2.0 as σ st /σ0 increases. Correspondingly, A2ω /A02ω is decreasing as shown in Fig. 4.13. On the other hand, the vertical COD in the case that kT a = 5.2
shows different behavior as we can see from Fig. 4.12. The waveform of the vertical COD becomes similar to the case that kT a = 2.0 as σ st /σ0 increase from σ st /σ0 = 0.0 to 0.4. Then,amplitude of the vertical COD decreases with a little phase shift. Due to this fact, there is a peakof A2ω /A02ω around σ st /σ0 = 0.2 in Fig. 4.14 although the amplitude of the vertical COD in thecase that σ st /σ0 = 0.0 is largest. From the above, the behavior of 2nd higher-harmonic generation
is significantly influenced by f in and σ st /σ0 . The behavior of A3ω /A03ω is similar in both cases asshown in Figs. 4.13 and 4.14.
Next, time histories of horizontal COD at the center point of a penny-shaped crack subjected to
an incident plane SV wave for various σ st /σ0 are shown in Fig. 4.15 when kT a = 2.0. In thissimulation, φ and v̂ are given by φ = T V and v̂ = (1, 0, 0), respectively. The normalized Fourier

64

4.4. Numerical results
5
4

0.0
0.2
0.4
0.6
0.8

3
2
1
0
0

5

10

15

20

25

Figure 4.11.: Time variations of vertical COD at the center point of a penny-shaped crack subjected
to an incident plane P wave varying σ st /σ0 when kT a = 2.0.
5
4

0.0
0.2
0.4
0.6
0.8

3
2
1
0
0

2

4

6

8

10

Figure 4.12.: Time variations of vertical COD at the center point of a penny-shaped crack subjected
to an incident plane P wave varying σ st /σ0 when kT a = 5.2.

amplitude corresponding to the 3rd higher-harmonic component A3ω /A03ω is also plotted in Fig.
4.16. The slip between crack faces does not occur in the case that about σ st /σ0 > 1.35 becauseµs = 0.74. From Fig. 4.15, the amplitude of horizontal COD decreases, and the waveform becomessimilar to the square wave as σ st /σ0 increases. A3ω /A03ω peaks at about σ st /σ0 = 0.6 and decreases
after the peak as shown in Fig. 4.16. The comparatively large 3rd higher-harmonic wave is generated
in the case that σ st /σ0 = 1.2 although σ st /σ0 is near the nonslip condition σ st /σ0 ≃ 1.35. Thisis because µs > µd , and the crack faces are hard to stop once they begin to slip. Conversely,
the 3rd higher-harmonic component decreases gently if µs = µd , and this fact has been presented
from solving the antiplane shear wave problems by Meziane et al. [59]. The normalized frequency
spectrum in the case that σ st /σ0 = 0.6, when a most large 3rd higher-harmonic wave is generated,
is shown in Fig. 4.17. From the small window of Fig. 4.17, the waveform of ΩT V /(au0 ) is similar
to the triangular wave. It is also confirmed from the frequency spectrum in Fig. 4.17 that the 3rdhigher-harmonic wave is generated.
Finally, numerical results for the oblique incidence of P wave with σ st /σ0 = 0.0 are presented.

65Normalized Fourier amplitude

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

Normalized Fourier amplitude

Figure 4.13.: Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kT a = 2.0.
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.2

0.4

0.6

0.8

1.0

Figure 4.14.: Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kT a = 5.2.

Time histories and normalized frequency spectra of ΩL /(au0 ) varying θin in the case that kT a = 2.0
are presented in Figs. 4.18 and 4.19, respectively. The interval of incident angle is 15◦ . In order
to compare them with the different frequency cases, the corresponding numerical results in the casethat kT a = 5.2 are also shown in Figs. 4.20 and 4.21. From Figs. 4.18 and 4.19, the amplitude ofΩL /(au0 ) and the higher-harmonic components decrease as the θin increases. On the other hand,from Fig. 4.20, it can also be seen that the amplitude of ΩL /(au0 ) decreases as the θin increases.
However, the amplitude is very small when θin is 45◦ or more, and the waveforms of ΩL /(au0 )become comparatively flat. Therefore, it is confirmed that there are many low frequency components
in the frequency spectra of Fig. 4.21. In addition, the locations of peak near kT a = 5.2 are shifted
because the scattered waves are distorted and have their aftereffects as shown in Fig. 4.20. From
the above, behavior of scattered waves and their frequency spectra is significantly different among
different f in . Therefore, the size estimation of debonding area is considered as possible using thisbehavior.

66

4.4. Numerical results

4

0.0
0.2
0.4
0.6
0.8
1.0
1.2

3
2
1
0
-1
-2
-3
-4
0

2

4

6

8

10

12

14

Figure 4.15.: Time variations of horizontal COD at the center of a penny-shaped crack subjected to
an incidence plane SV wave varying σ st /σ0 when kT a = 2.0.
7
6
5
4
3
2
1
0
0.0

0.2

0.4

0.6

0.8

1.0

1.2

Normalized Fourier amplitude

Figure 4.16.: Normalized Fourier amplitude corresponding to 3rd higher-harmonic component
A3ω /A03ω in far-field backscattered SV waves varying σ st /σ0 when kT a = 2.0.
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Figure 4.17.: Normalized frequency spectrum of far-field amplitude of backscattered SV wave
ΩT V /(au0 ) when kT a = 2.0 and σ st /σ0 = 0.6.

67

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
6
4
2
15°
30°
45°
60°
75°

0
-2
-4
-6
-8
0

5

10

15

20

25

Normalized Fourier amplitude

Figure 4.18.: Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying θinwhen kT a = 2.0 and σ st /σ0 = 0.0.
1.0
Incident wave

0.8

15°
30°
45°
60°
75°

0.6
0.4
0.2
0.0
0

2

4

6

8

10

Figure 4.19.: Normalized frequency spectra of far-field amplitude of backscattered P wave
ΩL /(au0 ) varying θin when kT a = 2.0 and σ st /σ0 = 0.0.

Debonding area on steel-aluminum interface
Numerical results for the scattering problems by a penny-shaped debonding area on the steelaluminum interface are presented. The material constants of steel and aluminum in Table 4.1 areused for DI and DII , respectively, and the friction coefficients in the bottom of Table 4.2 are used
for this simulation. In Fig. 4.22, the absolute values of the reflection and transmission coefficientsfor displacement are shown. Ra,b and T a,b indicate the reflection and transmission coefficients,
respectively, and the first and second superscripts denote the wave modes before and after its conversion, respectively. In Fig. 4.22, the rapid fluctuation can be seen at θin ≃ 65◦ , and this incident
angle is the critical angle for the P-P transmission. In this section, numerical results under the similar conditions to the previous section are presented in order to compare between the steel-steel andsteel-aluminum cases.
Time histories of vertical displacement on the top and bottom surfaces at the center point of aninterface crack are shown in Figs. 4.23 and 4.24 when kTI a = 2.0 and 5.2, respectively. Although the
behavior in the case that kTI a = 2.0 is similar to the one in Fig. 4.7, the amplitude of displacement is68

4.4. Numerical results
15
10
5

15°
30°
45°
60°
75°

0
-5
-10
-15
0

2

4

6

8

10

12

Normalized Fourier amplitude

Figure 4.20.: Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying θinwhen kT a = 5.2 and σ st /σ0 = 0.0.
1.4
1.2

Incident wave

1.0
15°
30°
45°
60°
75°

0.8
0.6
0.4
0.2
0.0
0.0

5.2

10 . 415 . 6

20 . 826 . 0Absolute value of coefficient

Figure 4.21.: Normalized frequency spectra of far-field amplitude of backscattered P wave
ΩL /(au0 ) varying θin when kT a = 5.2 and σ st /σ0 = 0.0.

3.0
2.5
2.0
1.5
1.0
0.5
0.0
0

10

20

30

40

50

60

70

80

90

Figure 4.22.: Absolute values of reflection and transmission coefficients for displacement when asemi-infinite and flat interface between steel and aluminum is subjected to an incident
plane P wave with incident angle θin .

69

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
3Top
Bottom

2
1
0
-1
-2
-3
0

5

10

15

20

25

Figure 4.23.: Time variations of vertical displacement on the center point of a penny-shaped interface crack subjected to an incident plane P wave when kTI a = 2.0 and σ st /σ0 = 0.
3

Top
Bottom

2
1
0
-1
-2
-3
0

2

4

6

8

10

Figure 4.24.: Time variations of vertical displacement on the center point of a penny-shaped interface crack subjected to an incident plane P wave when kTI a = 5.2 and σ st /σ0 = 0.

different due to reflection and transmission coefficients as shown in Fig. 4.23. For normal incidenceof P wave, the reflection and transmission coefficients are |RL,L | ≃ 0.44 and |T L,L | ≃ 1.44,
respectively. The amplitude of displacement in the case that kTI a = 5.2 is also influenced dueto this fact as shown in Fig. 4.24. However, the difference between Figs. 4.8 and 4.24 is large
compared with the case that kTI a = 2.0. This difference in behavior of waveform is caused by thedifference of wave velocities between steel and aluminum. It seems that this effect is significant in
the high frequency case.
The normalized frequency spectra of far-field amplitude of backscattered P wave ΩL /(au0 ) are
shown in Figs. 4.25 and 4.26 when kTI a = 2.0 and 5.2, respectively. They show almost same
behavior as the ones in Figs. 4.9 and 4.10.
Time histories of vertical COD at the center point of an interface crack for various static compressive stress σ st /σ0 are shown in Figs. 4.27 and 4.28 when kTI a = 2.0 and 5.2, respectively.
The normalized Fourier amplitudes corresponding to the 2nd and 3rd higher-harmonic components,
A2ω /A02ω and A3ω /A03ω , are also plotted in Figs. 4.13 and 4.14. Note that the static compressive70Normalized Fourier amplitude

4.4. Numerical results
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Normalized Fourier amplitude

Figure 4.25.: Normalized frequency spectrum of far-field amplitude of backscattered P wave
ΩL /(au0 ) when kTI a = 2.0 and σ st /σ0 = 0.
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0.0

5.2

10 . 415 . 6

20 . 826 . 0Figure 4.26.: Normalized frequency spectrum of far-field amplitude of backscattered P wave
ΩL /(au0 ) when kTI a = 5.2 and σ st /σ0 = 0.

stress σ st /σ0 given here is different from the ones in the case of steel-steel interface because thetransmission coefficient for stress is |TσL,L | ≃ 0.56 in this case. The opening of crack faces does
not occur if σ st /σ0 ≥ |TσL,L |. However, the behavior in the cases that kTI a = 2.0 and 5.2 are very
similar to the cases of steel-steel interface.
Next, time histories of horizontal COD at the center point of a penny-shaped interface crack
subjected to an incident plane SV wave varying σ st /σ0 are shown in Fig. 4.31 when kTI a = 2.0.
The normalized Fourier amplitude corresponding to 3rd higher-harmonic components in the farfield backscattered SV wave ΩT V /(au0 ) is also plotted in Fig. 4.32. In this case, the transmission
coefficient for stress corresponding to SV-SV transmission is significant, and |TσT V,T V | ≃ 0.51.
Therefore, the slip between crack faces does not occur if σ st /σ0 > |TσT V,T V |/µs ≃ 0.84. Similarly
to the case of steel-steel interface, the amplitude of horizontal COD decreases, and the waveform
becomes similar to the square wave as σ st /σ0 increases in Fig. 4.31. As shown in Fig. 4.32, thepeak of A3ω /A03ω is located about at σ st /σ0 = 0.4. Thus, the normalized frequency spectrum inthe case that σ st /σ0 = 0.4 is shown in Fig. 4.33. From Fig. 4.33, the waveform of ΩT V /(au0 ) is71

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material
5
4

0.0
0.1
0.2
0.3
0.4
0.5

3
2
1
0
0

5

10

15

20

25

Figure 4.27.: Time variations of vertical COD at the center point of a penny-shaped interface crack
subjected to an incident plane P wave varying σ st /σ0 when kTI a = 2.0.
5

0.0
0.1
0.2
0.3
0.4
0.5

4
3
2
1
0
0

2

4

6

8

10

Figure 4.28.: Time variations of vertical COD at the center point of a penny-shaped interface crack
subjected to an incident plane P wave varying σ st /σ0 when kTI a = 5.2.

similar to the triangular wave, and it is confirmed that the 3rd higher-harmonic wave is generated.
Finally, numerical results for the oblique incidence of P wave with σ st /σ0 = 0.0 are presented.
Time histories and normalized frequency spectra of far-field amplitude of backscattered P wave
ΩL /(au0 ) varying θin in the case that kTI a = 2.0 and 5.2 are presented in Figs. 4.34–4.21. Weshould pay attention to the behavior near the critical angle which is about 65◦ here. Hence, theresults with θin = 65◦ are added into Figs. 4.34–4.37. From Figs. 4.34 and 4.36, the amplitudesof ΩL /(au0 ) are smaller than those in Figs. 4.18 and 4.20 due to the reflection and transmission
coefficients. It is remarkable that the amplitude of ΩL /(au0 ) in the case that θin = 65◦ is muchlarger than those with θin = 60◦ and 70◦ . This is because the reflection and transmission coefficientsare much larger than those with other θin , and the large COD is then generated. However, thefeatures of frequency components of ΩL /(au0 ) are similar to the ones with θin = 60◦ and 70◦ as
shown in Figs. 4.35 and 4.37.

72

Normalized Fourier amplitude

4.5. Concluding remarks
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.1

0.2

0.3

0.4

0.5

Normalized Fourier amplitude

Figure 4.29.: Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kTI a = 2.0.
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0

0.1

0.2

0.3

0.4

0.5

Figure 4.30.: Normalized Fourier amplitude of ΩL /(au0 ) corresponding to 2nd and 3rd higherharmonic components, A2ω /A02ω and A3ω /A03ω , in backscattered P waves varying
σ st /σ0 when kTI a = 5.2.

4.5. Concluding remarks• In Chapter 4, we have applied the CQ-FMBEM to the nonlinear ultrasonic simulation for thedebonding area on the interface of two semi-infinite elastic solids. We have conducted somenumerical simulations for the steel-steel and steel-aluminum interfaces and investigated thecharacteristic of higher-harmonic generation due to the static compressive stress. Moreover,the numerical results have been compared between the cases of steel-steel and steel-aluminum
interfaces.
• We have successfully reproduced the clapping motion and friction on the crack faces andobserved the higher-harmonic generation using the presented contact model of crack faces.
The clapping motion and dynamic friction of crack faces have generated the every order ofhigher-harmonic waves and odd order of ones, respectively. Therefore, it is considered thatthe presented model is valid.

734. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

4
3

0.0
0.2
0.4
0.6
0.8

2
1
0
-1
-2
-3
-4
0

2

4

6

8

10

12

14

Figure 4.31.: Time variations of horizontal COD at the center of a penny-shaped interface crack
subjected to an incidence plane SV wave varying σ st /σ0 when kTI a = 2.0.
7
6
5
4
3
2
1
0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized Fourier amplitude

Figure 4.32.: Normalized Fourier amplitude corresponding to 3rd higher-harmonic component
A3ω /A03ω in far-field backscattered SV waves varying σ st /σ0 when kTI a = 2.0.
1.0

Incident wave
Backscattered wave

0.8
0.6
0.4
0.2
0.0
0

2

4

6

8

10

Figure 4.33.: Normalized frequency spectrum of far-field amplitude of backscattered SV wave
ΩT V /(au0 ) when kTI a = 2.0 and σ st /σ0 = 0.4.

74

4.5. Concluding remarks
3
2
15°
30°
45°
60°
65°
75°

1
0
-1
-2
-3
-4
0

5

10

15

20

25

Normalized Fourier amplitude

Figure 4.34.: Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying θinwhen kTI a = 2.0 and σ st /σ0 = 0.0.
1.0

Incident wave

0.8

15°
30°
45°
60°
65°
75°

0.6
0.4
0.2
0.0
0

2

4

6

8

10

Figure 4.35.: Normalized frequency spectra of far-field amplitude of backscattered P wave
ΩL /(au0 ) varying θin when kTI a = 2.0 and σ st /σ0 = 0.0.

• In the high frequency cases, i.e. kT a = 5.2, the scattered waves sometimes have been distorted strongly. Then, many low frequency components are contained in their frequency spectra. However, we could not observe the sub-harmonic generation.
• In the case of the steel-aluminum interface, the reflection and transmission coefficients significantly effect in the results. For the normal incidence of P and SV waves, the amplitudeof COD greatly depends on these coefficients. Moreover, scattered waves obtained by using
the incident angle near the critical one have much larger amplitude than the others. However,the behavior of frequency spectra is not very different from others in the presented numericalresults.

75

4. 3-D Simulation of Higher-harmonic Waves Due to an Interface Crack of Bi-material

8
6
15°
30°
45°
60°
65°
75°

4
2
0
-2
-4
-6
-8
0

2

4

6

8

10

12

Normalized Fourier amplitude

Figure 4.36.: Time variations of far-field amplitude of backscattered P wave ΩL /(au0 ) varying θinwhen kTI a = 5.2 and σ st /σ0 = 0.0.
1.4

Incident wave

1.2

15°
30°
45°
60°
65°
75°

1.0
0.8
0.6
0.4
0.2
0.0
0.0

5.2

10 . 415 . 6

20 . 826 . 0Figure 4.37.: Normalized frequency spectra of far-field amplitude of backscattered P wave
ΩL /(au0 ) varying θin when kTI a = 5.2 and σ st /σ0 = 0.0.

76Chapter 5

2-D Simulation of Nonlinear UltrasonicWaves Due to Interior and Surface
Breaking Cracks
In this chapter, the boundary element formulations and numerical results for 2-D simulation ofhigher- and sub-harmonic generation are presented. We consider the scattering problems by cracks
in an infinite elastic solid and surface breaking cracks in an elastic half-space. The frequency response characteristic of linear systems is important for the sub-harmonic simulation, and we consider complex shape and arrangement of cracks. Therefore, the Galerkin method is used for spatial
discretization in this simulation. In general, the Galerkin BEM requires expensive computational
cost and produces accurate solutions compared with the collocation BEM. Moreover, although thecollocation hypersingular BIE is known to require a C 1,α continuity for displacement on boundaries,
usual C 0 basis functions are allowed to use in the variational one [12, 40].
In this chapter, the hypersingular integral (p.f. integral) in the BIE for crack problems is regularized only to strong singular one (p.v. integral), where p.f. denotes the finite part of a divergent
integral. This is because it is not very difficult to analytically calculate the integration of the staticpart of fundamental solutions over a straight-line element for 2-D elastodynamics.

5.1. Problem statement and BIE
In this section, we present the problem statements and formulation of BIE for the scattering problems by cracks in an infinite domain and surface breaking cracks in a half-space.

5.1.1. Scattering problem by cracks in infinite domain
Scalar wave scattering problems by a crack have been come down to the hypersingular BIE by
Martin and Rizzo [50]. Then, the counterpart for elastodynamic problems has been numericallysolved by BEMs. The hypersingular BIE is effective in dealing with crack problems for infinite andsemi-infinite domains and also used in this study.

77

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Figure 5.1.: Analysis model for elastic wave scattering by a crack in a 2-D infinite domain.

Let S be a smooth curved crack surface in R2 as depicted in Fig. 5.1. D is a homogeneous,isotropic, and linearly elastic solid, and S consists of S + and S − called positive and negative sidesof crack face, respectively (S = S + ∪ S − and S + ∩ S − = ∅). n+ and n− are the unit normal
vectors to S + and S − , respectively. s+ is the unit tangential vector to S + . Disregarding the bodyforce, we consider the problem to find the solution u which satisfies the following equations:()
c2T ∇2 u(x, t) + c2L + c2T ∇∇ · u(x, t) = ü(x, t)
t+ (x, t) + t− (x, t) = 0 x ∈ S,()[u](x, t) ≡ u+ (x, t) − u− (x, t) = 0
scscu (x, 0) = u̇ (x, 0) = 0()
x ∈ D ≡ R2 \S ,

(5.1)
(5.2)

x ∈ ∂S,(5.3)

x ∈ D,(5.4)

where the superscript ± indicates the each side of crack face. Eq. (5.2) indicates traction continuity
over crack faces. Eq. (5.3) denotes that the COD at the crack tip ∂S is equal to zero, and thiscondition ensures the uniqueness of solution in crack problems. Eq. (5.4) is the initial condition
and indicates that there are no scattered waves at t ≤ 0.
Considering the Sommerfeld radiation condition for scattered waves and initial and boundary
conditions (5.4) and (5.2), the scattered displacement fields can be expressed in the following integral representation:u (x, t) = −

∫ t∫

sc

0

S+

T + (x, y, t − τ ) · [usc ](y, τ )dSy dτx ∈ D.

(5.5)

Then, substituting usc = u − uin into Eq. (5.5) and considering the continuity of uin over crack
faces yieldu(x, t) = uin (x, t) −

∫ t∫
0

S+

T + (x, y, t − τ ) · [u](y, τ )dSy dτx ∈ D.

(5.6)

The traction operator with respect to x is multiplied from the left in both hand sides of Eq. (5.6).
Then, taking the limit that x ∈ D → x ∈ S + , the collocation hypersingular BIE can be obtained asfollows:
∫ t
∫
+
in;+
t (x, t) = t(x, t) −p.f.
W + (x, y, t − τ ) · [u](y, τ )dSy dτ x ∈ S + ,
(5.7)
0

78

S+

5.1. Problem statement and BIE

Figure 5.2.: Analysis model for wave scattering by a surface breaking crack in a 2-D elastic halfspace.

where W + is the hypersingular kernel corresponding to n+ and given as follows:++
Wbj(x, y, t) = n+a (x)nk (y)Cbaic Cjkpq∂ ∂
Uip (x, y, t).
∂xc ∂yq(5.8)

Note that the hypersingular kernel W for 2-D problems has singularity of order 1/r2 as r ↓ 0. Multiplying the weight function ψ and integrating over the surface S + with respect to x, the variational
hypersingular BIE can be derived as follows:∫∫
ψ(x)t+ (x, t)dSx =ψ(x)tin;+ (x, t)dSx
+
+
S
S
∫
∫ t∫
ψ(x)p.f.
W + (x, y, t − τ ) · [u](y, τ )dSy dSx dτ.
−
0

S+

(5.9)

S+

Both collocation and variational BIEs contain p.f. integrals as the result of taking of the limit to theboundary. Eq. (5.9) is the BIE which is used for the boundary element analysis in this study.

5.1.2. Scattering problem by surface breaking cracks in half-space
Next, we show the variational formulation of BIE for the scattering problem by a surface breaking
crack in an elastic half-space. In this study, the full-space fundamental solutions are used for solving
the initial and boundary value problem. Then, the scattered far-fields are evaluated by means ofapproximated Green’s functions for an elastic half-space. The similar boundary element analyses
for the traction-free crack have been implemented in frequency-domain by Zhang and Achenbach[102, 103].
Let Ss and Sc be flat free surface and smooth curved crack face, respectively in R2 as depicted
in Fig. 5.2. We assume that the governing equation and boundary conditions on crack face Sc aresame as Eqs. (5.1)–(5.4). In addition, the boundary condition on the free-surface Ss is given byt(x, t) = 0

x ∈ Ss .

(5.10)

Therefore, we can obtain the following integral representation of scattered waves in D by means of79

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
a similar manner to the crack problems in an infinite domain.
u (x, t) = −∫ t∫

T (x, y, t − τ ) · usc (y, τ )dSy dτ

sc

−

0

Ss

0

Sc+

∫ t∫

T + (x, y, t − τ ) · [u](y, τ )dSy dτx ∈ D,(5.11)

where the scattered wave is defined as the disturbance of free-field ufree by a crack. Multiplying thetraction operator corresponding to x from the left in both sides of Eq. (5.11) and taking the limitthat x ∈ D → x ∈ Ss , Sc+ , the hypersingular BIE for the surface breaking crack is obtained asfollows:
∫ t
∫
sc;+
t
(x, t) = −p.f.
W + (x, y, t − τ ) · usc (y, τ )dSy dτ
0

−

Ss

∫ t

∫
p.f.

Sc+

0

W + (x, y, t − τ ) · [u](y, τ )dSy dτ x ∈ Ss , Sc+ .

(5.12)Taking the free-field of traction tfree (≡ t−tsc ) into account, we can obtain the following expression:+

free;+t (x, t) = t−(x, t) −

∫ t

p.f.
Sc+

0

W + (x, y, t − τ ) · usc (y, τ )dSy dτ

p.f.
0

∫

∫ t

∫
Ss

W + (x, y, t − τ ) · [u](y, τ )dSy dτ x ∈ Ss , Sc+ ,(5.13)

where tfree consists of incident and reflected waves and satisfies governing equation (5.1) andboundary condition (5.10). The variational formulation corresponding to Eq. (5.13) can be derived
using the weight function ψ as follows:∫Sc+∫ψ(x)t+ (x, t)dSx =
−
−

∫ t∫

Sc+

ψ(x)tfree;+ (x, t)dSx∫
W + (x, y, t − τ ) · usc (y, τ )dSy dSx dτψ(x)p.f.
0

Ss ∪Sc+0

Ss ∪Sc+

Ss

∫ t∫

∫
ψ(x)p.f.

Sc+

W + (x, y, t − τ ) · [u](y, τ )dSy dSx dτ.

(5.14)The scattered wave usc is used for the variable on free-surface, while the COD [u] is the one on
crack face as shown in Eq. (5.14). Considering appropriate boundary conditions on crack face, Eqs.
(5.9) and (5.14) can be solved numerically. In order to accurately evaluate the p.f. integral whichcontains the hypersingular kernel, the regularization technique is used for relaxing the singularity.

80

5.2. Regularization of BIE

5.2. Regularization of BIE
In this section, the regularization of hypersingular BIE is presented. The superscript + is hereinafter
omitted in this chapter.
According to [93], the second term in the right-hand side of Eq. (5.9) can be regularized fromp.f. integral to p.v. one as follows:∫∫ t∫

Wbj (x, y, t − τ )[uj ](y, τ )dSy dSx dτ∫ t∫∫∂[uj ](y, τ )= ede eck Cbaic
ψ(x)na (x)p.v. Σikj (x, y, t − τ )ne (y)dSy dSx dτ
∂yd
0
S
S
∫ t∫
∫
− ρCbaic
ψ(x)na (x) Üij (x, y, t − τ )nc (y)[uj ](y, τ )dSy dSx dτ
0
S
S
∫ t∫
∫
]
∂ [
Σikj (x, y, t − τ )[uj ](y, τ ) dSy dSx dτ, (5.15)− eck Cbaic
ψ(x)na (x) sd (y)
∂yd
0
S
S
ψ(x)p.f.

0

S

S

where eij is a component of the 2-D permutation tensor. Σ is the fundamental solution with respect
to stress and has an singularity of order 1/r as r ↓ 0 (so-called strong singularity). Note that thissingularity is not related with stress singularity at crack tips and sometimes called non-physicalsingularity. The third term in the right-hand side of Eq. (5.15) contains the tangential derivative
sd (y)∂/∂yd and vanishes when [u] is equal to zero at the edges of boundary S as described in Eq.
(5.3). It therefore vanishes in the case that crack is located on the inside of material.
Now, we show the regularized BIEs corresponding to Eqs. (5.9) and (5.14). The regularized BIE
for the scattering problem by cracks in an infinite domain can be obtained as follows:∫S∫ψ(x)tb (x, t)dSx =ψ(x)tinb (x, t)dSx
S∫∫ t∫∂[uj ](y, τ )ψ(x)na (x)p.v. Σikj (x, y, t − τ )ne (y)dSy dSx dτ+ ede eck Cbaic
∂yd
0
S
S
∫ t∫
∫
− ρCbaic
ψ(x)na (x) Üij (x, y, t − τ )nc (y)[uj ](y, τ )dSy dSx dτ,(5.16)
0S

S

where boundary condition (5.3) is used. On the other hand, the regularized BIE for the scattering
problem by surface breaking cracks in an elastic half-space can be written as follows:815. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
Center of element
Nodal point

Tip element

Tip element

Crack tip

Crack tip

Figure 5.3.: Spatial shape functions with respect to displacement and traction for a crack in an infinite domain.

∫∫
ψ(x)tfreeb (x, t)dSx

ψ(x)tb (x, t)dSx =Sc+ ede eck Cbaic− ρCbaic∫ t∫− eck Cbaic

0

0

Ss

∫

Ss ∪Scsd (y)Ss

Ss ∪Sc∫]∂ [Σikj (x, y, t − τ )usc(y,τ)
dSy dSx dτj∂yd
Σikj (x, y, t − τ )ne (y)ψ(x)na (x)p.v.
Sc∫∂[uj ](y, τ )dSy dSx dτ∂yd

Üij (x, y, t − τ )nc (y)[uj ](y, τ )dSy dSx dτψ(x)na (x)∫ t∫0

ψ(x)na (x)Ss ∪Sc∂usc
j (y, τ )dSy dSx dτ∂yd

Üij (x, y, t − τ )nc (y)usc
j (y, τ )dSy dSx dτψ(x)na (x)Ss ∪Sc∫ t∫

Σikj (x, y, t − τ )ne (y)
Ss

∫

∫ t∫

+ ede eck Cbaic0∫ψ(x)na (x)p.v.

∫ t∫0− eck Cbaic

Ss ∪ScSs ∪Sc0− ρCbaic

Sc

∫ t∫

Sc

∫

ψ(x)na (x)sd (y)Sc]∂ [Σikj (x, y, t − τ )[uj ](y, τ ) dSy dSx dτ. (5.17)∂yd

5.3. Discretization of BIE
Next, the time and spatial discretization using the IRK based-CQM [48] and Galerkin method,respectively, is presented. In this study, we use straight-line elements for division of surfaces, S,
Sc , and Ss . For interpolation of quantities, different shape functions between displacement andtraction are taken because we consider the continuity and discontinuity of displacement and traction
between two adjacent elements, respectively. Specifically, the piecewise linear interpolation and tip
element are used for [u] and usc , and t is interpolated by the piecewise constant shape function. The
shape functions with respect to these variables for a crack in an infinite domain are presented in Fig.
5.3. Using the Galerkin method, the weight functions ψ are taken to be same as the corresponding
shape functions of variable considered in each BIE. Therefore, ψ is different between BIEs fordisplacement and traction.

82

5.3. Discretization of BIE

5.3.1. Scattering problem by cracks in infinite domainFirst, we discretize the BIE for scattering problems by cracks in an infinite domain. If the boundary S in Eq. (5.16) is divided into Me straight-line elements consisting of Md nodal points, thediscretized BIEs for [u] and t at n-th and i-th sub-step in time can be written as follows:
Md
m ∑
∑

ij;0
WIJ
· [u]j;n
J =−

Me
∑

∫
t
ti;n
α ϕα

α=1

j=1 J=1

supp(ϕu
I )∩SαϕuI (x)dSx∫+
supp(ϕuI)

ϕuI (x)tin (x, (n + ci )∆t)dSx −∫t (x, (n + ci )∆t)dSx −dSx = ϕtαSαij;n−k
WIJ· [u]j;kJ ,(5.18)

ij;n−kW̃αJ· [u]j;kJ ,(5.19)

k=0 j=1 J=1

∫

t t
ti;n
α ϕα ϕα

Md
n−1
m ∑
∑∑

in

Sα

Md
n ∑
m ∑
∑
k=0 j=1 J=1

where subscripts I and J indicate the indices of nodal point. In addition, supp(ϕ) denotes the∫
support of ϕ, and Sα dSx in the left hand side of Eq. (5.19) is equal to the length of α-th element.
i;nThe specific expressions of [u]i;n
I and tα are[u]i;nI = [u](xI , (n + ci )∆t),ti;nα = t(xα , (n + ci )∆t),

ij;κ
ij;κ
where xI is the position vector of I-th nodal point. The influence functions WIJ
and W̃αJ
in
Eqs. (5.18) and (5.19) are given by



m (
) ∫
∑
ij;κ
−1 
WIJ = F̃lκEβ (ζl )



ij

β=1


supp(ϕu
I)

∫
m (
)
∑
ij;κ
−1 
t
W̃αJ = F̃lκEβ (ζl ) ϕα
ij

β=1

Sα

ϕuI (x)WJβ;l (x)dSx  ,(5.20)



WJβ;l (x)dSx  ,(5.21)

where()
WJβ;l (x)bj∫
= ede eck Cbaic na (x)p.v.

supp(ϕu
J)

∫
( )2
l
−ρ λβ Cbaic na (x)∫−eck na (x)Cbaic

supp(ϕuJ)Σ̂ikj (x, y, λlβ )ne (y)supp(ϕuJ)sd (y)∂ϕuJ (y)dSy∂ydÛij (x, y, λlβ )nc (y)ϕuJ (y)dSy]∂ [Σ̂ikj (x, y, λlβ )ϕuJ (y) dSy .
∂yd(5.22)In Eq. (5.22), Û and Σ̂ are the 2-D fundamental solutions corresponding to displacement and stressin Laplace-domain. Note that the underlined term in the right-hand side of Eq. (5.22) is canceledby boundary condition (5.3) for a crack on the inside of materials.
In this study, we use two types of special shape functions for displacement ϕuT1 and ϕuT2 on the tip
elements (see Fig. 5.3) in order to describe the proper behavior of COD at the crack tip. If y1 and83

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
y2 are the left and right edges of element, respectively, the specific expressions of ϕuT1 and ϕuT2 areshown as follows:
√
√
|y
−
y
|
|y − y2 |1
ϕuT1 (y) =, ϕuT2 (y) =.
|y2 − y1 ||y1 − y2 |Note that the derivatives of shape functions on tip elements have additional singularity of order√
1/ r corresponding to the stress singularity as y → ∂S. Therefore, the coordinate transformation
is used for the first term in the right-hand side of Eq. (5.22) before the numerical integration using
the standard Gaussian quadrature.
We summarize the procedure of transformation as follows:
1. The interval of integration over the line element is transformed from [y1 , y2 ] to [0, 1]. If thekernel function is A(y), the integral is transformed as follows:
∫ y2

∫ 1
A(y)dSy = ley1

A(l)dl,(5.23)

0where le = |y2 − y1 |, and y = y1 + (y2 − y1 )l.
2. The integral variable l is changed to appropriate one l∗ considering the location of crack tip.
When the crack tip is located at l = 0, the relation between l and l∗ is taken asl = (l∗ )2 ,J ∗ (l∗ ) = 2l∗ ,where J ∗ is the Jacobian corresponding to coordinate transformation as l to l∗ . On the otherhand,
l = 1 − (1 − l∗ )2 ,J ∗ (l∗ ) = 2 (1 − l∗ )for the case that the crack tip is located at l = 1. Then, the Jacobian corresponding to each l∗√can cancel the singularity of order 1/ r, and Eq. (5.23) is rewritten as follows:
∫ 1

∫ 1
le

A(l)dl = le0A(l∗ )J ∗ (l∗ )dl∗ .

(5.24)

0

3. In order to use the standard Gaussian quadrature, l∗ is transformed to f with interval [−1, 1]
as follows:
∫ 1
∫ 1
1
∗ ∗ ∗
∗
le
A(l )J (l )dl = leA(f )J ∗ (f )df,
(5.25)
20−1
where f is given by l∗ = (f + 1)/2. Using Eq. (5.25), the integrals of Σ̂ for tip elements canbe evaluated accurately.
In this study, all integrals of static parts of Û and Σ̂ are analytically calculated.

84

5.3. Discretization of BIE

(a)

(b)

Figure 5.4.: (a) Double node model and (b) triple node model for connection point between surface
breaking crack and free-surface.

5.3.2. Scattering problem by surface breaking cracks in half-space
Second, we discretize the BIE for scattering problems by a surface breaking crack in an elastic halfspace. For the connection point between the surface breaking crack and free-surface, we need toconsider discontinuity of displacement. The double node model as depicted in Fig. 5.4 (a) is usually
utilized for this type of problem, where square nodal points are in the same position. If we use thedouble node model, boundaries Ss and Sc are considered as continuous and closed. Therefore,
the fourth and seventh terms in the right-hand side of Eq. (5.17) can be canceled. However, thetriple node model as depicted in Fig. 5.4 (b) is utilized in this study because the manageability of
nonlinear iterative calculation becomes much easier than the case of double node model.
If the boundaries Sc and Ss in Eq. (5.16) is divided into Mc and Ms straight-line elements
consisting of Md nodal points, the discretized BIEs for [u] and usc , and t at n-th and i-th sub-stepin time are written as follows:
Md
m ∑
∑

∫

Mc
∑

ij;0
WIJ
· vJj;n = −

t
ti;n
α ϕα

α=1

j=1 J=1

supp(ϕu
I )∩SαϕuI (x)dSx∫+
supp(ϕuI)

ϕuI (x)tfree (x, (n + ci )∆t)dSx −∫dSx = ϕtαSαij;n−k
WIJ· vJj;k ,(5.26)

ij;n−kW̃αJ· vJj;k .

(5.27)

k=0 j=1 J=1

∫
t t
ti;n
α ϕα ϕα

Md
m ∑
n−1
∑∑

free

t
Sα

(x, (n + ci )∆t)dSx −

Md
n ∑
m ∑
∑
k=0 j=1 J=1

In Eqs. (5.26) and (5.27),{
vIi;n =[u]i;nI
usc;i;nI}{
when I-th nodal point belongs to

Sc
Ss

}
.

For the calculation of influence functions in Eqs. (5.26) and (5.27), the underlined term in theright-hand side of Eq. (5.22) has to be evaluated. This term can be modified into the following85

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
expression and easily evaluated for 2-D problems.
∫]∂ [Σ̂ikj (x, y, λlβ )ϕuJ (y) dSy∂ydsupp(ϕu
J)
∫
]
∑
∂ [
= eck na (x)Cbaicsd (y)
Σ̂ikj (x, y, λlβ )ϕuJ (y) dSy∂ydSe
Se ∈supp(ϕu
J)
[
]y=y2
∑
= eck na (x)Cbaic
Σ̂ikj (x, y, λlβ )ϕuJ (y),(5.28)

eck na (x)Cbaicsd (y)

y=y1

Se ∈supp(ϕuJ)

where Se is an element area and has the interval [y1 , y2 ].

5.4. Nonlinear boundary conditions and numerical procedure
The nonlinear boundary conditions on crack faces are different from the ones for an interface crack
in Section 4.2.1 because this problem is 2-D, and the formulations and variables are different. Therefore, the nonlinear boundary conditions and numerical procedure are briefly presented in this section.
In order to simultaneously solve BIEs (5.18) and (5.19), the appropriate boundary conditions
should be selected at each time step. We also consider three types of state in a similar way to
Section 4.2.1. The boundary conditions for separation, stick, and slip states are summarized asfollows:
t=0: separation,(5.29a)
˙ ·s=0[u] · n = ug , [u]: stick,(5.29b)()

˙ · s µd t · n [u] · n = ug , t · s = sgn [u]: slip.
(5.29c)As shown in Eq. (5.29c), the slip direction in the 2-D problem is positive or negative, and its
evaluation is easier than the 3-D case.
The transition conditions for the iterative calculation are summarized as follows:[u] · n ≥ ug
: separation → stick,





t·n>0: stick or slip → separation,|t · s| ≥ µs |t · n|(
) ( prev )
˙ · s / [u]
˙
 [u]
·s ≤0

(5.30a)(5.30b): stick → slip,(5.30c): slip → stick.

(5.30d)Eq. (5.30a) indicates that the phase is changed from separation to stick when the crack faces overlap.
Eq. (5.30b) means that the normal tensile force to the crack face opens it. The right-hand side of Eq.
(5.30c) corresponds to the maximum static friction force. The crack faces begin to slide relatively in
the Coulomb friction law when the absolute value of tangential force exceeds the maximum static
˙ · s = 0 in practice.
friction one. The transition condition from slip to stick is considered as [u]However, it is difficult to satisfy this condition in numerical simulation. Therefore, we assume that86

5.5. Numerical results
Straight-line
crack

Figure 5.5.: A straight-line crack subjected to an incident plane wave.

the direction change of relatively tangential movement is ”stop condition” as shown in Eq. (5.30d).
If the left inequalities in Eq. (5.30) are satisfied, the corresponding right transition occurs.
Using boundary conditions (5.29), discretized BIEs (5.18) and (5.19) can be simultaneously
solved. The iterative algorithm is similar to the case of interface crack in Section 4.3. In addition, discretized BIEs (5.18) and (5.19) are modified for the velocity of displacement tangential tothe crack face as well as Section 4.2.2. The scattering problems by surface breaking cracks are also
solved in a similar way to the one by cracks in an infinite elastic solid.

5.5. Numerical results
In this section, numerical results of the proposed IRK-based CQ-BEM are presented. Before thesub-harmonic generation, the linear problem is first solved in order to verify the proposed method
for the crack problem. Second, the higher-harmonic simulation is conducted to validate the contact
model of crack faces. Finally, numerical results of sub-harmonic generation due to cracks in an
infinite domain and a surface breaking crack in a half-space are presented. The non-accelerated
CQ-BEM with RK2 is used for all numerical simulations in this section.

5.5.1. Linear problem
In advance of the nonlinear ultrasonic simulation, the numerical results for the traction-free crack
are shown in order to verify the proposed IRK-based CQ-BEM. We compute the stress intensity
factor (SIF) of mode I for straight-line crack subjected to an incident plane P wave as depicted
in Fig. 5.5, where a is the half length of a crack. The incident plane P wave is defined by themonotonically increasing function as follows:cL
u (x, t) = −u0 dL
a
in

(
) (
)
p · (x − χ)
p · (x − χ)
t−
H t−
.
cL
cL

(5.31)

p, dL , and χ are given by p = dL = (sin θin , cos θin ) and χ = (−a, 0) in this simulation, respectively. The crack surface S is divided into 40 boundary elements, time increment ∆t is taken87

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
1.4
1.2
1.0
0.8
0.6

This work
This work
Thau and Lu (1971)

0.4
0.2
0.0
0

2

4

6

8

10

12

Figure 5.6.: Time variations of dynamic SIF of mode I when θin = 30◦ .

as cT ∆/a = 0.05, and Poisson’s ratio ν is given by ν = 0.25. The mode I dynamic SIF can becalculated from COD near the crack tip as follows:√1µ 2π
lim √ [u](ε, t) · n(ε),
KI (t) = −4(1 − ν) ε→0 ε(5.32)

where ε is the distance from the crack tip.
Time histories of dynamic SIF of mode I corresponding to the crack tips at x1 /a = −1.0 and
1.0 are shown in Fig. 5.6 when a straight-line crack is subjected to an incident plane P wave with
θin = 30◦ . The dynamic SIFs shown here are normalized by the exact solution of static problems√
KIst = σ22 πa. The short time exact solutions for this problem have been obtained by Thau andLu [94], and they are plotted in the same figure for comparison. Both numerical solutions are ingood agreement with the exact ones and converge to the static exact ones over time. Therefore, thescattering problem by a crack can be accurately solved using the proposed IRK-based CQ-BEM.

5.5.2. Higher-harmonic simulation
In order to validate the contact model of crack face, the numerical results of higher-harmonic simulation are presented in this section. The analysis model in Fig. 5.5 is also used for this simulation,
and we confirm that our model can reproduce the clapping motion and friction on crack faces. θin isgiven by θin = 0◦ here because fundamental motions on crack faces are investigated. The following
sinusoidal burst wave with nin cycles is used as the incident wave for this simulation.

( in
)
in

 u (x, t) = u0 dφ sin (2πΛφ ) H (Λφ ) H n − Λφ ,()
p · (x − χ)
in

t−
,
 Λφ = fcφ88(5.33a)(5.33b)

5.5. Numerical results
where φ = L or T because of the 2-D problem. The far-field approximated scattered wave isused for the spectrum analysis of scattered wave (see Appendix B). The time increment is taken as
f in ∆t = 0.05, and Poisson’s ratio of base material is given by ν = 0.3 supposing steel. The friction
coefficients are thus given by the top row of Table 4.2. nin is taken as nin = 30 because we conducttransient analysis for a sufficiently long time and investigate nearly stationary vibration.
First, some numerical results of P wave incidence are presented when θin = 0◦ , kT a = 3.0, andug /u0 = 0.0. Time histories of vertical COD at the center point of a crack is shown in Fig. 5.7(a). On the other hand, time histories and normalized frequency spectra of far-field amplitude of√backscattered P wave ΩL /(u0 a) in Fig. 5.7 (b), and (c), respectively. In thses figures, ”linear”
and ”nonlinear” indicate the results obtained for the traction-free crack and the crack with contact
boundary conditions, respectively. The small windows in the figures show the enlarged view of
early parts. As shown in Fig. 5.7 (a), the vertical COD obtained by the presented model cannot take√the negative value due to contact. From Fig. 5.7 (b), the waveform of ΩL /(u0 a) is distorted by
the clapping motion. Therefore, its frequency spectrum contains the higher harmonic components
as shown in Fig. 5.7 (c). It is known that all (2nd, 3rd, 4th, etc.) orders of higher-harmonic waves
are generated for the normal incidence of P wave [89], and that is reproduced by the presented
numerical simulation.
Second, numerical results of S wave incidence are presented when θin = 0◦ , kT a = 3.0, andug /u0 = 0.0. In this case, the static compressive stress σ st is given over a crack uniformly in orderto generate the dynamic friction force. σ st is given by σ st /σ0 = 1.0 here. If σ st does not cause
COD due to contact of crack faces, σ st is only added into tin or tfree in BIEs similarly to Chapter 4.
Therefore, if we give ug /u0 as a parameter, σ st /σ0 is equal to zero in this study. The reverse is also
true.
Time histories of tangential traction at the center point of a crack is shown in Fig. 5.8 (a). On theother hand, time histories and normalized frequency spectra of far-field amplitude of backscattered√
S wave ΩT /(u0 a) in Fig. 5.8 (b), and (c), respectively. As shown in Fig. 5.8 (a), the tangential
traction becomes similar to square wave due to dynamic friction force. Therefore, the waveform√of ΩT /(u0 a) is distorted and the amplitude becomes small as shown in Fig. 5.8 (b). From Fig.
5.8 (c), the 3rd higher-harmonic component is contained in the nonlinear case. It is known that
odd order of higher-harmonic waves are generated due to pure shear motion with contact [89], andthat is also reproduced. From the above, we can confirm that the presented numerical method canaccurately simulate the clapping motion and dynamic friction on crack faces.
Finally, we investigate the characteristic of 2nd higher-harmonic generation varying pre-opening
displacement g or static compressive stress σ st when a straight-line crack is subjected to normal
incidence of plane P wave. Normalized Fourier amplitudes corresponding to 2nd higher-harmonic
wave A2ω /A02ω for each kT a varying ug /u0 and σ st /σ0 are shown in Fig. 5.9 (a) and (b), respectively. The behavior of A2ω /A02ω varying ug /u0 is different among various kT a as shown in Fig.
5.9 (a). However, the maximum value of A2ω /A02ω for each kT a is located at ug /u0 = 0.0. Thisis because contact of crack faces is strongly caused if ug /u0 is small. On the other hand, behaviorof A2ω /A02ω varying σ st /σ0 is different. The maximum value of A2ω /A02ω in the case that about895. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
6(a)LinearNonlinear

4
2
0
-2
-4
0

50

100

150

200

8

(b)

6

LinearNonlinear

4
2
0
-2
-4
-6

Normalized Fourier amplitude

0

50

100

150

1.0

(c)
0.8

LinearNonlinear

0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 5.7.: Time variations of (a) COD at the center point of a crack and (b) far-field amplitudeof backscattered P wave when a straight-line crack is subjected to an incident plane P
wave. (c) is normalized frequency spectrum of far-field amplitude.

kT a < 1.2 is located at σ st /σ0 = 0.0 as we can see Fig. 5.9 (b). However, the maximum valueof A2ω /A02ω for about 1.3 < kT a is located at σ st /σ0 > 0.0. In addition, the maximum value ofA2ω /A02ω is located at about σ st /σ0 = 0.4 when 3.0 < kT a < 5.5. In order to investigate the√detail, the time histories of far-field amplitude of backscattered P wave ΩL /(u0 a) in the case that90

5.5. Numerical results
0.8

(a)

0.6
0.4

Nonlinear

0.2
0.0
-0.2
-0.4
-0.6
-0.8
0

20

40

60

80

100

10

(b)

5

LinearNonlinear

0
-5
-10

Normalized Fourier amplitude

0

20

40

60

80

100

1.0

(c)
0.8

LinearNonlinear

0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 5.8.: Time variations of (a) tangential traction at the center point of a crack and (b) far-field
amplitude of backscattered S wave when a straight-line crack is subjected to an incident
plane S wave, and σ st /σ0 = 1.0. (c) is normalized frequency spectrum of far-field
amplitude.

√
kT a = 5.0 are shown in Fig. 5.10. As shown in Fig. 5.10, the amplitude of ΩL /(u0 a) in thecase that σ st /σ0 = 0.4 is comparatively small because the compressive static stress prevents crack
faces from opening. However, the waveform is strongly distorted, and it is expected to contain many91

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

(a)

(b)

Figure 5.9.: Fourier amplitude corresponding to 2nd higher-harmonic wave normalized by the one
when ug /u0 = σ st /σ0 = 0.0 for (a) various kT a and ug /u0 , and (b) various kT a andσ st /σ0 .
10
8
6
4
2
0
-2
-4
-6
0

20

40

60

80

100

Figure 5.10.: Time variations of far-field amplitude of backscattered P wave when kT a = 5.0.

higher-harmonic components.
From the above, there is optimal static compressive stress for the 2nd higher-harmonic generation when kT a is comparatively large. The branch point kT a ≃ 1.2 is almost same as the firstmode quasi-eigenfrequency in the linear system. It has been pointed out by some researchers that
there is optimal static compressive stress for the 2nd higher-harmonic generation [15, 70, 87]. However, the presented 2-D numerical simulation revealed that this phenomenon has relation to quasieigenfrequency of the linear system containing a crack.

5.5.3. Simulation of sub-harmonic waves due to interior cracks
Some numerical results of sub-harmonic generation by cracks in an infinite domain are shown here.
We consider two types of model for the numerical simulations as depicted in Fig. 5.11. The first is92

5.5. Numerical results
Half-sine-shaped
crack

A

Straight-line
crack

B

(a)

(b)

Figure 5.11.: Elastic wave scattering by (a) a half-sine-shaped crack and (b) two straight-line cracks
arranged in parallel.

an incident wave scattering by a half-sine-shaped crack (Fig. 5.11 (a)), and the second is that by two
cracks arranged in parallel (Fig. 5.11 (b)). In practice, it is not sure that cracks with the shape andarrangement presented here exist. However, our objective is to implement numerical simulations
for the system in which the resonance is caused easily. In order to investigate the sub-harmonic
generation in the case that contact boundary conditions on crack faces are added into such a system,
the analysis models in Fig. 5.11 are selected in this study.
The time increment is taken as f in ∆t = 0.05, and Poisson’s ratio of base material is given byν = 0.3 supposing steel. The friction coefficients are thus given by the top row of Table 4.2. The
cycle number of the incident wave nin is given by nin = 30.

Sub-harmonic generation by a curved crack
First, numerical results obtained by solving the scattering problem by a curved crack as depicted inFig. 5.11 (a) are shown. Fourier amplitude ratios of sub-harmonic component to fundamental one
in backscattered P wave A0.5ω /Aω for various kT a, ug /u0 , and θin are shown in Figs. 5.12–5.16
when height h/a is varying. The area enclosed by a white line in Fig. 5.16 indicates the condition
under which we can observe the sub-harmonic generation. Therefore, we could not observe thesub-harmonic generation when 0.0 ≤ h/a ≤ 1.5 as shown in Figs. 5.12–5.15. For the case thath/a = 2.0 in Fig. 5.16, the sub-harmonic generation is caused under certain conditions except thesymmetric case, i.e. θin = 0◦ . For instance of the sub-harmonic generation, the time history andnormalized frequency spectrum of far-field amplitude of backscattered P wave are shown in Fig.
5.17 when kT a = 1.2, ug /u0 = 2.0, h/a = 2.0, and θin = 60◦ . As shown in Fig. 5.17 (a),large and small amplitude is periodically repeated, and this motion becomes stable from the middle.
This waveform is typical one which includes sub-harmonic components, and there is a clear peakcorresponding to the sub-harmonic component in Fig. 5.17 (b). In this study, we define that subharmonic wave can be observed if such a waveform and frequency spectrum are confirmed. On
the other hand, we could not observe the sub-harmonic generation even if A0.5ω /Aω is large in the93

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Figure 5.12.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when ahalf-sine-shaped crack is subjected to an incident plane P wave with nin = 30 (h/a =
0.0).

94

5.5. Numerical results

Figure 5.13.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when ahalf-sine-shaped crack is subjected to an incident plane P wave with nin = 30 (h/a =
0.5).

95

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Figure 5.14.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when ahalf-sine-shaped crack is subjected to an incident plane P wave with nin = 30 (h/a =
1.0).

96

5.5. Numerical results

Figure 5.15.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when ahalf-sine-shaped crack is subjected to an incident plane P wave with nin = 30 (h/a =
1.5).

97

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Sub-harmonic
generation

Figure 5.16.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when ahalf-sine-shaped crack is subjected to an incident plane P wave with nin = 30 (h/a =
2.0).

98

5.5. Numerical results
4

(a)

3
2
1
0
-1
-2
-3
-4
0

100

200

300

400

500

Normalized Fourier amplitude

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

1.2

2.4

3.6

4.8

6.0

Figure 5.17.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a curved crack is subjected to an incident plane P wave
(kT a = 1.2, ug /u0 = 2.0, h/a = 2.0, and θin = 60◦ ).

outside of white line. As an example of that, the time history and normalized frequency spectrum of
far-field amplitude of backscattered P wave are shown in Fig. 5.18 when kT a = 2.5, ug /u0 = 0.0,√
h/a = 2.0, and θin = 60◦ . As shown in Fig. 5.18 (a), the waveform of ΩL /(u0 a) is strongly
distorted, and the value of Aω is considered as small. Although A0.5ω /Aω is comparatively large
due to this fact, there is no peak corresponding to the sub-harmonic component as shown in Fig.
5.18 (b). In order to investigate the behavior of sub-harmonic generation varying ug /u0 , A0.5ω /Aω
for various ug /u0 is shown in Fig. 5.19 when kT a = 1.2, h/a = 2.0, and θin = 60◦ . As shownin Fig. 5.19, A0.5ω /Aω changes continuously without rapid fluctuation when ug /u0 is varying.
In addition, the chaotic vibration, which has been observed in [88, 34], has never seen from thenumerical results in this thesis.
In order to investigate the vibration around a crack when the sub-harmonic wave is generated,the displacement fields in the case that kT a = 1.2, ug /u0 = 2.0, h/a = 2.0, and θin = 60◦ fromf in t = 20.0 to 22.0 are presented in Fig. 5.20. The color depth of crack line indicates the phaseof contact conditions as shown in the upper right of Fig. 5.20. These displacement fields are in the99

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
3

(a)
2
1
0
-1
-2
-3

Normalized Fourier amplitude

0

50

100

150

200

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

2.5

5.0

7.5

10 . 012 . 5Figure 5.18.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a curved crack is subjected to an incident plane P wave
(kT a = 2.5, ug /u0 = 0.0, h/a = 2.0, and θin = 60◦ ).
0 . 250 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

Figure 5.19.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 1.2, h/a = 2.0, and θin = 60◦ .

100

5.5. Numerical results: separation: stick: slip
6.0

4.0

2.0

0.0

Crack
Torsional
vibration
C4(0.0,1.5 a )C3(0.0, a )C2(0.0,0.5 a )C1(0.0,0.0)

Figure 5.20.: Displacement fields around a curved crack from f in t = 20.0 to 22.0 when h/a = 2.0,
θin = 60◦ , kT a = 1.2, and ug /u0 = 2.0.

101

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
3

3

(a)

C1
C2

2

C3
C4

2

1

1

0

0

-1

-1

-2

-2

-3

(b)

-3
-4

-2

0

2

4

-4

-2

0

2

4

Figure 5.21.: Histories of displacement at points (a) C1 and C2, and (b) C3 and C4 in the lower rightof Fig. 5.20 from f in t = 20.0 to 22.0 when h/a = 2.0, θin = 60◦ , kT a = 1.2, andug /u0 = 2.0.
3

3

C1
C2
C3
C4

2
1

(a)

C1
C2
C3
C4

2
1

0

0

-1

-1

-2

-2

-3

(b)

-3
-4

-2

0

2

4

-4

-2

0

2

4

Figure 5.22.: Histories of displacement at points C1–C4 in the lower right of Fig. 5.20 from f in t =20.0 to 21.0 when h/a = 2.0, θin = 60◦ , kT a = 1.2, and (a) ug /u0 = 0.5 and (b)ug /u0 = 3.0.

nearly stationary vibration after the transient analysis is proceeded for a sufficiently long time. It
can be obviously seen that the vibration period is double of 1/f in , especially in the inner part of acurved crack. Focusing there, the large displacement to right is generated at f in t = 20.0, and thefriction is caused on the right side of crack at around f in t = 20.2 to 20.4. Then, the inner part isdeformed to left with comparatively small amplitude at around f in t = 21.0. This part begins to
be deformed to right again at around f in t = 21.6, and the state of displacement fields eventually
returns to the first one at f in t = 22.0. From the above, it seems that the torsional vibration with the
double period of incident wave as shown in lower right of Fig. 5.20 is generated by the contact of
crack faces.
Histories of displacement at points C1–C4 in lower right of Fig. 5.20 from f in t = 20.0 to 22.0
are shown in Fig. 5.21. The circle in the figure indicates the displacement at f in t = 20.0. As shownin Fig. 5.21 (a), the displacement orbits at points C1 and C2 are distorted, and the locations returnto the positions at f in t = 20.0 spending double of 1/f in . The cause of these phenomena is thecontact of crack faces because the displacement orbit of harmonic vibration must be ellipse with the102

5.5. Numerical results
period 1/f in . As shown in Fig. 5.21 (b), the torsional vibration like the lower right of Fig. 5.20
is observed at points C3 and C4. In addition, the direction of displacement at point C4 is rapidlychanged because it is near the crack face. On the other hand, displacement orbits in the cases thatthe sub-harmonic generation could not be observed are presented in Fig. 5.22 for the time range
from f in t = 20.0 to 21.0. Fig. 5.22 (a) shows the case that ug /u0 = 0.5, and it can be seen that thedisplacement orbits are distorted by contact of crack faces. However, all of these periods are 1/f inand different from ones in Fig. 5.21. The displacement orbit is almost ellipse with 1/f in period in
the case that ug /u0 = 3.0 as shown in Fig. 5.22 (b) because contact of crack faces hardly occurs.
From the above, the torsional vibration is generated as sub-harmonic resonance under certain
conditions for the model in Fig. 5.11 (a). As a result, the sub-harmonic wave is contained in the
scattered wave.

Sub-harmonic generation by two straight-line cracks arranged in parallel
Second, numerical results obtained by solving the scattering problem by two straight-line cracks inparallel as depicted in Fig. 5.11 (b) are shown. Fourier amplitude ratios of sub-harmonic component
to fundamental one in backscattered P wave A0.5ω /Aω for various kT a, ug /u0 , and θin are shownin Figs. 5.23–5.25 when the interval between two cracks h/a is varying. We could not observe thesub-harmonic generation when h/a = 1.5 as shown in Fig. 5.25. For the cases that h/a = 0.5
and 1.0 in Figs. 5.23 and 5.24, respectively, the sub-harmonic generation is caused under certain
conditions except the case of parallel incidence to cracks, i.e. θin = 90◦ . For instance of the subharmonic generation, the time history and normalized frequency spectrum of far-field amplitude ofbackscattered P wave are shown in Fig. 5.26 when kT a = 2.5, ug /u0 = 2.0, h/a = 0.5, and√θin = 0◦ . As shown in Fig. 5.26, the waveform of ΩL /(u0 a) is the typical one which contains
the sub-harmonic component, and there is a clear peak of A0.5ω /Aω in the frequency spectrum.
A0.5ω /Aω for various ug /u0 is shown in Fig. 5.27 when kT a = 2.5, h/a = 0.5, and θin = 0◦ . In
this case, A0.5ω /Aω also changes continuously as shown in Fig. 5.27.
In order to investigate the vibration around cracks when the sub-harmonic wave is generated, thedisplacement fields in the case that kT a = 2.5, ug /u0 = 2.0, h/a = 0.5, and θin = 0◦ fromf in t = 20.0 to 22.0 are presented in Fig. 5.28. It can be obviously seen that the vibration period
is 2/f in , especially at the part between two cracks. Focusing there, the large displacement appears
upward at f in t = 20.0, and the displacement becomes small at around f in t = 20.2 to 21.0. Then,contact of crack faces occurs on the lower crack at f in t = 21.2, and upward large displacement isgenerated by diffracted waves at f in t = 21.6 to 21.8. Finally, the displacement state returns to thefirst one at f in t = 22.0. From the above, it seems that the bending vibration with the double period
of incident wave as shown in lower right of Fig. 5.28 is generated by the contact of crack faces.
Histories of displacement at points D1–D4 in lower right of Fig. 5.28 from f in t = 20.0 to 22.0
are shown in Fig. 5.29. As shown in Fig. 5.29 (a) and (b), all of the displacement orbits are strongly
distorted, and the displacement returns to the position at f in t = 20.0 spending double of 1/f in .
From Fig. 5.29 (a), one upward and downward motion spends 2/f in , and it is therefore confirmed

103

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Sub-harmonic
generation

Figure 5.23.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when two
straight-line cracks arranged in parallel are subjected to an incident plane P wave with
nin = 30 (h/a = 0.5).

104

5.5. Numerical results

Sub-harmonic
generation

Figure 5.24.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when two
straight-line cracks arranged in parallel are subjected to an incident plane P wave with
nin = 30 (h/a = 1.0).

105

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Figure 5.25.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when two
straight-line cracks arranged in parallel are subjected to an incident plane P wave with
nin = 30 (h/a = 1.5).

106

5.5. Numerical results
6

(a)
4
2
0
-2
-4
-6
0

50

100

150

200

Normalized Fourier amplitude

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

2.5

5.0

7.5

10.0

12.5

Figure 5.26.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when two straight-line cracks arranged in parallel are subjected
to an incident plane P wave (kT a = 2.5, ug /u0 = 2.0, h/a = 0.5, and θin = 0◦ ).
0 . 400 . 35
0 . 300 . 250 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

Figure 5.27.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 2.5, h/a = 0.5, and θin = 0◦ .

107

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks: separation: stick: slip
3.5
3.0

2.0

1.0

0.0

Bending vibration

D4( a ,0.25 a )Crack D3(0.75 a ,0.25 a )D1(0.25 a ,0.25 a )D2(0.5 a ,0.25 a )Figure 5.28.: Displacement fields around two cracks arranged in parallel from f in t = 20.0 to 22.0
when h/a = 0.5, θin = 0◦ , kT a = 2.5, and ug /u0 = 2.0.

108

5.5. Numerical results
3

3

(a)

2

2

1

1

0

0

-1

-1

-2

-1.0

-0.5

(b)

-2

D1
D2

-3

D3
D4

-3
0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

Figure 5.29.: Histories of displacement at points (a) D1 and D2, and (b) D3 and D4 in the lower
right of Fig. 5.28 from f in t = 20.0 to 22.0 when h/a = 0.5, θin = 0◦ , kT a = 2.5,
and ug /u0 = 2.0.
3

3

(a)

2

2

1

1

0

0

-1

-1

-2
-3
-1.0

D1
D2
D3
D4
-0.5

-2
-3
0.0

0.5

1.0

-1.0

(b)

D1
D2
D3
D4
-0.5

0.0

0.5

1.0

Figure 5.30.: Histories of displacement at points D1–D4 in the lower right of Fig. 5.28 from f in t =20.0 to 21.0 when h/a = 0.5, θin = 0◦ , kT a = 2.5, and (a) ug /u0 = 1.0 and (b)ug /u0 = 3.0.

that the bending vibration is generated. On the other hand, displacement orbits in the cases that the
sub-harmonic generation could not be observed are presented in Fig. 5.30 for the time range fromf in t = 20.0 to 21.0. Fig. 5.30 (a) shows the case that ug /u0 = 1.0, and it can be seen that thedisplacement orbits are distorted by contact of crack faces. However, all of these periods are 1/f inand different from the ones in Fig. 5.29. The displacement orbit is almost ellipse with 1/f in period
in the case that ug /u0 = 3.0 as shown in Fig. 5.30 (b) because contact of crack faces hardly occurs.
From the above, the bending vibration is generated as sub-harmonic resonance under certain
conditions for the model in Fig. 5.11 (b).
Relation between sub-harmonic generation and characteristic of frequency
response
From numerical results of sub-harmonic generation by a curved crack and two straight-line cracks
arranged in parallel, it seems that the sub-harmonic resonance such as torsional and bending vibrations, depends on the shape and arrangement of cracks. Therefore, it is presumed that the fre-109

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
10

10

(a)

(b)

8

8

6

6

4

4

2

2

0

0
0

1

2

3

4

5

0

10

1

2

3

4

5

10

(d)

(c)
8

8

6

6

4

4

2

2

0

0
0

1

2

3

4

5

0

1

2

3

4

5

10

(e)
8
6
4
2
0
0

1

2

3

4

5

Figure 5.31.: Frequency responses for the model in Fig. 5.11 (a) using the frequency-domain BEM
and assuming that crack faces are always separated. The norm ||[u]||/u0 at point A in
Fig. 5.11 (a) for various θin and kT a is plotted when (a) h/a = 0.0, (b) 0.5, (c) 1.0,(d) 1.5, and (e) 2.0.

quency response characteristic of the linear system containing cracks without contact of crack faces
is important. That is investigated by solving the scattering problem for the same models using
the frequency-domain BEM. In this study, the norm of COD ||[u]|| is considered as the factor of
frequency response. The norm || · || is defined by||v|| ≡√∑

vi vi .

(5.34)i

The norm of COD ||[u]||/u0 at points A in Fig. 5.11 (a) varying kT a is shown in Fig. 5.31 forvarious h/a and θin . All lines are smooth in Fig. 5.31 (a), (b), and (c) compared with the ones in Fig.
5.31 (d) and (e). ||[u]||/u0 in Fig. 5.31 (d) shows a little rapid fluctuation at about kT a = 0.7 when110

5.5. Numerical results
30

30

(a)

(b)

25

25

20

20

15

15

10

10

5

5

0

0
0

1

2

3

4

5

0

1

2

3

4

5

30

(c)
25
20
15
10
5
0
0

1

2

3

4

5

Figure 5.32.: Frequency responses for the model in Fig. 5.11 (b) using the frequency-domain BEM
and assuming that crack faces are always separated. The norm ||[u]||/u0 at point B in
Fig. 5.11 (b) for various θin and kT a is plotted when (a) h/a = 0.5, (b) 1.0, and (c)1.5.

30◦ ≤ θin ≤ 90◦ . As shown in Fig. 5.31 (e), ||[u]||/u0 has rapid fluctuation at about kT a = 0.5
when 15◦ ≤ θin ≤ 90◦ . This fluctuation corresponds to the first mode quasi-eigenfrequency andseems to be important for the probability of sub-harmonic generation because the sub-harmonic
generation could be observed when h/a = 2.0 and 15◦ ≤ θin ≤ 90◦ . In addition, the sub-harmonic
generation could be observed in the case that kT a is a little larger than the double of kT a at the
rapid fluctuation as shown in Fig. 5.16. h/a and θin are significant to the characteristic of frequency
response, and they are therefore important for the sub-harmonic generation.
Next, the norm of COD ||[u]||/u0 at points B in Fig. 5.11 (b) varying kT a is shown in Fig. 5.32
for various h/a and θin . ||[u]||/u0 in Fig. 5.32 (a) shows rapid fluctuation at about kT a = 0.8 when0◦ ≤ θin ≤ 75◦ . As shown in Fig. 5.32 (a)–(c), the smaller h/a is, the more rapid the fluctuation of||[u]||/u0 is. However, ||[u]||/u0 is gentle in the case that θin = 90◦ because the multiple reflection
hardly occurs between two cracks. It also seems that this fluctuation is important for the probabilityof sub-harmonic generation because the sub-harmonic generation could be observed only whenh/a = 0.5 and 1.0 and 0◦ ≤ θin ≤ 75◦ . In addition, the the sub-harmonic generation could beobserved in the case that kT a is larger than the double of kT a at the rapid fluctuation as shown in
Figs. 5.23 and 5.24.

111

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
Free surface

Straight-line
crack

Figure 5.33.: Elastic wave scattering by a surface breaking crack in an elastic half-space.

5.5.4. Simulation of sub-harmonic waves due to a surface breaking crack
The numerical results of sub-harmonic generation by a surface breaking crack in an elastic halfspace as depicted in Fig. 5.33 are presented. We assume that a straight-line crack with length a isconnected to the free surface with the angle θc . An incident plane wave described in Eq. (5.33) with
φ = L (P wave) and nin = 15 is used in this simulation. The far-field scattered waves and freefields are evaluated as presented in Appendices B and C, respectively.
Fourier amplitude ratios of sub-harmonic component to fundamental one in backscattered P wave
A0.5ω /Aω for various kT a, ug /u0 , and θin are shown in Figs. 5.34–5.37 when the crack angle θc
is varying. We could not observe the sub-harmonic generation when 0◦ ≤ θc ≤ 15◦ in Figs. 5.34
and 5.35. However, for the case that θc = 30◦ in Fig. 5.36, the sub-harmonic generation is slightly
caused around kT a = 3.7 and ug /u0 = 0.0. As shown in Fig. 5.37, the sub-harmonic generation
can be obviously observed in the case that θc = 45◦ compared with the case that θc = 30◦ . For
instance of the sub-harmonic generation, the time history and normalized frequency spectrum of
far-field amplitude of backscattered P wave are shown in Fig. 5.38 when kT a = 3.7, ug /u0 = 0.0,
θc = 45◦ , and θin = 75◦ . The typical sub-harmonic waveform and clear peak of A0.5ω /Aω can beobserved in Fig. 5.17 (a) and (b), respectively. A0.5ω /Aω for various ug /u0 is shown in Fig. 5.39
when kT a = 3.7, θc = 45◦ , and θin = 75◦ . It can be seen from Fig. 5.39 that A0.5ω /Aω changes
gently. However, the behavior is different from the cases of interior cracks, and the maximum valueof A0.5ω /Aω is observed at ug /u0 = 0.0.
In order to investigate the vibration around a surface breaking crack when the sub-harmonic
wave is generated, the displacement fields in the case that kT a = 3.7, ug /u0 = 0.0, θc = 45◦ ,and θin = 75◦ from f in t = 10.0 to 12.0 are shown in Fig. 5.40. It can be obviously seen that thevibration period is double of 1/f in , especially around the crack. Focusing on the upper left of asurface breaking crack, the displacement to left at f in t = 10.0 is observed, and the contact of crack
faces is caused at f in t = 10.2 to 10.4. Then, this part is deformed from upper left to upper rightwith large amplitude at f in t = 10.8 to 11.6 without contact of crack faces. This part begins to be
deformed to left again at around f in t = 11.8, and the state of displacement eventually returns to
the first one at f in t = 12.0. From the above, it seems that the bending vibration of the convex part112

5.5. Numerical results

Figure 5.34.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when asurface breaking crack is subjected to an incident plane P wave with nin = 15 (θc =0◦ ).

with the double period of incident wave as shown in lower right of Fig. 5.40, is generated due to thecontact of crack faces.
The history of displacement at point E in lower right of Fig. 5.40 from f in t = 10.0 to 12.0
is shown in Fig. 5.41. The circle in the figure indicates the displacement at f in t = 10.0. As
shown in Fig. 5.41, the displacement orbit is distorted, and the displacement returns to the position
at f in t = 10.0 spending double of 1/f in . Therefore, the bending vibration of the convex part isgenerated as sub-harmonic resonance under certain conditions.
In order to investigate the relation with frequency responses, the norm of COD ||[u]||/u0 at the113

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Figure 5.35.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when asurface breaking crack is subjected to an incident plane P wave with nin = 15 (θc =15◦ ).

origin O in Fig. 5.33 varying kT a is shown in Fig. 5.42 for various θc . As shown in Fig. 5.42 (d),||[u]||/u0 has rapid fluctuation at about kT a = 1.1 when 60◦ ≤ θin ≤ 75◦ compared with others
in Fig. 5.42 (a), (b), and (c). It also seems that this fluctuation is important for the probability of
sub-harmonic generation. However, the sub-harmonic generation is caused in the case that kT a islarger than the triple of kT a at the rapid fluctuation as shown in Fig. 5.37. In addition, the optimal
ug /u0 for the sub-harmonic generation is ug /u0 = 0.0 in this case. Therefore, the behavior of
sub-harmonic generation is different between interior cracks and surface breaking crack. This may
be because the one side of convex part is free surface without contact.

114

5.6. Concluding remarks

Sub-harmonic
generation

Figure 5.36.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when asurface breaking crack is subjected to an incident plane P wave with nin = 15 (θc =30◦ ).

5.6. Concluding remarks• In Chapter 5, we have applied the IRK-based CQ-BEM without any acceleration techniques
to the 2-D elastic wave scattering problems by cracks in an infinite elastic solid and surface
breaking cracks in an elastic half-space. Moreover, the contact boundary conditions on crack
faces are considered, and the phenomena of higher- and sub-harmonic generation are investigated by the presented numerical method.
• The IRK-based CQ-BEM produces accurate solutions for the linear crack problems. We

115

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks

Sub-harmonic
generation

Figure 5.37.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
wavenumber kT a, pre-opening displacement ug /u0 , and incident angles θin when asurface breaking crack is subjected to an incident plane P wave with nin = 15 (θc =45◦ ).

have also numerically solved the fundamental nonlinear ultrasonic problems such as normal
incidence of P and S waves to a straight-line crack in order to validate the presented contact
model. From these results, it could be confirmed that the proposed IRK-based CQ-BEM canaccurately solve the crack problems with contact boundary conditions.
• We conducted the higher-harmonic simulation for a straight-line crack subjected to an incident plane P wave varying ug /u0 or σ st /σ0 . From the numerical results, the 2nd higherharmonic wave is generated most strongly when ug /u0 = σ st /σ0 = 0.0 if the center frequency of incident wave is comparatively low. However, the 2nd higher-harmonic wave is116

5.6. Concluding remarks

2.0

(a)

1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0

Normalized Fourier amplitude

0

10

20

30

40

50

60

70

80

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

3.7

7.4

11 . 114 . 818 . 5Figure 5.38.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a surface breaking crack is subjected to an incident plane
P wave (kT a = 3.7, ug /u0 = 0.0, θc = 45◦ , and θin = 75◦ ).
0 . 35
0 . 300 . 250 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

Figure 5.39.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when kT a = 3.7, θc = 45◦ , and θin = 75◦ .

117

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks: separation: stick: slip
4.0
3.0
2.0
1.0
0.0

E (-0.35 a ,-0.175 a )Figure 5.40.: Displacement fields around a surface breaking crack from f in t = 10.0 to 12.0 whenθc = 45◦ , θin = 75◦ , kT a = 3.7, and ug /u0 = 0.0.

118

5.6. Concluding remarks

4

2

0

-2

-4
-4

-2

0

2

4

Figure 5.41.: History of displacement at point E in the lower right of Fig. 5.40 from f in t = 10.0 to
12.0 when θc = 45◦ , θin = 75◦ , kT a = 3.7, and ug /u0 = 0.0.
12

12

(a)

(b)

10

10

8

8

6

6

4

4

2

2

0

0
0

1

2

3

4

5

0

12

1

2

3

4

5

12

(c)

(d)

10

10

8

8

6

6

4

4

2

2

0

0
0

1

2

3

4

5

0

1

2

3

4

5

Figure 5.42.: Frequency responses for the model in Fig. 5.33 using the frequency-domain BEM andassuming that crack faces are always separated. The norm ||[u||/u0 at the origin O in
Fig. 5.33 for various θin and kT a is plotted when (a) θc = 0◦ , (b) 15◦ , (c) 30◦ , and (d)
45◦ .

119

5. 2-D Simulation of Nonlinear Ultrasonic Waves Due to Interior and Surface Breaking Cracks
generated most strongly under some compressive static stress if the center frequency of incident wave is comparatively high. The border frequency of this phenomenon is near the firstmode quasi-eigenfrequency of the linear system (about kT a = 1.2).
• We have dealt with the scattering problems by a half-sine-shaped crack and two straight-line
cracks arranged in parallel in an infinite elastic solid in this chapter. The torsional and bending
vibrations with the double period of the incident wave are generated as the nonlinear subharmonic resonance. In addition, the characteristic of frequency response greatly depends onthe crack shape and incident angle, and it seems that this characteristic is significantly related
to the probability of sub-harmonic generation.
• We also implemented the 2-D simulation of nonlinear ultrasonic due to a surface breaking
crack in an elastic half-space. The bending vibration of convex part with the double periodof the incident wave is generated as the nonlinear sub-harmonic resonance. Similarly to thecase of interior cracks, the characteristic of frequency response seems to be important for the
sub-harmonic generation. However, the frequency behavior of sub-harmonic generation isdifferent from the case of the interior crack.

120

Chapter 6

3-D Simulation of Nonlinear UltrasonicWaves Due to Interior Cracks
In this chapter, the boundary element formulations and numerical results for the 3-D simulation of
sub-harmonic generation are presented. We consider the scattering problems by cracks in an infinite
domain. The fast CQ-BEM proposed in Chapter 3 is extended to the 3-D crack problems withcontact boundary conditions. The Galerkin method is used for spatial discretization here, similarly
to Chapter 5.
For 2-D problems, the regularization of hypersingular integrals to the strong singular one isenough to use the tip element as shown in Chapter 5. However, the analytical integration overelements at the crack front in 3-D problems is much more complicated than that in 2-D. Therefore,
we regularize the hypersingular integrals to weakly singular ones, and all of the integrations overelements are evaluated numerically. Regularization techniques for the hypersingular BIE have beendeveloped by many researchers [93]. For instance, the weakly singular formulations have been developed for the 3-D acoustic and elastic wave problems by Liu and Rizzo [44, 45]. Also, the direct
numerical integration method of hypersingular integral has been developed by Guiggiani et al. [27].
In this study, we use the regularization method proposed by Nishimura and Kobayashi [61]. Theirmethod is based on the stress function representation of differentiated fundamental solution and canrelax the hypersingular integrals to weakly ones.
First, the formulation of time-domain BIE for crack problem is briefly explained. Second, theregularization and discretization are presented, and some remarks related with the application of
FMM are summarized. Then, the computational cost and accuracy of the proposed CQ-FMBEM
are demonstrated by solving linear crack problems. Finally, we implement 3-D sub-harmonic simulations and discuss the phenomena through obtained numerical results.

6.1. Problem statement and BIE
We show the initial and boundary conditions for 3-D crack problems in time-domain. Let S be asmooth curved crack surface in R3 bounded by a smooth edge ∂S as depicted in Fig. 6.1. D is ahomogeneous, isotropic, and linearly elastic solid, and S consists of S + and S − called positive and121

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks

Figure 6.1.: Analysis model for elastic wave scattering by a crack in a 3-D infinite domain.

negative sides of crack face, respectively (S = S + ∪ S − and S + ∩ S − = ∅). n is the unit normal
vector to the crack face, and the superscript ± indicates its each side. Disregarding the body force,the problem is to find the solution u which satisfies the following equations:()
c2T ∇2 u(x, t) + c2L + c2T ∇∇ · u(x, t) = ü(x, t)
t+ (x, t) + t− (x, t) = 0 x ∈ S,()[u](x, t) ≡ u+ (x, t) − u− (x, t) = 0
scscu (x, 0) = u̇ (x, 0) = 0()
x ∈ D ≡ R3 \S ,

(6.1)
(6.2)

x ∈ ∂S,x ∈ D.

(6.3)
(6.4)

Similarly to 2-D case in Chapter 5, the variational formulation of hypersingular BIE with weightfunction ψ can be derived as follows:∫∫+
ψ(x)t (x, t)dSx =ψ(x)tin;+ (x, t)dSx
+
+
S
S
∫
∫ t∫
ψ(x)p.f.
W + (x, y, t − τ ) · [u](y, τ )dSy dSx dτ.
(6.5)
−
0

S+

S+

The superscript + is hereinafter omitted. In Eq. (6.5), W is the 3-D hypersingular kernel in timedomain. Similarly to the 2-D case, it can be written as follows:
Wbj (x, y, t) = na (x)nk (y)Cbaic Cjkpq∂ ∂
Uip (x, y, t).
∂xc ∂yq(6.6)

6.2. Regularization of BIE
In this section, the regularization of variational BIE (6.5) is presented. First, the stress function
representation of differentiated fundamental solution in Laplace-domain is shown because we use
the CQM and evaluate the integration of hypersingular kernel in Laplace-domain. According to[61], the stress function representation for differentiated fundamental solutions in Laplace-domain

1226.2. Regularization of BIE
elastodynamic problems is shown as follows:
Cbaic Cjkpq∂ ∂ ∂ ∂
∂ ∂Ûip (x, y, s) = eacm ebds ejpn ekqt
Φ̂msnt (x, y, s)∂xc ∂yq∂xc ∂xd ∂yp ∂yq+Ψ̂abjk (x, y, s) + Dabjk δ(x − y),(6.7)

where Φ̂ is a function of order r as r ↓ 0, and Ψ̂ is a function which has an integrable singularity
at x = y. In addition, D is a constant tensor, and this term is canceled in the BIE. The specificexpressions of Φ̂ and Ψ̂ for isotropic media are given as follows:
}
1
µ{
(λ
+
2µ)
(δ
δ
+
δ
δ
)
2λδ
δ
+
ms
nt
mt
sn
st
mn
4π(λ + µ) s2T( −sL r)ee−sT r
×
−
,
(6.8)
r
r
[
)
{
} ( s2 e−sL r
s2T e−sT r
12LµsT λδab δjk µ (δaj δbk + δak δbj )− 2
Ψ̂abjk (x, y, s) =4π(λ + µ)
s2T r
sL r
{
− λ2 δab δjk δpq + µ2 (δap δjq δbk + δbp δkq δaj + δap δkq δbj + δbp δjq δak ))]} ∂ ∂ ( e−sL r
e−sT r−+ 2λµ (δab δjp δkq + δjk δap δbq ). (6.9)∂xp ∂xqrr

Φ̂msnt (x, y, s) = −In addition, this decomposition has been extended to anisotropic elastodynamics by Becache et al.
[11].
Taking into account the application of CQM, we consider the following integral containing the
Laplace-domain hypersingular kernel Ŵ corresponding to the second term in the right-hand side ofEq. (6.5):
∫
IˆW =

∫
ψ(x)p.f.

S

Ŵbj (x, y, s)ϕ(y)dSy dSx ,(6.10)

S

where ϕ is the spatial shape function for [u]. Substituting Eq. (6.7) into Eq. (6.10) yields thefollowing expression:∫IˆW =ψ(x)na (x)∫∂ ∂ ∂ ∂
Φ̂msnt (x, y, s)dSy dSx
× p.f. ϕ(y)nk (y)eacm ebds ejpn ekqt
∂xc ∂xd ∂yp ∂yq
S∫∫+ψ(x)na (x) Ψ̂abjk (x, y, s)ϕ(y)nk (y)dSy dSx .
(6.11)
S

S

S

123

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks

case 1case 2

Figure 6.2.: Two types of triangular tip element for crack front.

Using Stokes’ theorem and taking Eq. (6.3) into account, Eq. (6.11) can be regularized as follows:
∫
IˆW =

∫
∂ψ(x)
∂ϕ(y)
∂ ∂
eacm na (x)ekqt nk (y)ebds ejpn
Φ̂msnt (x, y, s)dSy dSx
∂xc S
∂yq
∂xd ∂yp
S
I
∫
∂ ∂
∂ϕ(y)
ebds ejpn
Φ̂msnt (x, y, s)dSy dxm−
ψ(x) ekqt nk (y)∂yq∂xd ∂yp∂SS∫∫+
ψ(x)na (x) Ψ̂abjk (x, y, s)ϕ(y)nk (y)dSy dSx .
(6.12)SS

The second term in Eq. (6.12) includes the contour integral over the crack front with respect to x.
If we select the weight function ψ as ψ = 0 on ∂S, this term vanishes. The second derivative of Φ̂in the first term of Eq. (6.12) has weak singularity of order 1/r as r ↓ 0.

6.3. Discretization of BIE
The IRK-based CQM [48] is used for time discretization. For spatial discretization, the piecewise
linear interpolation is used for [u] except elements which are located at the crack front. We utilize
two types of tip element as depicted in Fig. 6.2 in order to accurately calculate COD near the crack
front using flatly triangular elements. As shown in Fig. 6.2, m1 and m2 are the unit tangential andnormal vectors to the crack front, respectively. The specific expressions of shape function ϕuI for
the two types of tip element are shown as follows:case 1√ϕu1 (y) =m2 · (y − y2 ),m2 · (y1 − y2 )ϕu2 (y) = ϕu3 (y) = 0.

case 2
m1 · (y − y3 )ϕu1 (y) =m1 · (y1 − y3 )m1 · (y − y1 )ϕu3 (y) =m1 · (y3 − y1 )

124

√
√

m2 · (y − y2 ),m2 · (y1 − y2 )m2 · (y − y2 ).
m2 · (y3 − y2 )ϕu2 (y) = 0,

6.3. Discretization of BIE√
Note that the derivatives of shape function for tip element have additional singularity of order 1/ r
corresponding to stress singularity at the crack front.
On the other hand, the piecewise constant shape function ϕt is used for the interpolation of t inorder to describe the discontinuity of traction between two adjacent elements. Therefore, the weightfunctions for displacement ψ u and traction ψ t are different because the Galerkin method is usedin this study. For simplicity, all of the weight functions ψ u are however taken as piecewise linear
functions. Thus, ψ u ̸= ϕu on tip elements.
If the m-stage Radau IIA method [33] is used for the IRK-based CQM, and the boundary S is
divided into Me flatly triangular elements composed of Md nodal points, the discretized BIEs for[u] and t at n-th and i-th sub-step in time are written as follows:
Md ∑
m [
∑

]
ij;0
WIJ
· [u]j;n
J

=−

∫

Me
∑

t
ti;n
α ϕα

supp(ψIu )∩Sα

α=1

J=1 j=1∫+
supp(ψIu )ψIu (x)tin (x, (n + ci )∆t)dSx −

∫
t t
ti;n
α ϕα ϕαψIu (x)dSx

Md ∑
n−1
m
∑∑

ij;n−k
WIJ· [u]j;kJ ,(6.13)

k=0 J=1 j=1∫

dSx = ϕtαtin (x, (n + ci )∆t)dSx

Sα

Sα

−

Md ∑
m
n ∑
∑

ij;n−kW̃αJ· [u]j;kJ ,(6.14)

k=0 J=1 j=1∫
ij;κ
where Sα dSx in the left hand side of Eq. (6.14) is equal to the area of α-th element. WIJ
andij;κ
W̃αJ in Eqs. (6.13) and (6.14) are influence functions and given as follows:
ij;κ
−1
WIJ
= F̃lκ

(
)
β;l
ZIJ

[ m
∑(

∫
bj

]
)
β;l
Eβ (ζl ) ZIJ,=
supp(ψIu )eacm na (x)∫×supp(ϕuJ)∫+supp(ψIu )

ij;κ
−1
W̃αJ
= F̃lκ

(
)
β;l
Z̃αJ

[ m
∑(

∂ψIu (x)∂xcekqt nk (y)∫ψIu (x)na (x)∂ϕuJ (y)∂ ∂ebds ejpn
Φ̂msnt (x, y, λlβ )dSy dSx∂yq∂xd ∂ypsupp(ϕuJ)Ψ̂abjk (x, y, λlβ )ϕuJ (y)nk (y)dSy dSx ,])β;lEβ (ζl ) Z̃αJ ,I∫= −ϕtα∂Sαsupp(ϕuJ)ekqt nk (y)

∫

+ ϕtα

na (x)
Sα∫(6.16)(6.17)

ij

β=1

bj

(6.15)ij

β=1∂ϕuJ (y)∂ ∂ebds ejpn
Φ̂msnt (x, y, λlβ )dSy dxm∂yq∂xd ∂ypsupp(ϕuJ)Ψ̂abjk (x, y, λlβ )ϕuJ (y)nk (y)dSy dSx ,(6.18)

125

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks

2 3

1
3
2

Crack front

1
1

3

2

Subdivision

3

1

3

2

1

1

3

2

1

1

2

1

Coordinate transformation

Figure 6.3.: Schematic of element subdivision and coordinate transformation for case 1 tip elementin Fig. 6.2.

Table 6.1.: Relation between coordinate systems (d1 , d2 ) and (d∗1 , d∗2 ) for sub-elements E1 , E2 , andE3 in Fig. 6.3.
E1
d1 = (d∗1 )2
d2 = 1 − (1 − d∗2 )2E2
d1 = d∗1
d2 = 1 − (1 − d∗2 )2E3
d1 = 1 − (1 − d∗1 )2
d2 = 1 − (1 − d∗2 )2where ∂Sα indicates the edge of Sα . Note that there are only weakly singular integrals in Eqs.
(6.16) and (6.18). Therefore, applying the element subdivision [43] and the transformation of local
coordinate system, we can accurately evaluate all of the integrals in Eqs. (6.16) and (6.18) using thestandard Gaussian quadrature.

126

6.4. Numerical integration over element surfaces

6.4. Numerical integration over element surfaces
In this study, we use the element subdivision [43] and the transformation of local coordinate system√
in order to cancel the weak singularities of order 1/r and 1/ r. The similar regularization technique
has been presented in [21]. For instance, we present regularization for the case 1 in Fig. 6.2 whenfield point x is located on the same element. The integrals with respect to x in Eqs. (6.16) and (6.18)
are evaluated by the standard Gaussian quadrature after the calculation of integrals with respect toy.
We summarize the procedure of the regularization as follows:
1. The element subdivision method is applied in order to regularize the singularity of order 1/rat the point x = y. The definitions of sub-elements E1 , E2 , and E3 are shown in the aboveof Fig. 6.3. Then, the coordinate system of each sub-element is transformed into the local
one (d1 , d2 ). We select (d1 , d2 ) for each element as shown in the middle of Fig. 6.3, and theranges of d1 and d2 for each element are [0, 1]. If the kernel function is A(y), the integral istransformed as follows:∫∫ 1∫ 1
A(d)Jd dd1 dd2 (i = 1, 2, or 3),(6.19)
A(y)dSy =Ei

00

where Jd is the Jacobian of the coordinate transformation and can cancel the singularity oforder 1/r at d2 = 0 in the middle of Fig. 6.3.
√
2. In order to regularize the additional singularity of order 1/ r caused by the derivatives ofshape function ϕu at the crack front, the additional transformation of local coordinate system is introduced. The coordinate system (d1 , d2 ) is changed to (d∗1 , d∗2 ) using the relations
between old and new ones in Table 6.1 for each element as follows:∫ 1∫ 1∫ 1∫ 1A(d∗ )Jd Jd∗ dd∗1 dd∗2 ,(6.20)
A(d)Jd dd1 dd2 =
0

0

0

0

where Jd∗ is also the Jacobian of the coordinate transformation. Consequently, the singularityof each sub-element is canceled by each Jd∗ .
3. The coordinate system (d∗1 , d∗2 ) is transformed to (f1 , f2 ), where the intervals of f1 and f2 are[−1, 1]. Then, the integrations over each sub-element are evaluated by the standard Gaussian
quadrature.
The other cases of integration such as case 2 tip element or the case that x is located at the edge ofelement, can be regularized in a similar way.

6.5. Application of FMM
The computational time and required memory often become significant issues for solving 3-D problems. Therefore, the proposed CQ-BEM is accelerated by the FMM as presented in Chapter 3.

1276. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
The multipole and local expansions for non-regularized hypersingular kernel are used because the
FMM for the regularized one requires much more computational time [100, 101]. The differenceof accuracy between the FMM with and without regularization is not significant because the FMM
calculation is implemented only for far-field contributions. In Chapter 3, we explained the computational algorithm of the CQ-FMBEM. Therefore, the multipole and local expansions and some
remarks for this problem are presented here.
i;k;β;li;k;β;lIn order to derive the multipole and local expansions, Wb;IJ
and W̃b;αJ
are defined by[∫i;k;β;lWb;IJ=[i;k;β;lW̃b;IJ=]∫

supp(ψIu )∫ψIu (x)supp(ϕu
J)

∫

ϕtα
Sα

supp(ϕuJ)Ŵbh (x, y, λlβ )ϕuJ (y)dSy dSx [uh ]i;k;β;l,J(6.21)]

Ŵbh (x, y, λlβ )ϕuJ (y)dSy dSx[uh ]i;k;β;l,J(6.22)

where[uh ]i;k;β;l
=
J

m (
)
∑
Eβ (ζl ) [uh ]j;k
J .
ij

j=1

i;k;β;li;k;β;lThe scaled multipole expansions for Wb;IJ
and W̃b;αJ
without regularization are given byi;k;β;l
Wb;IJ
=

[

∞ ∑
v
∑

Av

{

∫
Cbcad

v=0 w=−v{+ Bv eoce∂2L
nc (x)ψIu (x)(x − y0 )dSxϱO∂xa ∂xd v,w
supp(ψIu )

∫
Cbcad

∞ ∑
v
∑

supp(ψIu )nc (x)ψIu (x)∂2T(x − y0 )dSx
ϱO
∂xd ∂xe v,w

[

{
∫
t
Av Cbcad ϕα}}

L ;i;k;β;l× ϱ MW
(y0 )
v,w;J
]

WT ;i;k;β;l(y0 )ϱ Mo,v,w;J

,
s=λlβ

(6.23)}∂2
WL ;i;k;β;lL(y0 )ϱ Ov,w (x − y0 )dSx ϱ Mv,w;J
∂x
∂x
a
d
S
α
v=0 w=−v]{}∫∂2
WT ;i;k;β;lt
T+ Bv eoce Cbcad ϕαnc (x),(6.24)(x − y0 )dSx ϱ Mo,v,w;J (y0 )ϱO∂xd ∂xe v,wSαli;k;β;lW̃b;IJ=nc (x)s=λβ

L ;i;k;β;l
T ;i;k;β;lwhere ϱ MW
and ϱ MWare the scaled multipole moments with respect to P and Sv,w;Jo,v,w;Jwaves, respectively, and given by]∂L(y − y0 ) ϕJ (y)dSy , (6.25)
Chf pq nf (y)ϱ F̄∂yp ∂yq v,w
supp(ϕJ )][∫∂WT ;i;k;β;li;k;β;lT(y − y0 ) ϕJ (y)dSy .
(y0 ) = [uh ]Jeoup Chf pq nf (y)ϱ F̄ϱ Mo,v,w;J∂yq ∂yu v,wsupp(ϕJ )∫[WL ;i;k;β;l(y0 ) = [uh ]i;k;β;lϱ Mv,w;JJ(6.26)

128

6.6. Nonlinear boundary conditions and numerical procedure
Note that the double integrals with respect to x and y are separated, and therefore, far-feild contributions can be rapidly evaluated using the FMM. The same translation theorems as Ui;k;β;lb;αγ andi;k;β;lTb;αγ in Chapter 3 can be used for M2M, M2L, and L2L calculations.
However, the modified fast multipole algorithm for the Galerkin BEM (see Chapter 3 in [101]),
has to be taken in order to ensure the following two identities because we canceled some terms in
the regularization procedure from Eq. (6.11) to Eq. (6.12) using Eq. (6.3) and ψ u = 0 on ∂S.
∫∫supp(ψIu )ψIu (x)p.f.

supp(ϕuJ)∫=eacm na (x)supp(ψIu )∫×supp(ϕu
J)

∫
+
∫
ϕtα

supp(ψIu )∂ψIu (x)∂xcekqt nk (y)∫ψIu (x)na (x)

∫

p.f.
Sα

Ŵbj (x, y, s)ϕuJ (y)dSy dSx

supp(ϕuJ)∫∂ϕuJ (y)∂ ∂ebds ejpn
Φ̂msnt (x, y, s)dSy dSx∂yq∂xd ∂ypsupp(ϕu
J)

∂Sα

∫
+ ϕtα

supp(ϕuJ)ekqt nk (y)∫na (x)
Sα(6.27)

Ŵbj (x, y, s)ϕuJ (y)dSy dSx∫= −ϕtαΨ̂abjk (x, y, s)ϕuJ (y)nk (y)dSy dSx ,supp(ϕuJ)∂ϕuJ (y)∂ ∂ebds ejpn
Φ̂msnt (x, y, s)dSy dxm∂yq∂xd ∂ypΨ̂abjk (x, y, s)ϕuJ (y)nk (y)dSy dSx .

(6.28)

Eq. (6.27) holds if ψIu and ϕuJ are equal to zero at the edges of supp(ψIu ) and supp(ϕuJ ), respectively.
Therefore, a set of multipole moments corresponding to ϕuJ is calculated for one y0 as shown in Fig.
6.4 (a). The coefficients of local expansion have to be calculated in a similar way. Then, the far-field
contributions can be evaluated by using the fast multipole algorithm. Correspondingly to this fastmultipole algorithm, the near-field contributions have to be evaluated as node to node. Similarly,
Eq. (6.28) holds if ϕuJ = 0 at the edges of supp(ϕuJ ). Also in this case, a set of multipole moments
corresponding to ϕuJ has to be calculated for one y0 as shown in Fig. 6.4 (b). As a result, the nearfield contributions have to be evaluated as node to element. Taking into account these calculation
ways and two identities (6.27) and (6.28), the counterpart of direct matrix-vector product can beevaluated by using the FMM. Other numerical procedures of CQ-FMBEM are same as the ones in
Chapter 3.

6.6. Nonlinear boundary conditions and numerical procedure
The nonlinear boundary conditions and numerical procedure are briefly summarized in this section. In order to simultaneously solve discretized BIEs (6.13) and (6.14), the appropriate boundary
conditions should be selected at each time step. We consider three types of state in a similar way
to Section 4.2.1. The boundary conditions for separation, stick, and slip states are summarized as129

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks

L2L

M2M
M2L

(a)

L2L

M2M
M2L

(b)

Figure 6.4.: Schematic of potential calculation to ensure (a) Eq. (6.27) and (b) Eq. (6.28).

follows:

t=0





 [u] · n = ug ,[u] · n = ug ,: separation,(6.29a)[u˙s ] = 0: stick,(6.29b)[u˙s ]
ts = µd (t · n)[u˙s ]: slip,(6.29c)where the vectors with the subscript s indicate the ones which consist of the tangential components
to the crack face. The transition conditions for the iterative calculation are also summarized asfollows:[u] · n ≥ ug
: separation → stick,(6.30a)






t·n>0
: stick or slip → separation,(6.30b)|ts | ≥ µs |t · n|
: stick → slip,(6.30c)

prev

˙
˙

( stop )[us ] · [us ]: slip → stick,(6.30d)
prev < cos θ [u˙ ] [u˙ ]sswhere θstop is given by 90◦ . If the left inequalities in Eq. (6.30) are satisfied, the corresponding
right transition occurs.

1306.7. Numerical results

Penny-shaped crack

Figure 6.5.: A penny-shaped crack with radius a subjected to an incident plane P wave.

Figure 6.6.: Mesh structure of a penny-shaped crack when Me = 656 and Md = 361.

Using boundary conditions (6.29), discretized BIEs (6.13) and (6.14) can be simultaneously
solved. The iterative algorithm is similar to the case of interface crack of bi-material in Section
4.3. In addition, discretized BIEs (6.13) and (6.14) are modified for the velocity of COD tangential
to the crack face as well as Section 4.2.2.

6.7. Numerical results
In this section, numerical results of the proposed CQ-FMBEM for 3-D crack problems are presented. First, the linear problem is numerically solved in order to verify the accuracy and to demonstrate the computational efficiency. Second, the higher-harmonic simulation is carried out to validate the contact model of crack faces. Finally, numerical results of sub-harmonic generation due tocracks in an infinite domain are shown and compared with the ones for the 2-D case.

6.7.1. Linear problem
In advance of the nonlinear ultrasonic simulation, the numerical results for the traction-free crack
are shown. The time histories of COD and SIF are computed for the accuracy verification when a131

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
5

(a)
4

3

2

1

0
0.0

0.2

0.4

0.6

0.8

1.0

1.4

(b)
1.2
1.0
0.8
0.6

This work
Hirose and Achenbach ( 1989 )Reference

0.4
0.2
0.0
0

2

4

6

8

10

Figure 6.7.: (a) Vertical COD at each several time-steps and (b) time variations of normalized dynamic SIF when a penny-shaped crack is subjected to normal incidence of plane P
wave.

penny-shaped crack is subjected to normal incidence of plane P wave as depicted in Fig. 6.5. The
incident plane P wave is given byuin
i (x, t) = −u0 δi3cL t − x3
H (cL t − x3 ) .
a(6.31)

The crack surface is divided into 656 flatly triangular elements as depicted in Fig. 6.6, and timeincrement ∆t is given by cT ∆t/a = 0.1. Poisson’s ratio of base material ν is given by ν = 0.25 inorder to compare with reference solutions, and the CQ-FMBEM with RK3 is used here.
Vertical COD at several time steps on the positive part of x1 -axis is shown in Fig. 6.7 (a). The
vertical COD at x1 /a = 0.0 is constantly raised until about cT t/a = 0.6 because the diffracted P
wave arrives at about cT t/a = 0.58. From cT t/a = 0.9 to 1.2, the behavior of COD around the132

6.7. Numerical results
106

Computational time ( sec )CQ - FMBEM

105

104

103
102

103

104

105

Number of elements

Figure 6.8.: Computational time of CQ-FMBEM using RK3 for 3-D crack problems varying thenumber of elements Me when N = 128.

Computational time ( sec )

106

105

CQ - FMBEM

104

103
101

102

103

104

Number of time steps

Figure 6.9.: Computational time of CQ-FMBEM using RK3 for 3-D crack problems varying thenumber of time steps N when Me = 1464.

center of the crack is greatly different from an early time due to diffracted surface waves.
Next, the dynamic SIF of mode I is computed by the present method. According to [84], thedynamic SIF corresponding to mode I can be calculated by√µ 2π
1KI (t) = −lim √ [u](ε, t) · n(ε),
4(1 − ν) ε→0 ε(6.32)

133

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
where ε indicates the distance from the crack front. We normalize the dynamic SIF using the exact√
solution of static problems KIst = 2σ33 a/π. The time history of dynamic SIF is presented inFig. 6.7 (b). The solutions obtained by Hirose and Achenbach [35] are plotted in the same figure
for comparison. They have used the conventional TD-BEM in which the collocation method is used
for time and spatial discretization. On the other hand, this problem in frequency-domain can become to a Fredholm integral equation of the second kind using the axisymmetric property, and it
has been numerically solved by Rahman [68]. The transient solutions constructed by the inverse
Fourier transform, which are called the ”reference solutions”, are also presented in Fig. 6.7 (b).
The reference solutions have slight oscillation at about cT t/a = 2.1 and 4.3, when the waveform
is rapidly fluctuated. However, the reference solutions are considered as most accurate in Fig. 6.7(b). The behavior of solutions obtained by the proposed method and Hirose and Achenbach arevery similar and both solutions converge to the static solution. At about cT t/a = 2.1 and 4.3, thesolutions obtained by the proposed method seem to be more accurate.
The computational efficiency of the proposed method is confirmed by solving the same problem
for different N and Me . The computational time varying the number of elements Me is shown in
Fig. 6.8 when N = 128. The solid line in the figure indicates the regression one of numerical results.
As shown in Fig. 6.8, the computational complexity for Me seems to be about O(Me log Me ). On
the other hand, the computational time varying the number of time steps N is shown in Fig. 6.9
when Me = 1464. When about N < 200, the computational complexity seems to be O(N ).
However, the order of computational time seems to be O(N log N ) for comparatively large N . Thisis because calculation of the influence functions for near-field contribution is most time-consumingwith small N , and its complexity is O(N Me ). Consequently, it seems that the total computational
complexity of the proposed method is about O(N Me log N log Me ).

6.7.2. Higher-harmonic simulation
In order to validate the contact model of crack faces for 3-D problems, the numerical results ofhigher-harmonic simulation are presented here. Hereinafter, the CQ-FMBEM with RK2 is used for
all nonlinear ultrasonic simulations. The analysis model in Fig. 6.5 is also used for this simulation,
and we confirm that our model can reproduce the clapping motion and friction on crack faces similarly to the 2-D case in Section 5.5.2. The following sinusoidal burst wave with nin cycles is usedas the incident wave for the nonlinear ultrasonic simulation.

( in
)
in

(6.33a) u (x, t) = u0 dφ sin (2πΛφ ) H (Λφ ) H n − Λφ ,()
p · (x − χ)
in

t−
.
(6.33b) Λφ = fcφ
We assume that dT H = (0, 1, 0) and dT V = dT H ×p. The far-field approximation of scattered wave
(see Appendix B) is used for the spectrum analysis. The time increment is taken as f in ∆t = 0.05,
and Poisson’s ratio of base material is given by ν = 0.3 supposing steel. The friction coefficientsare thus given by the top row of Table 4.2. nin is given by nin = 15 here.

1346.7. Numerical results
6

(a)
4

LinearNonlinear

2
0
-2
-4
-6
0

20

40

60

80

100

15

(b)
10

LinearNonlinear

5
0
-5
-10
-15

Normalized Fourier amplitude

0

20

40

60

80

1.0

(c)
0.8

LinearNonlinear

0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 6.10.: Time variations of (a) COD at the center point of a crack and (b) far-field amplitudeof backscattered P wave when a penny-shaped crack is subjected to an incident plane
P wave. (c) is normalized frequency spectrum of far-field amplitude.

First, numerical results of P wave incidence are presented when θin = 0◦ , kT a = 3.0, andug /u0 = 0.0. Time histories of vertical COD at the center point of a crack are shown in Fig. 6.10(a). Time histories and normalized frequency spectra of far-field amplitude of backscattered P wave
are also presented in Fig. 6.10 (b), and (c), respectively. As shown in Fig. 6.10 (a), the vertical COD

135

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
0.8

(a)

0.6
0.4

Nonlinear

0.2
0.0
-0.2
-0.4
-0.6
-0.8
0

10

20

30

40

50

30

(b)
20

LinearNonlinear

10
0
-10
-20
-30

Normalized Fourier amplitude

0

10

20

30

40

50

1.0

(c)
0.8

LinearNonlinear

0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 6.11.: Time variations of (a) tangential traction at the center point of a crack and (b) far-field
amplitude of backscattered SV wave when a penny-shaped crack is subjected to an
incident plane SV wave, and σ st /σ0 = 1.0. (c) is normalized frequency spectrum of
far-field amplitude.

in the nonlinear case cannot take negative values due to contact conditions. From Fig. 6.10 (b), thewaveform of ΩL /(au0 ) is distorted by the clapping motion. Therefore, the frequency spectrum ofΩL /(au0 ) in the nonlinear case contains higher-harmonic components as shown in Fig. 6.10 (c).

136

6.7. Numerical results

Saddle-shaped crack

F

Penny-shaped crack

G

(a)

(b)

Figure 6.12.: Elastic wave scattering by (a) a saddle-shaped crack and (b) two penny-shaped cracks
arranged in parallel.

Also in the 3-D case, it is confirmed that all (2nd, 3rd, 4th, etc.) orders of higher-harmonic waves
are generated.
Second, numerical results of SV wave incidence are presented when θin = 0◦ , kT a = 3.0,
and ug /u0 = 0.0. In this case, the static compressive stress σ st has a positive value over a crack
uniformly in order to generate the dynamic friction force. σ st is given by σ st /σ0 = 1.0 here. Timehistories of tangential traction at the center point of a crack are shown in Fig. 6.11 (a). Timehistories and normalized frequency spectra of far-field amplitude of backscattered SV wave are alsopresented in Fig. 6.11 (b), and (c), respectively. As shown in Fig. 6.11 (a), the tangential traction
becomes similar to square wave due to dynamic friction force, and its amplitude corresponds to the
dynamic friction coefficient µd . From Fig. 6.11 (b), the waveform of ΩT V /(au0 ) is distorted dueto the friction force, and the amplitude is smaller than the linear one. Therefore, the 3rd higherharmonic component is contained in the nonlinear case as shown in Fig. 6.11 (c). It is known that
odd order of higher-harmonic waves are generated due to pure shear motion with contact [89], andthat is also reproduced. From the above, we can confirm that the presented numerical method canaccurately simulate the clapping motion and dynamic friction on crack faces.

6.7.3. Sub-harmonic simulation
In order to confirm that the sub-harmonic generation can be caused in the 3-D model, we present
some numerical results in this section. The similar models to the ones in Section 5.5.3 is used forthe 3-D sub-harmonic simulation. Therefore, a saddle-shaped crack and two penny-shaped cracks
arranged in parallel in an infinite elastic solid depicted in Fig. 6.12 are considered as the analysis
models. The time increment is taken as f in ∆t = 0.05, and the material constants are same as thehigher-harmonic simulation in the previous section. The incident plane wave is assumed to be P137

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
16

(a)

14
12
10
8
6
4
2
0
0

1

2

3

4

5

16

(b)

14
12
10
8
6
4
2
0
0

1

2

3

4

5

Figure 6.13.: Frequency responses for the model in Fig. 6.12 (a) using the frequency-domain BEM
and assuming that crack faces are always separated. The norm ||[u]||/u0 at point F in
Fig. 6.12 (a) for various θin and kT a is plotted when (a) w/a = 1.0 and (b) 2.0.

wave, and φ = L in Eq. (6.33).
Sub-harmonic generation by a saddle-shaped crack
First, we presented numerical results of wave scattering by a saddle-shaped crack in an infinite
elastic solid as depicted in Fig. 6.12 (a). In the 2-D simulation, the nonlinear torsional vibration
is caused at the inner part of a curved crack, and sub-harmonic waves are generated under certain
conditions as presented in Section 5.5.3. Therefore, the similar model with finite width w in 3-D isconsidered. Only the cases that h/a = 2.0 and θin = 60◦ are presented here because sub-harmonic
waves are generated in a comparatively wide range of kT a and ug /u0 in the 2-D case as shown in
Fig. 5.16.
In advance of the sub-harmonic simulation, the frequency response analysis varying w/a is conducted in order to predict the probability of sub-harmonic generation. The norm ||[u]||/u0 at point
F in Fig. 6.12 (a) for various kT a and θin is plotted in Fig. 6.13. From Fig. 6.13 (a), the peak of||[u]||/u0 is located at about kT a = 1.2 in the case that w/a = 1.0. As shown in Fig. 6.13 (b),the results in the case that w/a = 2.0 are similar to the 2-D ones in Fig. 5.31 (e), and the peak138

6.7. Numerical results
0 . 25(a)
0 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

0 . 25(b)
0 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

Figure 6.14.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various preopening displacement ug /u0 when a saddle-shaped crack is subjected to an incident
plane P wave (w/a = 1.0, θin = 60◦ , and nin = 15).

of ||[u]||/u0 is located at about kT a = 0.7. Comparing Fig. 6.13 (a) and (b), ||[u]||/u0 is moregentle in the case that w/a = 1.0 than w/a = 2.0. Therefore, it is expected that the sub-harmonic
generation occurs in the case that w/a = 2.0 similarly to the 2-D case although that may be notcaused in the case that w/a = 1.0.
It is difficult to conduct the same number of numerical simulations as the 2-D case because of thecomputational cost for the 3-D simulation. Therefore, we present numerical results varying the preopening displacement ug /u0 for several center wavenumbers of the incident wave kT a. A0.5ω /Aω
for various ug /u0 in the case that w/a = 1.0 is presented in Fig. 6.14. kT a is changed from 1.0 to
5.0 with increments of 1.0 in Fig. 6.14 (a), and we could not confirm the sub-harmonic generation.
According to the tendency of the 2-D results in Section 5.5.3, the sub-harmonic generation occurs at
kT a which is a little larger than the double of the first peak of frequency response. In this case, thefirst peak of frequency response is located at about kT a = 1.2, so that kT a is changed from 2.2 to
2.8 with fine increments in Fig. 6.14 (b). Although we could not find the sub-harmonic generation
here, A0.5ω /Aω in the case that kT a = 2.8 becomes very large as shown in Fig. 6.14 (b). Therefore,
A0.5ω /Aω only in this case is plotted in Fig. 6.15 with a different vertical axis. The maximum value139

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
1.0
0.8
0.6
0.4
0.2
0.0
0

1

2

3

4

5

Figure 6.15.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various preopening displacement ug /u0 when a saddle-shaped crack is subjected to an incident
plane P wave (kT a = 2.8, w/a = 1.0, θin = 60◦ , and nin = 15).

4

(a)

3
2
1
0
-1
-2
-3
-4

Normalized Fourier amplitude

0

20

40

60

80

100

(b)

2.0
1.5
1.0
0.5
0.0
0.0

2.8

5.6

8.4

11 . 214 . 0Figure 6.16.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane P
wave (kT a = 2.8, ug /u0 = 3.4, w/a = 1.0, θin = 60◦ , and nin = 15).

of A0.5ω /Aω is about 0.95 as we can see from Fig. 6.15. In order to investigate what happened here,
the time history and frequency spectrum of far-field amplitude of backscattered P wave are shown140

6.7. Numerical results
0 . 25(a)
0 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

0 . 25(b)
0 . 200 . 150 . 100 . 050 . 00
0

1

2

3

4

5

Figure 6.17.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various preopening displacement ug /u0 when a saddle-shaped crack is subjected to an incident
plane P wave (w/a = 2.0, θin = 60◦ , and nin = 15).

in Fig. 6.16 when kT a = 2.8, ug /u0 = 3.4. The frequency spectrum is normalized by Aω . From
Fig. 6.16 (a), the waveform of ΩL /(au0 ) is strongly distorted, and the amplitude is very small inthe middle. The frequency spectrum is also distorted, and the peak is unstable and shifted as shownin Fig. 6.16 (b). As a result, Aω is small, and A0.5ω /Aω becomes large.
Next, A0.5ω /Aω for various ug /u0 in the case that w/a = 2.0 is presented in Fig. 6.17. In Fig.
6.17 (a), kT a is changed from 1.0 to 5.0 with increments of 1.0, and we could not confirm the
sub-harmonic generation from these cases. The first peak of frequency response is located at about
kT a = 0.7, and therefore, kT a is changed from 1.2 to 1.8 with fine increments in Fig. 6.17 (b).
There are two slight peaks at around kT a = 1.2 and ug /u0 = 2.6, and kT a = 1.4 and ug /u0 = 1.6
as indicated by arrows, and we can confirm the sub-harmonic generation under these conditions. For
instance of the sub-harmonic generation, the time history and normalized frequency spectrum of farfield amplitude of backscattered P wave are shown in Fig. 6.18 when kT a = 1.4 and ug /u0 = 2.6.
As shown in Fig. 6.18, the large and small amplitude is periodically repeated, and the slight peakof A0.5ω /Aω is confirmed. However, they are not clear because nin = 15. In order to show moreclear sub-harmonic generation, the waveform and frequency spectrum in the case that nin = 30 are

141

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
15

(a)
10
5
0
-5
-10
-15

Normalized Fourier amplitude

0

50

100

150

200

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

1.4

2.8

4.2

5.6

7.0

Figure 6.18.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane P
wave (kT a = 1.4, ug /u0 = 2.6, w/a = 2.0, θin = 60◦ , and nin = 15).

presented in Fig. 6.19. It can be confirmed from Fig. 6.19 (a) that the waveform becomes stable inthe state that large and small amplitude is repeated periodically. The more clear peak of A0.5ω /Aω
can be seen in Fig. 6.19 (b) than in Fig. 6.18 (b). A0.5ω /Aω in this case is much smaller than thatin the 2-D case. However, it is expected that A0.5ω /Aω becomes large as w/a increases because thecase that w/a → ∞ is equivalent to the 2-D one. The sub-harmonic generation can be caused forthe model in Fig. 6.12 (a) with sufficiently large w/a.
Sub-harmonic generation by two penny-shaped cracks arranged in parallel
Second, we present numerical results of the wave scattering by two penny-shaped cracks arrangedin parallel in an infinite elastic solid as depicted in Fig. 6.12 (b). In the 2-D simulation, the nonlinear
bending vibration is caused at the part between two straight-line cracks, and sub-harmonic waves
are generated under certain conditions as presented in Section 5.5.3. Therefore, it is expected thatsimilar bending vibrations are generated in the 3-D case for the model in Fig. 6.12 (b). Only thenormal incidence is considered here because the sub-harmonic waves can be observed in the 2-D
case except θin = 90◦ .
In advance of the sub-harmonic simulation, the frequency response analysis varying h/a is con-142

6.7. Numerical results
15

(a)
10
5
0
-5
-10
-15

Normalized Fourier amplitude

0

50

100

150

200

250

300

350

400

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0.0

1.4

2.8

4.2

5.6

7.0

Figure 6.19.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when a saddle-shaped crack is subjected to an incident plane P
wave (kT a = 1.4, ug /u0 = 2.6, w/a = 2.0, θin = 60◦ , and nin = 30).

30
25
20
15
10
5
0
0

1

2

3

4

5

Figure 6.20.: Frequency responses for the model in Fig. 6.12 (b) using the frequency-domain BEM
and assuming that crack faces are always separated. The norm ||[u]||/u0 at point G in
Fig. 6.12 (b) for various h/a and kT a is plotted when θin = 0◦ .

ducted in order to predict the probability of sub-harmonic generation. The norm ||[u]||/u0 at point
G in Fig. 6.12 (b) for various kT a is plotted in Fig. 6.20. From Fig. 6.20, the results are very similar143

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
0 . 16(a)0 . 140 . 120 . 100 . 080 . 060 . 040 . 020 . 00
0

1

2

3

4

5

0 . 16(b)

0 . 140 . 120 . 100 . 080 . 060 . 040 . 020 . 00
0

1

2

3

4

5

0 . 16(c)
0 . 140 . 120 . 100 . 080 . 060 . 04
0

1

2

3

4

5

Figure 6.21.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when two penny-shaped cracks arranged in parallel
are subjected to an incident plane P wave (h/a = 0.5, θin = 0◦ , and nin = 15).

to the 2-D ones in Fig. 5.32. Therefore, it is expected that sub-harmonic waves are generated in thecases that h/a = 0.5 and 1.0.
A0.5ω /Aω for various ug /u0 in the case that h/a = 0.5 is presented in Fig. 6.21. In Fig. 6.21(a), kT a is changed from 1.0 to 5.0 with increments of 1.0, and a peak of A0.5ω /Aω is observed
around kT a = 3.0 and ug /u0 = 3.2. In order to investigate the transition of A0.5ω /Aω varying

144

6.7. Numerical results
15

(a)
10
5
0
-5
-10
-15

Normalized Fourier amplitude

0

20

40

60

80

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 6.22.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when two penny-shaped cracks arranged in parallel are subjected
to an incident plane P wave (kT a = 3.0, ug /u0 = 3.2, h/a = 0.5, θin = 0◦ , andnin = 15).

kT a, A0.5ω /Aω for kT a = 2.2 to 2.8 and 3.2 to 3.8 with fine increments are shown in Fig. 6.17(b) and (c), respectively. There are several peaks of A0.5ω /Aω in Fig. 6.17 (b) and (c), specificallyaround kT a = 2.8 and ug /u0 = 3.4, kT a = 3.2 and ug /u0 = 2.8, kT a = 3.4 and ug /u0 = 3.2,
and kT a = 3.6 and ug /u0 = 2.8 as indicated by arrows. Under these conditions, the sub-harmonic
generation can be confirmed. For instance of the sub-harmonic generation, the time history andnormalized frequency spectrum of far-field amplitude of backscattered P wave are shown in Fig.
6.22 when kT a = 3.0, ug /u0 = 3.2. As shown in Fig. 6.22, the large and small amplitude isperiodically repeated around the last part, and the peak of A0.5ω /Aω is confirmed. However, they
are not clear because nin = 15. In order to show more clear sub-harmonic generation, the waveformand frequency spectrum in the case that nin = 30 are presented in Fig. 6.23. From Fig. 6.23 (a), it isconfirmed that the waveform becomes stable in the state that large and small amplitude is repeatedperiodically. The more clear peak of A0.5ω /Aω can be seen in Fig. 6.23 (b) than in Fig. 6.22 (b).
Next, A0.5ω /Aω for various ug /u0 in the case that h/a = 1.0 is presented in Fig. 6.24. In Fig.
6.24 (a), kT a is changed from 1.0 to 5.0 with increments of 1.0, and we could not confirm the
sub-harmonic generation in these cases. The first peak of the frequency response is located at about145

6. 3-D Simulation of Nonlinear Ultrasonic Waves Due to Interior Cracks
15

(a)
10
5
0
-5
-10
-15

Normalized Fourier amplitude

0

50

100

150

1.0

(b)
0.8
0.6
0.4
0.2
0.0
0

3

6

9

12

15

Figure 6.23.: (a) Time variation and (b) normalized frequency spectrum of far-field amplitude ofbackscattered P wave when two penny-shaped cracks arranged in parallel are subjected
to an incident plane P wave (kT a = 3.0, ug /u0 = 3.2, h/a = 0.5, θin = 0◦ , andnin = 30).

kT a = 1.6, and therefore, kT a is changed from 3.2 to 3.8 with fine increments in Fig. 6.24 (b).
However, We could not find the sub-harmonic generation in these cases. As a result, it seems to beharder for the 3-D model in Fig. 6.12 (b) than the 2-D model in Fig. 5.11 (b) that the sub-harmonic
generation is caused. This may be because the width of cracks is finite, and the effect of multiple
reflection is weaker in the 3-D case than the 2-D case.

6.8. Concluding remarks• In Chapter 6, we have successfully applied the new CQ-FMBEM to the 3-D nonlinear ultrasonic simulation for cracks in an infinite elastic solid. The validity of our proposed method
was confirmed through numerical results of normal incidence of P and S waves to a pennyshaped crack. We have also succeeded in the simulation of sub-harmonic generation due tointerior cracks with contact boundary conditions in 3-D.
• From numerical results for the linear problem, the accuracy and efficiency of the CQ-FMBEM
for crack problems were confirmed. The computational complexity of the CQ-FMBEM is146

6.8. Concluding remarks
0 . 16(a)0 . 140 . 120 . 100 . 080 . 060 . 040 . 020 . 00
0

1

2

3

4

5

0 . 16(b)
0 . 140 . 120 . 100 . 080 . 060 . 04
0

1

2

3

4

5

Figure 6.24.: Fourier amplitude ratio of sub-harmonic frequency to fundamental one for various
pre-opening displacement ug /u0 when two penny-shaped cracks arranged in parallel
are subjected to an incident plane P wave (h/a = 1.0, θin = 0◦ , and nin = 15).

O(N Me log N log Me ). The far-field contributions can be rapidly calculated for the Galerkinformulation because double integrals with respect to source and field points are separated in
the multipole expansion.
• For the case that a saddle-shaped crack is subjected to an incident plane P wave, we conducted the numerical simulation with two kinds of finite width w. We could observe the subharmonic generation in the case that w/a = 2.0 from numerical results but not for w/a = 1.0.
As a result, it has been confirmed that the sub-harmonic generation occurs if w is sufficiently
large.
• For the case that two penny-shaped cracks arranged in parallel are subjected to an incident
plane P wave, we conducted the simulation with two kinds of interval h. It could also befound that the sub-harmonic generation is caused if sufficiently small h similarly to the 2-D
case. However, we could not see the sub-harmonic generation in the case that h/a = 1.0.
Therefore, the finite width may also be significant in this case.

147

Chapter 7

Final Remarks
In this dissertation, we have investigated the acceleration of CQ-BEM for 3-D acoustic and elastic wave problems. The proposed fast CQ-BEM is accurate and efficient for the large-scale wave
problems especially dealing with the (semi-) infinite domain. The application of the fast CQ-BEM
to the nonlinear ultrasonic simulation has also been performed, and the behavior of nonlinear ultrasonic waves has been investigated through numerical results. The results obtained by this study aresummarized as follows:
Chapter 1 We have explained the background of this study about the NLUT based on CAN and
the TD-BEM. To investigate the behavior of higher-harmonic generation and the mechanismof sub-harmonic one, we focused numerical simulation as the theoretical approach. In theprevious researches, the behavior of higher-harmonic generation due to an interface crack has
not been investigated in full 3-D, and there are a few reports of sub-harmonic simulation.
Therefore, it is necessary to carry out these simulations and to quantitatively analyze them.
For this type of problems, the TD-BEM is considered as an appropriate numerical method because of its accuracy and efficiency. However, the conventional TD-BEM has some problems
such as instability in time-marching process and large computational complexity. Therefore,
we mentioned the importance of the stabilization and acceleration of the TD-BEM in thischapter.
Chapter 2 The IRK-based CQ-BEM formulation and its important aspects are presented in thischapter. This numerical method produces more accurate numerical solutions than the linear
multistep-based one although the required memory and computational time increase in thesame N and Me . However, it is sometimes difficult to satisfy accuracy requirements using
the linear multistep-based CQ-BEM. Therefore, we strongly recommend to use the IRK-based
CQ-BEM because of its high performance in accuracy.
Chapter 3 We have successfully accelerated the IRK-based CQ-BEM in this chapter. The originalcomputational complexity and required memory of the IRK-based CQ-BEM are O(m2 N 2 Me2 )even if we use an iterative solver. As the result of the acceleration based on the FMM andthe rapid convolution algorithm, the computational complexity and required memory can be149

7. Final Remarks
reduced to O(mN Me log N log Me ) and O(mN Me ), respectively. We have also presented
its applicability to the elastic wave and coupled acoustic-elastic wave problems.
Chapter 4 The 3-D simulation of nonlinear ultrasonic waves due to an interface crack with contact boundary conditions between two semi-infinite elastic solids, has been performed in thischapter. The appropriate boundary element formulation for this problem has been proposed inorder to accurately compute the tangential velocity of displacement. We have been carried out
some numerical simulations of the higher-harmonic generation and validated the numericalmodel of interface crack through obtained numerical results. For normal incidence of P and S
waves, the reflection and transmission coefficients between two materials significantly effectin the static compressive stress. For oblique incidence of P wave, the amplitude of scatteredwave rapidly fluctuates due to the behavior of reflection and transmission coefficients near thecritical angle.
Chapter 5 The long-time analyses of the nonlinear ultrasonic generation by interior and surface
breaking cracks in 2-D have been implemented by the non-accelerated CQ-BEM. Verification and validation of the proposed CQ-BEM have been conducted by solving the fundamental problems. From the simulation of higher-harmonic waves due to a straight-line crack, it
was found that there is optimal static compressive stress for the 2nd higher-harmonic generation related with the normalized wavenumber of the incident wave. On the other hand, we
have successfully conducted the numerical simulation of the sub-harmonic generation undercertain conditions. From numerical results, we have investigated the relation between thecharacteristic of frequency response and the probability of sub-harmonic generation. As aresult, the sub-harmonic generation is caused due to interior cracks with some pre-opening
displacements when the frequency of the incident wave is a little larger than the first modequasi-eigenfrequency of the linear system. However, the behavior of sub-harmonic generation due to a surface breaking crack is different, and that occurs when the frequency of theincident wave is about triple of the first mode quasi-eigenfrequency of the linear system.
Chapter 6 The 3-D simulation of nonlinear ultrasonic waves due to cracks in an infinite elastic
solid has been conducted by the fast CQ-BEM presented in Chapter 3. Therefore, the fast
CQ-BEM has been extended to the 3-D crack problem and verified by solving linear problems. We also presented the computational complexity through some numerical results. The
crack model has also been validated by solving the fundamental nonlinear problems such as apenny-shaped crack subjected to normal incidence of P and S waves. From the numerical results of the nonlinear ultrasonic simulation, it was confirmed that the sub-harmonic generation
is caused under the similar conditions to the 2-D cases.
In this study, we succeeded in reproducing the sub-harmonic waves due to the CAN in the 2-D
and 3-D numerical simulations as presented in Chapters 5 and 6. As a result, the sub-harmonic
generation greatly depends on the situation such as size, shape, and arrangement of cracks andexistence of free-surface. Therefore, it is difficult to use these phenomena for the NDE if there is150

no information of flaws in advance of measurement, and the applicability is limited. However, it isimportant to completely reveal the mechanism and to make the range of applicability clear. Also, it
may be expected that the sub-harmonic generation phenomenon is used for other engineering fields.
In the future, we are going to extend the proposed 3-D numerical method of nonlinear ultrasonicsimulation to the scattering problems by a surface breaking crack similarly to the 2-D case. However, for performing many numerical simulations, the proposed CQ-BEM is not sufficiently fast.
Therefore, we are going to continue the acceleration of the CQ-BEM. Also, we should compare
obtained numerical results with experimental data and consider the consistency between numericaland experimental approaches.
The FDM and FEM are usually used for the time-dependent (nonlinear) problems because of
their robustness, computational efficiency, and ease of formulation. On the other hand, the stabilityof TD-BEM has been improved and accelerated by some special techniques recently as shown inthis dissertation. The author hopes that the TD-BEM is more widely used in the engineering fields.

151

Appendix A

Translation Operators and Recursions
The translation operators for the fundamental solution of the modified Helmholtz equation (E|F)m,jn,l
are obtained using the analogous manner as the Helmholtz equation [23].
(M|M)b,da,c (y0 → y1 ) =

a+c
∑

a,c,e
(−1)d Ac,e W−b,d,b−d
Fe,d−b (y0 − y1 ),(A.1)

e=|a−c|(M|L)b,da,c (y1 → x1 ) =

a+c
∑

a,c,e
(−1)a+d Ac,e W−b,d,b−dOe,d−b (x1 − y1 ),(A.2)

e=|a−c|(L|L)b,da,c (x1 → x0 ) =

a+c
∑

a,c,e
(−1)d Ac,e W−b,d,b−d
Fe,d−b (x1 − x0 ),(A.3)

e=|a−c|a,c,ewhere Aa,b = (2a + 1)(2b + 1) and Wb,d,f
is expressed using Wigner-3j symbol ( ·· ·· ·· ) [57] asfollows:
(
)(
)
a c e
a c e
a,c,e
Wb,d,f =
.
(A.4)
0 0 0
b d f

The recursions of (E|F)ba,c are following [28]:
b
b
b
b
−
+
−
a+
ab (E|E)a+1,c = −aab (E|E)a−1,c + ac−1,b (E|E)a,c−1 + ac+1,b (E|E)a,c+1 ,

(A.5)

b+1
b+1
b
b
−
+
−
b+
a,−b (E|E)a+1,c = ba,−b (E|E)a−1,c + bc−1,−b (E|E)a,c−1 − bc+1,−b (E|E)a,c+1 ,

(A.6)

b
b
b
b
−
+
−
a+
ab (M|L)a+1,c = −aab (M|L)a−1,c − ac−1,b (M|L)a,c−1 − ac+1,b (M|L)a,c+1 ,

(A.7)

b+1
b+1
b
b
−
+
−
b+
a,−b (M|L)a+1,c = ba,−b (M|L)a−1,c − bc−1,−b (M|L)a,c−1 + bc+1,−b (M|L)a,c+1 ,

(A.8)

−
−
−
where coefficients a+
ab , aab , bab , and bab are defined as follows:√(a + b + 1)(a − b + 1),
2a + 1√(a − b + 1)(a − b + 2)
b+
,
ab =
2a + 1a+ab =√(a + b)(a − b),
2a + 1√(a + b)(a + b − 1)
b−
.
ab =
2a + 1
a−
ab =

(A.9)
(A.10)

153

A. Translation Operators and Recursions
The initial values for the recursion (E|F)00,c are given as follows:(M|M)00,c = (2c + 1)ic (sφ |y0 − y1 |),(A.11)(M|L)00,c = (2c + 1)kc (sφ |y1 − x1 |),(L|L)00,c = (2c + 1)ic (sφ |x1 − x0 |).

(A.12)(A.13)

Using the relations (E|F)ba,c = (E|F)−ba,c ,(E|E)ba,c =2c + 1(E|E)bc,a ,2a + 1(M|L)ba,c =2c + 1(−1)a+c (M|L)bc,a ,2a + 1(A.14)

and recursions (A.5)-(A.8) with appropriate computational procedure, the translation operators(E|F)ba,c can be calculated in O(Nt3 ) computational time. Then, the scaling of the translation operators (E|F)ba,c can be calculated to replace ic (z) and kc (z) in Eqs. (A.11)-(A.13) with ĭc (z) andk̆c (z), respectively. Notice that the plus or minus signs in these recursions are different from theones [28] for the Helmholtz equation.

154

Appendix B

Far-field Approximation of ScatteredWaves
We explain the calculation method of far-field approximation of scattered waves.

B.1. Inclusions and cracks in 2-D and 3-D
Replacing the fundamental solutions in the integral representation of the scattered wave with the
far-field approximated ones, the far-field scattered waves can be evaluated. Therefore, the far-field
scattered wave usc;far from inclusions with surface S and cracks with one side of crack faces S + in
an infinite elastic solid can be described as follows:∫ t∫
U far (x, y, t − τ ) · t(y, τ )dSy dτ
usc;far (x, t) =
0
S
∫ t∫
−
T far (x, y, t − τ ) · u(y, τ )dSy dτ,
(B.1)
0
S
∫ t∫
usc;far (x, t) = −T +;far (x, y, t − τ ) · [u](y, τ )dSy dτ,
(B.2)
0

S+

where the superscript ”far” indicates the far-field approximated function. Defining that x = |x| andx̂ = x/x and considering x ≫ |y| as the far-field condition, we can approximate as r ≃ x − x̂ · yand ∂r/∂xi ≃ x̂i . Consequently, the far-field approximated fundamental solutions corresponding
to displacement and traction for 2-D elastodynamic problems in time-domain can be respectively
obtained as follows:

(
)
√ 
x−x̂·y


2
H
t
−
cL
1
2 cT
√(
Uijfar (x, y, t) = √)()
2 x̂i x̂j

π
c
µ 8πx

x−x̂·y
t
1
 L
+
t
−
x
cL
cL

(
)

x−x̂·y
H t − cT+ (δij − x̂i x̂j ) √()() , (B.3)x−x̂·y 
t
1

+
t−
x

cT

cT

155

B. Far-field Approximation of Scattered Waves

√ 
)
(
(
)
x − x̂ · y1
2  c2L − 2c2T δik x̂j + 2c2T x̂i x̂j x̂kfar√
Tij (x, y, t) = − √)() nk (y)δ t − 3 (tcL
8πx π x−x̂·y1
 cLt − cLx + cL()δjk x̂i + δij x̂k − 2x̂i x̂j x̂kx − x̂ · y 
.
(B.4)
+ √(
)(
) nk (y)δ t −
cT
x−x̂·y
t
1

cT
+
t−
x

cT

cT

Similarly, the counterparts for 3-D problems are given by
[ 2
(
)
(
)]
cT
1
x − x̂ · y
x − x̂ · yx̂i x̂j δ t −+ (δij − x̂i x̂j ) δ t −,4πxµ c2L
cL
cT
{[
( 2
)
]
(
)
c2T
cL
1
x − x̂ · yfarx̂i 2 − 2 δjk + 2x̂i x̂j x̂k nk (y)δ̇ t −Tij (x, y, t) = −
4πx
cL
c3L
cT
(
)}
]
x − x̂ · y1[.
δik x̂j + δij x̂k − 2x̂i x̂j x̂k nk (y)δ̇ t −+cT
cT

Uijfar (x, y, t) =

(B.5)

(B.6)

Substituting Eqs. (B.3)–(B.6) into Eqs. (B.1) and (B.2) yields the following forms because thefar-field scattered wave can be completely separated into P, SV, SH wave components in 3-D and P
and SV (S) ones in 2-D.
usc;far (x, t) =
[
(
)
(
)
(
)]

1
x
x
x


x̂ΩL x̂, t −+ v̂ΩT V x̂, t −+ ĥΩT H x̂, t −for 3-D,


cL
cT
cT
 4πx
[
(
)
(
)]


1
x
x


x̂ΩL x̂, t −+ v̂ΩT x̂, t −
 √
cL
cT
8πx

(B.7)
for 2-D,

where v̂ and ĥ are the displacement vectors of SV and SH waves, respectively. Ωφ (φ = L, T V ,or T H) is the far-field amplitude and indicates the invariable waveform of the scattered wave at far√field. The scattered waves at far-field are attenuated by distance in order of O(1/ x) and O(1/x)
for 2-D and 3-D cases, respectively, as shown in Eq. (B.7).
If we use the CQM formulation for evaluation of the far-field scattered waves, the equivalentscattered far-field can be obtained by using the far-field approximated Laplace-domain fundamental
solutions. The approximated 2-D Laplace-domain fundamental solutions are obtained by using theasymptotic behavior of the modified Bessel function K0 as follows:√
K0 (sφ r) ∼[ππ ]exp −sφ x + i exp [sφ x̂ · y] .
2isφ x4(B.8)

On the other hand, the 3-D counterparts are obtained by using the approximations r ≃ x − x̂ · yand ∂r/∂xi ≃ x̂i .

156

B.2. Interface cracks between two semi-infinite elastic solids in 3-D
Interface crack

Figure B.1.: Interface crack of two-layered elastic solid.

B.2. Interface cracks between two semi-infinite elastic solids in
3-D
The far-field scattered waves by an interface crack are evaluated in Chapter 4. Therefore, the corresponding far-field approximated Green’s function is shown. The far-field scattered waves can beevaluated by replacing T + in Eq. (B.2) with Green’s function for the two-layered semi-infinite
elastic solid. The far-field approximated Green’s functions can be obtained from the integral representation of the fundamental solutions using the ray theory [2] and the stationary phase method[80]. The ray theory has been applied to the BEM in order to improve the accuracy and efficiency
by Kimoto and Hirose [41].
Let S I and S II be each side of crack faces as depicted in Fig. B.1. The integral form of far-field
scattered wave from interface cracks is given byusc;far;I(II)(x, t) = −∫ t∫
0S I(II)

H far;II(I)→I(II) (x, y, t − τ ) · [u](y, τ )dSy dτ,(B.9)

where H far;I→II indicates the far-field approximation of traction Green’s function for the twolayered elastic solid when the source and field points are located in DI and DII , respectively. Considering that inhomogeneous waves are generated over the critical angles, the scattered waves shouldbe evaluated in frequency-domain. Therefore, the approximations of frequency-domain Green’s
function are shown here. The far-field approximation of frequency-domain traction Green’s function for the two-layered elastic solid can be written as follows:
I(II)→II(I);farH̃ip(x, y, ω) = −∑∑nj (y)Cpjkq ζq±I(II)α=L,T V β=L,T V 4πxµ

I(II) cI(II)
α



 x
±
ζ · y ±α,βiω×A±α,β
d
iω
exp
−
i
k
I(II)
 cII(I)
cα 


β

c

nj (y)Cpjkq ζq±
I(II)

−

[

H,T H
A±T H,T H d±Tiω exp iω
i
I(II) k
I(II)
cT
4πxµ

{

ζ± · y
−
II(I)
I(II)
cT
cT
x

I(II)

I(II)

β

c
β

cα
ζ1 = cαII(I) x̂1 ,ζ2 = II(I) x̂2

}]

,
I(II)
I(II)
c
c
ζ1 = ITI(I) x̂1 ,ζ2 = T
II(I) x̂2
cT
cT

(B.10)

157

B. Far-field Approximation of Scattered Waves

Surface breaking crack

Figure B.2.: Surface breaking crack in elastic half-space.

where
(
)
I(II) 2
cT
|x̂ |

α,β
A±α,β
= TI(II)→II(I)(ζ ± ) I(II) II(I)
d±α .
k
ν k
cα cβ
3

(B.11)The superscripts on the material constants such as Cijkl and µ indicate each domain. In Eq. (B.11),α,βTI(II)→II(I)(ζ ± ) is the transmission coefficient with propagation vector of the incident wave ζ ±when the wave mode is changed from α to β, and the wave is propagating from DI(II) to DII(I) .
The propagation vector ζ ± is defined by
ζ ± = (ζ1 , ζ2 , ±ν) ,ν=√

1 − (ζ1 )2 − (ζ2 )2 ,(B.12)and d±α and d±α,β are the displacement vector before and after the transmission, respectively. The
sign ± has to be select to converge exp[·] in Eq. (B.10). The Laplace-domain counterpart of Eq.
(B.10) can be obtained by replacing ω with is. Only for y3 = 0, the closed form of time-domaincounterpart of Eq. (B.10) is available. Therefore, the time-domain expression can be used for thecase in Chapter 4.

B.3. Surface breaking cracks in 2-D elastic half-space
The far-field scattered waves by surface breaking cracks in an elastic half-space are evaluated in
Chapter 5. Therefore, the corresponding far-field approximated Green’s function is shown. Let Sc
be one side of crack faces as depicted in Fig. B.2. The integral form of far-field scattered wave fromsurface breaking cracks in an elastic half-space is given byusc;far(x, t) = −∫ t∫
0

H far (x, y, t − τ ) · [u](y, τ )dSy dτ,(B.13)

Sc

where H far indicates the far-field approximation of traction Green’s function for the elastic halfspace. The far-field approximation of frequency-domain traction Green’s function for the elastic

158

B.3. Surface breaking cracks in 2-D elastic half-spacehalf-space can be written as follows:∑∑nj (y)Cpjkq ζq±√µcα 8πkβ x
α=L,T V β=L,T V
[ {
}
]
x
ζ± · yiπ±α,β ∓α,β×Bk diiω exp iω
−
+
,
cβ
cα
4 ζ1 = cα x̂1

far
far
H̃ip
(x, y, ω) = T̃ip
(x, y, ω) −(B.14)cβ2±α,βα,β ± (cT ) |x̂3 | ±α
Bkd ,= R (ζ )cα cβ ν k

(B.15)

where T̃ far is the far-field approximation of the full-space traction fundamental solution. ζ ± and ν
for 2-D are different from the previous 3-D ones and they are defined by
ζ ± = (ζ1 , ±ν) ,ν=√

1 − (ζ1 )2 .

(B.16)

159

Appendix C

Free-field for two-layered elastic solid
We explain the calculation method of the free-field for a two-layered semi-infinite elastic solid
subjected to the incident plane wave as depicted in Fig. C.1. In this study, the nin cycles sinusoidal
burst plane wave with amplitude u0 such as Eq. (4.19), is utilized for the incident wave in the
nonlinear ultrasonic simulations. If the incident wave uin is emitted from DI , it can be written
using the center circular frequency ω in as follows: in)( I) ( I) (
in
I
in

 u (x, t) = u0 dφ sin Λφ H Λφ H 2πn − Λφ ,()
pin · (x − χ)
I
in

t−
.
 Λφ = ωcIφ(C.1a)(C.1b)

If the incident angle is smaller than the critical one, it is easy to evaluate the propagation vectors
after reflection and transmission using Snell’s law. In this case, the reflection and transmission
coefficients can be calculated easily from the given interface conditions and must be real values[73]. Consequently, the free-field can be calculated in time-domain directly. However, it is difficult to evaluate the free-field in time-domain if the incident angle is over the critical one. Therefore, we numerically construct the time-domain free-field using the inverse Fourier transform of thefrequency-domain one.
Taking into account that nin is integer, the Fourier transform of the Eq. (C.1) is given by[]
F uin (x, t) (x, ω) =∫ ∞uin (x, t)eiωt dt
−∞
}
{(
[ in
])
in
[ iω
]
ω
u0 din2n iπω
φin=
1 − exp
exp I p · (x − χ) ,ω incα
(ω in )2 − ω 2(C.2)

where ω is the circular frequency, and F indicates the Fourier transform. Assuming continuity ofdisplacement and traction on the interface, the following displacement fields of the reflected andtransmitted waves in frequency-domain can be obtained by solving the reflection and transmission
problem of an incident plane wave for a two-layered elastic solid.

161

C. Free-field for two-layered elastic solid
Transmitted SV wave

Transmitted P wave

Incident P wave
Reflected P wave
Incident SV wave
Reflected SV wave

Figure C.1.: Reflection and transmission of the incident plane wave on the interface between twoelastic solids.

[]
F uref;I (x, t) (x, ω) =∑in
u0 drefβ ωRφ,β (pin )in )2 − ω 2(ωβ=L,T V
{(
}
[ in
])
]
[ iω
2n iπω
ref1 − exp×
exp I pβ · (x − χ) ,
ω in
cβ
∑

u0 dtransω inββ=L,T V(ω in )2 − ω 2][
F utrans;II (x, t) (x, ω) =

{(

×

[

(C.3)

T α,β (pin )2nin iπω1 − exp
ω in

])
exp

[ iω

]

ptrans· (x − χ)
β
cII
β

}
,

(C.4)

where Rφ,β (pin ) and T φ,β (pin ) are the reflection and transmission coefficients, respectively, when
the incident plane wave is propagating to the pin direction and its mode is changed from φ to β.
The time-domain free-field can be evaluated by means of the inverse Fourier transform after the
calculation of Eqs. (C.3) and (C.4). For elastic half-space, the free-field can be evaluated in asimilar way.
We show the reflection and transmission coefficients for the steel-aluminum interface which isused in Chapter 4. Therefore, the material constants in Table 4.1 are used. The positive directionsof propagation and vibration are shown as the big and small arrows in Fig. C.1, respectively. The
detailed calculation method of the reflection and transmission coefficients can be found in e.g. [3,80, 73].
The real and imaginary parts of reflection and transmission coefficients for P wave incidence
are shown in Fig. C.2. As shown in Fig. C.2, there is a critical angle which corresponds to thetransmitted P wave when θin ≃ 65◦ . The similar graphs corresponding to SV wave incidence areshown in Fig. C.3. There are two critical angles which correspond to the transmitted and reflected
P waves when θin ≃ 29◦ and 32◦ , respectively. If the reflection and transmission coefficients arecomplex values, the corresponding phase shift is caused.

162

3

(a)
Real part

2
1
0
-1
-2
0

10

20

30

40

50

60

70

80

90

3

Imaginary part

(b)
2
1
0
-1
-2
0

10

20

30

40

50

60

70

80

90

Figure C.2.: (a) Real parts and (b) imaginary parts of reflection and transmission coefficients of P
wave incidence for various incident angles θin .
2(a)Real part

1

0

-1

-2
0

10

20

30

40

50

60

70

80

90

2

Imaginary part

(b)
1

0

-1

-2
0

10

20

30

40

50

60

70

80

90

Figure C.3.: (a) Real parts and (b) imaginary parts of reflection and transmission coefficients of SV
wave incidence for various incident angles θin .

163

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