رسالة دكتوراة بعنوان Efficient Finite Element Modelling of Ultrasound Waves in Elastic Media

رسالة دكتوراة بعنوان
Efficient Finite Element Modelling of Ultrasound Waves in Elastic Media
by
Mickael Brice Drozdz
A thesis submitted to the University of London for the degree of
Doctor of Philosophy
IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
University of London
Department of Mechanical Engineering
Imperial College of Science Technology and Medicine
Table of contents
Chapter 1
Introduction

1. Introduction . 18
2. Objectives 19
3. Outline of the thesis . 22
   Chapter 2
   Theoretical background
4. Introduction . 24
5. Theory of wave propagation in elastic media . 24
   2.1 Bulk wave propagation 24
   2.1.1 Bulk wave propagation in infinite isotropic elastic media 24
   2.1.2 Bulk propagation in a semi-infinite isotropic elastic medium . 26
   2.2 Guided wave propagation in a plate 28
6. Finite elements modelling of wave propagation . 34
   3.1 Explicit method 35
   3.2 Implicit method 37
   3.2.1 ABAQUS/Standard procedure . 38
   3.2.2 COMSOL Multiphysics procedure . 40
7. Conclusions . 42
   Chapter 3
   Modelling waves in unbounded elastic media using absorbing layers
8. Introduction . 43
9. Review of non-investigated techniques 45
   2.1 Infinite element methods . 45
   2.2 Non reflecting boundary condition . 47
10. Absorbing layer theory 47
    3.1 Concept 47
    3.2 Perfectly matched layer (PML) 48
    3.3 Absorbing layer using increasing damping (ALID) 50
11. Efficient layer parameters’ definition . 53
    4.1 Analytical model for bulk waves . 54
    4.1.1 General definition 55
    4.1.2 Validation procedure 55
    4.1.3 PML analytical model . 57
    4.1.4 ALID analytical model 61
    4.2 Analytical model for 2D guided wave cases 66
    4.2.1 Consideration for guided wave PML implementation 66
    4.2.2 Validation procedure 67
    4.2.3 PML analytical model for guided wave cases 68
    4.2.4 ALID analytical model 71
12. Demonstrators . 77
    5.1 Computational efficiency 77
    5.1.1 Bulk wave model 77
    5.1.2 Guided wave model 80Table of contents
    7
    5.2 Time reconstruction 81
    5.3 Wave scattering 84
    5.3.1 Single-mode reflection coefficient . 84
    5.3.2 Multi-mode reflection coefficient 86
13. Discussion 88
14. Conclusion 89
    Chapter 4
    On the influence of mesh parameters on elastic bulk wave velocities
15. Introduction . 90
16. Explicit solving 91
    2.1 Introduction . 91
    2.2 Linear quadrilateral elements . 92
    2.2.1 Square elements . 92
    2.2.2 Rectangle elements 100
    2.2.3 Rhombus elements 102
    2.2.4 Parallelogram elements . 104
    2.2.5 Conclusion 105
    2.3 Linear triangular elements . 106
    2.3.1 Equilateral triangle elements . 106
    2.3.2 Isosceles triangle elements 110
    2.3.3 Scalene triangle elements . 112
    2.3.4 Conclusion 114
    2.4 Modified quadratic triangular elements . 114
    2.4.1 Equilateral triangle elements . 114
    2.4.2 Isosceles triangle elements 119
    2.4.3 Scalene triangle elements . 120
    2.4.4 Conclusion 122
17. Implicit solving . 123
    3.1 Introduction 123
    3.2 Linear quadrilateral elements 123
    3.2.1 Square elements 123
    3.2.2 Rectangle elements 124
    3.2.3 Rhombus elements 125
    3.2.4 Parallelogram elements . 126
    3.2.5 Conclusion 127
    3.3 Quadratic quadrilateral elements 128
    3.3.1 Square elements 128
    3.3.2 Rectangle elements 129
    3.3.3 Rhombus elements 130
    3.3.4 Parallelogram elements . 131
    3.3.5 Conclusion 132
    3.4 Linear triangular elements . 133
    3.4.1 Equilateral triangle elements . 133
    3.4.2 Isosceles triangle elements 133
    3.4.3 Scalene triangle elements . 134
    3.4.4 Conclusion 135
    3.5 Quadratic triangular elements 136
    3.5.1 Equilateral triangle elements . 136
    3.5.2 Isosceles triangle elements 137
    3.5.3 Scalene triangle elements . 138Table of contents
    8
    3.5.4 Conclusion 139
    3.6 Modified quadratic equilateral triangular elements 139
    3.6.1 Equilateral triangle elements . 139
    3.6.2 Isosceles triangle elements 140
    3.6.3 Scalene triangle elements . 141
    3.6.4 Conclusion 142
18. Conclusions 142
    Chapter 5
    Accurate modelling of defects using Finite Elements
19. Introduction 146
20. Model definition 148
21. Reflection from a straight edge . 150
22. Reflection from a straight crack at an angle . 156
    4.1 Crack of unit length 158
    4.2 Crack of length 0.25 . 161
    4.3 Crack of length 4 165
    4.4 Conclusion . 168
23. Reflection from circular defects 168
    5.1 Hole of unit diameter . 170
    5.2 Hole of diameter 0.25 173
    5.3 Hole of diameter 4 . 176
    5.4 Conclusion . 178
24. Conclusions 179
    Chapter 6
    Local mesh refinement
25. Introduction 181
26. Fictitious domain technique . 182
    2.1 Review 182
    2.2 Presentation 182
    2.3 Conclusion . 183
27. Abrupt mesh density variation . 183
    3.1 1D wave propagation models 183
    3.1.1 Model definition 183
    3.1.2 L wave 1D model using theoretical material properties 185
    3.1.3 L wave 1D model with matched acoustic impedance 186
    3.1.4 L and S wave 1D model with matched acoustic impedance . 187
    3.1.5 L and S wave 1D model with varying acoustic impedance . 190
    3.1.6 L wave 1D model with different mesh ratio . 193
    3.2 2D wave propagation models 195
    3.2.1 Model definition 195
28. Gradual mesh density variation . 199
    4.1 1D wave propagation model . 199
    4.2 2D wave propagation model . 201
29. Conclusions 202Table of contents
    9
    Chapter 7
    Conclusions
30. Review of thesis 204
31. Summary of findings . 205
    2.1 Absorbing layers 205
    2.2 Influence of mesh parameters on the elastic bulk wave velocities 207
    2.3 Accurate modelling of complex defects using Finite Elements 208
    2.4 Local mesh refinement . 209
32. Future work 210
    3.1 Absorbing layers 210
    3.2 Influence of mesh parameters on the elastic bulk wave velocities 210
    3.3 Accurate modelling of complex defects using Finite Elements 211
    3.4 Local mesh refinement . 211
    References10
    List of figures
    Figure 1.1 a) 2D plane strain model of a plate including a defect, b) Time signal at
    the monitoring point 19
    Figure 2.1 Modes considered and their orientation 25
    Figure 2.2 Geometry of a 2D plate . 27
    Figure 2.3 Typical deformation caused by symmetric (a) and anti-symmetric (b)
    modes 28
    Figure 2.4 Illustration of the deformation of a plate caused by a) propagating, b)
    propagating evanescent, c) evanescent waves which have a) real, b) complex, c) imaginary wave numbers . 31
    Figure 2.5 Phase velocity against frequency.thickness for a 3mm thick steel plate .
    32
    Figure 2.6 Wave number against frequency for a 3mm thick steel plate . 32
    Figure 2.7 Example of S0 mode shapes for a free plate case at different frequencies
    shown for a 3mm thick steel plate 33
    Figure 2.8 Illustration of DL for a a) linear square element, b) linear triangle element
    and c) quadratic triangle element 36
    Figure 3.1 a) 2D plane strain model of a plate including a defect, b) Time signal at
    the monitoring point 42
    Figure 3.2 Illustration of use of infinite elements 44
    Figure 3.3 ABAQUS benchmark model: a) Model geometry, b) vertical displacement at point A, Extended model (reference): c) Model geometry, d) vertical displacement at point A 45
    Figure 3.4 Absorbing layer concept for 2D models: a) infinite medium, b) semi infinite medium, c) plate 46
    Figure 3.5 Variation of αx(x) and αy(y) in a 2D model 48
    Figure 3.6 Spatial spread of the reflection and transmission for a single layer (no
    mode conversion shown for simplicity) 54
    Figure 3.7 Illustration of extreme angles defining the range of angles to consider
    when dimensioning an absorbing layer 54
    Figure 3.8 FE model used to validate the analytical models a) normal incidence
    model, b) angled incidence model 56
    Figure 3.9 a) Reflection coefficient against αx b) Reflection coefficient against the
    number of elements per wavelength . 57
    Figure 3.10 Reflection coefficient for a given PML obtained with bulk wave analytical and FE models . 59
    Figure 3.11 Reflection coefficient for a given ALID obtained with bulk wave analytical and FE models 64
    Figure 3.12 FE model used to validate the guided wave analytical models 67
    Figure 3.13 Reflection coefficient for a given PML obtained with guided wave analytical and FE models . 70
    Figure 3.14 Definition of the multi layered system . 71
    Figure 3.15 Reflection coefficient for a given ALID obtained with guided wave analytical and FE models . 76
    Figure 3.16 a) bulk wave demonstrator, FE model: b) without absorbing layer, c) with
    ALID, d) with PML 77List of figures
    11
    Figure 3.17 Absolute displacement field for the bulk demonstrator with ALID at
    time: a)5msec b)10msec c)15msec d)20msec. Colour scale extends from
    0 (blue) to 0.1% (red) of the maximum absolute displacement. Grey indicates out of scale (0.1% to 100%). White dashed line indicates the boundary between area of study and ALID 78
    Figure 3.18 a) guided wave demonstrator, FE model: b) without absorbing layer, c)
    with ALID, d) with PML 79
    Figure 3.19 Absolute displacement field for the guided demonstrator with ALID at
    time: a)150msec b)300msec c)450msec d)600msec. Colour scale is varied and extends from 0 (blue) to 2% or 10% (red) of the maximum absolute displacement as indicated on the figure. Grey indicates out of scale
    (2% or 10% to 100%). White dashed line indicates the boundary between
    area of study and ALID 80
    Figure 3.20 Input preprocessing . 81
    Figure 3.21 Model geometry for time reconstruction case 81
    Figure 3.22 Normal displacement monitored 700mm away from the defect. a) Classical time domain analysis with ABAQUS, b) Frequency domain analysis
    with ABAQUS, c) Frequency domain analysis with COMSOL 82
    Figure 3.23 a) dispersion curve data used for input definition, b) input definition . 82
    Figure 3.24 Representation of model used for guided wave scattering validation 83
    Figure 3.25 Example of a typical spatial FFT curve 83
    Figure 3.26 Reflection coefficient against notch width . 84
    Figure 3.27 Energy reflection coefficient for A0 incident on a 2mm square notch in
    an 8mm thick aluminium plate from 140kHz to 500kHz . 86
    Figure 4.1 Definition of the main feature of the model . 90
    Figure 4.2 a) Longitudinal and b) shear wave excitation for a square element mesh
    and c) longitudinal and d) shear excitation for a triangular elements mesh
    91
    Figure 4.3 Schematic defining L0, L90, L45 and Lθ in a mesh of square elements .
    92
    Figure 4.4 a) Longitudinal and b) shear velocity errors against CFL for various mesh
    densities at 0 degrees 93
    Figure 4.5 Velocity error against mesh density for shear and longitudinal waves at 0
    degree with a CFL of 0.025 95
    Figure 4.6 Velocity errors against CFLX for various mesh densities at 0 degrees 95
    Figure 4.7 Velocity error against mesh density for shear and longitudinal waves at 0
    and 45 degree 97
    Figure 4.8 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various values of mesh density plotted
    in polar (a and b) and linear (c and d) plots 97
    Figure 4.9 Velocity errors against CFLX for various mesh densities at 45 degrees .
    98
    Figure 4.10 Velocity error against the scaled Courant number CFLX and mesh density N . 99
    Figure 4.11 a) Shape of the different rectangular elements used in the mesh; Variation
    of the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various R plotted in a polar (b and c) and linear (d
    and e) fashion. The coloured circles indicate the error prediction along
    the element side and diagonal . 100List of figures
    12
    Figure 4.12 a) Shape of the different rhombic elements used in the mesh; Variation of
    the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various shearing angle g plotted in a polar (b and
    c) and linear (d and e) fashion. The coloured circles indicate the error prediction along the element side and diagonal . 102
    Figure 4.13 a) Shape of the different parallelogramatic elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various shearing angle g plotted in a polar (b and c) and linear (d and e) fashion. The coloured circles indicate the
    error prediction along the element side and diagonal . 104
    Figure 4.14 Schematic defining L0, L90, L30 and Lq in a mesh of equilateral-triangular elements . 105
    Figure 4.15 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various mesh densities plotted in a linear (a and b) and polar (c and d) fashion 106
    Figure 4.16 Velocity error against mesh density for shear and longitudinal waves at 0
    and 30 degrees 107
    Figure 4.17 Velocity errors against CFLX for various mesh densities at a) 0 and b) 30
    degrees 108
    Figure 4.18 a) Shape of the different isosceles-triangular elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of f plotted in a polar (b
    and c) and linear (d and e) fashion. The coloured circles indicate the error
    prediction along the element side and diagonal 110
    Figure 4.19 a) Shape of the different scalene-triangular elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion. The coloured circles indicate the error
    prediction along the element side and diagonal 112
    Figure 4.20 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various mesh densities plotted in a polar
    (a and b) and linear (c and d) fashion . 114
    Figure 4.21 Schematic defining L0, L90, L30 and Lq in a mesh of quadratic equilateral-triangular elements 114
    Figure 4.22 Velocity error against mesh density for shear and longitudinal waves at 0
    and 30 degrees 115
    Figure 4.23 Velocity errors against CFLX for various mesh densities at a) 0 and b) 30
    degrees 117
    Figure 4.24 a) Shape of the different quadratic isosceles-triangular elements used in
    the mesh; Variation of the longitudinal (b and d) and shear (c and e) velocity error against the angle of incidence for various values of f plotted
    in a polar (b and c) and linear (d and e) fashion. The coloured circles indicate the error prediction along the element side and diagonal . 118
    Figure 4.25 a) Shape of the different quadratic scalene-triangular elements used in the
    mesh; Variation of the longitudinal (b and d) and shear (c and e) velocity
    error against the angle of incidence for various values of g plotted in a polar (b and c) and linear (d and e) fashion. The coloured circles indicate the
    error prediction along the element side and diagonal . 120
    Figure 4.26 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various values of mesh density plotted
    in linear (a and b) and polar (c and d) plots 123List of figures
    13
    Figure 4.27 a) Shape of the different rectangular elements used in the mesh; Variation
    of the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various values of R plotted in a polar (b and c) and
    linear (d and e) fashion . 124
    Figure 4.28 a) Shape of the different rhombic elements used in the mesh; Variation of
    the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various values of g plotted in a polar (b and c) and
    linear (d and e) fashion . 125
    Figure 4.29 a) Shape of the different parallelogramatic elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion 126
    Figure 4.30 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various values of mesh density plotted
    in polar (a and b) and linear (c and d) plots 127
    Figure 4.31 a) Shape of the different rectangular elements used in the mesh; Variation
    of the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various values of R plotted in a polar (b and c) and
    linear (d and e) fashion . 129
    Figure 4.32 a) Shape of the different rhombic elements used in the mesh; Variation of
    the longitudinal (b and d) and shear (c and e) velocity error against the
    angle of incidence for various values of g plotted in a polar (b and c) and
    linear (d and e) fashion . 130
    Figure 4.33 a) Shape of the different parallelogramatic elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion 131
    Figure 4.34 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various mesh densities plotted in a linear (a and b) and polar (c and d) fashion 132
    Figure 4.35 a) Shape of the different quadratic isosceles-triangular elements used in
    the mesh; Variation of the longitudinal (b and d) and shear (c and e) velocity error against the angle of incidence for various value of f plotted in
    a polar (b and c) and linear (d and e) fashion . 133
    Figure 4.36 a) Shape of the different scalene-triangular elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion 134
    Figure 4.37 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various mesh density plotted in a linear
    (a and b) and polar (c and d) fashion 135
    Figure 4.38 a) Shape of the different quadratic isosceles-triangular elements used in
    the mesh; Variation of the longitudinal (b and d) and shear (c and e) velocity error against the angle of incidence for various values of f plotted
    in a polar (b and c) and linear (d and e) fashion . 136
    Figure 4.39 a) Shape of the different scalene-triangular elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion 137List of figures
    14
    Figure 4.40 Variation of the longitudinal (a and c) and shear (b and d) velocity error
    against the angle of incidence for various mesh densities plotted in a linear (a and b) and polar (c and d) fashion 138
    Figure 4.41 a) Shape of the different quadratic isosceles-triangular elements used in
    the mesh; Variation of the longitudinal (b and d) and shear (c and e) velocity error against the angle of incidence for various values of f plotted
    in a polar (b and c) and linear (d and e) fashion . 139
    Figure 4.42 a) Shape of the different scalene-triangular elements used in the mesh;
    Variation of the longitudinal (b and d) and shear (c and e) velocity error
    against the angle of incidence for various values of g plotted in a polar (b
    and c) and linear (d and e) fashion 140
    Figure 5.1 (a) Longitudinal and (b) shear wave excitation for a square element mesh
    and (c) longitudinal and (d) shear excitation for a triangular element mesh
    147
    Figure 5.2 Straight edge model: a) with edge, b) without edge . 149
    Figure 5.3 a) square mesh at 0 degrees aligned with the edge, b) square mesh at 45
    degrees, c) triangular mesh . 149
    Figure 5.4 Implicit models for straight edge: Monitored absolute displacement for a
    longitudinal wave excitation using CPE4 and CPE4R meshes at 0 degrees, CPE4 and CPE4R meshes at 45 degrees and CPE3, CPE6 and
    CPE6M triangular elements. Thin red line is reference for N=30 for each
    case 151
    Figure 5.5 Implicit models for a straight edge: Monitored absolute displacement for
    a shear wave excitation using CPE4 and CPE4R meshes at 0 degrees,
    CPE4 and CPE4R meshes at 45 degrees and CPE3, CPE6 and CPE6M
    triangular elements. Thin red line is reference for N=30 for each case
    152
    Figure 5.6 Explicit models for a straight edge: Monitored absolute displacement for
    a longitudinal wave excitation using CPE4R meshes at 0 degrees, CPE4R
    meshes at 45 degrees and CPE3 and CPE6M triangular elements. Thin
    red line is reference for N=30 for each case . 153
    Figure 5.7 Explicit models for a straight edge: Monitored absolute displacement for
    a shear wave excitation using CPE4R meshes at 0 degrees, CPE4R meshes at 45 degrees and CPE3 and CPE6M triangular elements. Thin red line
    is reference for N=30 for each case 154
    Figure 5.8 Straight crack model: a) with crack, b) without crack 156
    Figure 5.9 Definition of unit long cracks with triangular and square meshes. Blue
    line shows modelled crack and red line theoretical crack (which is the
    same line with triangular element meshes but not with regular square element meshes) . 157
    Figure 5.10 Implicit models for a crack of unit length: Monitored absolute displacement for a longitudinal wave excitation using mesh made of CPE3,
    CPE6, CPE6M, CPE4 and CPE4R elements. Thin red line is reference for
    N=30 for each case . 158
    Figure 5.11 Implicit models for a crack of unit length: Monitored absolute displacement for a shear wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 158List of figures
    15
    Figure 5.12 Explicit models for a crack of unit length: Monitored absolute displacement for a shear and longitudinal wave excitation using mesh made of
    CPE3, CPE6M and CPE4R elements. Thin red line is reference for N=30
    for each case 159
    Figure 5.13 0.25 unit long crack definition with triangular and square meshes. Blue
    line shows modelled crack and red line theoretical crack (which is the
    same line with triangular element meshes but not with regular square element meshes) . 161
    Figure 5.14 Implicit models for a crack of length 0.25: Monitored absolute displacement for a longitudinal wave excitation using mesh made of CPE3,
    CPE6, CPE6M, CPE4 and CPE4R elements. Thin red line is reference for
    N=30 for each case . 162
    Figure 5.15 Implicit models for a crack of length 0.25: Monitored absolute displacement for a shear wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 162
    Figure 5.16 Explicit models for a crack of length 0.25: Monitored absolute displacement for a shear and longitudinal wave excitation using mesh made of
    CPE3, CPE6M and CPE4R elements. Thin red line is reference for N=30
    for each case 163
    Figure 5.17 4 unit long crack definition with triangular and square meshes. Blue line
    shows modelled crack and red line theoretical crack (which is the same
    line with triangular element meshes but not with regular square element
    meshes) . 165
    Figure 5.18 Implicit models for a crack of length 4: Monitored absolute displacement
    for a longitudinal wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 165
    Figure 5.19 Implicit models for a crack of length 4: Monitored absolute displacement
    for a shear wave excitation using mesh made of CPE3, CPE6, CPE6M,
    CPE4 and CPE4R elements. Thin red line is reference for N=30 for each
    case 166
    Figure 5.20 Explicit models for a crack of length 4: Monitored absolute displacement
    for a shear and longitudinal wave excitation using mesh made of CPE3,
    CPE6M and CPE4R elements. Thin red line is reference for N=30 for
    each case . 166
    Figure 5.21 Circular defect model: a) with circular defect, b) without circular defect
    168
    Figure 5.22 Unit diameter hole definition with triangular and square meshes 169
    Figure 5.23 Implicit models for a hole of unit diameter: Monitored absolute displacement for a longitudinal wave excitation using mesh made of CPE3,
    CPE6, CPE6M, CPE4 and CPE4R elements. Thin red line is reference for
    N=30 for each case . 170
    Figure 5.24 Implicit models for a hole of unit diameter: Monitored absolute displacement for a shear wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 170
    Figure 5.25 Explicit models for a hole of unit diameter: Monitored absolute displacement for a shear and longitudinal wave excitation using mesh made of
    CPE3, CPE6M and CPE4R elements. Thin red line is reference for N=30
    for each case 171List of figures
    16
    Figure 5.26 0.25 unit diameter hole definition with triangular and square meshes 173
    Figure 5.27 Implicit models for a hole of diameter 0.25: Monitored absolute displacement for a longitudinal wave excitation using mesh made of CPE3,
    CPE6, CPE6M, CPE4 and CPE4R elements. Thin red line is reference for
    N=30 for each case . 173
    Figure 5.28 Implicit models for a hole of diameter 0.25: Monitored absolute displacement for a shear wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 174
    Figure 5.29 Explicit models for a hole of diameter 0.25: Monitored absolute displacement for a shear and longitudinal wave excitation using mesh made of
    CPE3, CPE6M and CPE4R elements. Thin red line is reference for N=30
    for each case 174
    Figure 5.30 4 units diameter hole definition with triangular and square meshes . 175
    Figure 5.31 Implicit models for a hole of diameter 4: Monitored absolute displacement for a longitudinal wave excitation using mesh made of CPE3,
    CPE6, CPE6M, CPE4 and CPE4R elements. Thin red line is reference for
    N=30 for each casE . 176
    Figure 5.32 Implicit models for a hole of diameter 4: Monitored absolute displacement for a shear wave excitation using mesh made of CPE3, CPE6,
    CPE6M, CPE4 and CPE4R elements. Thin red line is reference for N=30
    for each case 176
    Figure 5.33 Explicit models for a hole of diameter 4: Monitored absolute displacement for a shear and longitudinal wave excitation using mesh made of
    CPE3, CPE6M and CPE4R elements. Thin red line is reference for N=30
    for each case 177
    Figure 6.1 Definition of 1D model . 183
    Figure 6.2 Reflection coefficient for longitudinal and shear waves against Young’s
    modulus . 190
    Figure 6.3 Reflection coefficient for longitudinal and shear waves against Young’s
    modulus and Poisson’s ratio 191
    Figure 6.4 a) Total reflection, b) Reflection due to the impedance change, c) and d)
    Reflection due to the tie (linear scale and log scale) 193
    Figure 6.5 2D model geometry 195
    Figure 6.6 Absolute displacement field in the top right corner of the 2D models.
    Longitudinal wave excitation with a) theoretical and b) adjusted material
    properties. c) Definition of wave packet positions. d), e), f) same with
    shear wave excitation 197
    Figure 6.7 Definition of 1D model with gradual mesh density change . 198
    Figure 6.8 Absolute displacement field for a) longitudinal and b) shear wave excitation models with a gradual change of mesh density at t=34 (longitudinal)
    and t=68 (shear) 199
    Figure 6.9 Gradual mesh density change for the 2D model . 200
    Figure 6.10 Absolute displacement field in the top right corner of the 2D model with
    a) theoretical and b) adjusted material properties . 20017
    List of Tables
    Table 6.1 Table of reflection coefficients due to the tie between two meshes in %
    . 197
    Table 6.2 Table of reflection coefficients due to the impedance difference between
    two meshes in % . 197

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