Bachelor of Engineering Thesis - UQ

UQ Engineering

Faculty of Engineering, Architecture and Information Technology

THE UNIVERSITY OF QUEENSLAND

Bachelor of Engineering Thesis

Investigating the Simulation Capabilities of ANSYS in

Modelling the Fundamental Antisymmetric Lamb Wave

Student Name: Jeffrey BARRETT

Course Code: MECH4500

Supervisor: Associate Professor Martin Veidt

Submission date: 22nd October 2018

A thesis submitted in partial fulfilment of the requirements of the Bachelor of Engineering

(Hons) degree in Mechanical and Aerospace Engineering

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Acknowledgments Firstly, I would like to express my sincerest gratitude to Associate Professor Martin Veidt

for his continuing assistance throughout the project. Without his guidance, this thesis would not

have been possible. This project was an invaluable opportunity to develop my skills as a

professional engineer and I was very fortunate to have Martin as a mentor.

I would also like to thank my family for their amazing support throughout the entirety of

my university studies. Without the support of my mother and father, Nola and Ian, and my three

sisters, Amanda, Sally and Lisa, I would never have made it.

A special thanks to my friends who have been with me all the way through to the end.

Without the support from Tim, Andy, Dragan, Matt and Dan (to name a few), these past few

years would have been so much more difficult.

Finally, thank you Aísling for supporting me through the tough times. I could never express

how much your love and support has helped me, I couldn’t have done it without you.

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Abstract Structural health monitoring (SHM) is the continuous, real-time monitoring of the integrity

of a component, with the primary aim of detecting the onset of material damage [1, 2]. Some

SHM systems utilise networks of imbedded sensors to emit and receive Lamb waves, which are

elastic waves that propagate in thin structures [2]. When Lamb waves encounter structural

damage, the reflected waves contain information about the size, location and nature of the

damage [3]. Finite element method software packages such as ANSYS provide cost-effective

options for engineers to simulate Lamb wave propagation [2]. There is a strong motivation to

develop accurate and reliable numerical models of Lamb waves and their complex interactions

with structural damage. These models provide a valuable tool in the design of SHM systems.

The aim of the thesis was to investigate the simulation capabilities of ANSYS in modelling

the fundamental antisymmetric (A0) Lamb wave. The simulation results were to be compared

against analytical solutions to validate ANSYS as a numerical tool for modelling the A0 Lamb

wave. The investigation aimed to deliver a proven methodology for ANSYS simulation of the

A0 mode which could be used in future works relating to SHM system design.

A 2D model of an aluminium 2024-T6 plate was developed in ANSYS Explicit Dynamics.

The model was meshed using 4-node solid elements with characteristic lengths ranging from

0.15 – 1.50 mm. The A0 Lamb mode was activated by a 100 kHz sinusoidal tone burst,

modulated by a Hanning window. ANSYS was shown to accurately model the dispersive

properties of the A0 mode. The energy-distribution approach for time of arrival was the most

reliable method for calculating group velocity. ANSYS was found to accurately model group

velocity, with a minimum numerical error of only 0.15% The accuracy of the simulations

improved as mesh element length was reduced. The 2D Fast Fourier Transform was used to

calculate phase velocity of the incident wave pulse, with a numerical error of only 0.19%.

A three-dimensional model of the aluminium plate was developed in ANSYS using 8-

node brick elements. The simulation results showed good agreement with both the 2D and

analytical models, with an average error of 3.47% in A0 mode group velocity. The excitation

frequency was varied from 25 kHz – 400 kHz and group velocity results were used to develop

an experimental dispersion curve which showed excellent agreement with the analytical curve.

The mesh element length criterion was found to be noncritical for accurate ANSYS simulations.

A surface notch was developed in the 2D model. ANSYS accurately captured conversion

between the A0 and S0 modes due to interactions with the notch. Error in the reflected A0 and S0

modes was just 1.18% and 1.44% respectively. The amplitude of the reflected S0 mode

increased consistently with notch depth, while a mid-thickness notch caused the largest

amplitude of the A0 mode. A mid-thickness void was developed in the model. Only the A0 mode

was reflected from the damage, which was attributed to the through-thickness location of the

void. The amplitude of the reflected wave pulse increased consistently with void length between

1 – 5 mm, however no discernible trend was established for void lengths between 5 – 30 mm.

The capabilities of ANSYS in accurately modelling the A0 mode and its interactions with

damage were demonstrated in this thesis. ANSYS is highly recommended as a viable tool for

future works relating to the design of Lamb wave based SHM systems and the methodology

outlined in this report may provide a useful reference for these investigations.

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Table of Contents

1 Introduction .................................................................................................................. 1

2 Aims of the thesis ........................................................................................................ 2

3 Project scope ................................................................................................................ 2

4 Literature review .......................................................................................................... 4

Fundamentals of Lamb waves .............................................................................. 4

Phase velocity and group velocity ........................................................................ 5

Dispersion of Lamb waves ................................................................................... 5

Lamb wave mode selection .................................................................................. 7

Excitation frequency selection............................................................................ 10

Modelling Lamb waves using the finite element method (FEM) ....................... 11

Element selection ................................................................................................ 14

Signal processing techniques .............................................................................. 15

Modelling structural damage in FEM ................................................................. 17

Conclusions from the literature review .............................................................. 18

5 Development of the two-dimensional ANSYS model ............................................... 19

Overview of the study ........................................................................................ 19

Methodology for constructing the FE model ...................................................... 20

Analysis system and system properties .............................................................. 21

Engineering material properties ......................................................................... 22

Geometry setup ................................................................................................... 23

Model setup ........................................................................................................ 23

Selection of the excitation frequency ................................................................. 26

Modelling the excitation frequency .................................................................... 28

Boundary constraints .......................................................................................... 29

Analysis settings ................................................................................................. 30

Data capture and exporting the results ............................................................... 31

6 Analysis of the two-dimensional ANSYS simulation ............................................... 33

Overview ............................................................................................................ 33

Verification of the excitation signal ................................................................... 33

Signal processing of the raw data ....................................................................... 37

Determination of the simulated wave pulse group velocity ............................... 44

6.4.1. Reference-amplitude approach for ToA ...................................................... 44

iv

6.4.2. Issues associated with the reference-amplitude approach for ToA ............. 45

6.4.3. Sensitivity of reference-amplitude ToA to amplitude threshold ................. 50

6.4.4. Sensitivity of reference-amplitude ToA to separation distance .................. 51

6.4.5. Energy distribution approach for wave pulse ToA ..................................... 52

6.4.6. Validation of the 2D simulation by group velocity ..................................... 56

6.4.7. Conclusions from the analysis of group velocity ........................................ 59

Determination of the simulated wave pulse phase velocity ............................... 60

6.5.1. Methodology for calculating phase velocity ............................................... 60

6.5.2. Influence of spatial resolution on the wavenumber-frequency domain ...... 61

6.5.3. Validation of the 2D ANSYS simulation by phase velocity ....................... 68

7 Development of the three-dimensional ANSYS model ............................................. 69


Analysis settings, material properties and geometry .......................................... 69

Model setup ........................................................................................................ 70

Simulation results ............................................................................................... 71

8 Analysis of the three-dimensional ANSYS simulation ............................................. 72

Signal processing of the raw data ....................................................................... 72

Model validation and comparison of results with the 2D model ........................ 74

Conclusions from the 3D ANSYS simulation .................................................... 76

9 Investigating model rigorousness across the low-frequency regime ......................... 77


Selection of the excitation frequencies ............................................................... 77

Analysis of the results ........................................................................................ 79

10 Interactions between the A0 mode and a surface notch ............................................. 82

Overview and significance of the study ............................................................. 82

Scope of the study .............................................................................................. 82

Results ................................................................................................................ 83

Analysis of the nodal displacement data ............................................................ 84

Influence of notch depth on the amplitude of reflected Lamb waves ................ 92

11 Interactions between the A0 mode and a mid-thickness void .................................... 95


Scope of the study .............................................................................................. 95

Model results ...................................................................................................... 96

Analysis of the nodal displacement data ............................................................ 97

v

Influence of void length on the amplitude of the reflected Lamb wave ........... 100

12 Recommendations for further works ....................................................................... 103

13 Conclusion ............................................................................................................... 105

14 References ................................................................................................................ 108

15 Appendices ............................................................................................................... 111

Appendix A: Signal processing Python code ................................................... 111

Appendix B: Python implementation of the 2D FFT ....................................... 120

Appendix C: Group velocity results for the 3D simulation .............................. 125

Appendix D: Reflected wave pulse group velocities ....................................... 125

vi

List of Figures

Figure 1: The symmetric Lamb mode (a) causes predominantly in-plane displacement of

particles, while the antisymmetric Lamb mode (b) causes predominantly out-of-plane

displacement [2]. ........................................................................................................................ 4

Figure 2: The antisymmetric Lamb mode in a 1 mm thick aluminium 2024 plate at a) 0 mm,

b) 250 mm, c) 500 mm from the excitation source. The dispersive nature of Lamb waves causes

the wave packet to spread out as it travels through the medium. ............................................... 6

Figure 3: Dispersion curves for aluminium 2024, generated using LAMSS Waveform

Revealer show (a) phase velocity and (b) group velocity as a function of frequency-thickness

for the first four antisymmetric and symmetric Lamb modes . .................................................. 6

Figure 4: The phase velocity dispersion curve for aluminium 2024 demonstrates the A0

mode’s shorter wavelength for a given frequency. For example, at 500 kHz-mm, 𝑐𝐴0 =

1880𝑚𝑠 while 𝑐𝑆0 = 5386𝑚𝑠. For a plate thickness of 1 mm, the wavelengths are therefore

𝜆𝐴0 = 3.76 𝑚𝑚 and 𝜆𝑆0 = 10.77 𝑚𝑚. ................................................................................... 8

Figure 5: Hayashi and Kawashima compared A0 and S0 modes in a composite laminate. It

was found that the A0 mode was sensitive to delaminations (pictured) at all through thickness

locations, while the S0 mode was not sensitive to the delaminations located between plies 2-3

and at the midplane [20]. ............................................................................................................ 9

Figure 6: Ng and Veidt used ANSYS to model the interaction between the A0 mode and a

delamination in a carbon/epoxy composite plate (a) [8]. Lasˇova´ used ABAQUS to conduct a

two-dimensional analysis of the propagation of the A0 and S0 modes in an aluminium plate [14].

.................................................................................................................................................. 12

Figure 7: Common elements used in ANSYS for modelling in two-dimensions and three-

dimensions are PLANE42 (a) and SOLID45 (b) respectively [32].......................................... 14

Figure 8: Hourglassing results in the non-physical deformation of finite elements [34]. .. 15

Figure 9: The Hilbert function reveals the energy distribution of the signal. The energy

envelope can be used to precisely identify the peak amplitudes within a signal that contains a

significant amount of noise, as shown from (a) to (b) [2]. ....................................................... 16

Figure 10: The 2D FFT can be used to reveal the Lamb wave dispersion curves (a) [37].

Costley used the 2D FFT to obtain the wavenumber-frequency dispersion curves of aluminium

(b) by measuring evenly spaced 50 displacement signals across the plate [36]. ...................... 17

Figure 11: Cracks are modelled in FEM by removing elements and ensuring that the

remaining surfaces are separated [1]. ....................................................................................... 18

Figure 12: The Explicit Dynamics Analysis System is available within the ANSYS 18.2

Workbench................................................................................................................................ 20

Figure 13: The Analysis System settings were configured for (a) two-dimensional geometry

analysis and (b) an explicit time integration scheme using the Autodyn solver. ..................... 21

Figure 14: The engineering material properties of aluminium 2024 were entered into the

material database in ANSYS and assigned to the 2D model. ................................................... 22

Figure 15: The ANSYS DesignModeler toolbox was used to create the geometry for the

2D cross section of a plate ........................................................................................................ 23

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Figure 16: The ANSYS Mechanical model tree contains the model parameters which define

the physics of the system. The material selection was defined in (a) Geometry, and Cartesian

coordinates were selected in (b) Coordinate System. ............................................................... 24

Figure 17: The 2D model of the aluminum 2024 plate was meshed using quadrilateral 4-

node solid elements. The mesh was defined by the characteristic element length, which is

0.75 mm in (a). The meshed plate is shown in (b). .................................................................. 25

Figure 18: Named selections were created at 30 mm intervals along the plate. This provided

16 equally spaced nodes at which the nodal displacement data was captured. ........................ 26

Figure 19: The phase velocity dispersion curves for Al 2024-T6 show that at an excitaton

frequency of 100 kHz, only the fundamental modes exist (a). The analytical solutions to the

dispersion curves show the A0 phase velocity is 1550 m/s (b). ................................................ 27

Figure 20: The group velocity dispersion plots for aluminium 2024-T6 show that at an

excitation frequency of 100 kHz, only the fundamental Lamb modes will exist (a). At this

excitation frequency the group velocity is 2621 m/s (b). ......................................................... 27

Figure 21: Out-of-plane (y direction) nodal displacements were applied to the mesh nodes

occurring in the 3 mm from the left-hand side of the 2D plate model. .................................... 28

Figure 22: The excitation signal was a 5-cycle sinusoidal tone burst modulated by a Hanning

window function. ...................................................................................................................... 29

Figure 23: The excitation displacement amplitude was entered into ANSYS as a function of

time. .......................................................................................................................................... 29

Figure 24: A fixed support was applied to the far edge of the model to constrain the model

in space. .................................................................................................................................... 30

Figure 25: The waveform was not accurately captured using 500 nodes per wavelength (a).

It was found that 5000 nodes per wavelength provided sufficient resolution to accurately

capture the wave pulse as it travelled across the plate (b). ....................................................... 31

Figure 26: The ANSYS results window provided a graphical output of the nodal

displacement data, which was used to qualitatively analyse the propagation of the wave and

make sense of the raw data. ...................................................................................................... 32

Figure 27: The raw displacement data captured at the excitation location shows the

sinusoidal tone burst was accurately modelled in all FE models with a 1 µm amplitude and

50 µs period. ............................................................................................................................. 34

Figure 28: Comparison of the excitation signals of the simulated and analytical models

reveals good agreement in the overall waveform, despite a small offset in the beginning of the

wave packet. ............................................................................................................................. 35

Figure 29: The displacement results were transformed from the time domain to the

frequency domain to reveal the frequency spectrum of the excitation signals for (a) 0.15 mm

mesh and (b) 1.50 mm mesh. .................................................................................................... 36

Figure 30: The energy envelopes of the (a) 0.15 mm mesh and (b) 1.50 mm mesh were

plotted against the analytical model, showing a high level of agreement in both models. ...... 36

Figure 31: Nodal displacement results at x = 300 mm show the incident and reflected Lamb

wave. Dispersion was accurately captured in the simulation with velocity differences between

the high and low frequencies within the wave pulses. ............................................................. 38

Figure 32: Close-up view of the incident wave packet indicates that the 0.1µs data-capture

provided good temporal resolution of the propagating Lamb wave’s displacement amplitude.

.................................................................................................................................................. 38

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Figure 33: Wave dispersion is evidenced by pulse widening between nodes located (a) 60

mm, (b) 120 mm, (c) 180 mm, (d) 240 mm from the excitation source. .................................. 40

Figure 34: Comparison of the nodal displacements at 300 mm shows mesh density impacts

the amplitude and speed of the simulated wave pulse. The raw data indicates convergence

toward the analytical solution as mesh length decreases. ......................................................... 41

Figure 35: An algorithm was developed to normalise the nodal displacement data using the

local maximum rather than the global maximum. .................................................................... 42

Figure 36: Wave pulses were normalised to allow for comparison between mesh sizes and

with the analytical solutions. The 0.15 mm mesh was normalised using the local maximum (a)

and shows good agreement to the analytical solution (b). ........................................................ 43

Figure 37: ToA at 30 mm from the excitation source was determined using a cut-off

threshold of 1% at 11.4µs. ........................................................................................................ 44

Figure 38: ToA of the analytical and simulated Lamb waves, at 300 mm from the excitation

source, was determined using a cut-off threshold of 1% at 107.3µs and 107.5µs respectively.

.................................................................................................................................................. 44

Figure 39: The reference-amplitude approach for ToA was used to calculate the wave pulse

group velocity. Using a threshold of 5% shows that the finite element solution converged to the

analytical solution as the mesh length was decreased. ............................................................. 45

Figure 40: The reference-amplitude approach for ToA resulted in numerous outlying

datapoints, which were attributed to limitations in the methodology and wave dispersion. .... 46

Figure 41: Attenuation and wave pulse widening resulted in different wave peaks being used

as the reference point for ToA. The second peak reached the 5% threshold in (a) and (b), while

the third peak was measured in (c) and (d). .............................................................................. 47

Figure 42: Wave pulse group velocity was found to increase as element length was reduced

(a). Since all wave speeds exceeded the cg of 2621 m/s, this meant numerical error increased on

average as the mesh resolution improved (b). .......................................................................... 48

Figure 43: Spectral leakage causes high frequency components to exist within the wave

pulse. ......................................................................................................................................... 49

Figure 44: The reference-amplitude approach was highly sensitive to the user-defined

threshold at which point ToA was defined. This was due to the amplitude response of high

frequency components being measured when the threshold was low. ..................................... 51

Figure 45: The reference-amplitude approach for ToA was highly sensitive to the separation

distance over which group velocity was calculated. Increasing separation distance resulted in a

net reduction in numerical error across all models. .................................................................. 52

Figure 46: Energy distribution of the measured signals at (a) 30 mm and (b) 150 mm reveal

the incident and reflected Lamb wave pulses. .......................................................................... 53

Figure 47: The ToA was approximated by averaging the time over which the amplitude

exceeded the ToA reference amplitude. At (a) 30 mm the ToA is 36.0 µs and at (b) 150 mm the

ToA is 82.2 µs. ......................................................................................................................... 54

Figure 48: Sensitivity analysis of methodologies for calculating ToA based on (a) amplitude

threshold and (b) Hilbert function, reveal that the energy-distribution based approach is

significantly less-sensitive to separation distance. ................................................................... 55

Figure 49: Sensitivity analysis of methodologies for calculating ToA based on (a) amplitude

threshold and (b) Hilbert function, reveal that the energy-distribution based approach is

significantly less-sensitive to reference amplitude. .................................................................. 56

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Figure 50: Group velocity was calculated over a separation distance of 90 mm to ensure that

the influence of dx on the measured 𝑐𝑔 was minimised. ......................................................... 57

Figure 51: Calculated group velocities (a) reveal erroneous data points at the far boundary

of the model (a). This was caused by interactions between the incident and reflected wave

resulting in ToA error (b). ........................................................................................................ 57

Figure 52: Using the energy-distribution of the wave pulse for ToA, the group velocities of

the various FE models showed excellent agreement with the analytical value of 2621 m/s (a).

The general trend of the data was a reduction in numerical error as the finite element length

became shorter, which was consistent with expected outcomes (b)......................................... 58

Figure 53: Reducing characteristic mesh element length was found to consistently improve

model accuracy. The minimum error was 0.13% in the 0.15 mm mesh model, while the

maximum error was 5.64% in the 1.50 mm model. ................................................................. 59

Figure 54: The nodal responses were extracted from the model at evenly spaced points and

were amalgamated in a 2D matrix in preparation for the 2D FFT. .......................................... 61

Figure 55: Closeup view of the wavenumber-frequency plot reveals a range of uncertainty

which is attributed to the spatial resolution. ............................................................................. 63

Figure 56: A spatial resolution of less than 1 node per wavelength resulted in the indication

of non-physical bahviour of the Lamb wave. This occurred for separation distances of (a)

40 mm, (b) 30 mm, and (c) 20 mm. .......................................................................................... 64

Figure 57: A spatial resolution of more than 1 node per wavelength indicatated physical

bahviour of the Lamb wave. This occurred for separation distances of (a) 15 mm, (b) 12.5 mm

, (c) 10 mm, (d) 7.5 mm, (e) 5 mm, and (f) 2.5 mm. ................................................................ 66

Figure 58: The influence of spatial resolution on (a) average frequency and (b) phase

velocity is unclear. This may be attributed to the scope of the testing covering an insufficiently

fine spatial resolution. .............................................................................................................. 68

Figure 59: The FE model properties were set to 3D to capture the propagation of Lamb

waves through the x-z plane (a). The model was a square plate with dimensions 400 mm ×

400 mm × 3 mm (b). ................................................................................................................. 70

Figure 60: The characteristic mesh element length was 1 mm to provide an acceptable

compromise between accuracy and computational time. ......................................................... 71

Figure 61: The nodal displacement in the thickness direction was measured to capture the

antisymmetric Lamb wave as it propagated along the plate. .................................................... 72

Figure 62: Reflections from the side boundaries of the plate introduced complexity into the

3D model which was not seen the 2D model. This required more deliberate selection of the

simulation time to avoid noise in the displacement data. ......................................................... 72

Figure 63: The raw data captured at 60 mm from the excitation reveals the incident and

reflected wave pulses (a). The wave pulse was normalised and compared with the analytical

solution, revealing excellent agreement overall (b).................................................................. 73

Figure 64: The nodal displacement data captured at 200 mm (a) reveals the simulated wave

pulse travelled with a lower velocity as indicated by the lag between wave packets (b). ........ 74

Figure 65: Comparison of the wave pulses at (a) 40 mm and (c) 180 mm reveals an overall

consistency in the shape of the Lamb waves simulated in the 3D and 2D models. The energy

distributions of the wave pulses (b) and (d) show that there was aliasing seen in the 3D model

which was attributed to numerical error. .................................................................................. 75

x

Figure 66: The Hilbert transform reveals a shorter excitation pulse period at higher f0 (a),

while the Fast Fourier Transform of the excitation signal reveals a narrower frequency

bandwidth at lower f0 (b). ......................................................................................................... 79

Figure 67: The dispersion curve generated from the simulation results shows strong

agreement with the theoretical curve for aluminium 2024-T6. ................................................ 80

Figure 68: The increased wave duration for the 25 kHz model resulted in less separation

between incident and reflected wave pulses (a), which may have introduced numerical error not

seen in higher frequency models such as 400 kHz (b). ............................................................ 81

Figure 69: Increasing excitation frequency resulted in reduced displacement amplitude. . 81

Figure 70: The notch was modelled geometrically by removing mesh elements from the

model, extending in the thickness direction from the surface of the plate. .............................. 82

Figure 71: Interaction between the incident Lamb wave and the surface notch resulted in a

reflected wave propagating back toward the excitation source. ............................................... 83

Figure 72: The nodal displacement response at 210 mm from the excitation source reveals

the symmetric Lamb mode is reflected from the notch and arrives earlier than the A0 mode. 84

Figure 73: The nodal displacement response at 210 mm from the excitation source reveals

the antisymmetric Lamb mode is reflected from the notch and arrives later than the S0 mode.

.................................................................................................................................................. 84

Figure 74: The y direction nodal displacement at 300 mm from the excitation source reveals

the wave pulses reflected off the structural damage, along with significant boundary noise. . 85

Figure 75: The Hilbert transform of the y direction nodal displacement data was used to

distinguish the reflected wave pulses and determine ToA. ...................................................... 86

Figure 76: Comparison of the y displacement data between the damaged and undamaged

plate confirms the nature of the wave peaks as the reflected A0 and S0 modes only appear due

to interaction with the notch. .................................................................................................... 86

Figure 77: Closeup view of the y displacement data shows the S0 mode is detected earlier

than the A0 mode as the symmetric mode travels at a higher group velocity, as indicated in the

aluminum 2024-T6 dispersion curves. ..................................................................................... 87

Figure 78: Measurement of the x direction nodal displacements improves detection of the

incident and reflected symmetric Lamb modes. ....................................................................... 88

Figure 79: The energy envelope of the x direction nodal displacement data provides

enhanced identification of the symmetric mode and was used to calculate group velocity. .... 88

Figure 80: The x direction nodal displacement data was used to distinguish the structural

damage reflections from the boundary reflections (a). The boundary reflections are clearly

identified from the undamaged plate (b). ................................................................................. 89

Figure 81: The reflected A0 and S0 wave pulses (a) were distinguished from the boundary

noise by comparison of the x direction signal response with that captured for the undamaged

plate (b). .................................................................................................................................... 89

Figure 82: Mode conversion is clearly evident between the antisymmetric and symmetric

modes through measurement of x displacement (a). The A0 amplitude is much greater than that

of the S0 mode in the y direction, due to the out-of-plane perturbation (b). ............................. 90

Figure 83: Simulation results published by Alkassar (a), capturing the x direction nodal

displacement after the A0 Lamb wave interaction with a vertical surface crack [10]. Results

published by Su (b), showing mode conversion between the S0 and A0 modes after interaction

with structural damage [2]. ....................................................................................................... 91

xi

Figure 84: The ToA of the reflected S0 and A0 wave pulses were determined, and the

corresponding group velocities were calculated. The range of data was averaged to determine

the average group velocity. ....................................................................................................... 92

Figure 85: The x direction nodal displacement data indicates that increased notch depth

resulted in larger amplitude of the reflected S0 Lamb wave pulse, measured at (a) 120 mm, (b)

210 mm, (c) 330 mm, (d) 420 mm. .......................................................................................... 93

Figure 86: The y direction nodal displacement data indicates that a mid-thickness notch

depth results in the largest amplitude of the reflected A0 Lamb wave pulse, measured at (a) 120

mm, (b) 210 mm, (c) 330 mm, (d) 420 mm. ............................................................................ 94

Figure 87: A horizontal void was modelled in the centre of the plate, with varying lengths

ranging between 1 – 30 mm. .................................................................................................... 96

Figure 88: The ANSYS simulation results reveal only the A0 Lamb mode was reflected from

the material damage. ................................................................................................................. 97

Figure 89: The normalised x directional nodal displacement data (a) and corresponding

energy envelope (b) at 210 mm reveals a damage-reflected A0 mode. No S0 mode was generated

due to interaction with 5 mm long damage. ............................................................................. 98

Figure 90: The normalised y directional nodal displacement data (a) and corresponding

energy envelope (b) at 210 mm clearly shows the reflected A0 mode from the 5 mm long

centrally located void................................................................................................................ 99

Figure 91: In-plane (x) displacement at 240 mm reveals no mode conversion between the

A0 and S0 modes as a result of interaction with the 5 mm void. This is because the void is located

in the centre of the plate where the shear stress is zero. ........................................................... 99

Figure 92: Stress data generated using the software Disperse shows the shear stress

distribution across the thickness of the plate, with the shear stress being zero at the centre. 100

Figure 93: Out-of-plane (y) displacement at 240 mm shows the A0 wave pulse reflected off

the 5 mm horizontal void. The wave pulse has an approximate ToA of 500 𝜇𝑠. .................. 100

Figure 94: The y direction nodal displacement data indicates that increased void length (up

to 5 mm) resulted in larger amplitude of the reflected A0 Lamb wave pulse, measured at (a) 120

mm, (b) 210 mm, (c) 330 mm, (d) 420 mm. .......................................................................... 101

Figure 95: The y direction nodal displacement data reveals the relationship breaks down at

void lengths larger than 5 mm, measured at (a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm.

No discernible trend was identified between amplitude and void lengths from 5 – 30 mm. . 102

Figure 96: Ng and Veidt modelled the propagation of the A0 mode in a composite laminate

using ANSYS LS-DYNA. Development of a 3D composite model in ANSYS Explicit

Dynamics could provide a viable tool for SHM system design [8]. ...................................... 103

Figure 97: Shen discussed a methodology for modelling non-reflective boundaries in

ANSYS [41]. .......................................................................................................................... 104

Figure 98: The group velocity calculated across the 3D FE model shows good agreement

with the 2D model (a), however the numerical error showed an increasing trend across the

length of the plate (b).............................................................................................................. 125

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List of Tables

Table 1: Scope of the thesis. ................................................................................................. 3

Table 2: Comparison between the symmetric and antisymmetric Lamb modes. ............... 10

Table 3: Primary roles of the key components of an FE model. ........................................ 20

Table 4: Engineering material properties of aluminium 2024-T6 [40]. ............................. 22

Table 5: Geometrical properties of the 2D plate model. ..................................................... 23

Table 6: Time of arrival measurements at 150 mm along the 2D plate. ............................. 45

Table 7: Average group velocity and associated error at different f0 frequencies. ............. 49

Table 8: Summary of the group velocity data captured in the 2D simulation. ................... 59

Table 9: Scope of the spatial resolution sensitivity analysis. ............................................. 62

Table 10: Influence of spatial resolution on the average frequency and average phase

velocity of the simulated Lamb wave. ...................................................................................... 66

Table 11: Comparison of average group velocities calculated in the 2D and 3D simulations.

.................................................................................................................................................. 75

Table 12: Excitation frequencies and associated wave speeds. .......................................... 77

Table 13: Average group velocity measurements across the frequency range. .................. 79

Table 14: Summary of tested notch geometries. ................................................................. 83

Table 15: Summary of the tested void lengths.................................................................... 96

Table 16: Calculated group velocities of the reflected A0 mode from the surface notch. 125

Table 17: Calculated group velocities of the reflected S0 mode from the surface notch. . 126

1

1 Introduction

The detection of structural damage within an engineering component is highly important to

prevent unexpected failure and potentially catastrophic consequences in safety-critical

applications. The prevalence of composite materials is rapidly increasing with the rising

performance requirements of modern structures in many industries. Composites such as carbon

fibre reinforced plastics have favourable material properties including high specific strength

and stiffness, low weight, good fatigue performance and resistance to corrosion [4]. Recent

advances in manufacturing processes have reduced the production costs of composite materials

significantly which has led to widespread use in the aerospace, automotive, military and

transportation industries [4]. Composite materials have also introduced unique challenges

related to the detection of structural damage within a component. Due to the complexity of

many composite structures, conventional mechanical testing procedures are insufficient to

accurately gauge the properties of the structure [5]. This has led to the development of numerous

non-destructive evaluation (NDE) techniques such as visual inspection, magnetic particles,

modal characteristics and ultrasonic inspection [1, 6].

Structural health monitoring (SHM) is the continuous, real-time monitoring of the structural

integrity of a component utilising a network of imbedded sensors [1, 2]. The overall objective

of SHM is to detect the onset of damage within a material, thus allowing a component to be

repaired or removed from service prior to failure [2]. Conventional NDE techniques often

require costly and time-consuming maintenance programs for operators, as components may

need to be removed from service at regular intervals for inspection and those with complex

geometry may require disassembly [1]. Structural health monitoring using Lamb waves has

been a significant focus of research since the 1980s [2]. Lamb waves are elastic waves which

propagate in thin plate structures [2]. Lamb waves can travel long distances within a material

without a significant decrease in amplitude and when a discontinuity such as damage is

encountered, waves will be reflected [1, 3]. The reflected waves carry information about the

discontinuity which can be extracted via signal processing techniques to discern the size,

location, type and nature of damage within a structure [3]. Lamb wave-based SHM is highly

attractive as it allows for real-time structural monitoring of components using an array of

carefully positioned transducers [2]. This is particularly useful for structures which have high

surface areas or complex geometries, which would be significantly more difficult to inspect via

an alternative NDE technique.

Guided waves have been shown, both in experimental and numerical simulations, to provide

an accurate and reliable method for detecting damage within both isotropic and anisotropic

materials [2, 7-10]. Finite element method (FEM) modelling is the most cost-effective method

for simulating the propagation and wave scattering behaviour of Lamb waves [2].

Commercially available software packages such as ABAQUS and ANSYS have been shown to

successfully model Lamb waves in metallic and composite materials [8, 10].

It is important to accurately model the propagation of Lamb waves within a structure, as it

allows engineers to predict the highly complex behaviour of Lamb wave scattering at a

discontinuity. Realistic numerical modelling of Lamb waves could significantly reduce the need

for experimental testing and allow for more rapid, flexible design of SHM solutions. By

accurately modelling the scattering characteristics of Lamb waves at a discontinuity, engineers

2

can determine the optimal transducer array design to best detect damage within a structure. This

can improve safety by reducing the likelihood of reflected wave signals going undetected in

complex anisotropic composite materials and decrease costs by reducing the scope of

experimental validation.

2 Aims of the thesis

The overall aim of the thesis was to investigate the simulation capabilities of ANSYS in

modelling the fundamental antisymmetric Lamb wave. The project aimed to deliver a validated

methodology for modelling the propagation of the A0 Lamb wave in ANSYS. The simulation

results were to be verified against analytical solutions to determine the viability of ANSYS as

a numerical tool for modelling the A0 Lamb mode. The investigation aimed to explore the

performance of the software in both the two-dimensional and three-dimensional simulation

environments. In addition, the capabilities of the software in modelling the interactions between

the incident Lamb wave and structural damage were to be explored.

The research goals were satisfied through meeting the following objectives over the course

of the investigation;

• The fundamental concepts relating to guided waves, finite element method and

signal processing were consolidated.

• A comprehensive review of the literature was carried out, which formed the

foundations of the investigation.

• A baseline methodology for activating, measuring and processing the antisymmetric

mode in ANSYS was established.

• A two-dimensional numerical model of the A0 Lamb wave was developed, and the

simulation results were validated against analytical solutions.

• A three-dimensional numerical model of the A0 Lamb wave was developed, and the

simulation results were validated against analytical solutions.

• The rigorousness of the model was explored across the low-frequency regime.

• Interactions between the A0 Lamb mode and structural damage were explored, and

the simulations were validated against results published in the literature.

3 Project scope

It was important to clearly define the scope of each goal to ensure that the overall aim of the

thesis was achieved within the research period. The primary limitation on project scope was the

time available to complete the investigation. The finite element simulations and associated

analysis of results were highly time-consuming. It was therefore necessary to ensure that all

work was pertinent to the aim. The scope of the thesis objectives is presented in Table 1 below.

3

Table 1: Scope of the thesis.

Major objective Topic Scope

Review of the

background

theory

In scope: Fundamentals relating to wave propagation, finite element modelling and signal

processing were reviewed. This provided the baseline knowledge for understanding and

the literature as well as forming the overall aims of the thesis.

Out of scope: More advanced understanding of these fields was unattainable due to the

limited time available to establish project scope and thesis aims.

Literature

review

In scope: The most relevant publications relating to finite element modelling of Lamb

waves and signal processing techniques were reviewed.

Out of scope: Supplementary publications which could provide further context were

generally omitted from scope due to time limitations.

Baseline

methodology

In scope: A simple 2D model was developed using arbitrary frequency, excitation and

material properties to determine how to activate the A0 mode and receive useful data.

Out of scope: While this stage was highly critical to the overall thesis, some of the

obtained results were not published due to their lack of relevance to the overall aim.

2D finite element

model

In scope: The analysis involved a fixed plate geometry and isotropic material properties

for simplicity. The excitation frequency was kept constant across the several FE mesh

resolutions which were compared to determine the influence on the results. Qualitative

validation of the results was limited to visual comparisons to analytical solutions.

Quantitative validation of the results was limited to determination of group velocity and

phase velocity of the simulated wave pulses and comparison to analytical results.

Out of scope: Variations in plate geometry and excitation frequency were omitted due to

time constraints. Multiple element types were also omitted from scope due to primarily

focusing on the influence of mesh element length. Analysis of boundary reflections was

avoided due to the non-physical modelling of plate boundaries in the FE model.

3D finite element

model

In scope: A single finite element mesh was to be tested and model accuracy was to be

established by determination of group velocity. The material properties were the same as

the 2D model to allow for direction comparison of the obtained results.

Out of scope: Anisotropic material properties were unable to be explored due to the

extensive learning curve required to implement such a model in ANSYS. Variation in

element length was not explored due to this being covered in the 2D model. Phase velocity

was not calculated for the 3D model due to the extensive time required for the analysis.

Frequency

investigation

In scope: The frequency range was limited to the low-frequency regime to avoid complex

higher-order modes. Nine simulations were to be carried out across 25 kHz – 400 kHz.

Out of scope: Incremental variation in frequency was out of scope due to the

computational resources required for the simulations. Analysis of higher order modes.

Interactions with

structural

damage

In scope: 2D analysis of the interactions between the A0 mode and two types of material

damage: a surface notch and mid-thickness void.

Out of scope: 3D analysis of more complex damage types (delaminations, non-symmetric

cracks) were out-of-scope due to the complex FE modelling required for such analyses.

4

4 Literature review

Fundamentals of Lamb waves

Elastic waves are mechanical waves that propagate within a structure due to a perturbation

[3]. Sources of elastic waves induce volume (compression or extension) or shape (shear)

deformations which excite particles increasingly distant from the source as the wave propagates

[3]. Elastic waves induce elastic deformation only, meaning the particles have no net

displacement after excitation. There are numerous modalities of elastic waves which are defined

by their characteristic particle motion. These include longitudinal waves, shear waves, Rayleigh

waves, Lamb waves, Stonely waves and Creep waves [2].

Lamb waves are elastic waves which propagate in structures having planar dimensions

much greater than thickness, such as a plate [2]. Lamb waves are guided by the upper and lower

free surface boundaries of the medium, hence the term guided wave [2]. Lamb waves result

from the superposition of many longitudinal (P) and shear (SV) waves which, as they travel

through a structure, undergo continuous reflections and mode conversions at the free boundaries

[3, 11]. These reflected waves constructively and destructively interfere, and the resultant wave

packet is the Lamb wave [11]. It has amplitude and phase information that is the sum of all the

individual ultrasonic waves [11]. Due to the superposition of both longitudinal and shear

waves, Lamb waves induce particle displacement in the thickness direction while wave motion

extends radially from the source of the excitation [3].

There are two Lamb modes, symmetric Si and antisymmetric Ai, which are characterised by

the displacement behaviour of the particles [3]. Figure 1 is a graphical representation of the

particle perturbation associated with the two Lamb modes. The symmetric mode causes

predominantly in-plane displacement of particles, resulting in compression of the plate, while

the antisymmetric mode causes primarily out-of-plane displacement of particles, resulting in

flexural plate bending [2]. Both symmetric and antisymmetric Lamb waves have infinite modes,

denoted S0, S1, S2… and A0, A1, A2… respectively, with S0 and A0 being the lowest-order

fundamental modes [2]. Many modes may exist simultaneously, with higher order modes

appearing at higher excitation frequencies [2]. Each mode travels at a different velocity and

wavelength which is a result of the dispersive nature of Lamb waves (see Section 4.3) [11].

a)

b)

Figure 1: The symmetric Lamb mode (a) causes predominantly in-plane displacement of particles, while the

antisymmetric Lamb mode (b) causes predominantly out-of-plane displacement [2].

5

Phase velocity and group velocity

Elastic waves are characterised by various parameters which form the fundamental tools in

deriving the analytical solutions of Lamb waves. In a wave packet, the wavenumber k describes

the spatial frequency of perturbations while the wavelength λ describes the spatial period of

perturbations [3]. The propagation of Lamb waves is described by phase velocity c and group

velocity cg [2]. Phase velocity is the relationship between the spatial frequency k and the

temporal frequency ω, and describes the propagation speed of the phase for a particular

frequency in a wave packet [2, 3]. Phase velocity is given by ( 1 ) [3].

𝑐 =𝜔

𝑘=

𝜔

2𝜋𝜆 ( 1 )

The group velocity is the speed at which the overall wave packet propagates through a

medium [2]. Group velocity is generally defined by ( 2 ).

𝑐𝑔 =𝜕𝜔

𝜕𝑘 ( 2 )

Dispersion of Lamb waves

The velocity at which a guided wave propagates through a structure depends on the

excitation frequency and the thickness of the medium [2, 11]. This dependency is known as

dispersion and occurs because the energy within a wave packet propagates at different speeds

depending on the frequency [12]. This causes a wave packet to effectively spread out as it

propagates through a structure. Figure 2 shows the simulated time history displacement data

for the propagation of the antisymmetric Lamb mode in a 1 mm thick aluminium 2024 plate

(𝐸 = 72.4 𝐺𝑃𝑎, 𝜌 = 2780𝑘𝑔

𝑚3 𝜈 = 0.33). The plots were generated using LAMSS® Waveform

Revealer, which is an analytical tool for generating theoretical waveforms and dispersion curves

based on arbitrary engineering data, developed by the Laboratory for Active Materials and

Smart Structures (LAMSS) at the University of South Carolina [13]. The excitation signal used

to generate the Lamb wave is a 5-cycle Hanning windowed tone burst with a centre frequency

of 100 kHz. Figure 2 (a), (b) and (c) show the amplitude response at 0 mm, 250 mm and

500 mm from the excitation source respectively. The dispersive nature of the antisymmetric

Lamb wave is clearly demonstrated with wave packet widening as it propagates through the

medium. Widening of the wave pulse is due to the dispersive relationship between velocity and

frequency. This behaviour can make it difficult to identify the boundary of the wave envelope,

which is usually defined by a certain cut off threshold, and can lead to complications when

attempting to measure the time of arrival (ToA) of a signal [11, 12, 14].

6

Figure 2: The antisymmetric Lamb mode in a 1 mm thick aluminium 2024 plate at a) 0 mm, b) 250 mm, c) 500 mm from

the excitation source. The dispersive nature of Lamb waves causes the wave packet to spread out as it travels through the

medium.

The Rayleigh-Lamb equations describe the dispersive characteristics of Lamb waves [3].

Ostachowicz, Kudela, Krawczuk and Zak provide an extensive derivation in Guided Waves in

Structures for SHM [3]. Using the Rayleigh-Lamb equations, the dispersion curves for a

material can be solved numerically. Dispersion curves are used to relate the group velocity or

phase velocity of a Lamb mode to the excitation frequency and the thickness of the medium.

a) b)

Figure 3: Dispersion curves for aluminium 2024, generated using LAMSS Waveform Revealer show (a) phase velocity

and (b) group velocity as a function of frequency-thickness for the first four antisymmetric and symmetric Lamb modes .

Figure 3 shows the phase and group velocity dispersion curves for aluminium 2024

generated numerically by the software LAMSS Waveform Revealer. Figure 3 shows that there

is a frequency below which only the fundamental antisymmetric and symmetric modes will

exist. The energy associated with excitation signals in this low-frequency region is insufficient

to activate the higher order Lamb modes [2]. Hence, the minimum frequency required to excite

a higher order mode is known as the cut-off frequency [2]. From Figure 3 (a-b) the lowest cut-

off frequency for the first order modes is that of the A1 mode at approximately 1660 kHz-mm.

Thus, excitation signals below this cut-off frequency will induce only the A0 and S0 modes.

Excitation of only the fundamental Lamb modes is a commonly used practice by many authors

investigating long range Lamb wave NDE [1, 8-11, 14, 15]. Generally this is to aid in signal

processing, which can become significantly more complicated due to the presence of multiple

higher order modes within a response signal [15].

7

Lamb wave mode selection

Alleyne and Cawley have summarised the main criteria for mode selection of Lamb waves

in NDE applications [16]. The core requirements are as follows: limited dispersion, low

attenuation, defect sensitivity, appropriate excitation, detectability and selectivity [11, 16].

Generally, highly dispersive Lamb modes are undesirable as the spreading of the wave

packet reduces the resolution that can be obtained when detecting the signal [12]. By

considering energy conservation and neglecting losses, it can be shown that the amplitude of

the wave packet will decrease proportional to the square root of the increase in time duration of

the wave packet [12]. Thus, the wave spreading associated with dispersion leads to decreasing

amplitude as the wave packet propagates through the medium. This can lead to difficulties when

detecting a response signal as the amplitude of the wave packet may decrease below the

sensitivity threshold of the receiver [12]. Comparing the dispersion characteristics of the

symmetric and antisymmetric modes, the S0 mode has a higher group velocity than the A0 mode

in the low frequency-thickness domain. Due to its lower velocity, the A0 mode has been shown

highly useful in pulse-echo NDE scenarios as the reflected signals are more easily

distinguishable due to the increased time separation between the sent and received signals [11].

Attenuation is the dissipation of Lamb wave energy, resulting in the gradual reduction of

amplitude magnitude [2]. Attenuation occurs due to a combination of two primary interactions.

Due to the viscoelasticity of the medium through which the Lamb wave travels, some energy is

lost when particles are disturbed and interact with one another [17]. Energy dissipation is also

attributed to energy leakage out of a structure and into the surrounding medium (unless in a

vacuum) as the mechanical waves will propagate in fluids such as air, water or oil [18].

Attenuation is more significant for the antisymmetric Lamb mode than the symmetric mode

due to the out-of-plane displacement of particles on the surface of the structure [2]. Hence, this

issue is primarily attributed to the ‘leaky’ energy dissipation source. However, the severity of

the attenuation is strongly dependent on the surrounding fluid and is less pronounced in air than

in other more conductive mediums such as water and soil [2].

The third consideration for guided wave mode selection is the capability of the Lamb wave

to detect material damage such as a crack in an isotropic plate or delamination in a composite

material [16]. In the low-frequency domain, the phase velocity of the A0 Lamb mode is lower

than that of the S0 mode. Hence, the A0 Lamb wave has a shorter wavelength for the same

frequency. This effect is demonstrated in Figure 4, which shows the large difference in phase

velocity, and hence wavelength, between the A0 and S0 modes in aluminium 2024 at a frequency

of 500 kHz. This is a particularly important characteristic for detection of damage, as a shorter

wavelength means that the A0 mode is more sensitive to small defects [11]. This has been a

primary factor in the mode selection for numerous authors investigating Lamb wave damage

detection techniques [8, 11, 19].

8

Figure 4: The phase velocity dispersion curve for aluminium 2024 demonstrates the A0 mode’s shorter wavelength for a

given frequency. For example, at 500 kHz-mm, 𝑐𝐴0= 1880

𝑚

𝑠 while 𝑐𝑆0

= 5386𝑚

𝑠. For a plate thickness of 1 mm, the

wavelengths are therefore 𝜆𝐴0= 3.76 𝑚𝑚 and 𝜆𝑆0

= 10.77 𝑚𝑚.

The fundamental Si and Ai modes have almost uniform in-plane and out-of-plane

displacement respectively, hence both types are theoretically capable of damage detection [11].

Alkassar investigated the suitability of the A0 and S0 Lamb modes for damage detection in

aluminium, via the pitch-catch NDE technique [10]. Using a 2D FEM model, it was shown that

both the symmetric and antisymmetric modes were capable of detecting the damage at any

arbitrary depth [10]. However, the symmetric mode has been shown to be ineffective for

detection of delaminations in unidirectional and cross ply composite laminates [8]. Guo and

Cawley investigated this behaviour in 2D FEM simulations in which it was found that the S0

mode was not capable of detecting delaminations at through-thickness locations with zero shear

stress [9, 10]. Hayashi and Kawashima explored the antisymmetric mode using 2D strip-

element (semi-analytical) FEM analysis [20]. It was shown that the A0 mode is sensitive to

delaminations at all through-thickness locations within a composite laminate, as shown in

Figure 5 [8, 20]. It was also found that the S0 mode was not sensitive to delaminations located

at the interface where the shear stress was zero, in accordance with the findings of Guo and

Cawley [20]. This strong dependency on defect location limits the application of the S0 mode

in many industries, such as aerospace and military, which are increasingly moving toward

composite materials. Hence, many authors have chosen to study primarily the A0 Lamb wave

for damage detection as the conclusions are potentially relevant to a wider range of applications

[8, 11, 15].

9

Figure 5: Hayashi and Kawashima compared A0 and S0 modes in a composite laminate. It was found that the A0 mode was

sensitive to delaminations (pictured) at all through thickness locations, while the S0 mode was not sensitive to the delaminations

located between plies 2-3 and at the midplane [20].

The S0 Lamb mode can require a complicated transducer arrangement to obtain the

symmetric signal [2, 11, 21]. In contrast, the antisymmetric mode is comparatively simple to

obtain via the excitation of a single piezoelectric transducer [16]. A piezoelectric transducer

mounted to the surface of a structure generates a vertical force through expansion of the

piezoelectric element, thus inducing an antisymmetric normal stress and activating the A0 mode

[11]. Excitation of the A0 mode via this method is highly attractive as the amplitude of the S0

mode which is also generated is typically an order of magnitude lower than the A0 response

[11]. Hence, the energy transferred to the symmetric mode is considerably less than that of the

antisymmetric mode [11]. It is typically undesirable to have the amplitudes of both the A0 and

S0 modes in the same order of magnitude. This is because the S0 mode induces a limited amount

of out-of-plane displacement, as does the A0 mode induce limited in-plane displacement. Hence,

a signal response may become significantly noisy if the amplitude response of the other mode

is large. This can lead to unnecessary complications during signal processing and hence should

be avoided.

Table 2 summarises the advantages and disadvantages of the fundamental symmetric and

antisymmetric Lamb modes. As can be interpreted from the comparison, the antisymmetric

mode provides several distinct advantages over the symmetric mode for the purposes of this

study. Most notably, the lower group velocity of the A0 mode makes it significantly easier to

distinguish from its reflections, hence aiding in signal processing of the results. Additionally,

its lower phase velocity and sensitivity to damage at all through-thickness locations makes the

A0 mode more versatile when exploring the interactions with material damage. As mentioned

previously, the most notable downside of the A0 mode is the increased susceptibility to

attenuation, however this effect can be mitigated by selection of the appropriate medium.

10

Table 2: Comparison between the symmetric and antisymmetric Lamb modes.

Symmetric mode Antisymmetric mode

−

Higher 𝑐𝑔 for a given frequency means the S0

mode can be more difficult to distinguish from

its boundary reflections, particularly in

structures containing multiple damages.

+

Lower 𝑐𝑔 for a given frequency means the A0

mode is more easily distinguished from its

reflections, particularly in structures containing

multiple damages.

+ In-plane particle displacement means the S0

mode experiences less attenuation. −

Out-of-plane particle displacement means the A0

mode experiences more attenuation.

−

Longer wavelength for a given frequency

means the S0 mode is less sensitive to small

defects.

+ Shorter wavelength for a given frequency means

the A0 mode is more sensitive to small defects.

−

The S0 mode is not sensitive to delaminations

at through-thickness locations where the shear

stress is zero.

+ The A0 mode is sensitive to damage at all

through-thickness locations.

− A complex transducer arrangement is often

required to obtain the S0 mode. +

Activation of the A0 mode is comparatively

simple using a piezoelectric transducer.

− Measuring in-plane displacement to detect the

symmetric mode is often difficult. +

Measuring out-of-plane displacement to detect

the antisymmetric mode is comparatively simple

using a strain-gauge.

Excitation frequency selection

Staudenmann outlines the three main criteria around which the characteristics of an

excitation signal are defined for generating guided waves in thin structures [5]. The

characteristic wavelength λ0, which is driven by the central excitation frequency f0, must be

large compared to the thickness of the structure [5]. This criterion guides frequency selection

based on the geometry of the structure. Secondly, the pulse must be distinguishable from

reflections which are generated from the interaction with boundaries and/or discontinuities [5].

This criterion defines the spatial position of the excitation signal to ensure that the captured

response is not distorted by noise from boundary reflections. Finally, the length of the pulse

must remain short compared to the planar dimensions of the structure through which it

propagates [5]. This is to ensure that incident and reflected signals remain easily distinguishable

from one another in the time domain. This criterion defines the bandwidth of the excitation

pulse in the frequency domain as well as the duration of the pulse in the time domain.

Staudenmann discusses that wave separation is best maintained when the frequency spectrum

of a wave pulse is narrow-banded [5]. This is because the phase velocities within a narrow-

banded pulse differ less, resulting in decreased wave spreading due to dispersion [5].

Staudenmann identifies that the ideal signal to satisfy the third criterion is one which has a

short duration in the time domain and also has a narrow-banded spectrum in the frequency

domain [5]. This compromise is achieved by modulating the excitation signal using a Hanning

window function [5]. The Hanning window function reduces the effect of spectral leakage in

the frequency domain [22]. Alleyne and Cawley outlined that it is necessary to control the

excitation bandwidth in the frequency and wavenumber domains [23]. It was noted that this is

best achieved using a tone burst enclosed in a Hanning or Gaussian window [23]. The formula

of the Hanning window function is given by ( 3 ) [5].

11

ℎ(𝑡) =𝑎

2(1 − cos (

2𝜋𝑓0𝑡

𝑁))

( 3 )

Typically, most authors follow the recommendations laid out by Alleyne and Cawley,

utilising a sinusoidal tone burst modulated by a Hanning window [1, 5, 8, 10-12, 19, 22, 24-

26]. Using ( 3 ), the amplitude of the excitation signal is therefore given by ( 4 ).

𝐴(𝑡) =𝑎

2(1 − cos (

2𝜋𝑓0𝑡

𝑁)) sin(2𝜋𝑓0𝑡)

( 4 )

A low number of cycles within a pulse defines a wide frequency spectrum, while high values

of N will define very narrow frequency spectrums [5]. In accordance with Staudenmann’s

guidelines for excitation pulse selection, a higher number of cycles within a pulse will result in

reduced pulse-widening and better wave separation. However, the duration of the excitation

pulse T is defined by the central frequency and the number of cycles (𝑇 =𝑁

𝑓0) [5]. Thus, the

duration of the pulse will increase as the number of cycles increases for a given frequency. This

means that a compromise exists numerically between N and T to achieve a useful signal.

Alkassar et al. used a central frequency of 100 kHz in a 5-count Hanning windowed sinusoidal

tone burst in their simulations of the S0 and A0 modes in aluminium [10]. In a FEM study of

Lamb waves in quasi-isotropic laminates, Ng and Veidt used a 140 kHz narrow-band 6-cycle

sinusoidal tone burst signal to generate the A0 Lamb mode. Common values of N found within

the full literature search typically range between 3 – 6 [1, 5, 8, 10-12, 19, 22, 24-26]. The central

frequency f0 commonly used in experiments typically falls between 2 – 200 kHz [5].

Modelling Lamb waves using the finite element method (FEM)

Su discusses the common approaches used for modelling Lamb waves numerically, these

include the finite element method (FEM), boundary element method (BEM), finite strip element

method (FSM) and the spectral element method [2]. The most cost-effective and convenient

approach is typically FEM modelling due to several commercially available products such as

ANSYS, ABAQUS and Patran [2]. As such, many authors have simulated Lamb waves using

FEM in a wide range of studies (see Figure 6) [1, 8, 10, 11, 14, 19, 20, 26-29]. Su outlined the

two main requirements for modelling Lamb waves via FEM; activation of the wave pulse and

acquisition of the response [2]. Lamb waves can be activated by application of a nodal

constraint, such as a nodal displacement or nodal force, at the position of the actuator [2]. The

S0 mode can be activated by a radial in-plane nodal constraint, while the A0 mode is activated

by an out-of-plane nodal constraint [2]. Acquisition of the wave pulse is achieved by measuring

the dynamic response which is typically nodal displacement or strain [2].

12

a)

b)

Figure 6: Ng and Veidt used ANSYS to model the interaction between the A0 mode and a delamination in a

carbon/epoxy composite plate (a) [8]. Lasˇova´ used ABAQUS to conduct a two-dimensional analysis of the propagation of

the A0 and S0 modes in an aluminium plate [14].

The fundamental assumption for FEM modelling of Lamb waves is the linear elastic

interactions between nodes [29]. Under this assumption, the equations of motion for the system

under some dynamic load is given by ( 5 ) [29].

𝑀 �̈� + 𝐶�̇� + 𝐾𝑢 = 𝐹𝑎 ( 5 )

Where u is displacement, M is the mass matrix of the structural elements, C is the damping

matrix, K is the stiffness matrix and Fa is the applied loads. The principal behind FEM

simulation is that ( 5 ) is populated with the material properties of the structure, the initial

conditions of the nodes and the dynamics of the load. By solving the equations of motion of the

system through numerical integration, the displacements of the mesh nodes are acquired. The

method of integration, time-step and other analysis settings are dependent on software selection,

user specification and/or the physics of the problem.

Leckey et al. compared several numerical codes (ABAQUS, ANSYS and COMSOL) in

simulating guided ultrasonic waves in composite laminates [26]. In this study a 3-cycle Hanning

windowed sine wave was used to actuate the A0 Lamb mode [26]. The ANSYS Mechanical

14.5 implicit solver was selected, using a Newton-Raphson time integration scheme to solve (

5 ), with a fixed time-step of 0.1 μs [26]. Leckey et al. found that all tested numerical codes

were adequate for simulating the propagation of guided waves if configured correctly [26].

Triangular or tetrahedral elements were found to produce the most uniform mesh in ANSYS

[26]. The typical simulation times in ANSYS were generally longer than either ABAQUS or

COMSOL which was attributed to the higher number of degrees of freedom for each element

[26].

One major difficulty highlighted in the work by Leckey et al., is the significantly high

computational times required for ANSYS implicit computations. For example, one such

simulation took 170 hours [26]. It is therefore important to ensure that the appropriate

integration scheme is selected when using FEM to model Lamb wave propagation. Duczek et

al. discuss the differences between explicit and implicit time integration and their applicability

to FEM modelling of Lamb waves [30]. Explicit time integration uses a central difference

method which relies solely on the results from the previous time step [30]. Explicit time

13

integration schemes are conditionally stable meaning there is a critical time-step above which

convergence to a solution is not guaranteed [30]. The critical time-step for the explicit central

difference scheme is defined by the central excitation frequency, as shown by ( 6 ) [27, 30].

Δ𝑡𝑐𝑟 =2

𝑓0

(√1 − 𝜁2 − 𝜁)

( 6 )

Where f0 is the excitation frequency (Hz), and ζ is damping. With no damping the critical

time-step is defined purely by f0. Implicit time integration methods depend only on the

excitation frequency and are unconditionally stable, meaning convergence to a solution is

guaranteed for any time-step [30]. However, the disadvantage associated with implicit schemes

is that the current time-step and previous time-step must both be evaluated [30]. Duczek et al.

compared the simulation results of Lamb wave propagation in ABAQUS using both implicit

and explicit integration solvers [30]. Similar accuracy was found between the two integration

solvers, however the explicit solution was much faster at 605 seconds compared to 74,454

seconds for the implicit solver [30]. The significantly lower computation time required for the

explicit solution is due to the simplified matrix operations required in each time step. Although

implicit integration schemes are unconditionally stable, it was found that the time-step was

similar between the two methods (approximately 0.01 μs) as it was driven by the physics of the

problem rather than criterion for convergence in the explicit solver [30]. Duczek concluded that

explicit integration solvers are typically recommended for most SHM-related simulations of

guided waves in FEM [30]. The advantage of a slightly larger time-step in implicit solutions is

generally outweighed by the increased computational effort required in each step [30].

Moser investigated 2D modelling of Lamb waves in ANSYS 5.3 with particular focus on

verifying the criteria for convergence [29]. Moser recommends that the critical time increment

Δtcr required for convergence should allow for at least 20 points per cycle, which is expressed

mathematically by ( 7 ) [29]. This criterion is more conservative than ( 6 ) and may explain why

implicit and explicit solvers showed little variation in time-step. ABAQUS/Explicit dynamics

solver also recommends stability for the integration timestep is given by ( 7 ) [19].

Δ𝑡𝑐𝑟 ≤1

20𝑓0

( 7 )

The temporal behaviour of a simulated Lamb wave is controlled by the time-step given by

( 7 ) [29]. It must also be ensured that the propagating wave is resolved spatially by discretising

the structure with a sufficient mesh density [29]. It is recommended that the mesh allows for 10

– 20 nodes per wavelength to ensure sufficient spatial resolution [29]. The more conservative

criterion is given by ( 8 ) [29].

𝑙𝑒 ≤𝜆𝑚𝑖𝑛

20

( 8 )

Generally, the higher the number of mesh nodes per wavelength, the better the spatial

resolution and overall accuracy of the simulation [29]. However, as the EOM’s of the system

become more complex with increasing mesh nodes, discretising the system with smaller le

results in longer computational times. Moser found that the spatial criterion given by ( 8 ) was

not overtly critical [29]. Good agreement between simulated and analytical results occurred

14

even when the element lengths were greater than le [29]. However, it was also found that the

temporal requirement is highly important to the accuracy of the simulation. The numerical

solution was found to worsen as the ratio of 1/(Δ𝑡𝑐𝑟𝑓0) approached the minimum of 20 [29].

Element selection

As with selection of the integration timestep and element length, the element type is another

consideration when modelling Lamb waves via FEM. Authors have shown it is possible to

simulate Lamb waves in 2D and 3D FEM simulation environments. The typical element type

used by most authors for 2D simulations is a 4-node quadrilateral structural element [10, 14,

19, 28, 29]. In ANSYS this element type is the PLANE42 2D Structural Solid, which assumes

plane strain or plane stress and has a total of eight degrees of freedom (DOF) (each node is

capable of translation in the x/y directions) [31, 32]. These element types also have uniform

mass across the area, which is required for realistic Lamb wave propagation results [29].

Generally, authors recommend that elements are assumed to be in plane strain in the z-direction

when simulating 2D Lamb wave propagation [10, 19, 29]. The PLANE42 Structural Solid

element is shown below in Figure 7 (a), which was sourced from the ANSYS Theory Reference

Release 5.6 (user manual) [32].

Volume elements are required when simulating Lamb wave propagation in three-

dimensions. These are typically eight-node brick elements such as SOLID45 in ANSYS [8, 27].

This element type has eight nodes (one at each corner), each having 3 degrees of freedom (x/y/z

translation) [31]. Brick elements have been shown highly successful in simulating Lamb waves

in FE studies focused on both metallic and composite materials [8, 27]. A visualisation of this

element type is provided in Figure 7 (b) [32].

a)

b)

Figure 7: Common elements used in ANSYS for modelling in two-dimensions and three-dimensions are PLANE42 (a)

and SOLID45 (b) respectively [32].

Shell elements are special elements which occupy a midground between 2D area elements

and 3D volume elements. Shell elements are two dimensional in nature, but can be curved to

fit a three-dimensional surface [31]. These element types are very effective for modelling thin

structures and because they have 4 nodes, each with 6 DOF, the computation time and file size

associated with FE analysis using shell elements is typically more efficient than solid elements

[31]. Shell elements were used by Liu to model Lamb waves in aluminium in a 1D simulation

with good agreement to the analytical solution [33].

15

It is important that selected FEM software controls numerical errors which can be

introduced when using volume elements; notably shear locking and hourglassing. Shear locking

is a phenomenon which causes elements to become overly stiff in bending applications [31].

This occurs in fully integrated first order brick elements as the ‘edges’ between nodes cannot

bend under an applied moment, which causes a non-physical shear stress to be introduced [31].

In many FE codes, reduced integration schemes are introduced to avoid shear locking, which

involve a single integration point within a solid element rather than eight [31]. However,

reduced integration also introduces the issue of hourglassing, which is the tendency of elements

to deform in non-physical ways. This occurs when an element is deformed without generating

strain energy within the element, thus leading to zero-order energy modes [31]. Figure 8 shows

three possible ‘hourglass’ modes that may occur in a finite element mesh of 4-node quadrilateral

elements [34]. The existence of such deformations results in non-physical behaviour of the

structure. This often occurs in course meshes and can lead to a structure having unrealistic

flexibility [31]. ANSYS (among other FE codes) provides inbuilt hourglassing control which

monitors the existence of these zero-order energy modes to ensure that results are physical.

Figure 8: Hourglassing results in the non-physical deformation of finite elements [34].

Signal processing techniques

Signal processing techniques are required to analyse the nodal displacement data captured

in a FEM simulation of Lamb wave propagation. Measurement of wave speed is a fundamental

tool in verifying the accuracy of numerically simulated Lamb waves. Staudenmann discusses

the difficulties associated with measurement of the group velocity of a wave pulse [5]. Due to

the dispersive nature of Lamb waves, wave packet widening makes it difficult to accurately

determine the time of arrival of the wave pulse. Several possible points of measurement exist

within the wave packet; these being the beginning of the pulse, middle (or average) of the pulse,

end of the pulse, the point of maximum amplitude or the centre of the pulse [5]. Staudenmann

concluded that of these potential measurement points, only the beginning of the pulse can be

used to determine the group velocity as all other measurements generate errors due to distortion

of the pulse. Using the beginning of the wave packet as a point of reference, the group velocity

can be resolved by ( 9 ) [5];

𝑐𝑔 =Δ𝑥

Δ𝑡

( 9 )

Su discussed the primary tools available for time domain analysis of digital signals [2]. One

such tool is the Hilbert transform, which converts the displacement signal of a Lamb wave into

its energy distribution as a function of time [2]. The Hilbert transform is given by ( 10 ) [2].

16

𝐻(𝑡) =1

𝜋∫

𝑓(𝜏)

𝑡 − 𝜏𝑑𝜏

∞

−∞

( 10 )

The result of computing the Hilbert transform is an envelope which depicts the energy

distribution of the signal in the time domain [2]. The Hilbert transformation is demonstrated in

Figure 9, which shows a signal response in (a) and the corresponding energy distribution in (b)

[2]. The energy distribution helps to identify Lamb wave pulses within signals that contain

noise, as shown in (a). The energy distribution of the signal (b) clearly distinguishes the wave

pulses from the noise within the signal. Murat used the Hilbert transform to calculate the group

velocity of the A0 Lamb mode based on displacement data generated from a 3D FEM simulation

of a composite plate [11]. By taking the Hilbert transform of the signal, the peak of the signal

envelope provides a consistent point of reference from which the group velocity of the wave

pulse could be calculated between two locations [11].

a) b)

Figure 9: The Hilbert function reveals the energy distribution of the signal. The energy envelope can be used to precisely

identify the peak amplitudes within a signal that contains a significant amount of noise, as shown from (a) to (b) [2].

Staudenmann also discussed a numerical methodology to determine the phase velocity of a

signal, by taking two measurements at different locations and using the change in phase angle

to produce a numerical expression for c given by ( 11 ) [5].

𝑐 =2𝜋𝑓

𝑘= 2𝜋Δ𝑥

𝑓

Δψ

( 11 )

The phase of a signal is given by the imaginary component of its Fast Fourier Transform

(FFT). The FFT is a numerical tool which converts a signal from the time domain to the

frequency domain [2]. Murat used this methodology to calculate the phase velocity of a Lamb

wave signal with good agreement between analytical and simulated results [11].

Another signal processing tool commonly used to analyse Lamb waves is the 2-dimensional

Fast Fourier Transform (2D FFT) [2, 14, 29, 35, 36]. Alleyne and Cawley investigated the 2D-

FFT for measurement of propagating Lamb wave signals [35]. It was noted that because Lamb

waves are sinusoidal in both the frequency and spatial domains, the temporal FFT can be carried

out in the time domain, followed by the spatial FFT in the space domain, to resolve the

amplitude magnitudes of the signal at discrete wavenumbers and frequencies [35]. By taking

17

the magnitude of the output of the 2D FFT, the amplitudes can be plotted against wavenumber

and frequency in a contour plot to reveal the wavenumber-frequency dispersion curves, as

shown in Figure 10 (a) [37].

Costley investigated the dispersion properties of laser-generated Lamb waves in aluminium

using the 2D Fast Fourier Transform [36]. By applying the 2D FFT on the displacement data

of 50 equally spaced points along the plate, the wavenumber-frequency dispersion curves were

revealed. The experimental dispersion curves are presented in Figure 10 (b). The contour plot

demonstrates the differences in resolution which are obtained depending on the spatial

resolution at which displacement data is recorded.

a) b)

Figure 10: The 2D FFT can be used to reveal the Lamb wave dispersion curves (a) [37]. Costley used the 2D FFT to

obtain the wavenumber-frequency dispersion curves of aluminium (b) by measuring evenly spaced 50 displacement signals

across the plate [36].

The two-dimensional Fast Fourier Transform is given by ( 12 ) [2]. Costley noted that in

order to satisfy spatial sampling criterion, the distance between each spatial measurement Δx

must be sufficiently small to prevent aliasing [36]. In addition, Δx must be evenly spaced and

the time-step (Δt) must be constant [36].

𝐻(𝑘, 𝑓) = ∫∫ 𝑢(𝑥, 𝑡)𝑒−𝑖(𝑘𝑥−𝜔𝑡)𝑑𝑥 𝑑𝑡

( 12 )

The wavenumber-frequency dispersion curves generated by the 2D FFT were used by

Lasˇova´ in a numerical study aimed at solving the group velocity of Lamb waves in an

aluminium plate [14]. By taking the time-history displacement of 4096 equally spaced nodes

along the plate and arranging the data ‘column-wise’ in the matrix u(x,t), the 2D FFT was

computed to reveal the wavenumber-frequency dispersion curves. Using ( 2 ) the group velocity

of the wave pulse was calculated numerically with very good agreement to analytical results

[14]. It was found that the accuracy of the dispersion curves was highly dependent on the

number of spatial time-signals used in the matrix u(x,t), with more positional data improving

the resolution of the dispersion curves [14].

Modelling structural damage in FEM

The main purpose of structural health monitoring is to effectively detect the onset of

structural damage. Common sources of damage within engineering materials are structural

18

fatigue, excessive load, impact and corrosion [2, 19]. Structural damage in metals commonly

manifests in the form of a crack, notch, pitting, crevice, exfoliation or inclusion [2, 19].

Composite materials have more complicated damage modes due to the complex interactions

between layers. Common forms of damage include delaminations, matrix cracking, fibre

breakage and interfacial debonding [2]. Structural damages have been modelled using FEM in

various studies [1, 2, 8, 10, 11, 19, 20, 38]. Su outlines the main techniques for modelling

structural damage within metallic structures [2]. Cracks and notches usually propagate

perpendicular to the surface of a plate and may extend partially or through the full thickness

[2]. Typically this is modelled in FEM by removing elements at the place of damage and

keeping the remaining surfaces apart [2]. Palmos studied the scattering behaviour of guided

waves in thin aluminium plates using horizontal cracks [1]. The aforementioned methodology

described by Su was used to develop a finite element model of the damage in ANSYS as shown

by Figure 11 below [1].

Figure 11: Cracks are modelled in FEM by removing elements and ensuring that the remaining surfaces are separated [1].

Conclusions from the literature review

A comprehensive review of the literature was carried out to consolidate the fundamental

theory required for the thesis investigation. Section 4.4 discussed the considerations involved

in the selection of Lamb wave modes for NDE applications. The fundamental antisymmetric

Lamb wave has been selected for this study based on the results of the literature review. This is

because it has a shorter wavelength than the S0 mode for a given frequency, meaning it is more

sensitive to small damages within a structure [5]. Additionally, its lower group velocity means

it is more easily distinguished from its reflections than the S0 mode [5]. Su proposed that the A0

mode can be activated in an FEM model by an out-of-plane nodal displacement. As such, to

activate the A0 mode in this investigation a nodal displacement constraint was applied to the

surface of the model.

Previous versions of ANSYS have been used in the simulation of Lamb waves. Alkassar et

al. used ANSYS 15 to perform 2D simulation of Lamb waves, while Leckey et al. explored the

capabilities of ANSYS 14.5 in 3D simulations [10, 26]. The results published in these studies

provided justification for using ANSYS in this thesis. The latest version of ANSYS (18.2) was

consequently selected for the study.

Section 4.5 presented the literature relevant to excitation frequency selection. It was found

that a sinusoidal tone burst, modulated by a Hanning window function, was the most appropriate

19

signal excitation for this study. The amplitude of the nodal displacement constraint was

therefore derived from Equation (4). The displacement amplitude was then applied in the

through-thickness direction of the ANSYS model to activate the A0 mode.

Publications relating to FEM modelling of Lamb waves are presented in Section 4.6. It was

concluded that FEM simulations of Lamb waves are more efficiently solved using explicit time

integration methods. Consequently, ANSYS Explicit Dynamics was selected as the analysis

environment for modelling the A0 mode due to its inbuilt explicit integration solver, Autodyn.

Two major criteria controlling the integration time-step and element length were presented by

Equations (7) and (8). Due to recommendations in the literature, the time-step was selected to

be program controlled. Several mesh resolutions were tested to investigate the influence of

element length on the simulation results. Mesh resolutions were selected as low as 6 nodes-per-

wavelength and as high as 33 nodes-per-wavelength to investigate element criterion (see 9.2).

Section 4.7 discussed the common element types used in FEM modelling of Lamb waves.

It was found that typical elements used for 2D plane strain analysis are 4-node quadrilateral

elements. 8-node structural solid brick elements are generally used for 3D analysis. It was

therefore concluded that PLANE42 (2D) and SOLID45 (3D) elements would be most

appropriate for the FEM simulations in ANSYS.

Finally, several signal processing techniques were discussed in Section 4.8. Two methods

were presented for calculating Lamb wave group velocity. Staudenmann recommended the

reference-amplitude approach, while Murat used the energy distribution [5, 11]. Both

methodologies were investigated in this thesis and the advantages and disadvantages of each

approach were to be discussed. The Hilbert transform was to be used to reveal the energy

distribution of the wave pulse, as proposed by Su [2]. Based on the work conducted by Lasˇova´,

the 2D FFT was to be used to determine the phase velocity of the simulated Lamb mode.

5 Development of the two-dimensional ANSYS model

Overview of the study

A two-dimensional finite element model was developed in ANSYS with the objective of

validating the software’s capability of accurately simulating the propagation of the fundamental

antisymmetric Lamb mode. A 2D model was selected for this investigation due to the relative

simplicity in building the model geometry, more simplified finite element mesh, faster

computation times and easier data extraction. This investigation provided a baseline capability

of the software in modelling Lamb wave propagation in the simplest scenario.

The material was selected as aluminium 2024-T6 as these material properties were available

in the analytical software LAMSS Waveform Revealer. The analytical solutions calculated by

this software package were used as the baseline to which the simulated numerical results were

compared.

Six finite element models with varying element length were developed and the results were

compared to investigate the influence of element resolution on model accuracy. The accuracy

of the obtained data was determined both qualitatively, through comparison with the analytical

solutions, and quantitatively, by calculation of group velocity and phase velocity. Two methods

of calculating group velocity were explored and a sensitivity analysis was carried out to

20

compare the reliability of each model. The influence of spatial resolution on the accuracy of

calculated phase velocities was also investigated.

Methodology for constructing the FE model

The model was developed in the ANSYS 18.2 Workbench, which is a graphical interface

that operates as a link between the various analysis environments available in the software. The

Analysis System is the starting point when developing an FE model. It defines the overall nature

of the analysis as well as the simulation capabilities of the software which are available to the

user. Once the analysis system is selected, the model tree is then available in the working

environment, as shown in Figure 12. The model tree details the necessary components which

need to be defined in order to produce a working FE model. Each component of the model is

necessary to create a physical and realistic FE model of a real-world system.

Figure 12: The Explicit Dynamics Analysis System is available within the ANSYS 18.2 Workbench

The seven components of the model tree capture all the individual elements which must be

considered in the development of a finite element model. Table 3 briefly summarises each of

these components and describes their primary function in the complete finite element model.

Table 3: Primary roles of the key components of an FE model.

Component Relevant properties to be defined

Analysis system Defines the analysis capabilities of the ANSYS solver as well as the design options

available to the user.

Engineering data Defines the engineering properties of the materials to be used in the analysis. Necessary

properties for elastic analysis include Young’s modulus, density and Poisson’s ratio.

Geometry Defines the geometry of the system. Original geometries can be created from within the

ANYSYS DesignModeler environment using basic technical drawing tools.

Model Defines the Finite Element discretisation of the system by meshing the volume/area of the

object with mesh elements. Elements can be selected and named for specific analysis.

Setup Defines the nature of the perturbation (displacement, force, strain etc.), the initial

conditions of the system and boundary constraints.

Solution Defines the system analysis settings which include: numerical time-step and integration

scheme, the type and quantity of recorded data, numerical damping factors, hourglassing

controls and system solution types.

Results Defines the nature of the data which is extracted in the post processing, and where

measurements are taken. Provides a graphical visualisation of the simulated results.

21

The flow chart below provides a graphical description of the main steps involved in developing

a finite element model, based on the primary components of the model tree. The order of the

steps reflects the steps taken to produce the 2D model.

Analysis system and system properties

As suggested in the Literature Review, the ANSYS Explicit Dynamics analysis system was

selected for this investigation. The system properties were then defined which govern the

overall physical nature of the system as well as the complexity of the analysis. The key system

property selections for the 2D model are shown in Figure 13.

The geometry analysis type was selected to be ‘2D’. This automatically constrains the

system to have principal directions and material properties defined in a two-dimensional plane

(default is the x-y plane). Additionally, this selection defines the 2D behaviour of the system in

the z-direction as either plane stress or plane strain depending on the user’s selection.

Explicit Dynamics analysis using the ANSYS Autodyn solver was defined in the analysis

properties. Autodyn is the ‘in-house’ explicit integration solver built into the ANSYS Explicit

Dynamics analysis environment. By default, Autodyn uses a Lagrangian integration scheme to

efficiently simulate solid elastic systems which undergo extremely high speed deformation such

as a shock [39]. Two Euler solver schemes are also available within Autodyn however these

are primarily associated with fluid flow and when simulating extreme plastic deformation [39].

a) b)

Figure 13: The Analysis System settings were configured for (a) two-dimensional geometry analysis and (b) an explicit

time integration scheme using the Autodyn solver.

Define the

Analysis

System

Define the

engineering

material data

Define the

geometry of

the model

Select and apply

the appropriate

mesh

Create named

selections

Export

results

Run the

solver

Add solution

probes to named

selections

Define the

perturbation and

apply constraints

Specify

analysis

settings

22

Engineering material properties

Selection of material properties for the FE model was driven by the requirement to validate

the accuracy of the captured data through comparison with analytical solutions in LAMSS

Waveform Revealer. In addition, it was also necessary to select a material which had been

explored within the literature to provide a baseline comparison for the physical nature of the A0

Lamb wave. Considering these criteria, the 2D FE model was developed using the material

properties of aluminium 2024-T6 (Al-2024). Al 2024-T6 is a versatile aluminium alloy used

primarily in the aerospace industry due to its high strength to weight ratio and fatigue resistance

[4]. The material properties of Al 2024-T6 were available in LAMSS Waveform Revealer,

meaning the analytical solutions of the A0 Lamb mode could be acquired. Additionally, the

propagation of Lamb waves in aluminium alloys have been thoroughly studied within the

literature by such authors including, but not limited to, Alkassar, Wilcox and Gresil [10, 12, 14,

19, 27].

The engineering material properties of the model were defined in the Engineering Data

toolbox. Finite element analysis of the elastic deformation of a material requires only three

engineering properties: Young’s modulus, density and Poisson’s ratio. This is because the

analysis is limited to the region of the material’s stress-strain curve in which the relationship

between stress and strain is approximately linear. The proportionality between stress and strain

is governed by the Young’s modulus, which represents the stiffness of the material.

The engineering material properties of aluminium 2024-T6 were sourced from the U.S.

Department of Defence MIL Handbook [40]. The material properties are summarised in Table

4 below.

Table 4: Engineering material properties of aluminium 2024-T6 [40].

Property Value

Density (𝝆) 2780 𝑘𝑔/𝑚3

Young’s modulus (𝑬) 72.4 𝐺𝑃𝑎

Poisson’s ratio (𝝂) 0.33

The properties of Al 2024-T6 were entered into the material database in ANSYS as shown in

Figure 14. The Bulk and Shear moduli are calculated automatically based on the primary

Young’s modulus and Poisson’s ratio. Once the material is stored in the database, the

engineering properties can then be assigned to the model geometry.

Figure 14: The engineering material properties of aluminium 2024 were entered into the material database in ANSYS and

assigned to the 2D model.

23

Geometry setup

The geometry of the 2D model consisted of a rectangular surface with length 500 mm and

thickness 3 mm. The coordinate system was such that the planar and thickness directions were

aligned to the x and y axes respectively. In this configuration, the 2D model represented that of

the cross-sectional geometry of a three-dimensional plate viewed on the x-y plane. The length

(x) and thickness (y) were modelled physically, while the width (z) was accounted for under the

assumption of the plane strain in the z direction, as recommended in the literature [10, 19, 29].

The 2D model was developed in the ANSYS DesignModeler toolbox, which is a 3D

interface for geometry creation. Figure 15 shows the graphical interface used to sketch the

model as well as the technical details of the surface body. The rectangular cross section of the

plate was created using the sketching and dimension tools. The rectangular surface was then

generated within the cross-sectional boundary using the ‘surfaces from edges’ tool. The result

was a 2D surface to which the engineering material properties were assigned, thus yielding a

physical model of a plate.

Figure 15: The ANSYS DesignModeler toolbox was used to create the geometry for the 2D cross section of a plate

The geometrical properties of the 2D model are summarised in Table 5 below.

Table 5: Geometrical properties of the 2D plate model.

Geometrical Property Measurement

Thickness (𝒕) 3 mm

Length (𝒍) 500 mm

Width (𝒘) 0 mm

Surface area (𝑨) 1,500 mm2

Body Type Surface Body

Model setup

The ANSYS Academic Research Mechanical and CFD license was necessary to conduct

the simulations within this investigation due to the unconstrained number of mesh nodes

allowed within license compared to other versions, which place limits on model complexity.

This became a highly constraining factor when moving into models with higher-resolution

surface meshes. The model setup defines the overall nature and purpose of the simulation as

24

well as the underlying physics involved in the given scenario. These model parameters are

defined in ANSYS Mechanical, which is the main interface for carrying out explicit

simulations. Figure 16 shows the Geometry and Coordinate Systems selections within the

model tree of the 2D Al 2024-T6 plate. In the Geometry options, the properties of Al 2024 were

assigned to the model by selecting the body and assigning ‘Aluminium’ from the material

library. The 2D behaviour of the model was set to plane strain which was consistent with

recommendations in the literature. The global coordinate system was selected as the default

Cartesian coordinate settings, with the x axis aligned with the principal lengthwise direction of

the plate and the y axis aligned in the thickness direction [10, 19, 29].

a)

b)

Figure 16: The ANSYS Mechanical model tree contains the model parameters which define the physics of the system.

The material selection was defined in (a) Geometry, and Cartesian coordinates were selected in (b) Coordinate System.

The surface of the 2D model was meshed using quadrilateral 4-node elements as observable

in Figure 17 (b). Each node was restricted to 2 degrees of freedom, being translation in the x

and y directions. The nodes were constrained in the z direction due to the assumption of plane

strain conditions. The mesh elements were defined as a function of the characteristic element

length as can be seen in Figure 17 (a). In the example images below, the characteristic element

length is 0.75 mm, hence over the thickness of 3 mm there are a total of 4 elements. In addition

to the pictured 0.75 mm mesh model, other models were developed having characteristic

element lengths of 1.5 mm, 1.0 mm, 0.5 mm, 0.25 mm and 0.15 mm. Simulations using

identical perturbations and analysis settings were carried out for each of the models to determine

the impact of mesh resolution on the accuracy of the simulated A0 Lamb mode. In addition, the

influence of mesh resolution on the computational efficiency of the simulation was investigated

such that the compromise between accuracy and computation speed could be better understood.

25

a)

b)

Figure 17: The 2D model of the aluminum 2024 plate was meshed using quadrilateral 4-node solid elements. The mesh

was defined by the characteristic element length, which is 0.75 mm in (a). The meshed plate is shown in (b).

To conduct a quantitative analysis of the Lamb waves generated in each simulation,

numerous nodal displacements were exported from the model. To export nodal data from

ANSYS this requires Named Selections to be defined at the particular nodes of interest to which

solution probes are then mapped. Additionally, named selections are also useful in defining the

nodes to which applied loads, or other constraints such as a fixed boundary, are applied.

In the 2D Al 2024-T6 model, the separation distance (dx) between points at which

displacement data was captured was primarily driven by the characteristic lengths of the tested

meshes. The mesh lengths were 1.5 mm, 1.0 mm, 0.75 mm 0.5 mm, 0.25 mm, and 0.15 mm,

meaning there were a number of possible separation distances which would provide whole-

number division with all mesh sizes. Initially, the spacing between points of data capture was

100 mm, making a total of 5 captured data points along the length of the plate. However, due

to the low number of data points, this meant that the variations seen in group velocity (due to

limitations in the methodology) made it difficult to establish any meaningful conclusions from

the quantitative analysis. See section 6.4.2 for a detailed discussion of the issues related to

calculation of group velocity. It was concluded that a higher spatial resolution would improve

the rigorousness of the quantitative analysis. However, due to manually selecting mesh nodes

based on x,y,z location, the creation of these named selections was found to be exceedingly time

consuming. Additionally, this process had to be repeated each time a new mesh was generated.

The spatial resolution along the length of the Al 2024-T6 plate was established at 16 equally

spaced nodes with a separation distance of 30 mm. This provided a reasonable compromise

between data resolution and time constraints for the numerous simulations. The named

selections were created on the top surface of the 2D plate as can be seen in Figure 18.

26

Figure 18: Named selections were created at 30 mm intervals along the plate. This provided 16 equally spaced nodes at

which the nodal displacement data was captured.

Selection of the excitation frequency

The centre frequency of the excitation signal was selected based on the dispersive properties

of aluminium 2024-T6, which are characterised by the phase and group velocity curves

presented in Figure 3. It was necessary to select a centre frequency such that only the

fundamental A0 Lamb mode was activated (along with the S0 mode). This is because the

presence of higher order Lamb modes in the high frequency regime can cause interference in

the measured displacement behaviour of the mesh nodes. As stated in the literature, the typical

frequency band is 2 – 200 kHz, hence the centre frequency for the 2D model testing was chosen

to be within this range [5].

Figure 20 (a) shows the phase velocity curves for Al 2024-T6. Below approximately

1714 kHz-mm only the fundamental Lamb modes exist, which is the cut-off frequency defining

the upper bound of the low-frequency regime. Taking the centre of the recommended frequency

range, at f0 equal to 100 kHz, only the fundamental modes exist. Additionally, the S0 mode

clearly travels at a much higher velocity when excited by a frequency of 100 kHz. This is

advantageous when investigating a single mode as it allows the S0 and A0 modes to be clearly

distinguished from one another. When the displacement response of a mesh node shows two

disturbances, the S0 mode would induce the first disturbance, and the A0 mode would induce

the later response. As such, a centre frequency of 100 kHz satisfied the requirements for the

phase velocity dispersive properties of Al 2024-T6. The phase velocity of the A0 Lamb mode

at an excitation frequency-thickness of 300 kHz-mm can be resolved from Figure 20 (b) at

1550 m/s.

27

a) b)

Figure 19: The phase velocity dispersion curves for Al 2024-T6 show that at an excitaton frequency of 100 kHz, only

the fundamental modes exist (a). The analytical solutions to the dispersion curves show the A0 phase velocity is 1550 m/s (b).

The excitation frequency must also satisfy the aforementioned criteria for the group velocity

dispersive characteristics of Al 2024-T6. Figure 20 (a) shows the group velocity dispersion plot

for Al 2024 from which it can be identified that the cut-off frequency, under which purely

fundamental Lamb mode propagation occurs, is approximately 1660 kHz-mm. Thus, at an

excitation frequency of 100 kHz (f-t of 300 kHz-mm) only the A0 and S0 Lamb modes will

occur. Observing Figure 20 (b), the group velocity of the fundamental antisymmetric Lamb

mode is approximately 2621 m/s at the centre frequency of 100 kHz.

a) b)

Figure 20: The group velocity dispersion plots for aluminium 2024-T6 show that at an excitation frequency of 100 kHz,

only the fundamental Lamb modes will exist (a). At this excitation frequency the group velocity is 2621 m/s (b).

Staudenmann identified that the maximum wavelength of the excitation signal, relative to

the thickness of the structure, is a key element within the criteria for Lamb wave propagation

[5]. He stated that the characteristic wavelength 𝜆0 must be sufficiently large compared to the

thickness of the structure [5]. At an excitation frequency of 100 kHz, the phase velocity of a

propagating Lamb wave in Al 2024-T6 is 1550 m/s (from Figure 3 (a)). The wavelength is

therefore calculated by;

𝜆0 =𝑐

𝑓=

1550 𝑚/𝑠

100,000 𝐻𝑧= 15.5 × 10−3𝑚 = 15.5 𝑚𝑚

28

As the thickness of the 2D plate is 3 mm, the ratio of wavelength to thickness is therefore

calculated by;

𝜆0

𝑡= 5.1

Thus, the ratio of wavelength to thickness is sufficiently high to validate the selection of the

central excitation frequency at 100 kHz.

Modelling the excitation frequency

The excitation signal was a nodal displacement in the thickness direction, aligned with the

y axis in global coordinates. The theoretical transducer size was arbitrarily selected at 3 mm in

diameter and was chosen to be placed on the top surface of the left-hand side of the 2D plate as

shown in Figure 21. To implement the perturbation in ANSYS, the nodal displacement loading

type was selected and applied to all surface nodes occurring within the first 3 mm from the left-

hand side of the plate. As the simulations were carried out for several mesh sizes, the number

of elements to which the nodal displacements were applied was varied accordingly. Figure 21

shows the mesh nodes (indicated with yellow labels) to which the nodal displacements were

applied for the 0.75 mm mesh size.

Figure 21: Out-of-plane (y direction) nodal displacements were applied to the mesh nodes occurring in the 3 mm from the

left-hand side of the 2D plate model.

The excitation signal was a 5-cycle sinusoidal tone burst modulated by a Hanning window

function. The signal was generated by implementing the mathematical expression given by

Equation (4) (shown again below) in a numerical algorithm using Python.

𝐴(𝑡) =𝑎

2(1 − cos (

2𝜋𝑓0𝑡

𝑁)) sin(2𝜋𝑓0𝑡)

The amplitude factor (𝑎) of the modulating function was selected small enough such that it

induced purely elastic deformation of the aluminium plate and remained realistic compared

with real-world PWAS transducers. The selected factor was 𝑎 = 1 µ𝑚, which resulted in a

propagating A0 Lamb wave with an amplitude magnitude consistent with results published by

Alkassar [10]. The centre frequency of the function was selected at 𝑓0 = 100 𝑘𝐻𝑧. The number

of cycles was selected in order to balance the pulse duration T and the width of the excitation

signal frequency spectrum. As discussed by Staudenmann, a higher number of cycles within a

pulse will result in reduced pulse-widening and better wave separation, however as 𝑇 =𝑁

𝑓0,

29

increasing the number of cycles increases the duration of the excitation [5]. This can lead to

difficulties in deciphering the incident and reflected waves if the period is too long. Hence, the

practices used in the literature were used as a guide to select N. Alkassar et al. used 𝑁 = 5

and 𝑓0 = 100 𝑘𝐻𝑧 in their simulations of the S0 and A0 modes in aluminium with excellent

agreement between analytical and simulated results [10]. Hence, the number of cycles was

selected to be 5 (𝑁 = 5) for the excitation of the A0 Lamb wave mode in the 2D plate.

The Python algorithm was developed to return a tabular response of the time and amplitude

generated by the function over the period of the excitation divided into 2000 steps. The period

of the excitation is 𝑇 =𝑁

𝑓0=

5

100,000 𝐻𝑧= 50𝜇𝑠. Hence the time-step over which the signal was

plotted is 𝑑𝑡 = 0.025𝜇𝑠. Each time step was used in Equation (4) to produce the instantaneous

amplitude of the excitation signal. The signal was then plotted in Figure 22 below.

Figure 22: The excitation signal was a 5-cycle sinusoidal tone burst modulated by a Hanning window function.

The excitation signal was modelled in ANSYS using the Tabular Data input option in the

Nodal Displacement settings. The displacement signal excitation in the ANSYS environment

is shown below in Figure 23. The tabular data was copied from the Python output and inserted

into ANSYS with the incremental variable being time. The analysis was set to last for 500 𝜇𝑠

to provide sufficient time for the incident Lamb wave to travel across the length of the plate,

reflect off the opposite boundary and return to the origin of excitation.

Figure 23: The excitation displacement amplitude was entered into ANSYS as a function of time.

Boundary constraints

A fixed support constraint was applied to the right-hand edge of the 2D plate model as

shown in Figure 24. The configuration of the model was that of a cantilever beam. This was

necessary to ensure that translation in the y direction, as a result of the excitation nodal

30

displacement, would be prevented by the reactionary force at the far end of the plate.

Consequently, when the excitation displacement is enacted on the left-hand side of the plate,

this configuration results in elastic bending.

Figure 24: A fixed support was applied to the far edge of the model to constrain the model in space.

Analysis settings

The ANSYS analysis settings control important parameters used in the solver’s explicit

integration scheme. The hourglass control was set to the Autodyn standard hourglassing

method, which is recommended for most simulation analyses. The simulation End Time was

set to 500 𝜇𝑠 to provide enough time for the propagation of the Lamb wave across the full

length of the plate and the reflection back to the excitation point. The Initial, Minimum and

Maximum Time Step settings were all set to be Program Controlled. Initially the time-step was

fixed to the critical time-step outlined in Equation (7), however the solver encountered an

“Unexpected Error” and failed to complete the analysis. Hence it was concluded that use of the

ANSYS default time-step settings would be required achieve a successful outcome.

Output controls were defined such that position data of the selected nodes would be saved

at 5000 equally spaced points within the 500 𝜇𝑠 simulation time. This meant data would be

captured at 0.1 𝜇𝑠 increments, which was found to be a reasonable compromise between data

resolution and computational efficiency of the simulation. The number of points was initially

chosen to be 500, with the resultant wave pulse shown in Figure 25 (a). Inspection of the results

clearly showed that this spatial resolution was too low to accurately capture the shape of the

Lamb wave as it passed the mesh node of interest. It can be seen from (a) that the peaks of the

wave pulse are non-physically rendered, which makes determination of certain properties such

as time of arrival more difficult. The same simulation with 5000 captured nodes is shown in

Figure 25 (b), which shows the shape of the wave pulse was accurately captured because of the

higher spatial resolution.

While 5000 points may have exceeded the minimum necessary to achieve sufficient

accuracy, the size of the simulation output was not overly constraining for the analysis. As such,

it was it was considered reasonable to be overly conservative and extract more data than

necessary, to ensure that the Lamb wave pulses were accurately captured. Additionally, the

frequency of the excitation was to be varied during the analysis, thus altering wave speed and

providing justification for using a more conservative dataset.

31

a) b)

Figure 25: The waveform was not accurately captured using 500 nodes per wavelength (a). It was found that 5000 nodes

per wavelength provided sufficient resolution to accurately capture the wave pulse as it travelled across the plate (b).

Data capture and exporting the results

The displacement-time data was captured at each of the equally spaced surface nodes, at

30 mm intervals along the plate, by assigning a Directional Deformation probe to each of the

named selections. The direction of measured deformation was set to be the y axis as the

asymmetric Lamb mode causes predominantly out-of-plane nodal displacement. The solution

results were then evaluated at each of the nodes and the results were exported to Microsoft

Excel. Since the results consistent of purely time and displacement points, the data export

involved simply copying and pasting from ANSYS into Excel.

Figure 26 provides the ANSYS results window, showing a graphical depiction of the

simulated antisymmetric Lamb mode at four points over the 500 𝜇𝑠 simulation. The data was

captured at 180 mm from the excitation. The displacement results were scaled by 30000 to see

the propagating Lamb wave, as the actual nodal displacements were in the order of 1 𝜇𝑚.

Figure 26 (a) shows the nodal displacement as the incident A0 Lamb mode first propagates

from the excitation source. Figure 26 (b) shows the nodal displacement at approximately

140 𝜇𝑠 as it nears the far end of the plate. Figure 26 (c) and (d) show the Lamb wave reflected

off the far boundary of the plate and the captured signal in the displacement results. The second

reflection is also captured as indicated by the third wave packet in the raw displacement data.

Overall, the captured data appeared realistic and physical in a qualitative sense. In all tested

mesh sizes, the solver converged to a valid solution and the simulated Lamb wave propagated

along the plate with realistic, physical behaviour. The wave travelled along the plate and

reflected off the boundary in all cases with little variation between any of the tested mesh sizes.

This indicated the differences between mesh resolution were not large enough to cause

noticeable, or largely non-physical, behaviour of the simulated Lamb wave.

The raw displacement data was exported by copying the data into an Excel spreadsheet. A

specific template was used for the data, which was arranged row-wise by time-incremental

displacement and sheet-wise by the location along the plate. The standardised format was

created in order to automate the analysis tools developed in Python for future models.

32

a)

b)

c)

d)

Figure 26: The ANSYS results window provided a graphical output of the nodal displacement data, which was used to

qualitatively analyse the propagation of the wave and make sense of the raw data.

33

6 Analysis of the two-dimensional ANSYS simulation

Overview

The analysis of the data captured in the 2D simulation involved several key elements.

Firstly, the excitation signal applied in the FE simulation was verified against the analytical

model to ensure that the perturbation was accurately modelled. The raw displacement data

captured in the simulations was then processed to produce useable data from which group

velocity and phase velocity could be determined.

Two methodologies for calculating the wave pulse group velocity were explored. The

‘reference-amplitude’ and ‘energy distribution’ approaches for time of arrival were compared

and the advantages and disadvantages of each were explored. A sensitivity analysis was carried

out to explore the reliability of each model. Using the selected approach, the group velocity of

the incident Lamb wave simulated in each of the FE models was calculated. The influence of

mesh element length on the accuracy of the simulation results was explored by comparing the

results of each model with the theoretical wave speed.

The phase velocity of the propagating Lamb wave was calculated using the two-dimensional

Fast Fourier Transform. The influence of spatial resolution on the accuracy of the obtained

results was investigated.

Where possible in the subsequent analysis, plots are presented with data pertaining to the

various FE models of different mesh resolutions. However for simplicity, many of the plots

provide the data of only one model as an example of the methodology or to demonstrate a

notable feature. The same analysis was carried out on all results to determine the influence of

mesh element resolution on the accuracy of the finite element simulations.

Verification of the excitation signal

The raw data exported from the ANSYS FE model was a tabular report of nodal

displacement as a function of time, 𝑢(𝑡). Each report contained 5000 measurements at equal

time increments recorded over the course of the simulation. The integration time step-size was

controlled by the ANSYS solver and is defined by the characteristic length of the mesh

elements. Consequently, the instantaneous time at which each measurement was taken was

variable across the six different 2D models.

Python was selected as the most capable and versatile numerical tool to process the raw

ANSYS data in semi-automated algorithm. The aim of such analysis was to yield useable results

from which the dispersive properties of the propagating Lamb wave could be understood. The

time-displacement data of nodes at which the excitation displacement was applied (𝑥 = 0 𝑚𝑚)

is shown below in Figure 27. Note, the reference point for 𝑥 = 0 𝑚𝑚 was the last excitation

node in each model, located 3 mm from the left-hand side of the beam. It can be seen that the

nodal sinusoidal tone burst shows good agreement to the mathematical expression shown in

Figure 22. It should be noted that because the excitation signal has zero displacement after

the 50 𝜇𝑠 oscillation period, the excitation nodes then become fixed in space and do not see any

further displacement when the reflected Lamb wave returns to the excitation position. This is

an inherently non-physical behaviour as the reflected signal should interact with the particles at

this location. In any case, the focus of the investigation was primarily on the incident and first

34

reflected (off the far boundary) Lamb wave. The second reflection was not considered in the

analysis, hence this non-physical behaviour could be neglected.

Figure 27: The raw displacement data captured at the excitation location shows the sinusoidal tone burst was accurately

modelled in all FE models with a 1 µm amplitude and 50 µs period.

It was first necessary to establish that the excitation applied to the finite element model was

consistent with the sinusoidal tone burst used to generate the analytical solutions in the software

LAMSS Waveform Revealer. It was necessary to confirm this agreement prior to investigating

the accuracy of the FE models to ensure that the ‘gold standard’ Lamb wave solutions were

generated from the same excitation signal as the numerical models. If the excitations were found

to differ significantly, any comparison between the numerical and analytical results could be

inaccurate, which would reduce the overall significance of the investigation.

Figure 28 presents the excitation nodal displacement results of the six tested mesh sizes in

addition to the excitation displacement used in the analytical model. As the displacements

measured in the simulation were in the order of 1 𝜇𝑚, the amplitudes were scaled in order to

compare to the analytical model (details of the normalisation algorithm are discussed in the

subsequent section). Figure 28 demonstrates the strong agreement between the excitation

signals used in the simulation and analytical environments. The overall shape of the sinusoidal

tone burst is highly similar, and the number of cycles is consistent. It can be identified that the

analytical and simulated results do not coincide in the time domain, with a small lag of

approximately 5 𝜇𝑠 between the incident excitations. This lag can be accounted for by simply

offsetting one of the measured signals in the time domain. However, since quantitative

comparison of the analytical and simulated waves focused primarily on wave velocities, which

are calculated using the time difference between measured signals, this lag has no impact on

the analysis of the signals.

35

Figure 28: Comparison of the excitation signals of the simulated and analytical models reveals good agreement in the

overall waveform, despite a small offset in the beginning of the wave packet.

To validate the excitation signal using a more rigorous methodology, the properties of the

wave packet were analysed in the frequency spectrum and the energy distributions were plotted.

An FFT algorithm was developed in Python to transform the nodal displacement data from the

time domain to the frequency domain. The resulting frequency spectrum is shown in Figure 29

(a) for the 0.15 mm mesh. Here it can be seen that the dominant frequency, at which the

amplitude of the curve is maximum, occurs at the centre frequency of 100 kHz. The amplitude

of the spectrum tapers off to zero due to the modulating effect of the Hanning window function.

The frequency bandwidth is relatively wide as indicated by the bell-shaped spectrum which

increases in amplitude from a lower bound of 60 kHz to an upper bound of 140 kHz. This means

that the excitation signal is a summation of all frequencies within the spectrum. However, it

was expected that the characteristics of the simulated Lamb wave would be driven primarily by

the dominant centre frequency of 100 kHz. Overall there is good agreement between the

analytical and simulation spectrums with both having peak amplitudes at approximately

100 kHz. Notably, some significant difficulties were encountered when plotting the frequency

spectrum of the 1.50 mm mesh, with the peak amplitude occurring well above the 100 kHz

centre frequency. This is shown in Figure 29 (b) below. It was concluded that the significant

offset was due to the methodology used to plot the frequency array which forms the horizontal

axis of the plot. The frequency increment Δ𝑓 is determined by the average time-step over all

measurements. If the time-step is inconsistent, due to the integration solver varying Δ𝑡 in order

to converge to a stable solution, this would result in the average time-step being an inaccurate

baseline to calculate Δ𝑓 as this increment would be variable. This issue was only seen in the

1.50 mm results and since the excitation signal response was qualitatively consistent (Figure

28) with the other models, it was considered a negligible limitation of the numerical analysis.

36

a) b)

Figure 29: The displacement results were transformed from the time domain to the frequency domain to reveal the

frequency spectrum of the excitation signals for (a) 0.15 mm mesh and (b) 1.50 mm mesh.

The energy distributions of the excitation signals were also plotted using the Hilbert

Function in an algorithm developed in Python. The excitation signals of the 0.15 mm and

1.50 mm meshes are shown below in Figure 30 (a) and (b) respectively. Here it can be seen

that the maximum energies occur at the centres of the wave packets at 25 𝜇𝑠. The amplitudes

of the energy envelopes return to zero after the pulses end at 50 𝜇𝑠. In both cases the simulated

responses are highly consistent with the analytical solution. Additionally, the strong agreement

between the energy distributions of the 1.50 mm model and the analytical model is supportive

of the previous claim that the offset FFT envelope was primarily attributed to numerical error.

a) b)

Figure 30: The energy envelopes of the (a) 0.15 mm mesh and (b) 1.50 mm mesh were plotted against the analytical

model, showing a high level of agreement in both models.

Qualitative comparison of the excitation responses in both time and frequency domains

showed there is a high level of agreement between the analytical and numerical models. The

results of this investigation provided justification for the baseline comparison between the

analytical and simulation results as evidence of simulation accuracy. Since the nodal

perturbation was consistent with the analytical model, comparison of Lamb wave properties

such as group velocity and phase velocity were used to characterise the accuracy of the

simulation results.

37

Signal processing of the raw data

The raw ANSYS output results provide the displacement data for each of the measured

surface nodes over the 500 𝜇𝑠 simulation time. Figure 31 is an example of the raw data captured

in the simulation of the 0.15 mm characteristic mesh length model. Figure 31 plots the nodal

displacement results measured at 𝑥 = 300 𝑚𝑚 from the excitation source, with time (𝜇𝑠) and

displacement amplitude (𝑚) plotted along the x and y axes respectively. It can be identified

from the captured data that the surface node captured three distinct wave packets over the

simulation period. Since the nodal displacement was measured along the thickness direction,

the Lamb wave packets are the antisymmetric mode, which causes primarily out-of-plane nodal

displacement. The time of arrival of the incident wave packet occurs at approximately 100 𝜇𝑠,

followed by the arrival of the reflection off the far edge of the plate at approximately 240 𝜇𝑠,

and finally the second reflection off the left edge of the plate at 440 𝜇𝑠.

The incident and reflected wave pulses clearly demonstrate that the dispersive nature of

Lamb waves was captured in the simulation. It can be observed that the beginning of the wave

pulse oscillates at a higher frequency than the tail of the wave pulse, which oscillates at a lower

frequency. This is because the high frequency components of the excitation signal (up to

140 kHz) arrive earlier, hence resulting in the high frequency oscillation at the beginning of the

wave pulse. Meanwhile, the lower frequency components of the excitation function (down to

60 kHz) arrive later, at the tail of the wave pulse. This behaviour is consistent with the

dispersive nature of Lamb waves, with higher frequencies travelling at a higher velocity than

lower frequencies (see dispersion curves provided in Figure 3). The dependency between

velocity and frequency also resulted in the wave-pulse widening captured between the incident

and reflected wave pulses in Figure 31. As the wave travelled along the plate, the velocity

difference between the high frequency and low frequency components resulted in the overall

widening of the wave pulse. This behaviour is also consistent with the results established in the

literature (see Figure 2).

Analysis of this data shows that the solver captured not only the incident Lamb wave, but

also the reflections from the boundaries. This highlights the fact that the local maxima or

minima of the incident wave packet may not be the global maxima or minima, depending on

the amplitude of the reflected waves. The implications of this consideration are discussed in

further detail below.

38

Figure 31: Nodal displacement results at x = 300 mm show the incident and reflected Lamb wave. Dispersion was

accurately captured in the simulation with velocity differences between the high and low frequencies within the wave pulses.

Figure 32 presents a close-up view of the nodal displacement shown in Figure 31, showing

the amplitude of the incident wave packet at 𝑥 = 300 𝑚𝑚 for the 0.15 mm mesh model.

Exporting 5000 data points at a 0.1 𝜇𝑠 increment resulted in a high temporal resolution of the

amplitude response of the wave pulse. The individual captured data points are plotted on the

graph by scatter points, with the solid line connecting the data points being a ‘best fit’ calculated

by the Python Matplotlib tool. It was important to achieve a high level of temporal resolution

for the signal processing of the captured Lamb waves, particularly for performing the FFT and

Hilbert transformations of the data.

Figure 32: Close-up view of the incident wave packet indicates that the 0.1µs data-capture provided good temporal

resolution of the propagating Lamb wave’s displacement amplitude.

High frequency

components Low frequency

components

39

The capability of ANSYS in modelling dispersion of the A0 Lamb wave is clearly evidenced

by observing the incident wave pulse at consecutive locations across the plate. Figure 33 shows

the displacement histories of four nodes at (a) 60 mm, (b) 120 mm, (c) 180 mm and (d) 240 mm

from the excitation source. It can be observed that the A0 Lamb wave experienced pulse

widening as it travelled across the plate, thus indicating wave dispersion. This phenomenon can

be evidenced quantitatively through determination of the period of the wave pulse at each of

these locations. The period of the wave is calculated by taking the time difference between the

beginning and end of the wave pulse, 𝑡𝑝𝑢𝑙𝑠𝑒 = 𝑡𝑒𝑛𝑑 − 𝑡𝑏𝑒𝑔. Determining these two locations

was difficult due to the small amplitudes of the high frequency oscillations at the beginning and

low frequency oscillations at the tail of the wave pulse. An algorithm was developed in Python

to capture the time data during which the nodal displacement exceeded a cut-off threshold of

1% deviation relative to the maximum nodal displacement. The initial and final times were then

used to calculate the period of the wave pulse and are indicated approximately in Figure 33.

The wave pulse period can be seen to increase consistently as the Lamb wave travels further

along the plate, with measurements of (a) 58 𝜇𝑠 at 60 mm, (b) 65 𝜇𝑠 at 120 mm, (c) 76 𝜇𝑠 at

180 mm and (d) 90 𝜇𝑠 at 240 mm. The increasing trend in period is a quantitative indication

that antisymmetric Lamb waves simulated in ANSYS Explicit Dynamics exhibit pulse

widening and, consequently, wave dispersion. However, the accuracy of these properties

compared to the analytical solutions defined by the Rayleigh-Lamb equations is yet to be

established.

a)

Δ𝑡 ≈ 58 𝜇𝑠

𝑡 ≈ 20 𝜇𝑠 𝑡 ≈ 78 𝜇𝑠

40

b)

c)

d)

Figure 33: Wave dispersion is evidenced by pulse widening between nodes located (a) 60 mm, (b) 120 mm, (c) 180

mm, (d) 240 mm from the excitation source.

Mesh size was found to have significant influence on the overall shape and speed of the

simulated A0 Lamb wave as shown below in Figure 34, which plots the nodal displacement


Δ𝑡 ≈ 40 𝜇𝑠 Δ𝑡 ≈ 105 𝜇𝑠


𝑡 ≈ 60 𝜇𝑠 𝑡 ≈ 136 𝜇𝑠


𝑡 ≈ 78 𝜇𝑠 𝑡 ≈ 168 𝜇𝑠

41

results at 𝑥 = 300 𝑚𝑚. The simulation results indicate that wave pulse group velocity

increased as mesh size was refined. The numerical results converged toward the analytical

solution consistently as mesh element length was decreased. The amplitude of the captured

displacement data also showed significant variation across the tested mesh sizes. The wave

pulse amplitude increased consistently with decreasing finite element length.

Figure 34: Comparison of the nodal displacements at 300 mm shows mesh density impacts the amplitude and speed of the

simulated wave pulse. The raw data indicates convergence toward the analytical solution as mesh length decreases.

The amplitude of a Lamb wave is directly related to the amplitude magnitude of the

excitation signal, which was defined at 1 𝜇𝑚. For this reason, the amplitudes of the measured

wave pulses were normalised to eliminate the influence of the excitation frequency amplitude

on the measured displacement data. This allowed for a more generalised analysis to be carried

out that was independent of amplitude, and purely a function of the frequency of the excitation

signal. The analytical solutions of the Rayleigh-Lamb equations were solved using LAMSS

Waveform Revealer which outputs normalised displacement data irrespective of excitation

amplitude. Since mesh size was found to directly affect the magnitude of the measured nodal

displacement, it was also convenient to normalise the signals to allow for direct comparison of

all mesh sizes to the analytical solutions, as opposed to scaling the analytical solutions to suit

each model.

Normalisation of the Lamb wave amplitude was carried out in accordance with Su’s

recommended methodology, which is to normalise the signal using the maximum magnitude of

its amplitude [2]. This is to say;

𝑢�̅� =𝑢𝑖

|𝑢𝑚𝑎𝑥|

Where 𝑢�̅� is the normalised nodal displacement, 𝑢𝑖 is the captured nodal displacement, and

𝑢𝑚𝑎𝑥 is the maximum magnitude nodal displacement captured within the wave pulse. As the

42

phase velocity and group velocity of Lamb waves are not equal in the low frequency domain,

the location of the amplitude maximum moves as the wave travels along the plate. Hence,

depending on where the measurement is taken, the maximum amplitude may be positive or

negative, which is why the absolute value was used to normalise the displacement data.

An algorithm was developed in Python to normalise the nodal displacements captured in

the ANSYS simulations. Determination of 𝑢𝑚𝑎𝑥 was challenging due to the appearance of the

reflected Lamb waves in the captured displacement data. Figure 35 shows the nodal

displacement data for the 0.15 mm mesh at 30 mm from the excitation. The incident Lamb wave

has a maximum amplitude of approximately −0.70 × 10−6 𝑚. However, in this case the local

maximum is not equal to the global maximum of approximately 0.80 × 10−6 𝑚, which occurs

within the reflected wave pulse at approximately 410 𝜇𝑠. As a result of this phenomena, a

functionality was required within the normalisation algorithm to differentiate the local

maximum from the global maximum when these points were not equal. This was implemented

by setting a maximum time which captured the incident Lamb wave signal but neglected the

signal of the reflected wave. This time occurs at 𝑡𝑙𝑜𝑐𝑎𝑙,𝑚𝑎𝑥 in Figure 35, which captures the

incident Lamb wave local maximum 𝑢𝑙𝑜𝑐𝑎𝑙,𝑚𝑎𝑥 and neglects the reflected global maximum

𝑢𝑔𝑙𝑜𝑏𝑎𝑙,𝑚𝑎𝑥. After the maximum amplitude is determined, the captured nodal displacements are

normalised to produce a Lamb wave pulse of maximum amplitude 1. This allowed direct

comparison with the analytical solutions.

Figure 35: An algorithm was developed to normalise the nodal displacement data using the local maximum rather than the

global maximum.

Figure 36 shows the results of the normalised wave pulse for the 0.15 mm mesh at 300 mm

from the excitation source. The maximum amplitude of the normalised incident wave packet is

unity which allows for direct comparison between the numerical and analytical results.

Qualitatively, there is strong agreement between amplitude responses of the normalised wave

packet and the analytical solution with minor deviation occurring at the tail end of the wave

pulse. The accuracy of the simulation can be measured quantitatively by determination of the

time of arrival and group velocities of the propagating Lamb waves. Figure 37 and Figure 38

𝑢𝑙𝑜𝑐𝑎𝑙,𝑚𝑎𝑥

𝑡𝑙𝑜𝑐𝑎𝑙,𝑚𝑎𝑥

𝑢𝑔𝑙𝑜𝑏𝑎𝑙,𝑚𝑎𝑥

43

show the ToA of each wave at 𝑥 = 30 𝑚𝑚 and 𝑥 = 300 𝑚𝑚 respectively, based on an

amplitude threshold cut-off of 1% relative to the maximum.

a) b)

Figure 36: Wave pulses were normalised to allow for comparison between mesh sizes and with the analytical solutions.

The 0.15 mm mesh was normalised using the local maximum (a) and shows good agreement to the analytical solution (b).

The ToA can be determined by measuring the time at which the amplitude of the wave first

reaches the cut-off threshold of 1%. It can be seen from Figure 37 that at 𝑥 = 30 𝑚𝑚 the ToA

of both simulated and analytical A0 waves are equal at approximately 11.4 𝜇𝑠. Figure 38 shows

that the two waves are travelling unequal velocities which is indicated by a difference in ToA

at 𝑥 = 300 𝑚𝑚 which was not seen at 𝑥 = 30 𝑚𝑚. The ToA of the analytical Lamb wave is

approximately 107.3 𝜇𝑠 while the ToA of the simulated wave pulse is approximately 107.5 𝜇𝑠.

The ToA measurements indicate a high level of agreement between the simulated and analytical

results with a relative error of approximately 0.18%.

By taking the difference of the ToA at the two points along the plate, the approximate group

velocities of the wave pulses can be calculated. The group velocity of the analytical model was

calculated at 2815.4 𝑚/𝑠 while the simulated wave pulse was calculated at 2809.5 𝑚/𝑠.

Hence, there was strong agreement between the analytical and simulation results. However, it

was identified that the calculated group velocities were significantly greater than the theoretical

100 kHz A0 group velocity in aluminium, at 2621 𝑚/𝑠. This suggested there were limitations

in the reference-amplitude methodology used to calculate group velocity, which was

investigated in section 6.4.

44

Figure 37: ToA at 30 mm from the excitation source was determined using a cut-off threshold of 1% at 11.4µs.

Figure 38: ToA of the analytical and simulated Lamb waves, at 300 mm from the excitation source, was determined using

a cut-off threshold of 1% at 107.3µs and 107.5µs respectively.

Determination of the simulated wave pulse group velocity

6.4.1. Reference-amplitude approach for ToA

To validate the ANSYS simulation of the A0 mode, the group velocity of the incident wave

pulse was calculated and compared with the theoretical wave speed. The group velocity was

calculated using Staudenmann’s recommended methodology. This was to take ToA

measurements at various points of known separation and using the time difference to determine

the velocity of the propagating wave pulse [5]. Staudenmann recommended that the beginning

of the wave pulse provides the most reliable point of reference for ToA of the propagating wave

[5]. This methodology was incorporated into the signal analysis Python code by storing the

instantaneous time when the normalised amplitude of the wave pulse exceeded a user-defined

reference amplitude. This process is shown graphically in the example below, which shows the

signal amplitudes at 𝑥 = 150 𝑚𝑚 for each of the tested mesh sizes. It can be seen from Figure

39 that the level of agreement between the simulation and analytical models improves as the

mesh density increases. The developed Python algorithm stores the ToA when the amplitude

𝑇𝑜𝐴

𝑇𝑜𝐴0.15𝑚𝑚 𝑇𝑜𝐴𝑎

45

reaches the reference amplitude, which was selected at 5% relative to the maximum in the

example below. These times are shown by the arrows in the figure below.

Figure 39: The reference-amplitude approach for ToA was used to calculate the wave pulse group velocity. Using a

threshold of 5% shows that the finite element solution converged to the analytical solution as the mesh length was decreased.

It is clear that there was significant variation in the measured times of arrival across the

tested mesh sizes. These are presented in Table 6, along with the relative error compared with

the analytical model, which had a ToA equal to 58.2 𝜇𝑠. Comparison of the mesh sizes reveals

the error discrepancy between the simulated and analytical models decreases with reduced mesh

element length. This finding suggested that reducing mesh element length had the direct

positive effect of improving model accuracy. The maximum ToA error occurred in the 1.5 mm

mesh at 8.9%, while the minimum error occurred in the 0.15 mm mesh at 0.2%. These findings

are in agreement with the qualitative analysis established previously.

Table 6: Time of arrival measurements at 150 mm along the 2D plate.

Mesh size (mm) Time of Arrival (µs) Percentage Error (%)

1.50 63.4 8.9

1.00 60.4 3.8

0.75 59.4 2.1

0.50 58.8 1.0

0.25 58.4 0.3

0.15 58.3 0.2

6.4.2. Issues associated with the reference-amplitude approach for ToA

The reference-amplitude approach for ToA was carried out on all the measured mesh nodes

along the length of the plate. This produced an array of ToA figures which were then used to

calculate wave pulse group velocity by dividing the spatial difference by the time difference for

each measurement along the plate. The algorithm is expressed mathematically by;

𝑐𝑔 =𝑥𝑖+1 − 𝑥𝑖

𝑡𝑖+1 − 𝑡𝑖

𝑖 = 0,1,2… . 𝑛

46

The cut-off threshold, at which point the ToA was defined, was set to 5% of the maximum

amplitude. The distance, 𝑑𝑥, over which the speeds were calculated was 30 mm (equal to the

distance between each measured mesh node). The calculated group velocities were then plotted

for each of the mesh sizes in addition to the analytical model in Figure 40. There are a number

of outlying datapoints within the calculated group velocities where the velocity was found to

be significantly lower than the associated mesh average. Additionally, the first calculated group

velocity was consistently higher than the average in each of the models. For example, between

mesh nodes at 𝑥 = 180 𝑚𝑚 and 𝑥 = 210 𝑚𝑚, the group velocity calculated for the 0.15 mm

mesh was 1795 m/s. The overall average for all calculated group velocities was 2890 m/s,

representing a 37.8% deviation from the mean. Similarly, the first calculated group velocity

was 3852 m/s, thus equating to a 33.2% increase from the mean group velocity. The source of

these large discrepancies in velocity were investigated for the 0.15 mm mesh.

Figure 40: The reference-amplitude approach for ToA resulted in numerous outlying datapoints, which were attributed to

limitations in the methodology and wave dispersion.

The source of the large deviation in group velocity was explored within Figure 41 below.

Observing Figure 41 (a) and (b), which are measured at 𝑥 = 150 𝑚𝑚 and 𝑥 = 180 𝑚𝑚

respectively, it can be seen that the amplitude threshold tolerance of 5% is reached during the

second peak in the incident Lamb wave pulse. The time between (a) and (b) is approximately

10 𝜇𝑠. It can be observed from (a) – (d) that the amplitude of the second wave peak was

decreasing with time. This phenomenon is a result of attenuation and wave pulse widening

which were accurately modelled in the FE simulation. Now observing the wave pulse in (c), it

is evident that the decrease in amplitude resulted in the reference peak used in (a) and (b) being

below the 5% threshold. The ToA in this case is measured at the subsequent peak and results

in a time difference of approximately 16.7 𝜇𝑠 between (b) and (c). This increase in time

consequently resulted in the lower calculated group velocity of 1795 m/s (𝑐𝑔 =30 𝑚𝑚

16.7𝜇𝑠=

1795𝑚

𝑠). Finally, observing the measurements taken at (c) and (d) it can be seen that both are

taken on the same wave peak, resulting in a more consistent time difference of 10.8 𝜇𝑠. The

attenuative effect of decreasing amplitude results in these ‘low-velocity’ measurements in each

47

of the FE models as well as the analytical solution. This represents a significant limitation in

the reference-amplitude based approach for ToA.

a) b)

c)

d)

Figure 41: Attenuation and wave pulse widening resulted in different wave peaks being used as the reference point for

ToA. The second peak reached the 5% threshold in (a) and (b), while the third peak was measured in (c) and (d).

To investigate the accuracy of the obtained group velocities across the range of tested mesh

sizes, the percentage error associated with each of the calculated values was plotted in Figure

42. To compensate for the aforementioned deviations in velocity due to attenuation, an

algorithm was developed within the Python code to identify and remove outliers in the data

sets. The criteria for outlying data points was based on the interquartile range (IQR). If a data

point fell outside of 1.5(IQR) below the lower quartile or 1.5(IQR) above the upper quartile, it

was considered an outlier and was removed from the data set. The remaining data points were

plotted in Figure 42 (a), and the datasets were averaged to determine the overall group velocity

for each FE model.

The percentage error in group velocity was calculated relative to the theoretical group

velocity of the A0 mode in aluminium 2024-T6. Referring to the group velocity dispersion curve

in Figure 3 (b), the group velocity of the A0 mode, propagating in a 3 mm thick plate and with

a centre frequency of 100 kHz, is 2621 m/s. Hence, the percentage error of each calculated

group velocity was calculated relative to the theoretical value of 2621 m/s. The percentage error

in group velocity was then plotted as a function of position for each of the FE models in Figure

42 (b). The overall average group velocity calculated for each FE model is presented in Table

7.

48

Observation of Figure 42 (a) reveals that group velocity increased (on average) as the mesh

element length was decreased. The minimum wave speed was calculated for the 1.5 mm mesh

model at 2704 m/s (3.15% error), while the maximum wave speed was calculated for the

analytical model at 2988 m/s (14% error). Since the lowest calculated group velocity was

greater than the theoretical 100 kHz value (2621 m/s), the trend of the data seemed to suggest

that reducing the mesh element length results in greater numerical error. This was counter to

the expected trend based on the mesh element criterion (refer Equation 8). It was hypothesised

that reducing mesh element length would improve the accuracy of FE numerical simulations.

As such, the tools used to determine group velocity were investigated more closely.

a) b)

Figure 42: Wave pulse group velocity was found to increase as element length was reduced (a). Since all wave speeds

exceeded the cg of 2621 m/s, this meant numerical error increased on average as the mesh resolution improved (b).

The conclusions from the analysis initially suggested that the ANSYS simulations produced

erroneous data, as the observed trend between mesh element length and numerical error was

counter to the established literature. However, upon further inspection, this seemingly

erroneous relationship was in fact attributed to the methodology used to define the wave pulse

time of arrival. The amplitude threshold at which ToA was defined was 5% of the maximum

amplitude of the wave pulse. This meant that wave pulse ToA was defined by the first instance

of the wave packet arriving at the point of measurement. Due to the dispersive nature of Lamb

waves, higher frequency signals travel at a higher velocity through a structure, resulting in wave

pulse widening. Hence, by measuring the wave pulse ToA at 5%, the higher frequency

components of the wave pulse were measured as these frequencies travelled through the

aluminium plate at a higher speed.

The FFT of the excitation signal was provided in Figure 29, showing the centre frequency

equal to that of the ideal f0 at 100 kHz. To control spectral leakage of the excitation signal in

the frequency domain, the sinusoidal tone burst was modulated with a Hanning window

function. However, closer inspection of the frequency domain revealed spectral leakage at

higher frequencies up to 200 kHz, as shown in Figure 43. The presence of these higher

frequency components within the excitation signal could explain the convergence of the FE

models to a speed much greater than the expected 2621 m/s.

49

Figure 43: Spectral leakage causes high frequency components to exist within the wave pulse.

Table 7 presents the average group velocities for each of the mesh sizes, along with the

percentage error relative to two theoretical wave speeds; the group velocity associated with the

centre frequency (100 kHz) and the group velocity associated with the high frequency

component (190 kHz). The aluminium 2024-T6 dispersion curve was used to resolve the group

velocity at 190 kHz, at 2992 m/s. Using this wave speed, the percentage errors of the

experimentally determined group velocities were calculated.

Table 7: Average group velocity and associated error at different f0 frequencies.

Mesh size (mm) Average group velocity

(m/s)

Percentage error (%)

relative to f0 100 kHz

Percentage error (%)

relative to f0 190 kHz

1.50 2704 3.2 9.6

1.00 2876 9.7 3.9

0.75 2922 11.5 2.3

0.50 2957 12.8 1.2

0.25 2980 13.7 0.4

0.15 2982 13.8 0.3

Analytical 2988 14.0 0.1

The data presented in Table 7 supports the claim that the experimentally determined group

velocities corresponded to that of the higher-frequency components within the wave pulse. This

was a direct consequence of using the beginning of the wave pulse as a point of reference for

ToA, as the faster high-frequency components were measured prior to the centre frequency

components. The trend between mesh size and model accuracy was highly consistent with the

expected outcome when the 190 kHz group velocity was used as the ideal value.

It is clear from this analysis that defining ToA of the wave pulse based on an amplitude

threshold has a large influence on the calculated wave speed. By using a low amplitude point

of reference, the unwanted higher frequency components were measured instead of the 100 kHz

centre frequency. In order to accurately validate the group velocity of the wave pulse against

analytical results, only the 100 kHz centre frequency can be used. This is because the exact

frequency of a particular measured point in the wave pulse is difficult to accurately establish.

The percentage error of the calculated group velocity was very low compared with the 190 kHz

50

wave speed, however this is a non-rigorous approach for validating the simulation results.

Hence, this presented a major limitation in the reference-amplitude based approach for ToA

determination.

There were two primary user-defined parameters defined within the analysis; the separation

distance between nodes over which group velocity was calculated, and the amplitude threshold

at which ToA was defined. Due to the findings of this analysis, the influence of both parameters

was explored more rigorously through the sensitivity analyses below.

6.4.3. Sensitivity of reference-amplitude ToA to amplitude threshold

The amplitude threshold at which ToA is defined was varied between 10% and 90% to

investigate its impact on the calculated wave pulse group velocity. The separation distance was

fixed at 300 mm, which was found to produce the least error in group velocity for the high-

resolution meshes (discussed in section 6.4.4 below). The percentage error of the average group

velocity was calculated relative to that of the excitation frequency (100 kHz) at 2621 m/s. The

group velocity percentage error for each FE model was then plotted as a function of ToA

reference amplitude in Figure 44. Through analysis of the data presented in Figure 44, it is

clearly evident that amplitude-threshold had a significant impact on the calculated wave pulse

velocity.

Below a ToA reference amplitude of 50% all mesh models show highly chaotic variation in

group velocity accuracy. This is likely due to measuring the amplitude response triggered by

the higher-frequencies within the wave pulse. These frequencies primarily appear at the

beginning of the wave pulse due to their higher group velocity. Similarly, they induce relatively

smaller displacement amplitudes due to their lower amplitude in the frequency domain

compared to the centre frequency.

Above ToA reference-amplitudes of 50%, the 0.15 mm and 0.25 mm mesh models

converge to consistent percentage errors of 1.4% and 1.5% respectively. Similarly, at a ToA

reference amplitude of 70-90% the 0.50 mm and 0.75 mm mesh models converge to a stable

error of 2.3% and 2.4% respectively. In these regions, the reference point for ToA was near the

maximum of the wave pulse. This resulted in the group velocity being calculated primarily for

the 100 kHz centre frequency, hence the calculated group velocity was more accurate.

Additionally, at the centre of the wave pulse, the amplitudes of the high and low frequencies

are dominated by that of the centre frequency. There are insufficient data points to determine

whether the remaining FE models would converge in a similar manner.

51

Figure 44: The reference-amplitude approach was highly sensitive to the user-defined threshold at which point ToA was

defined. This was due to the amplitude response of high frequency components being measured when the threshold was low.

6.4.4. Sensitivity of reference-amplitude ToA to separation distance

The effect of separation distance was then explored to investigate its impact on the

calculation of group velocity. This involved varying the distance between mesh nodes over

which the group velocity was calculated. By increasing separation distance, the time difference

between measurements increases. However, the number of averaged group velocity data points

is reduced because the number of possible steps across the plate is reduced. The errors

associated with the average velocities were then calculated (relative to 100 kHz) and plotted in

Figure 45. It can be observed that the error largely decreases with separation distance in all of

the models, however the minimum error occurs at varying Δ𝑥. The 0.15 mm, 0.25 mm and

0.50 mm mesh models, and the analytical model, show the strongest agreement, with the

minimum error occurring at the maximum separation distance of 300 mm. This is effectively

taking samples at a spatial frequency of half the plate length and averaging the velocity. Notably

the 0.75 mm and 1.00 mm mesh models have the minimum error at a spatial separation of

270 mm, while the 1.50 mm mesh shows the highest accuracy at 90 mm.

Overall the trend of the data suggests that increasing separation distance results in an

improvement in the accuracy of the calculated group velocity. This is likely explained by the

fact that as Δ𝑥 increases, the time difference Δ𝑡 also increases. The numerical error associated

with variations in the captured amplitude peak for ToA is therefore reduced as the increase in

Δ𝑡 dampens its impact on the calculated velocity.

It is clearly evident that the calculated group velocity is highly sensitive to the separation

distance between mesh nodes. By varying the separation distance from 30 mm to 300 mm the

average change in percentage error across all models is approximately 7.2%. Thus, by selecting

the separation distance arbitrarily the calculated 𝑐𝑔 could vary, on average, by up to 7.2%. This

suggests a significant limitation in the methodology as small variations in Δ𝑥, which is often

selected arbitrarily, could vastly impact the calculation of 𝑐𝑔, leading to erroneous conclusions

when validating the simulated Lamb waves in ANSYS.

52

Figure 45: The reference-amplitude approach for ToA was highly sensitive to the separation distance over which group

velocity was calculated. Increasing separation distance resulted in a net reduction in numerical error across all models.

Through analysis of the data presented in Figure 42, Figure 44 and Figure 45 it is clearly

evident that there are significant limitations in the methodology for defining ToA based on a

reference amplitude. ToA reference amplitudes between 10-90% were shown to have a highly

significant and chaotic influence on group velocity. This was mainly attributed to higher-

frequency responses being measured when the amplitude threshold was below 50%. Separation

distance was also found to have a significant impact on the accuracy of the calculated group

velocity, suggesting that arbitrary selection of Δ𝑥 could vastly alter the measurement of group

velocity.

The highly variable and nonlinear influence of ToA reference amplitude and separation

distance led to the conclusion that arbitrary selection of these parameters would greatly reduce

the significance of the Lamb wave analysis. It was shown through the sensitivity analysis that

selection of these parameters has a significant impact on the results, and hence conclusions,

drawn from the analysis. It was therefore necessary to develop a more rigorous and reliable

methodology for measurement of the group velocity such that the accuracy of the ANSYS

simulation could be validated with confidence.

6.4.5. Energy distribution approach for wave pulse ToA

The reference amplitude approach for ToA was found to be highly sensitive to user-defined

variables such as amplitude threshold, separation distance and outlier-refinement. This meant

that there was a high level of uncertainty in any conclusions drawn from the analysis, as the

methodology used in processing the data had a significant impact on the results. Consequently,

a more robust methodology for processing the data was developed. The Hilbert function reveals

the energy distribution of the wave in the time domain. It has been shown in at least one study,

by Murat, to provide a robust tool for determining ToA [11].

A new algorithm was implemented within the signal processing Python script which took

the Hilbert function of the displacement signal at each of the measured mesh nodes. Each of the

transformed signals then represented the energy distribution of the wave as a function of time.

Figure 46 (a) and (b) provide the energy distributions of the Lamb waves, simulated using the

53

0.15 mm mesh, at 𝑥 = 30 𝑚𝑚 and 𝑥 = 150 𝑚𝑚 respectively. Observing the energy envelope

of the signal response reveals the incident Lamb wave as well as the reflections from the plate

boundaries. Similar to the methodology used in normalising the signal response, care was taken

when analysing the energy envelopes to ensure that measurements were taken at the incident

wave rather than the reflections, which in some cases had a greater amplitude

a) b)

Figure 46: Energy distribution of the measured signals at (a) 30 mm and (b) 150 mm reveal the incident and reflected

Lamb wave pulses.

Figure 47 (a) and (b) provide close-up views of the incident waves seen in Figure 46. The

energy-envelope based approach for ToA determination is illustrated in (a) and (b). The

methodology involved calculating the average time at which the energy envelope exceeded a

user-defined reference amplitude. In the below cases the threshold was 90%. By taking two

measurements on either side of the peak of the distribution, the times were averaged to

determine the approximate centre of the wave pulse. This approach provided a significantly

higher consistency of reference point for ToA as there were no issues with peak selection as

identified in Figure 41. At (a) 30 mm the two reference times are 𝑡1 = 30.8 𝜇𝑠 and 𝑡2 =

41.2 𝜇𝑠, meaning the average time is 𝑡𝑎𝑣𝑔 = 36.0 𝜇𝑠. Similarly, at (b) 150 mm the threshold

reference times are 𝑡3 = 77.1 𝜇𝑠 and 𝑡4 = 87.3 𝜇𝑠, meaning the average time is 𝑡𝑎𝑣𝑔 =

82.2 𝜇𝑠. The group velocity was then calculated by 𝑐𝑔 =𝑑𝑥

𝑑𝑡=

(150−30) 𝑚𝑚

(82.2−36.0)𝜇𝑠= 2597 𝑚/𝑠. The

percentage error compared with the theoretical velocity of 𝑐𝑔 = 2621 𝑚/𝑠 is therefore 0.90%,

which represents a high level of accuracy. This confirms the energy-distribution approach

accurately captured the velocity of the centre frequency components of the wave pulse at

100 kHz.

54

a) b)

Figure 47: The ToA was approximated by averaging the time over which the amplitude exceeded the ToA reference

amplitude. At (a) 30 mm the ToA is 36.0 µs and at (b) 150 mm the ToA is 82.2 µs.

To test the rigorousness of the energy-distribution based approach for ToA determination,

a sensitivity analysis was carried out to investigate the effect of separation distance on the error

of the calculated group velocity. The amplitude reference threshold was selected at 95% for this

study. The results are presented in Figure 48 below, which shows the relationship between

separation distance and error for the reference-amplitude methodology for ToA in (a) and the

energy-distribution methodology for ToA in (b). It is clearly evident that measuring ToA via

the Hilbert function is significantly less sensitive to separation distance than shown in (a).

Varying the separation distance between 30 mm and 90 mm causes approximately 0.5%

deviation in percentage error in all the numerical models excluding the 1.5 mm mesh. However,

increasing the separation distance from 90 mm to 300 mm resulted in almost constant relative

error with negligible deviation in all the FE models excluding the 1.5 mm mesh.

Since the analytical solution shares none of the inherent numerical error which may exist in

the FE generated data, it provides the best validation for the methodology. Observation of (b)

shows that the analytical solution is less sensitive to separation distance than all the FE models,

with a total deviation in percentage error of 0.1%. Comparing this behaviour to the reference

amplitude approach for ToA, which caused a net decrease in error of 7.8%, the energy-

distribution based approach is clearly more reliable and less sensitive to separation distance.

The significance of this attribute is highly important as the separation distance, which is

often arbitrarily selected, could affect the conclusions drawn when analysing the signal

response in applications such as SHM. Incorrect calculation of the group velocity could yield

𝑡3 𝑡4 𝑡1 𝑡2

55

inaccurate determination of material properties, false identification of damage within a structure

or inaccurate FE simulation of real-world behaviour.

a) b)

Figure 48: Sensitivity analysis of methodologies for calculating ToA based on (a) amplitude threshold and (b) Hilbert

function, reveal that the energy-distribution based approach is significantly less-sensitive to separation distance.

A second sensitivity analysis was then performed with the variable being reference

amplitude. This is the amplitude of the energy-envelope at which point time of arrival was

recorded. Figure 49 (a) and (b) present the relationship between error in calculated 𝑐𝑔 as a

function of separation distance for the reference-amplitude and energy-distribution approaches

respectively. Comparison of the two approaches reveals the energy-distribution approach was

significantly less sensitive to amplitude threshold. The maximum deviation in the error of the

analytical model was 0.5% compared with a net change of 4.0% in error in the reference-

amplitude approach for ToA. All tested FE models, excluding the 1.5 mm mesh, experienced

lower variation in percentage error with a maximum net change of 1.1% using the energy-

distribution. Using the alternative approach, the maximum net change was significantly higher

at approximately 3.7% in the 1 mm mesh. The 1.5 mm mesh shows the highest sensitivity to

amplitude threshold with a deviation of 2.3% using the energy-envelope approach. This was

still significantly lower than the original methodology, which had a net change of 4.3%.

In addition to the decreased sensitivity of the calculated 𝑐𝑔 to amplitude, the energy-

distribution approach for ToA determination also provides a significantly more consistent trend

in the data compared with the original method. While there is some deviation in the order of

accuracy when comparing the models by mesh element length, with the 0.15 mm, 0.25 mm and

0.50 mm meshes overlapping at some points, there is a significantly more consistent trend using

the energy envelope. The analytical model shows the least 𝑐𝑔 percentage error, followed by the

0.15 mm and 0.25 mm meshes, which show similar average 𝑐𝑔 percentage error. The 0.75 mm,

1.00 mm and 1.50 mm FE models then consistently increase in average error across the range

of tested reference amplitudes.

The consistent trend suggests the energy-distribution model is significantly more reliable

than the original reference amplitude approach for calculation of ToA. Having a low sensitivity

to reference amplitude is a highly important attribute of the model as arbitrary user selected

parameters should not affect the overall conclusions of the results. For example, in Figure 49

56

(a) if the user selects a reference amplitude below 50%, the order of accuracy of the FE models

is significantly different to that if the amplitude was selected above 50%. This could lead to the

same issues as discussed previously and an overall misinterpretation of the data.

Calculation of group velocity using the energy-distribution of the wave pulse was

significantly more reliable than the reference-amplitude because it ensured that the point of

reference was the 100 kHz centre frequency. This methodology suppressed the influence of the

higher-frequency components within the A0 wave pulse, even at low amplitudes. The reduced

sensitivity of the energy-envelope approach for ToA to these user defined parameters provided

evidence for its use in validation of the FE model.

a) b)

Figure 49: Sensitivity analysis of methodologies for calculating ToA based on (a) amplitude threshold and (b) Hilbert

function, reveal that the energy-distribution based approach is significantly less-sensitive to reference amplitude.

6.4.6. Validation of the 2D simulation by group velocity

The energy-distribution approach was used to calculate the ToA of the incident wave pulse

at each of the measured mesh nodes. The separation distance, over which group velocity was

calculated, was selected at 90 mm. This ensured that the sensitivity of Δ𝑥 on the calculated 𝑐𝑔

was minimised, which is shown to occur when 90 𝑚𝑚 ≤ Δ𝑥 ≤ 300 𝑚𝑚, indicated by the flat

regions Figure 48 (b). The low boundary of this range (90 mm) was chosen since having a

smaller separation distance maximises the number of data points over which the calculated 𝑐𝑔

could be averaged. The incident wave pulse ToA was also calculated using a reference

amplitude of 90%. This was because the sensitivity of average 𝑐𝑔 to reference amplitude was

shown to be within 1.1% across the entire range for the higher resolution FE models. A

reference amplitude of 90% placed the average error approximately in the centre of this small

range for the majority of the FE models. Additionally, the peak of the wave pulse was the region

of maximum energy, hence is where the amplitude response of the centre frequency is

dominant. The configuration of the model and the calculation procedure is shown in Figure 50.

The Python code used to implement this methodology is provided in Appendix A (section 15.1).

57

Figure 50: Group velocity was calculated over a separation distance of 90 mm to ensure that the influence of dx on the

measured 𝑐𝑔 was minimised.

Figure 51 (a) provides the calculated group velocities of the tested FE models as a function

of position relative to the excitation. In all of the numerical models there was a outlier occurring

at the last calculation. This was a result of the small time-difference between the incident and

reflected waves measured near the plate boundary at 𝑥 = 480 𝑚𝑚. The effect is demonstrated

in Figure 51 (b) and causes interference between the two displacement signals making it

difficult to distinguish between the incident and reflected pulses. This limitation introduces

significant error into the determination of the ToA of the incident pulse. Consequently, the data

was filtered to identify outliers based on IQR.

a) b)

Figure 51: Calculated group velocities (a) reveal erroneous data points at the far boundary of the model (a). This was

caused by interactions between the incident and reflected wave resulting in ToA error (b).

Figure 52 (a) presents the filtered group velocity data from the 2D ANSYS simulation

plotted as a function of position from the excitation. Figure 52 (b) plots the error associated

with the group velocities relative to the theoretical value of 𝑐𝑔 = 2621 𝑚/𝑠. Overall the FE

models are highly consistent with the analytical solution, with the accuracy of the simulation

increasing with improved mesh resolution. This was in agreement with the established literature

and is supportive of the hypothesis. It can be identified from the plots that the numerical solution

converged toward the analytical solution as mesh element length was reduced. This is to say

y

𝑥

Δ𝑡 Δ𝑡 Δ𝑡

58

the order of agreement with the analytical solution, from best to lowest was: the 0.15 mm,

0.25 mm, 0.50 mm, 0.75 mm, 1.00 mm and 1.50 mm FE mesh models.

a) b)

Figure 52: Using the energy-distribution of the wave pulse for ToA, the group velocities of the various FE models showed

excellent agreement with the analytical value of 2621 m/s (a). The general trend of the data was a reduction in numerical

error as the finite element length became shorter, which was consistent with expected outcomes (b).

The data sets for each model were averaged to determine the average group velocity,

percentage error and the maximum range of error (maximum error subtract minimum error)

across all captured data points. This information is summarised in Table 8 and the average

group velocities are plotted in Figure 53.

The trend of the data clearly indicates that the accuracy of the group velocity of the

simulated antisymmetric Lamb wave increases with higher mesh resolution. At the maximum

characteristic element length of 1.50 mm the percentage error is 6.66%, while the at the

minimum 𝑙𝑒 the error is 1.15%. The range of error between the minima and maxima in the data

sets also follows the same trend, with the range of error decreasing with mesh element length.

Notably, the analytical solution had a percentage error of 1.02%. This represents the inherent

numerical error within the energy-envelope methodology used to determine 𝑐𝑔. This is the

baseline error which exists within all measurements taken using this calculation procedure. As

such, the minimum error attributed to the FE simulation of the A0 Lamb wave can be adjusted

by 1.02%, which is also shown in Table 8.

Computation time was highly dependent on mesh resolution with the fastest computation at

10.4 minutes in the 1.50 mm model and the longest at 84 minutes in the 0.15 mm model. This

was due to the increased number of mesh nodes in the high resolution meshes, and

consequently, the increased complexity of the FE matrix to be solved. Due to the 2D nature of

the simulation, the computation times remained practical compared with the 3D model

(discussed in section 8.2).

Overall there is excellent agreement between the high resolution meshes with the 0.15 mm,

0.25 mm and 0.50 mm models simulating the A0 Lamb wave with less than 1% numerical error,

at 0.13%, 0.29% and 0.65% respectively. The 0.75 mm and 1.00 mm element length models

also show strong agreement with the theoretical group velocity with 1.29% and 2.37%

numerical error respectively. The maximum error occurred in the 1.50 mm model at 5.64%

59

which is still strong agreement considering the thickness of the plate was modelled with only

two mesh elements.

The findings of the mesh size study indicate that the criterion for convergence governing

the characteristic mesh element length, 𝑙𝑒 ≤𝜆𝑚𝑖𝑛

20, is not critical for convergence in ANSYS

Explicit Dynamics. At a central excitation frequency of 100 kHz the Lamb wavelength was

calculated at 15.5 mm. The critical mesh element length is therefore calculated at 𝑙𝑒 ≤

0.78 𝑚𝑚, meaning the 1.0 mm and 1.5 mm fall outside this requirement. Despite failing to

meet the criteria, the ANSYS models successfully simulated the behaviour of the A0 Lamb

mode to a high degree of accuracy. The relaxed criteria governing mesh element length is

consistent with the findings of Moser [29].

Table 8: Summary of the group velocity data captured in the 2D simulation.

Mesh size (mm) Group velocity

(m/s)

Percentage error

(%)

Minimum FE

attributed error

(%)

Maximum range

of error (%)

1.50 2446 6.66 5.64 3.49

1.00 2532 3.39 2.37 2.52

0.75 2560 2.31 1.29 2.68

0.50 2577 1.67 0.65 2.56

0.25 2587 1.31 0.29 1.64

0.15 2591 1.15 0.13 1.21

Analytical 2594 1.02 - 1.15

Figure 53: Reducing characteristic mesh element length was found to consistently improve model accuracy. The minimum

error was 0.13% in the 0.15 mm mesh model, while the maximum error was 5.64% in the 1.50 mm model.

6.4.7. Conclusions from the analysis of group velocity

It was found that the reference amplitude approach, using raw nodal displacement data,

provides insufficient reliability for determination of the group velocity of a Lamb wave pulse.

The sensitivity of this methodology to user-defined thresholds for amplitude and separation

60

distance reduces the significance of the results. A methodology for ToA determination was

developed using the energy-distribution of the wave pulse which was calculated using the

Hilbert function. This methodology provided a significantly more consistent numerical tool

which was less sensitive to parameters such as reference amplitude and separation distance.

Using the energy-distribution approach for ToA the average group velocity of the simulated

antisymmetric Lamb wave was calculated for each of the FE models. It was found that ANSYS

Explicit Dynamics effectively models the group velocity behaviour of the A0 Lamb wave in 2-

dimensions to a high degree of accuracy. The minimum numerical error attributed to the FE

simulation was 0.13%, which was achieved using a characteristic mesh element length equal to

0.15 mm. The numerical error was found to increase as with mesh element length, with a

maximum error of 5.64% in the 1.50 mm model. The computation time increased with mesh

resolution from 10.4 minutes in the 1.50 mm model to 84 minutes in the 0.15 mm model. This

clearly demonstrates the compromise between accuracy and computation time when modelling

Lamb waves via FEA.

The conclusions from this analysis provide confirmation that ANSYS Explicit Dynamics

can effectively model the propagation of the antisymmetric Lamb wave in the 2D environment.

This is a highly significant finding as it confirms the validity of simulations which aim to model

the propagation of Lamb waves for applications such as SHM design.

Determination of the simulated wave pulse phase velocity

6.5.1. Methodology for calculating phase velocity

To further evidence the capabilities of ANSYS Explicit Dynamics in modelling the

antisymmetric Lamb wave, the phase velocity of the simulated wave pulse was determined.

Based on the recommendations of Lasˇova´, phase velocity was calculated using the 2D Fast

Fourier Transform. The nodal displacement data captured in the 0.25 mm mesh model was used

in this investigation.

The phase velocity of the Lamb wave pulse was solved by taking the 2D FFT of the nodal

displacement results captured in the ANSYS simulation. Figure 54 shows a diagram of the

ANSYS model of the 2D plate. The nodal responses at equally spaced points were extracted

from the results file. The nodal separation Δ𝑥 was varied to investigate the influence of spatial

separation on the resolution of the resultant wavenumber-frequency plots. The minimum Δ𝑥 at

which nodal data was extracted from the model was 2.50 mm. It is important to note that the

nodal displacement data extraction was limited to 400 mm from the excitation source. This was

to ensure that signal responses that experienced interactions between the incident and reflected

wave pulses were excluded from the analysis. This primarily occurred near the far boundary of

the model between 400 𝑚𝑚 ≤ 𝑥 ≤ 500 𝑚𝑚. It was necessary to exclude the reflected signals

from the data to ensure the Fast Fourier Transform was performed purely on the incident wave

pulse as to avoid distortion in the wavenumber-frequency plots.

61

Figure 54: The nodal responses were extracted from the model at evenly spaced points and were amalgamated in a 2D

matrix in preparation for the 2D FFT.

The results were sorted in an Excel Workbook and a Python module was developed to

implement the 2DFFT algorithm. The nodal displacement data sets were then amalgamated in

a single matrix 𝑢(𝑥, 𝑡) in row order as shown below. Note, the number of spatial points 𝑛 was

dependent on the selected Δ𝑥 which was varied from 2.5 mm to 40 mm. The maximum time 𝑡𝑘

was selected at 220 𝜇𝑠 to exclude the reflection from the incident wave pulse in all nodal

displacement results.

𝑢(𝑥, 𝑡) =

[ 𝑥1(𝑡0) 𝑥1(𝑡1) 𝑥1(𝑡2) ⋯ 𝑥1(𝑡𝑘)𝑥2(𝑡0) 𝑥2(𝑡1) 𝑥2(𝑡2) ⋯ 𝑥2(𝑡𝑘)𝑥3(𝑡0) 𝑥3(𝑡1) 𝑥3(𝑡2) ⋯ 𝑥3(𝑡𝑘)

⋮ ⋮ ⋮ ⋯ ⋮𝑥𝑛(𝑡0) 𝑥𝑛(𝑡1) 𝑥𝑛(𝑡2) ⋯ 𝑥𝑛(𝑡𝑘)]

The 2D FFT was then implemented on the matrix 𝑢(𝑥, 𝑡) to transform the displacement

results from the time-space domain to the wavenumber-frequency domain as shown below.

𝐻(𝑘, 𝑓) = ∫ ∫𝑢(𝑥, 𝑡)𝑒−𝑖(𝑘𝑥−𝜔𝑡)𝑑𝑥 𝑑𝑡

The resultant matrix was populated with complex numbers whose magnitude revealed the

Lamb wave pulse in the wavenumber-frequency domain. The magnitude of 𝑢(𝑥, 𝑡) was graphed

on a contour plot as a function of wavenumber (1/𝑚) and frequency (MHz). The complete

Python code of the 2DFFT implementation is provided in Appendix B (see section 15.1).

6.5.2. Influence of spatial resolution on the wavenumber-frequency domain

Spatial resolution is controlled by the separation distance Δ𝑥 between the nodes at which

displacement data is captured. A higher spatial resolution means the step size between mesh

nodes is smaller. The matrix 𝑢(𝑥, 𝑡) is therefore populated with more data which impacts the

resolution of the transformed response in the wavenumber-frequency domain. Costley noted

that a sufficiently high spatial resolution is necessary to avoid aliasing in the resultant

wavenumber-frequency domain [36]. The separation distance Δ𝑥 was varied from 2.5 mm to

40 mm to investigate the spatial requirements for the simulated antisymmetric Lamb wave in

ANSYS. The FE model having a characteristic mesh element length of 0.25 mm was used in

the sensitivity analysis as it provided a convenient integer division with the Δ𝑥 distances.

The scope of the testing is summarised in Table 9. Provided is the separation distance Δ𝑥,

the number of captured mesh nodes between 0 𝑚𝑚 ≤ 𝑥 ≤ 400 𝑚𝑚, the spatial resolution

normalised in nodes per 100 mm, and the spatial resolution in nodes per (nominal) wavelength

where 𝜆 = 15.5 𝑚𝑚.

Δ𝑥

y

𝑥

1 2 3 4 5 6 7 8 9 10 11… 𝑛

62

Table 9: Scope of the spatial resolution sensitivity analysis.

Separation distance

(mm)

Number of mesh nodes Mesh nodes per 100 mm Nodes per wavelength

40 11 2.75 0.39

30 14 3.59 0.52

20 21 5.25 0.78

15 27 6.92 1.03

12.5 33 8.25 1.24

10 41 10.25 1.55

7.5 53 13.59 2.07

5 81 20.25 3.1

2.5 161 40.25 6.2

The wavenumber-frequency contour plots for each tested Δ𝑥 are provided by Figure 56 and

Figure 57. These plots are useful as the phase velocity 𝑐 can be resolved directly by using the

relation 𝑐 = 𝜔/𝑘. The sensitivity analysis revealed that the resultant wavenumber-frequency

plot, and consequently the calculated phase velocity, is highly dependent on the spatial

resolution of the 2D FFT. Specifically, it was found that the number of nodes per (nominal)

wavelength had to be at least 1 to produce a wavenumber-frequency plot from which the

calculated phase velocity was physical.

The theoretical phase velocity was approximated from the aluminium 2024-T6 dispersion

curve at 1550 m/s. The methodology for calculating phase velocity using the wavenumber-

frequency plots is demonstrated below.

Figure 55 shows a closeup view of the k-f plot for Δ𝑥 = 2.5 𝑚𝑚. The colour of each contour

in the k-f plot represents its magnitude relative to the maximum amplitude. Blue contours

represent those with minimal or nil amplitude, green represents those with moderate relative

amplitude, and yellow indicates contours of maximum amplitude. The contour of maximum

amplitude in the k-f plot defines the frequency within the spectrum at which the wave energy is

maximum. This should theoretically occur at the central excitation frequency of 100 kHz.

As a consequence of using a finite number of spatial and time measurements in the 2D FFT,

aliasing occurs in the response which causes the k-f plot to have a coarse resolution. This

limitation in plot resolution introduces a range of uncertainty in both wavenumber and

frequency as indicated by the arrows in

Figure 55. It can be seen in the below example that the centre excitation frequency of

100 kHz lies within the bounds of the contour of maximum amplitude. However, due to the

limited resolution of the plot, there exists a range of valid frequencies for the contour. This

means there may be some degree of error between the apparent frequency of maximum wave

energy and actual frequency of the excitation.

63

Figure 55: Closeup view of the wavenumber-frequency plot reveals a range of uncertainty which is attributed to the spatial

resolution.

Wavenumber-frequency plots were generated for each of the tested spatial resolutions and

are provided below. Each of the various k-f plots could be easily discerned and classified in one

of two distinct groups, being physical and non-physical. When the spatial resolution was less

than 1 node per wavelength, the plots indicated non-physical behaviour of the simulated Lamb

wave. When the spatial resolution was greater than 1 node per wavelength, the data indicated

mostly physical behaviour of the Lamb wave. Hence, there was a clear discrepancy between

the data sets across this spatial criterion. For this reason, the k-f plots are separated between

Figure 56 and Figure 57. Figure 56 (a), (b) and (c) present the k-f plots for Δ𝑥 distances of

40 mm, 30 mm and 20 mm respectively. Figure 57 (a), (b), (c), (d), (e) and (f) present the k-f

plots for Δ𝑥 distances of 15 mm, 12.5 mm, 10 mm, 7.5 mm, 5 mm and 2.5 mm respectively.

The full k-f domain is shown in each plot, in addition to a close-up view of the contour of

maximum amplitude with the ranges of uncertainty in k and f indicated by arrows. The actual

excitation frequency is also plotted at 100 kHz.

a) b)

64

c)

Figure 56: A spatial resolution of less than 1 node per wavelength resulted in the indication of non-physical bahviour of

the Lamb wave. This occurred for separation distances of (a) 40 mm, (b) 30 mm, and (c) 20 mm.

a)

b) ``

65

c)

d)

e)

f)

66

Figure 57: A spatial resolution of more than 1 node per wavelength indicatated physical bahviour of the Lamb wave.

This occurred for separation distances of (a) 15 mm, (b) 12.5 mm , (c) 10 mm, (d) 7.5 mm, (e) 5 mm, and (f) 2.5 mm.

To account for the uncertainties in k and f in the wavenumber frequency plots, the minimum

and maximum frequency bounds were averaged to determine the approximate frequency of the

maximum energy of the wave pulse. This occurred at the contour of maximum amplitude. The

maximum and minimum wavenumbers were then used, along with the average frequency, to

approximate the bounds of uncertainty for the calculated phase velocity. The minimum and

maximum phase velocities were then averaged to determine the overall phase velocity of the

wave pulse. This procedure is demonstrated below for Δ𝑥 = 2.5 𝑚𝑚.

𝜔𝑎𝑣𝑔 =𝜔𝑚𝑖𝑛 + 𝜔𝑚𝑎𝑥

2=

(101.3 + 96.6)

2𝐻𝑧 = 99.0 𝐻𝑧

𝑐𝑚𝑖𝑛 =𝜔𝑎𝑣𝑔

𝑘=

99.0 𝑘𝐻𝑧

63.41𝑚

= 1561.4𝑚

𝑠

𝑐𝑚𝑎𝑥 =𝜔𝑎𝑣𝑔

𝑘=

99.0 𝑘𝐻𝑧

60.91𝑚

= 1625.5𝑚

𝑠

𝑐𝑎𝑣𝑔 =𝑐𝑚𝑖𝑛 + 𝑐𝑚𝑎𝑥

2= 1593.5

𝑚

𝑠

The results of the spatial resolution investigation are summarised in Table 10. The data

clearly indicates that a spatial resolution of less than 1 node per wavelength results in erroneous

calculation of phase velocity, with the percentage errors exceeding 100% in all cases. Notably,

there is no clear trend in the data as the spatial resolution is increased toward unity. The

percentage error is minimum at 0.52 nodes per wavelength and increases when the resolution

is altered in either direction. The significant error in the average phase velocity is primarily

attributed to the approximation of k as it is as a function of the spatial resolution, controlled by

Δ𝑥. As the frequency domain is dependent on the timestep, which was controlled by ANSYS

Autodyn, the calculated 𝑓𝑎𝑣𝑔 was significantly more accurate. The maximum percentage error

in the calculated 𝑓𝑎𝑣𝑔 was 1.35% across the test cases.

It can be identified from the data that a spatial resolution of more than 1 node per wavelength

results in highly accurate calculation of the incident Lamb wave’s phase velocity. Across all

tested spatial resolutions from 1.03 – 6.2 the maximum error was 4.14%. It is clear from the

trend in the data that the criterion of 1 node per wavelength is most critical for accurate

wavenumber-frequency data when performing the 2D FFT.

Table 10: Influence of spatial resolution on the average frequency and average phase velocity of the simulated Lamb wave.

Nodal

Separation (mm)

Nodes per

wavelength

Avg. frequency at

max. amplitude (kHz)

Percentage

error (%)

Avg. phase

velocity (m/s)

Percentage

error (%)

40 0.39 102.1 2.1 7613.5 391.19

30 0.52 99 1.05 3223.2 107.95

20 0.78 101.4 1.35 8031.4 418.15

15 1.03 101.4 1.38 1552.9 0.19

12.5 1.24 94.3 5.71 1557.7 0.5

10 1.55 99 1.04 1603.5 3.45

7.5 2.07 94.3 5.73 1533.4 1.07

5 3.1 101.4 1.37 1614.1 4.14

2.5 6.2 99 1.01 1593.5 2.81

67

Figure 58 (a) presents the effect of spatial resolution on the average frequency at which the

maximum wave energy was recorded in the k-f plots. One of the fundamental requirements for

k-f plot is agreement between the central excitation frequency and the frequency at which the

wave energy is maximised. Overall, the relationship between spatial resolution and average

phase velocity is unclear within the scope of the data. While the minimum error in frequency

occurs for the highest resolution model, at 1.01% for Δ𝑥 = 2.5 𝑚𝑚, the error in 𝑓𝑎𝑣𝑔 varies

chaotically as Δ𝑥 increases. The frequency range also varies significantly with resolution. When

the separation distance was 2.5 mm, 7.5 mm, and 12.5 mm, a single contour of maximum

amplitude was plotted hence resulting in a narrow frequency band. Models having separation

distance of 5 mm and 15 mm both showed a wider frequency band with 2 contours of equally

large magnitude. Notably the 10 mm model showed 4 nodes of maximum amplitude either side

of the 100 kHz central frequency, resulting in the widest frequency range.

Figure 58 (b) presents the effect of spatial resolution on the average phase velocity. Like

the average frequency, there is no discernible relationship between phase velocity and spatial

resolution seen within the scope of the testing. The most accurate approximations of phase

velocity occurred when Δ𝑥 = 15 𝑚𝑚 at 0.19% which was contrary to the expected outcome.

The accuracy of the calculated phase velocity varies nonlinearly with spatial resolution with the

maximum error occurring when Δ𝑥 = 5 𝑚𝑚.

Overall, no significant trends were noted between spatial resolutions greater than 1 node per

wavelength and the average frequency or phase velocity of the Lamb wave pulse. It is

hypothesised that this result was likely due to the scope of the testing covering an insufficient

spatial resolution range to see significant impact on the accuracy of the numerical model. In the

research carried out by Lasˇova´, the displacements of all 4096 mesh nodes were captured and

used in the 2D FFT when it was shown that increasing the resolution increased the accuracy of

the dispersion curves [14]. This level of resolution could not be reached in this analysis due to

time and computational constraints. It is possible that a significantly higher resolution needs to

be attained before spatial resolution begins to largely affect the accuracy of the generated k-f

plots.

Within the scope of the data gathered in the investigation, the relationships between spatial

resolution and average phase velocity and frequency are unclear. The data indicates that within

the tested range of Δ𝑥, spatial resolution has little influence on the average phase velocity. At

all spatial resolutions from 1.03 to 6.2 nodes per wavelength the calculated average frequency

and phase velocity were within 5.73% and 4.14% of the theoretical values respectively. This

finding suggests that future investigations requiring only approximations of phase velocity

could validly use spatial resolutions within the tested range. The most notable criteria for spatial

resolution found within the investigation is that at least 1 node per wavelength is required to

yield physical approximations of phase velocity.

68

a) b)

Figure 58: The influence of spatial resolution on (a) average frequency and (b) phase velocity is unclear. This may be

attributed to the scope of the testing covering an insufficiently fine spatial resolution.

6.5.3. Validation of the 2D ANSYS simulation by phase velocity

Through a sensitivity analysis it was found that the average frequency and phase velocity

were calculated to a high degree of accuracy when the spatial resolution was higher than 1 node

per wavelength. As a result of this conclusion, the 0.25 mm model was analysed further to

validate the accuracy of the ANSYS simulation of the fundamental A0 Lamb mode.

Table 10 presented the phase velocities calculated from each of the k-f plots along with the

percentage errors relative to the theoretical Lamb wave velocity. In aluminium 2024-T6, this is

1550 𝑚/𝑠. The calculated phase velocity with the minimum associated error was 𝑐 =

1552.9 𝑚/𝑠 at only 0.19% compared to the theoretical value. The maximum error occurred for

𝑐 = 1614.1 𝑚/𝑠 at 4.14%. These findings suggest that ANSYS Explicit Dynamics is highly

effective in accurately modelling the phase dispersion characteristics of the A0 Lamb wave. The

findings from the spatial resolution investigation suggest that nodal displacement data captured

in ANSYS can be reliably used to approximate Lamb wave phase velocity assuming at least 1

node exists per wavelength.

The numerical error in group velocity calculated for the 0.25 mm mesh ANSYS model was

somewhere between 0.35% and 1.37%, depending on the inherent error in the methodology.

Comparison of the group velocity error range to that of phase velocity, which is between 0.19%

and 4.14%, suggests an acceptable level of consistency between the two approaches. As the

phase velocity and group velocity are two independent characteristics of the Lamb wave, this

analysis further validates the capabilities of ANSYS in modelling the fundamental

antisymmetric Lamb wave.

The methodology used to calculate group velocity was more rigorously optimised than that

used to calculate phase velocity, which may explain why the error in 𝑐𝑔 is lower than the error

in 𝑐. Despite this, the degree of accuracy seen in the phase velocity was reasonably high

considering the maximum mesh node resolution was only 6.2 nodes per wavelength. It is

hypothesised that increasing the data resolution to capture all nodal displacement results would

improve the overall resolution of the wavenumber-frequency plots. Hence, this would yield

calculated phase velocities at an even higher degree of accuracy. Verification of this hypothesis

is suggested for future research investigations.

69

7 Development of the three-dimensional ANSYS model


Lamb wave-based SHM techniques typically utilise a network of transducers that emit and

detect guided waves along the surface of a structure. The sensor arrangement is often

multidirectional to ensure that the entire surface is captured by the network. Complex transducer

networks are often required to monitor composite structures due to their anisotropic nature and

hence more complicated wave propagation. This means that a finite element models of Lamb

waves must reach the complexity of three-dimensions to capture the real-world propagation

behaviour of Lamb waves. Three-dimensional FE models also allow more complex wave

scattering from boundaries such as edges and joins which exist in real-world structures.

To investigate the simulation capabilities of ANSYS in modelling the A0 Lamb mode in

three-dimensions, the analysis was performed using a 3D model of the aluminium 2024 plate.

This analysis provided validation of ANSYS Explicit Dynamics in modelling Lamb waves

using higher order mesh elements which allow propagation in the z direction. The results of this

study aim to validate the use of ANYS Explicit Dynamics to model the multidimensional

propagation of Lamb waves in SHM transducer networks.

Analysis settings, material properties and geometry

The 3D FE model was developed using predominantly the same analysis settings,

constraints and material properties as the 2D plate. This was to ensure that any significant

differences between the results of the 2D and 3D simulations could be attributed to specific

variables within the models with a high degree of confidence. Figure 59 (a) shows the model

properties for the 3D simulation with the Analysis Type set to ‘3D’. In setting this property the

z-direction plane strain criterion is turned off, thus allowing the numerical model to simulate

wave propagation in both x and z directions. This occurs by introducing a third degree of

freedom for displacement (z) to each of the FE mesh nodes. The x, y and z displacements are

solved by the FEM equations of motion, which are then more complex due to the added DOF.

The geometry of the 3D model is shown in Figure 59 (b). The model was a square plate with

equal side lengths of 400 mm in the x and z directions. The thickness of the plate was 3 mm in

the y direction. The thickness was kept equal to that of the 2D model due to its influence on

dispersive wave properties (𝑐 and 𝑐𝑔). The plate material was aluminium 2024-T6 and had the

same engineering properties as presented in Table 4.

70

a) b)

Figure 59: The FE model properties were set to 3D to capture the propagation of Lamb waves through the x-z plane (a).

The model was a square plate with dimensions 400 mm × 400 mm × 3 mm (b).

Model setup

The 3D plate geometry was imported into ANSYS Mechanical for the Model Setup. The

geometry was meshed using 8-node SOLID45 brick elements. Each of the mesh nodes had

3DOF, thus allowing translation in the x, y and z directions. The characteristic mesh element

length was selected based on two primary criteria: accuracy of the simulation results and

computation time. In the 2D investigation, all models produced relatively accurate simulation

results (highest FE error was 5.64%), however computation time ranged from 10.4 to 84

minutes depending on the mesh resolution. The 1 mm element length model was selected as

having the most favourable compromise between accuracy and efficiency, with an FE error of

2.37% and computation time of 52.8 minutes. A characteristic mesh element length of 1.0 mm

was therefore selected for the three-dimensional study. This resulted in a typical computation

time of around 8 hours for 3D simulations, which was the longest feasible time given the

number of test cases within the scope of the investigation. Figure 60 shows the 3D aluminium

plate meshed with 3 elements in the thickness direction, each having 𝑙𝑒 = 1.0 𝑚𝑚.

Figure 60 also shows the nodal displacement excitation which was applied to the centre

surface-node along the left-hand boundary of the pate. The excitation was a sinusoidal tone

burst modulated with a Hanning window function. The centre frequency was kept at 100 kHz

to provide a direct comparison between the 2D and 3D models. The far boundary of the plate

was constrained using a Fixed Support to provide the necessary reaction force against the nodal

displacement excitation.

The explicit integration solver was ANSYS Autodyn, with the settings kept the same as the

2D model to provide consistency. The time-step was set to computer controlled as were FE

variables such as hourglassing control and numerical damping. Named Selections were created

at equally spaced points along the length of the plate, with 𝑑𝑥 = 10 𝑚𝑚.

71

Figure 60: The characteristic mesh element length was 1 mm to provide an acceptable compromise between accuracy and

computational time.

Simulation results

The 100 kHz excitation frequency was applied to the surface of the 3D plate model and the

resultant Lamb wave propagation was captured over 220 𝜇𝑠. This simulation time allowed for

the incident wave to travel across the length of the plate to the far boundary. The nodal

displacement results at equidistant points along the plate were captured and the data was

exported for analysis in Python. Figure 61 shows the propagation of the fundamental

antisymmetric Lamb mode at 120 mm from the excitation source. The displacement is scaled

by a factor of 15,000 to provide a visual depiction of the wave packet. Here it can be seen that

the Lamb wave propagates in a radial direction from the source along the x-z plane. From a

qualitative standpoint, the simulated Lamb wave appears consistent with the 2D model and

behaves in a physically realistic manner. Figure 61 also shows nodal displacement in the y

direction captured in the ANSYS simulation at 𝑥 = 120 𝑚𝑚. The first excitation is the incident

Lamb wave packet which occurs at 43 𝜇𝑠. The second displacement excitation is due to

reflections from the side boundaries of the plate which meet at the centre at 160 𝜇𝑠. The

boundary reflections can be seen in Figure 62. Note that this reflected signal was not seen in

the 2D simulation, with the only reflection being that from the far boundary of the plate.

72

Figure 61: The nodal displacement in the thickness direction was measured to capture the antisymmetric Lamb wave as it

propagated along the plate.

The Lamb wave packets reflected from the side boundaries of the 3D model add complexity

to the signal processing which was not seen in the 2D model. This was the driving factor in

reducing the overall simulation time to 220 𝜇𝑠. After this point, significant noise was

introduced in the captured nodal displacement data due to subsequent interactions with the plate

boundaries. The model validation focused primarily on the dynamics of the incident Lamb

wave. This data was therefore omitted, thus reducing the overall simulation time required.

Figure 62: Reflections from the side boundaries of the plate introduced complexity into the 3D model which was not seen

the 2D model. This required more deliberate selection of the simulation time to avoid noise in the displacement data.

8 Analysis of the three-dimensional ANSYS simulation

Signal processing of the raw data

The nodal displacement data captured in the 3D simulation was exported from ANSYS for

signal processing in Python. The methodology for analysing the data was similar to that used

for the 2D model. The incident wave packet was first identified within each of the signals and

the maximum amplitude of the pulse was then used to normalise the entire data set. The

displacement histories were then plotted against the analytical solution to provide a visual

comparison of the wave pulse. Figure 63 (a) presents the raw data measured at 60 mm from the

73

excitation source. The incident and reflected waves can be clearly distinguished at this mesh

node. The incident wave pulse had a maximum amplitude 2.3 × 10−8 𝑚 which was then used

to normalise the amplitude, as shown in Figure 63 (b). Comparing the simulated wave pulse

with the analytical solution reveals very strong agreement between the shape of the A0 mode at

this location. The number of peaks within the wave pulse and the overall effect of wave pulse

widening as a result of dispersion is accurately captured by the simulation. The most notable

discrepancy occurs at the second peak in the wave pulse, which has a shallower amplitude in

the numerical model as compared with the analytical waveform.

The wave pulse period of the analytical model at 𝑥 = 60 𝑚𝑚 is 45.8 𝜇𝑠, while the period

is 46.6 𝜇𝑠 for the numerical wave pulse. The error in wave pulse period is therefore only 0.8 𝜇𝑠.

This suggests that wave dispersion was accurately modelled in the 3D simulation.

a) b)

Figure 63: The raw data captured at 60 mm from the excitation reveals the incident and reflected wave pulses (a). The

wave pulse was normalised and compared with the analytical solution, revealing excellent agreement overall (b).

Figure 64 (a) and (b) plot the raw and normalised data captured at 200 mm from the

excitation source. Here the separation between the incident and reflected waves is much lower,

which was considered during normalisation. Comparison of the numerical and theoretical wave

pulses reveals a noticeable lag of approximately 2 𝜇𝑠 which did not exist at 60 mm (Figure 63).

The earlier ToA of the analytical wave pulse indicates that the simulated A0 Lamb wave is

travelling at a lower group velocity than the analytical solution. Despite this, the dispersive

effect of wave pulse widening is still modelled highly accurately at 𝑥 = 200 𝑚𝑚, with pulse-

periods of 59.8 𝜇𝑠 and 61.4 𝜇𝑠 for the analytical and numerical models respectively. The error

between pulse-widths is therefore 1.6 𝜇𝑠 which is only a 0.8 𝜇𝑠 increase in error between

measurements at 𝑥 = 60 𝑚𝑚 and 𝑥 = 200 𝑚𝑚 from the excitation. This finding suggests that

ANSYS accurately models the wave-pulse widening effect of dispersion with an acceptable

level of numerical error. However, it also indicates that simulating Lamb wave propagation

over longer distances could lead to an increase in overall error in the shape of the wave pulse.

74

a) b)

Figure 64: The nodal displacement data captured at 200 mm (a) reveals the simulated wave pulse travelled with a lower

velocity as indicated by the lag between wave packets (b).

Model validation and comparison of results with the 2D model

The incident Lamb wave was plotted at each point along the plate and was normalised by

the maximum amplitude to provide a comparison between the 2D, 3D and analytical models.

The Hilbert transformation of each captured signal was taken to reveal the energy envelope of

the incident Lamb wave. The energy-distribution of the wave pulse was also used to compare

the models via group velocity. Figure 65 presents a comparison of the normalised Lamb wave

measured at two locations along the plate: (a) 40 mm from the excitation and (c) 180 mm from

the excitation. The Hilbert transformations are also plotted in (b) and (d).

There is strong consistency between the 2D and 3D models at both locations along the plate.

The number of peaks within the Lamb wave amplitude, along with the overall shape and period

of the wave pulse, indicates strong agreement between the two models. The time lag between

the 2D and 3D simulations remains consistent between the two measured points along the plate

with Δ𝑡40 = 1.6 𝜇𝑠 and Δ𝑡180 = 1.7 𝜇𝑠, indicating that the velocity difference between the two

wave pulses is not significant.

Analysis of the energy envelope of the 3D simulation results, reveals significant aliasing in

the response which was not seen in the 2D model results. The overall amplitude and location

of the energy distribution is consistent between the models, however there is a large amount of

noise within the amplitude of the 3D model. It was hypothesised that the introduction of a third

degree of freedom, being translation in the z-direction, resulted in an additional source of

numerical error into the displacement results. This is discussed in further detail below.

75

a) b)

c) d)

Figure 65: Comparison of the wave pulses at (a) 40 mm and (c) 180 mm reveals an overall consistency in the shape of

the Lamb waves simulated in the 3D and 2D models. The energy distributions of the wave pulses (b) and (d) show that there

was aliasing seen in the 3D model which was attributed to numerical error.

The energy distribution approach for ToA was used to calculate the group velocity of the

incident Lamb wave simulated in the ANSYS 3D model. Graphs plotting the calculated group

velocity at each measured mesh node along the length of the plate, for both the 2D and 3D

models, are provided in Appendix C (section 15.3). The group velocities calculated at all points

along the plate were averaged to determine the overall group velocity of the simulated

antisymmetric Lamb wave. Table 11 summarises the findings of the study. The average group

velocity of the incident Lamb wave simulated in the 3D model was 2503 m/s, which

corresponds to a 4.49% error compared with the theoretical value. By subtracting the baseline

error of 1.02% measured in the analytical model, the minimum error attributed to the FE

simulation was 3.47%. Comparing this with the 2D model, which had a minimum FE error of

2.37%, the two simulations yield highly consistent results. The difference in calculated group

velocities is 29 m/s (1.1%), with the wave pulse simulated in the 3D environment having the

lower velocity.

Table 11: Comparison of average group velocities calculated in the 2D and 3D simulations.

Model type Group velocity (m/s) Percentage error (%) Minimum FE attributed error (%)

2D 2532 3.39 2.37

3D 2503 4.49 3.47

Analytical 2594 1.02 -

76

The primary explanation of the increased error seen in the group velocity of the 3D Lamb wave

was the additional degree of freedom of each mesh node. This introduced numerical error and

energy losses due to translation of the mesh nodes in the z-direction. The 2D model assumed

plane strain in the z-direction, which is the ideal scenario and the simulation incurred less

numerical error and wave pulse energy losses as the lamb wave travelled along the plate. In the

3D simulation, mesh nodes interact in all three axes, meaning the accumulation of numerical

error may have resulted in energy losses within the FE mesh. This explanation is consistent

with the 3D simulated Lamb wave having a lower velocity, as the increased energy loss within

the FE mesh reduced the overall energy of the wave pulse as it travelled through the plate.

Additionally, it was expected that the 2D model would show stronger agreement with the

theoretical wave speed of 2621 m/s since this figure was solved from the dispersion curves.

Plane-strain conditions are a assumed when deriving the Rayleigh-Lamb equations from which

the Lamb wave dispersion curves are generated [3]. Hence, the analytical wave speed of

2621 m/s is generated under the assumption of plane strain conditions. Since the 2D model

incorporated the same simplifying assumption, while the 3D model did not, the small

differences in error were consistent with the expected outcome.

Conclusions from the 3D ANSYS simulation

The antisymmetric Lamb mode was excited in a 3D model of the aluminium 2024-T6 plate.

By analysing the incident wave pulse, the group velocity was calculated at 2503 m/s and the

minimum FE attributed error was estimated at 3.47%. Considering the model was meshed with

relatively coarse 1 mm length brick elements, the simulation results showed very strong

agreement with both the 2D model and the analytical solution. The study indicates that ANSYS

can be used to model the propagation of the antisymmetric mode with an acceptable degree of

error and without the need for a high-resolution mesh.

When simulating the A0 Lamb wave in the 3D model, it became apparent that the

computation resources required for such an analysis are extremely high. To run the 3D

simulation over 220 𝜇𝑠, the computation time was over 8 hours. Comparing this with the time

required to simulate Lamb wave propagation over 900 𝜇𝑠 in the 2D model, at just 52.8 minutes,

the scale of the required computation resources becomes apparent. The 2D model was shown

to accurately model Lamb wave propagation under the assumption of plane strain. Comparison

with the 3D model and the analytical solution validates this assumption.

The plane strain condition was shown to accurately model Lamb wave propagation, and

showed excellent agreement with 3D analyses, assuming the aspect ratio of the plate is at least

1 (square plate). It is therefore recommended that any relatively simple analysis which uses

isotropic materials and/or simple (plate) geometries, is performed in the 2D ANSYS

environment. This is because of the significantly shorter computation time required for

simulations as a consequence of the simplified FE matrix. However, it should be noted that the

2D environment cannot capture many of the complexities required for SHM design. Some of

these complexities include: scattering of reflected waves from material damage, propagation of

Lamb waves in composite materials, antisymmetric or non-regular component geometries, and

complex damage types. Only the 3D simulation environment can capture the complexity of

these analyses and ANSYS Explicit Dynamics is recommended in future works focused on

these topics.

77

9 Investigating model rigorousness across the low-frequency regime


The capabilities of ANSYS in simulating the fundamental antisymmetric mode have been

demonstrated using a centre frequency of 100 kHz. This frequency was arbitrarily selected

within the recommended range for Lamb wave propagation found in the literature. There are a

number of factors which may drive the selection of frequency for an NDT application. The

primary considerations which drive frequency selection include; the type of material damage,

the size of the damage relative to the wavelength, material properties, thickness of the structure,

and transducer type.

Excitation frequency selection can impact the stability and results of a simulation as

𝑓0 drives the spatial resolution for a fixed FE mesh. The consequence of increasing the

excitation frequency is a decrease in wavelength. When the wavelength of a pulse decreases,

the number of FE mesh nodes per wavelength also decreases. This could compromise the FE

mesh resolution criterion of at least 10-20 nodes per wavelength. Conversely, by decreasing 𝑓0

and hence increasing 𝜆0, the spatial resolution can be increased. It therefore may be possible to

yield more accurate models by using lower frequency excitations for a fixed FE mesh.

It is therefore important to investigate the rigorousness of the FE model across a range of

excitation frequencies to validate the conclusions drawn in the previous studies more broadly.

Nine simulations were carried out using various excitation frequencies ranging between 25 kHz

and 400 kHz. The simulations were run using the 3D ANSYS model of the aluminium 2024-

T6 plate. The model was meshed using the same 8-node solid brick elements with a fixed

characteristic mesh element length of 1 mm. All analysis settings and general model

configurations were kept the same as the previous investigation.

Selection of the excitation frequencies

Excitation of the fundamental antisymmetric Lamb mode was carried out through nodal

displacements centred around each of the frequencies provided in Table 12. The theoretical

group and phase velocities corresponding to each frequency are provided, along with the

associated wavelength which was sourced from the dispersion charts for aluminium 2024-T6

provided in Figure 3. The nodes per wavelength are also provided in Table 12 based on the

1 mm solid brick elements used to mesh the 3D aluminium plate.

Table 12: Excitation frequencies and associated wave speeds.

Excitation Frequency

(kHz-mm)

Theoretical 𝒄 (m/s) Theoretical 𝒄𝒈 (m/s) Wavelength

(mm)

Nodes per

wavelength

25 835 1642 33.4 33

40 1033 1960 25.8 26

50 1150 2122 23.0 23

75 1374 2418 18.3 18

100 1549 2621 15.5 15

150 1812 2869 12.1 12

200 2003 3006 10.0 10

300 2265 3130 7.6 8

400 2438 3163 6.1 6

78

The frequency range 25 kHz – 400 kHz was selected based on recommendations in the

literature as well as the dispersive characteristics of aluminium 2024-T6. The lower frequency

limit of 25 kHz was selected based on the commonly used frequency range of 0 – 200 kHz for

many Lamb wave based NDT studies [5]. The upper limit of 400 kHz was selected above this

frequency range to ensure that the conclusions drawn from the study are relevant and useful to

those falling outside this typical range (central frequencies of 300 – 400 kHz). The high end of

the excitation frequency was limited to 400 kHz to avoid higher order Lamb modes, which

begin to appear at excitation frequency thicknesses of 1660 kHz-mm (550 kHz for the 3 mm

aluminium plate). Due to spectral leakage above the centre frequency, f0 was limited to 400 kHz

to provide a 150 kHz buffer below the cut off frequency where these higher order modes appear.

All excitations were 5-cycle sinusoidal tone burst signals modulated by Hanning windowing

functions to ensure consistency between the test cases. Since the number of cycles were fixed,

this meant the period of the excitation decreased as the excitation frequency increased. Thus, a

higher 𝑓0 results in a shorter excitation wave pulse in the time domain. This is demonstrated in

Figure 66 (a) which plots the energy envelopes of the 25 kHz and 400 kHz excitation nodal

displacements calculated through the Hilbert transformation. Here it can be seen that the period

(𝑇 = 𝑁/𝑓0) of the low frequency signal is 200 𝜇𝑠 while the high frequency signal is 12.5 𝜇𝑠.

Conversely, by fixing the number of cycles and increasing the frequency, the overall bandwidth

of the excitation increases in the frequency domain. Figure 66 (b) plots the Fast Fourier

Transform of both signals and shows vastly different bandwidths between the two limits across

the frequency spectrum. The frequency bandwidths of the 25 kHz and 400 kHz excitation

signals are approximately 35 kHz (5 – 40 kHz) and 360 kHz (240 – 560 kHz) respectively, with

each range centred at the respective 𝑓0. Observing the 400 kHz spectrum in Figure 66 (b), it

can be seen that spectral leakage does result in some amplitude above 550 kHz. However, since

ToA was referenced using a 90% amplitude threshold on the energy distribution, which is where

the centre frequency is dominant, any higher order modes were to be avoided.

As discussed in the literature, an ideal wave pulse is both short in duration as well as narrow

in frequency bandwidth [5]. As the frequency was varied from 25 kHz to 400 kHz, both criteria

were seen in their extremes as shown in Figure 66. Hence, the compromise between pulse

duration and frequency bandwidth could be highly significant in the overall quality of the results

captured across the frequency range.

a) b)

79

Figure 66: The Hilbert transform reveals a shorter excitation pulse period at higher f0 (a), while the Fast Fourier

Transform of the excitation signal reveals a narrower frequency bandwidth at lower f0 (b).

Analysis of the results

The simulations were carried out in ANSYS Explicit Dynamics and the results analysed

using the same philosophy as the previous section. The energy enveloped based approach for

ToA was used to determine the group velocities at each of 40 measured mesh nodes across the

length of the aluminium plate. The distance 𝑑𝑥 between each measurement was 10 mm, and

the separation distance between mesh nodes over which 𝑐𝑔 was calculated was 90 mm. The

amplitude tolerance was kept at 90% to remain consistent with previous analyses. The group

velocities measured across the plate were averaged to produce the average 𝑐𝑔 of the incident

Lamb wave. The corresponding percentage error relative to the theoretical group velocity was

then determined. Table 13 provides a summary of the results captured across the simulations.

Table 13: Average group velocity measurements across the frequency range.

Excitation Frequency

(kHz-mm)

Average measured

𝒄𝒈 (m/s)

Theoretical 𝒄𝒈 (m/s) Percentage error (%)

25 1553 1642 5.42

40 1863 1960 4.95

50 2017 2122 4.95

75 2322 2418 3.97

100 2503 2621 4.50

150 2753 2869 4.04

200 2882 3006 4.13

300 3007 3130 3.93

400 3045 3163 3.73

Figure 67 plots the simulation results superimposed onto the group velocity dispersion

curve for aluminium 2024-T6. The theoretical group velocities are marked on the curve for each

frequency within the tested range. Overall there is strong agreement between the dispersive

properties of the simulated Lamb waves compared with the theoretical dispersion curve. As the

excitation frequency was increased, the group velocity of the incident Lamb wave increased in

a non-linear fashion with a similar proportionality as the theoretical curve. As frequency

increases from the lower limit of the spectrum, the A0 group velocity increases rapidly. The

gradient of the curve decreases toward the higher end of the frequency range which is consistent

with the theoretical curve.

80

Figure 67: The dispersion curve generated from the simulation results shows strong agreement with the theoretical curve

for aluminium 2024-T6.

The highest percentage error in average group velocity was 5.42% which was calculated for

the 25 kHz simulation. The lowest percentage error was calculated for the 400 kHz Lamb wave

at 3.73%. The overall trend of the results was a small decrease in the percentage error of 𝑐𝑔 as

the excitation frequency was increased. However, the range of error between the maximum and

the minimum was only 1.69%. Additionally, there were some inconsistencies within this trend,

with notable increases in error measured at 100 kHz and 200 kHz. Due to the limited change in

error, along with the inconsistencies identified within the trend, it is difficult to establish

whether increasing 𝑓0 caused the decrease in percentage error observed in the results. However,

the trend of the data does suggest that increasing 𝑓0 did not result in a net increase in the error

of the simulated results.

As the frequency was increased, the spatial resolution of the mesh decreased due to

shortening of the propagating Lamb wave. The maximum nodes per wavelength was 33 for the

25 kHz centre frequency and the minimum resolution was 6 nodes per wavelength at 400 kHz.

The generally accepted requirements for spatial resolution of 10 – 20 nodes per wavelength

were both compromised across the frequency range. The resolution dropped below 20 nodes

per wavelength at 75 kHz and below 10 nodes per wavelength at 300 kHz. The average group

velocity was shown to be consistently accurate in spite of both criteria being compromised. In

fact, the weak trend observed between frequency and error is directly the opposite what might

be expected from the spatial resolution criterion. The results of this analysis further suggest that

spatial resolution is not a critical criterion for the overall outcome of Lamb wave simulation in

ANSYS. The antisymmetric mode was simulated with 3.73% error with only 6 nodes per

wavelength. These results are in agreement with literature established by Moser [29].

The relative stability in the overall accuracy of the simulated Lamb waves also suggests that

the compromise between wave pulse duration and frequency bandwidth did not significantly

affect the outcome of the results. The 1.69% error discrepancy between the minimum and

maximum frequencies is likely attributed, at least partially, to numerical error within the

analysis model. However, the results may suggest that the negative effects of wave pulse

duration were more significant than that of a wider frequency bandwidth. Figure 68 shows the

81

25 kHz and 400 kHz raw mesh node results at 120 mm from the excitation source. Here it can

be seen that the increased pulse duration for the 25 kHz excitation results in less separation

between the incident and reflected wave pulses. While the analysis methodology did take the

reflected wave pulses into account, it is possible that some numerical error was introduced into

the model as a result of the boundary reflections.

a) b)

Figure 68: The increased wave duration for the 25 kHz model resulted in less separation between incident and reflected

wave pulses (a), which may have introduced numerical error not seen in higher frequency models such as 400 kHz (b).

Figure 68 also demonstrates the difference in displacement amplitude captured between the

25 kHz and 400 kHz simulations. To investigate the relationship between amplitude and

frequency, the maximum displacement at 𝑥 = 10 𝑚𝑚 was recorded for each of the simulations.

The displacement data was then graphed as a function of frequency by Figure 69. The trend of

the data demonstrates a clear relationship between excitation frequency and nodal displacement

amplitude. The effect of increasing frequency of the excitation, whilst keeping the amplitude

fixed, is a reduction in the amplitude of the propagating Lamb wave. This is an important

consideration in SHM practices, as the detection of wave pulses can be limited by the resolution

of the transducer. Hence, it should be understood that increasing frequency of the excitation

will reduce the overall amplitude of the measured response. This relationship should be

considered when designing SHM systems to ensure that the sensors are compatible with the

relative amplitude of the signal response.

Figure 69: Increasing excitation frequency resulted in reduced displacement amplitude.

82

10 Interactions between the A0 mode and a surface notch

Overview and significance of the study

Modelling the interactions between Lamb waves and material damage is a critical

application for SHM design. It is important to ensure that FE simulations accurately capture the

dispersive properties of the reflected of the Lamb wave as well as any mode conversion which

occurs due to interactions with the material damage. Accurate simulation of these interactions

could improve the design of the transducer networks used for detecting damage within a SHM

system. Accurate numerical simulation of Lamb wave interactions with material damage could

allow for more rapid design of sensor arrangement based on simulation data and eliminating

the need for extensive experimental testing. Accurate simulation of these interactions could also

aid in selecting the most appropriate sensor locations for detection of material damage. This

could be particularly useful for analysing anisotropic materials, such as composites, which have

significantly more complex scattering behaviour.

To investigate the capabilities of ANSYS in modelling the interaction between the

fundamental Lamb wave and material damage, a surface notch was incorporated in the 2D finite

element model. The interactions between the incident Lamb wave and the notch were observed

and the properties of the reflected waves were verified through determination of group velocity.

Scope of the study

The notch was modelled geometrically by removing a rectangular area from the 2D surface.

Several notches of varying depth were tested, which extended in the through-thickness direction

from the top surface of the plate. The geometry of the notch is shown in Figure 70 for the

1.5 mm depth notch. The leading edge of the notch was located at 750 mm from the left-hand

side of the plate.

Figure 70: The notch was modelled geometrically by removing mesh elements from the model, extending in the thickness

direction from the surface of the plate.

Table 14 summarises the scope of the tested notch geometries. The width of the notch in the

lengthwise direction of the plate was kept at a constant 0.5 mm, while the depth was varied

from 0.5 mm to 2.5 mm. The through-thickness notch depth therefore varied between 16.7%

83

and 83.3% relative to the 3 mm thick plate. The depth of the notch was varied to investigate the

influence of notch depth on the amplitude of the reflected Lamb waves.

Table 14: Summary of tested notch geometries.

Notch width (mm) Notch depth (mm) Through thickness depth (%)

0.5 0.5 16.7

0.5 1.0 33.3

0.5 1.5 50.0

0.5 2.0 66.7

0.5 2.5 83.3

The model was meshed using 4-node solid elements with a characteristic mesh element

length of 0.25 mm. The analysis was performed using the 2D model to reduce computational

complexity as well as to simplify the modelling of the notch. The mesh element length was

selected at 0.25 mm based on the acceptable compromise between computation speed and

model accuracy seen in section 6.4.6. A 0.25 mm mesh element length also provided an

adequate resolution to model the tested notch depths.

The excitation frequency was selected at 100 kHz and the plate properties were for

aluminium 2024-T6. This allowed for a direct comparison between the results captured from

the damaged plate and those captured from the undamaged plate in section 6.4.6.

Results

The simulations were carried out for the various notch depths and the results were extracted

from ANSYS for analysis. The ANSYS graphical output of the nodal displacement data clearly

shows that interaction between the incident wave pulse and the surface notch results in a

reflected wave pulse in the direction of the excitation source. This is shown for the 1.5 mm deep

notch in Figure 71.

Figure 71: Interaction between the incident Lamb wave and the surface notch resulted in a reflected wave propagating

back toward the excitation source.

Magnification of the nodal displacement results captured at 𝑥 = 210 𝑚𝑚 reveals a

symmetric wave reflected from the notch. This is shown in Figure 72 below. The wave pulse

was hypothesised to be the fundamental S0 mode generated through mode conversion due to

interactions with the notch. This hypothesis was to be confirmed through comparison of the

group velocity with the theoretical wave speed of the S0 mode. Arrival of the symmetric mode

Reflected wave pulse Incident wave pulse

84

was indicated by the distinct pattern of alternating ‘thick’ and ‘thin’ pulses which were

discussed in the literature (see Figure 1 (a)). The amplitude of the symmetric Lamb mode was

significantly lower in magnitude than the antisymmetric mode. Because of the large difference

in amplitude, the scale used in Figure 71 was insufficient to see the S0 mode. The symmetric

mode was also observed to arrive earlier than the antisymmetric mode. This indicated the wave

pulse travelled with a higher group velocity, which is consistent with the dispersive properties

of aluminium 2024-T6 (see Figure 3 (b)).

Figure 72: The nodal displacement response at 210 mm from the excitation source reveals the symmetric Lamb mode is

reflected from the notch and arrives earlier than the A0 mode.

The results captured at 𝑥 = 210 𝑚𝑚 also reveal the antisymmetric mode reflected from the

notch, as seen in Figure 73. The A0 mode was observed to arrive later than the S0 mode.

Figure 73: The nodal displacement response at 210 mm from the excitation source reveals the antisymmetric Lamb mode

is reflected from the notch and arrives later than the S0 mode.

Analysis of the nodal displacement data

Both x and y nodal displacements were captured in the ANSYS simulation and the data was

exported for quantitative analysis in Python. Nodal displacements in the y direction were

measured to capture the out-of-plane motion of the antisymmetric mode. The x direction nodal

displacement was measured to capture the symmetric mode, which primarily induces in-plane

motion. Figure 74 presents the y displacement data captured at 300 mm from the excitation

Reflected S0 mode

Reflected A0 mode

85

source, when the notch depth was 1.5 mm. The Hilbert transform of the signal provides the

energy envelope of the wave pulses, presented in Figure 75.

Figure 74: The y direction nodal displacement at 300 mm from the excitation source reveals the wave pulses reflected off

the structural damage, along with significant boundary noise.

The notable features within the captured nodal displacement dataset are labelled on each of

the plots. The incident A0 wave pulse was captured at approximately 152 𝜇𝑠 by the measured

mesh node at 300 mm from the excitation source. Interaction between the incident Lamb wave

and the 1.5 mm notch results in wave pulses reflected from the damage back in the direction of

the excitation source. The small peak indicating the reflected S0 Lamb wave can be identified

from Figure 75, occurring at 391 𝜇𝑠. The reflected A0 wave is more clearly identified in the

nodal displacement data as a result of its larger amplitude, with the approximate ToA at 485 𝜇𝑠.

The displacement of the A0 wave is larger in amplitude as it causes predominantly out-of-plane

nodal displacement, while the S0 mode is primarily in plane displacement. The remainder of the

captured signal is dominated by noise resulting from boundary reflections both from the

incident Lamb wave reflecting off the far boundary and the scattered wave reflecting off the

excitation boundary. The noise within this region demonstrates the difficulties which can be

experienced when analysing the nodal displacement data due to reflections off system

boundaries and other discontinuities. Careful attention must be maintained to ensure that the

correct amplitude peak for the scattered wave pulse is identified among the noise of the

measured data.

Incident wave pulse

Reflected S0 mode

Reflected A0 mode

Boundary reflections

86

Figure 75: The Hilbert transform of the y direction nodal displacement data was used to distinguish the reflected wave

pulses and determine ToA.

Figure 76 (a) and (b) present a comparison of the results between the damaged and

undamaged 2D plate models. The nodal displacement data in the y direction is plotted for each

scenario and helps to confirm the nature of the identified wave peaks. The incident Lamb wave

and the wave pulse reflected off the far boundary of the plate appear in the results of undamaged

plate, while the identified A0 and S0 modes scattered by the surface notch do not. This confirms

that the source of these wave peaks was indeed the interaction between the incident wave and

the material damage.

a) b)

Figure 76: Comparison of the y displacement data between the damaged and undamaged plate confirms the nature of the

wave peaks as the reflected A0 and S0 modes only appear due to interaction with the notch.

Close-up views of the reflected wave pulses are shown in Figure 77 (a) and (b) below. Since

Figure 77 plots the out-of-plane displacement, the amplitude of the reflected A0 mode is much

larger than the in-plane S0 mode. The times of arrival of the reflected wave pulses were

approximated using the wave pulse energy envelope method. The amplitude threshold over

which time was averaged was defined at 90% of the local maximum. The average ToA of the

Incident wave pulse

Reflected S0 mode

Reflected A0 mode


87

reflected S0 mode was approximately 391 𝜇𝑠, while the A0 mode was approximately 485 𝜇𝑠.

The approximate locations are marked in Figure 77 (b).

a) b)

Figure 77: Closeup view of the y displacement data shows the S0 mode is detected earlier than the A0 mode as the

symmetric mode travels at a higher group velocity, as indicated in the aluminum 2024-T6 dispersion curves.

The nodal displacement in the x direction, measured 300 mm from the excitation, is plotted

in Figure 78, along with the Hilbert transform of the signal in Figure 79. The plots present data

captured in the 1.5 mm notch simulation. The incident and reflected S0 wave modes are more

clearly discernible due to capturing the in-plane (x direction) nodal displacement. Observing

the incident wave pulses, it can be identified that the S0 mode arrives earlier than the A0 mode.

This is due to the S0 mode’s higher group velocity at 5385 m/s compared with the A0 mode

(2621 m/s) at a frequency thickness of 300 kHz-mm (see Figure 3 (b)). The antisymmetric mode

maintains a larger amplitude than the S0 mode in the x direction due to the out-of-plane nature

of the excitation nodal displacement. The excitation perturbation was specifically selected in

the through-thickness direction (y) to excite primarily the A0 mode. Hence, the S0 wave pulse

which is generated in the process has a significantly smaller amplitude.

Note, it can be observed from Figure 78 that there are two reflected S0 wave pulses in the

signal response. The first mode, observed at 230 𝜇𝑠, is the reflection produced when the

incident S0 mode encountered the material damage. The amplitude of this wave pulse is

comparatively low due to the out-of-plane excitation producing a low-energy incident S0 mode.

The second S0 wave pulse has a significantly larger amplitude and arrived later at 391 𝜇𝑠. This

is because this S0 mode was produced due to mode conversion when the A0 mode encountered

the material damage. Since this investigation focused primarily on the A0 Lamb mode, further

reference to the reflected S0 mode is specific to the wave pulse produced by mode conversion.

This is discussed in further detail below.

Reflected S0 mode

Reflected A0 mode Reflected S0 mode

Reflected A0 mode

88

Figure 78: Measurement of the x direction nodal displacements improves detection of the incident and reflected symmetric

Lamb modes.

The reflected S0 and A0 modes are indicated in Figure 79. The relative amplitude of the S0

mode is significantly pronounced than that seen in Figure 75. This makes determination of ToA

for the symmetric mode significantly more convenient and accurate using the energy envelope

method.

Figure 79: The energy envelope of the x direction nodal displacement data provides enhanced identification of the

symmetric mode and was used to calculate group velocity.

Figure 80 compares the nodal displacement data recorded in the undamaged plate and the

damaged plate (1.5 mm notch depth). The normalised displacement data are presented in (a)

from which the S0 and A0 wave pulses reflected from the material damage are clearly

distinguishable from the boundary reflections. Comparison with the undamaged plate reveals

Incident A0 mode

Reflected S0 mode

Reflected A0 mode


Incident S0 mode

Reflected S0 mode

Incident A0 mode

Reflected S0 mode

Reflected A0 mode


Incident S0 mode

Reflected S0 mode

89

the nature of the low amplitude wave pulses measured at 335 𝜇𝑠 and 458 𝜇𝑠. The first wave

pulse (1) is the reflection of the incident S0 pulse from the far boundary, while (2) is the second

reflection off the boundary located at the excitation origin. Due to its significantly higher

velocity, the S0 mode reflections occur prior to the arrival of the A0 mode at 663 𝜇𝑠 (3). These

wave peaks are labelled on the energy envelope plot provided in (b).

a) b)

Figure 80: The x direction nodal displacement data was used to distinguish the structural damage reflections from the

boundary reflections (a). The boundary reflections are clearly identified from the undamaged plate (b).

Closeup views of the reflected A0 and S0 wave pulses are provided in Figure 81. Through

measurement of the x nodal displacement the S0 mode amplitude is more clearly discernible

from the signal response. The times of arrival of the reflected S0 and A0 wave pulses were

determined at 386 𝜇𝑠 and 485 𝜇𝑠 respectively using the nodal y displacement data.

Observation of (b) confirms that the same results are yielded from the x nodal data, which

supports the methodology as the two datasets agree.

a) b)

Figure 81: The reflected A0 and S0 wave pulses (a) were distinguished from the boundary noise by comparison of the x

direction signal response with that captured for the undamaged plate (b).

The x and y nodal displacement results are provided in Figure 82 (a) and (b) respectively,

centred around the reflected wave pulses. The reflected S0 and A0 Lamb wave pulses are labelled

in Figure 82 (a). Comparison between the two plots reveals the large change in relative

amplitude of the S0 mode depending on the axis of displacement measured by the analysis. The

S0 mode is almost unable to be discerned in (b) while it is clearly identifiable in (a). The A0

(1)

(2)

(3)

Reflected A0 mode

Reflected S0 mode

Reflected A0 mode

Reflected S0 mode

90

mode has a significant amplitude in both plots as the excitation signal was an out-of-plane

displacement, hence exciting primarily the A0 mode.

Analysis of the plots also reveals the mode conversion between the A0 mode and the S0

mode, which occurred after interaction with the surface notch. Mode conversion occurs when

Lamb waves encounter a discontinuity [2, 5]. The interaction results in scattering of the

longitudinal (P) and shear waves (SV) waves, which interfere constructively and destructively

to form reflected A0 and S0 modes [11]. The mode conversion identifiable within the ANSYS

simulation data is highly consistent with results obtained by published authors (see below).

a) b)

Figure 82: Mode conversion is clearly evident between the antisymmetric and symmetric modes through measurement

of x displacement (a). The A0 amplitude is much greater than that of the S0 mode in the y direction, due to the out-of-plane

perturbation (b).

Alkassar investigated the propagation of the A0 mode and interactions with a surface crack

extending vertically from the surface of the plate [10]. Alkassar published Figure 83 (a) which

shows the captured x direction nodal displacements, showing the mode conversion between the

A0 and the S0 mode after encountering a crack [10]. Comparison between Figure 82 (a) and

Figure 83 (a) yield strong agreement between the behaviour of the Lamb wave after interaction

with damage. Mode conversion was observed in both cases after interaction with damage.

Additionally, the amplitude of the A0 is greater than the S0 mode in the x direction in both cases

as both involved excitation perturbations in the through-thickness direction.

Su discussed the mode conversion that occurs when Lamb waves interact with structural

damage [2]. Figure 83 (b) provides results published by Su, showing the mode conversion

between the S0 and A0 modes after interaction with damage [2]. The mode conversion shows

strong agreement with the results obtained in this investigation.

The agreement between the simulation results and the findings published in literature

provide evidence of the reliability of the data produced in the ANSYS simulation. Mode

conversion was captured after the Lamb wave encountered the notch. To provide a quantitative

validation of the reflected wave pulses, the group velocities were to be determined.

Mode conversion

S0 A0

91

a) b)

Figure 83: Simulation results published by Alkassar (a), capturing the x direction nodal displacement after the A0 Lamb

wave interaction with a vertical surface crack [10]. Results published by Su (b), showing mode conversion between the S0

and A0 modes after interaction with structural damage [2].

To quantitatively validate the reflected S0 and A0 Lamb wave pulses, the group velocities

were calculated using the energy envelope approach for ToA. The nodal separation was 90 mm

and the amplitude threshold for ToA was 90% to remain consistent with previous analyses.

Figure 84 demonstrates how the group velocities of the reflected S0 and A0 wave pulses were

determined. The A0 wave pulse is shown at two locations, (a) 180 mm and (b) 270 mm from

the excitation. The group velocity of the A0 wave pulse is therefore 2647 m/s as shown below.

Similarly, the S0 wave pulse is shown at (c) 180 mm and (d) 270 mm, with the corresponding

group velocity calculated at 5294 m/s. The average wave pulse group velocity was calculated

by averaging the calculated 𝑐𝑔 across all measured mesh nodes. The complete datasets of ToA

and corresponding group velocities are provided in Appendix D (see 15.4).

𝑐𝑔, 𝐴0=

270 − 180

532 − 498𝑚/𝑠 = 2647 𝑚/𝑠 𝑐𝑔, 𝑆0

=270 − 180

414 − 397𝑚/𝑠 = 5294 𝑚/𝑠

a) 𝑇𝑜𝐴 = 532 𝜇𝑠 b) 𝑇𝑜𝐴 = 498 𝜇𝑠

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c) 𝑇𝑜𝐴 = 414 𝜇𝑠 d) 𝑇𝑜𝐴 = 397 𝜇𝑠

Figure 84: The ToA of the reflected S0 and A0 wave pulses were determined, and the corresponding group velocities

were calculated. The range of data was averaged to determine the average group velocity.

The average group velocity of the reflected A0 wave pulse was calculated at 2583 m/s.

Compared with the theoretical wave speed of 2621 m/s, the error in the calculated results is

approximately 1.44%. Note, the incident A0 wave speed was calculated in the undamaged plate

at 2587 m/s. Comparison between incident and reflected waves reveals a slight drop in group

velocity. The scale of the velocity decrease is within the margins of numerical error, but may

also be due to energy losses in the wave pulse after interaction with the notch. This would occur

due to the scattering and reforming of the A0 mode after encountering the damage. The wave

speed may also have decreased due to the effects of attenuation, primarily energy leakage to

the surrounding environment or due to frictional losses. Overall, the reflected A0 mode is highly

consistent with the theoretical wave speed and previously calculated results.

The group velocity of the S0 pulse reflected from the surface notch was calculated at

5321 m/s. The theoretical group velocity at 300 kHz-mm was approximately 5385 m/s. The

percentage error associated with the calculated value was therefore only 1.18%. This low error

suggests the simulated mode conversion of the A0 mode to the S0 mode, which took place after

the incident wave encountered the material damage, was highly accurate. This analysis also

provides evidence of the capabilities of ANSYS in modelling the S0 Lamb mode with a high

degree of accuracy.

The calculated group velocities of the reflected A0 and S0 modes evidences the capabilities

of ANSYS in accurately modelling the interactions between the A0 Lamb wave and material

damage. The simulation accurately modelled the complex mode conversion which occurs due

to wave scattering and reformation after interaction with a discontinuity. The interactions

between the incident Lamb wave and the surface notch, including mode conversion and wave

pulse reflection, were found to be consistent with results published within the literature. The

group velocities of the reflected Lamb waves were found to strongly agree with the theoretical

wave speeds determined from the dispersion curves for aluminium 2024-T6.

Influence of notch depth on the amplitude of reflected Lamb waves

Being able to identify and determine the nature of damage within a component is integral

to applications of NDT in structural health monitoring. To determine if there was a relationship

between notch depth and the nature of the reflected Lamb wave pulses, simulations were carried

out using the various geometries provided in Table 14. The notch depth was varied between

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0.5 mm (16.7% plate thickness) and 2.5 mm (83.3% plate thickness). Both the x and y nodal

displacements were measured to capture the reflected S0 and A0 Lamb wave pulses. The data

was then exported and plotted for subsequent analysis.

The in-plane (x) nodal displacements at 120 mm, 210 mm, 330 mm and 420 mm from the

excitation source are plotted in Figure 85 (a), (b), (c) and (d) respectively. The plots capture the

S0 mode (labelled) produced due to mode conversion when the A0 mode interacted with the

material damage at the various notch depths. At all measured points along the plate, the

amplitude of the reflected S0 mode was found to increase as the notch depth was increased. The

greatest amplitude of the reflected S0 mode occurred for the 2.5 mm notch and the smallest

amplitude occurred for the 0.5 mm notch. The trend is more inconsistent for the A0 mode, which

saw the largest amplitude for a number of notch depths, depending on where the point of

reference was taken. For this reason, the x direction nodal displacement was not considered a

reliable source for establishing a trend for the A0 mode.

a) b)

c) d)

Figure 85: The x direction nodal displacement data indicates that increased notch depth resulted in larger amplitude of

the reflected S0 Lamb wave pulse, measured at (a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm.

The out-of-plane (y) nodal displacements at 120 mm, 210 mm, 330 mm and 420 mm from

the excitation source are plotted in Figure 86 (a), (b), (c) and (d) respectively. Measurement of

the y direction nodal displacement suggests a more consistent trend in the reflected A0 wave

pulse response than the in-plane displacement data. In all measured nodal responses, the 1.5 mm

notch (50% plate depth) was found to cause the largest amplitude of the reflected A0 wave pulse.

S0

A0

S0 A0

S0 A0 S0

A0

94

The 2 mm notch induced the second largest amplitude response, followed by the 1.0 mm and

2.5 mm notches, which produced similar responses. The 0.5 mm notch was found to induce a

reflected A0 Lamb wave with the smallest relative amplitude of all tested notch sizes. Notably,

the trend observed in the x nodal displacement for the S0 mode holds true for the y displacement

data. Increased notch depth consistently resulted in a reflected S0 wave pulse with a larger

relative amplitude.

a) b)

c) d)

Figure 86: The y direction nodal displacement data indicates that a mid-thickness notch depth results in the largest

amplitude of the reflected A0 Lamb wave pulse, measured at (a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm.

The results provided in Figure 85 suggest that the relative depths of multiple surface notches

could be predicted based on the amplitude response of the reflected S0 mode, measured in the x

direction. The trend of the data suggests that increasing notch depth consistently results in a

larger amplitude of the reflected S0 wave pulse. Due to inconsistent trends between x and y

nodal displacements, the x displacement data was found to be unreliable for the A0 mode.

The data presented in Figure 86 suggests that the maximum amplitude of the reflected A0

wave mode occurs for mid-thickness notch depths. The amplitude response was seen to

decrease (on average) as the notch depth tended away from the depth-to-thickness ratio of 0.5.

The out-of-plane nodal displacement data showed a consistent trend in S0 amplitude as was

observed from the in-plane displacement data presented in Figure 85.

A0

S0

A0

S0

A0

S0

A0

S0

95

This investigation highlights the fact that careful attention must be paid when interpreting

the amplitude response of damage-reflected Lamb waves. Depending on the direction of

displacement which is measured, different trends were established. The results suggested that

the amplitude of the S0 mode provides a consistent trend of increasing amplitude with notch

depth when both in-plane and out-of-plane displacement are measured. However, the same

consistency was not seen for the A0 mode, which showed no consistent trend for in-plane

displacement, while out-of-plane displacement suggested a mid-thickness notch depth causes

the largest amplitude response.

The scope of the tested notch depths was limited by time constraints, which meant that the

observations made in this study could not be evidenced more rigorously. There is insufficient

data to conclusively show that the relationship between amplitude and notch depth is generally

true. Hence it is recommended that future works focus on this relationship in order to validate

the observations made in this study. These works may use a more incremental variation in notch

geometry and a range of frequencies to investigate whether the trends observed in this report

hold true in a general sense.

11 Interactions between the A0 mode and a mid-thickness void


The capabilities of ANSYS in modelling the interaction with a notch have been investigated,

with high levels of agreement between literature and analytical results. To extend the relevance

of the conclusions, the model was modified to incorporate a horizontal void at the centre of the

plate. The interaction between the incident A0 Lamb wave and the material damage was to be

analysed and the results compared with the established literature.

Scope of the study

A rectangular void was modelled in the centre of the plate, located at 750 mm from the

excitation source. The thickness of the void was 0.5 mm and the length was varied between

1 mm and 30 mm (see Table 15). Figure 87 shows the horizontal void when the length was

5 mm. The model was meshed with 4-node solid elements and the material properties were that

of aluminium 2024-T6. The characteristic mesh element length remained at 0.25 mm, as this

resolution provided sufficient numerical accuracy whilst also resulting in simulations fast

enough to allow for the numerous test cases. The excitation frequency remained at 100 kHz and

the signal was a 5-cycle sinusoidal tone burst modified with a Hanning window function.

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Figure 87: A horizontal void was modelled in the centre of the plate, with varying lengths ranging between 1 – 30 mm.

Table 15 summarises the scope of the testing, including the geometries of the tested damages

along with their relative sizes compared with the theoretical A0 wavelength at a central

frequency of 100 kHz (15.5 mm).

Table 15: Summary of the tested void lengths.

Void thickness (mm) Void length (mm) Length relative to A0 wavelength (%)

0.5 1 6.5

0.5 1.5 9.7

0.5 2 12.9

0.5 3 19.4

0.5 4 25.8

0.5 5 32.3

0.5 7.5 48.4

0.5 10 64.5

0.5 15 96.8

0.5 20 129.0

0.5 30 193.5

Model results

The simulation results revealed a reflected A0 wave pulse was produced during the

interaction between the incident A0 Lamb wave and the void located in the centre of the

aluminium plate. The reflected antisymmetric mode is clearly visible in Figure 88 which shows

the simulation results for the 5 mm void. Upon closer inspection of the results, there was no

evidence of a reflected S0 wave being produced as a consequence of the interaction between the

incident wave and the material damage. This type of interaction was seen between the incident

Lamb wave and the surface notch. To investigate this observation more rigorously, both the x

and y nodal displacements were exported from the model and the results were analysed in

Python.

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Figure 88: The ANSYS simulation results reveal only the A0 Lamb mode was reflected from the material damage.

Analysis of the nodal displacement data

The normalised in-plane nodal displacement captured during the simulation of the 5 mm

length void is shown by Figure 89 (a) and the corresponding energy envelope of the data in (b).

The nodal data was captured at 210 mm from the excitation source. The displacement data for

the undamaged plate is also graphed to aid in distinguishing the damage-reflected wave pulses

from the incident and boundary-reflected wave pulses. Comparison between the damaged and

undamaged nodal data reveals that only one reflected wave pulse was produced by interaction

with the horizontal crack. This wave pulse was the A0 mode reflected from the material damage,

with a ToA of approximately 510 𝜇𝑠. The S0 wave pulses observed within the simulation data

(occurring at 69 𝜇𝑠, 352 𝜇𝑠, and 442 𝜇𝑠) were observed in both the damaged and undamaged

plates. Hence, these wave pulses were incident and boundary-reflected, and were not

consequences of interactions with the material damage.

a)

Reflected A0 wave pulse Incident A0 wave pulse

98

b)

Figure 89: The normalised x directional nodal displacement data (a) and corresponding energy envelope (b) at 210 mm

reveals a damage-reflected A0 mode. No S0 mode was generated due to interaction with 5 mm long damage.

The out-of-plane nodal displacement data at 210 mm from the excitation source, for the 5 mm

horizontal void, is shown in Figure 90 (a), with the energy envelope of the data presented in

(b). The nodal results clearly show the same reflected A0 Lamb wave at 510 𝜇𝑠 as seen in the x

nodal data. The wave pulse was produced as a consequence of interactions between the incident

A0 Lamb wave and the horizontal void located at 750 mm.

Notably, all tested void lengths produced similar displacement results, with only the A0 mode

being reflected from the material damage. The group velocity of the reflected Lamb wave was

calculated by measuring the ToA at various points along the plate and using the separation

distance between nodes to determine 𝑐𝑔. The average group velocity of the reflected A0 Lamb

wave due to interaction with the 5 mm void was 2571 m/s, thus a 1.89% error compared with

the theoretical wave speed. The high level of agreement between the reflected Lamb wave

group velocity and the analytical wave speed validates the accuracy of the simulation in a

quantitative sense. The reflected wave pulse produced due to interactions between the A0 Lamb

wave and the horizontal void was accurately simulated by the ANSYS Explicit Dynamics

solver, with a physically realistic wave speed consistent with analytical results.

a)

99

b)

Figure 90: The normalised y directional nodal displacement data (a) and corresponding energy envelope (b) at 210 mm

clearly shows the reflected A0 mode from the 5 mm long centrally located void.

Figure 91 provides a closeup view of the x direction nodal displacement data captured at

210 mm from the excitation source. The nodal displacement data (a) and energy envelope (b)

reveals the A0 mode reflected from the 5 mm void, and the S0 modes appearing in both the

damaged and undamaged plate. It is clear from the results that no significant mode conversion

took place between the A0 mode and S0 mode due to interaction with the material damage. This

is because the void was located in the centre of the plate, at which the shear stress is zero. As

discussed in the literature, Guo and Cawley determined the S0 mode was not capable of

detecting delaminations at through-thickness locations with zero shear stress [9]. Figure 92

presents the distribution of shear stress through the thickness of the 3 mm plate, generated using

the software package Disperse. The plot shows that the shear stress is minimized at the centre

and surfaces of the plate. Thus, as the void was located at the mid-thickness of the plate, the

shear stress at the damage was zero. This explains why the S0 mode was not produced as a

consequence of the interaction of the wave pulse with the damage. The simulation results

therefore agree with the established literature and evidence the capabilities of ANSYS Explicit

Dynamics in accurately modelling interactions between Lamb waves and material damage.

a) b)

Figure 91: In-plane (x) displacement at 240 mm reveals no mode conversion between the A0 and S0 modes as a result of

interaction with the 5 mm void. This is because the void is located in the centre of the plate where the shear stress is zero.

100

Figure 92: Stress data generated using the software Disperse shows the shear stress distribution across the thickness of the

plate, with the shear stress being zero at the centre.

A closeup view of the out-of-plane nodal displacement data is shown in Figure 93 centred

around the reflected A0 Lamb wave. Here it can be seen that the presence of the horizontal void

in the mid-plane of the plate resulted in a reflection of the A0 mode, with no significant mode

conversion to the S0 mode. The findings suggest that detection of a reflected A0 pulse, when the

incident Lamb wave is the A0 mode, may be used to characterise material damage occurring in

the midplane of the plate.

a) b)

Figure 93: Out-of-plane (y) displacement at 240 mm shows the A0 wave pulse reflected off the 5 mm horizontal void.

The wave pulse has an approximate ToA of 500 𝜇𝑠.

Influence of void length on the amplitude of the reflected Lamb wave

The length of the void was varied between 1 mm and 30 mm to investigate the influence of

damage length on the received A0 Lamb wave. The full range of damage geometries are

provided in Table 15. The results were graphed in two distinct datasets, those with a damage

length of 1 – 5 mm and those with damage length of 5 – 30 mm. The differences between these

datasets is discussed in the analysis below.

The out-of-plane nodal displacement data was captured at four locations along the plate;

120 mm, 210 mm, 330 mm, 420 mm from the excitation source. The energy distribution of each

101

amplitude response was then calculated, and the results were plotted in Figure 94 (a), (b), (c)

and (d). Figure 94 provides the amplitude response captured for the models containing void

lengths of 1 – 5 mm, centred at the damage-reflected A0 Lamb wave. A consistent trend was

observed across all captured mesh nodes (including those presented in Figure 94). It is clear

from the energy distribution results that increasing the length of the horizontal void resulted in

increasing amplitude of the reflected A0 Lamb wave. The largest relative amplitude of the

damage-reflected Lamb wave was seen in the 5 mm damage, while the smallest relative

amplitude was seen for the 1 mm void. As can be seen from the plots, the time of arrival of the

reflected A0 Lamb wave was consistent across void lengths of 1 – 5 mm.

a) b)

c) d)

Figure 94: The y direction nodal displacement data indicates that increased void length (up to 5 mm) resulted in larger

amplitude of the reflected A0 Lamb wave pulse, measured at (a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm.

The energy distribution results for void lengths of 5 – 30 mm are provided in Figure 95 at

(a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm from the excitation source. The

aforementioned trend between void length and the amplitude of the reflected A0 mode was not

discernible within this range. In this case, the largest amplitude occurred for the 5 mm void,

followed by the 15 mm void, while the 7.5 mm and 20 mm voids showed similar amplitude

response. Finally, the 10 mm and 30 mm void lengths resulted in the smallest relative amplitude

responses, with the latter case producing almost no discernible reflected wave pulse.

A0 A0

A0 A0

102

a) b)

c) d)

Figure 95: The y direction nodal displacement data reveals the relationship breaks down at void lengths larger than 5 mm,

measured at (a) 120 mm, (b) 210 mm, (c) 330 mm, (d) 420 mm. No discernible trend was identified between amplitude and

void lengths from 5 – 30 mm.

The relationship between void length and the reflected A0 Lamb wave amplitude was found

to be variable across the range of tested geometries (1 – 30 mm). However, a clear trend was

observed within the range of 1 – 5 mm, with increasing length resulting in a larger amplitude

of the reflected A0 Lamb wave. The results of the testing suggest that the amplitude of the

reflected A0 wave could be used to characterise the size of the material damage, within the range

of 1 - 5 mm (when the excitation frequency is 100 kHz). It is hypothesised that the ratio between

wavelength and void length may be a factor in the stability of this trend, as the relationship was

observed to break down when the damage size exceeded 32.3% of the wavelength. When the

length of the void increased above 32.3% of the central excitation wavelength (greater than

5 mm), the amplitude of the reflected A0 Lamb wave did not show any discernible relationship

with damage size.

Due to time constraints, the scope of the tested void length range was relatively limited.

There were insufficient data points collected within this study to conclusively validate the

trends observed between void length and wavelength. Hence, it is recommended that further

studies are carried out using a more sensitive incremental variation in notch length to validate

the hypothesis. The testing should involve a range of frequencies to confirm if the relationship

between void length and amplitude holds true across the frequency spectrum.

A0

A0

A0 A0

103

12 Recommendations for further works

ANSYS Explicit Dynamics was proven highly capable of modelling the A0 Lamb mode in

aluminium 2024-T6. The simulation results were validated against theoretical wave properties

and ANSYS was shown to be a viable tool for both 2D and 3D analyses. ANSYS is therefore

recommended in future works which focus on Lamb wave-based SHM systems. The

methodology outlined in this report could be used as a reference for simulating the A0 mode in

ANSYS. Through providing a validated methodology for simulating the A0 mode in ANSYS,

this thesis has opened a range of further works to explore. The topics discussed below are

mainly based on elements which were either out of the scope of this investigation due to time

constraints, or aim to extend upon particular observations made throughout the investigation.

Structural health monitoring systems are highly attractive for composite materials as they

exhibit more complicated damage modes than conventional materials. A valuable extension to

the work carried out in this investigation would be the development of a 3D composite laminate

in ANSYS. The propagation and scattering characteristics of the A0 Lamb wave could then be

explored. Due to time and resource constraints the scope of this investigation was limited to

isotropic material properties. However, the behaviour of Lamb waves in anisotropic materials

such as composites can be significantly more complex. Hence, this research extension would

provide further validation of the capabilities of ANSYS in modelling more complex behaviour.

Development of a viable numerical tool for modelling the propagation of Lamb waves in a

composite material could be highly useful in design of SHM systems. Ng and Veidt modelled

the A0 mode in a carbon-epoxy laminate using LS-DYNA [8]. This research is recommended

as a starting point for future investigations carried out in ANSYS Explicit Dynamics.

Figure 96: Ng and Veidt modelled the propagation of the A0 mode in a composite laminate using ANSYS LS-DYNA.

Development of a 3D composite model in ANSYS Explicit Dynamics could provide a viable tool for SHM system design [8].

This investigation involved the interaction between the A0 mode and a surface notch. The

amplitude of the reflected S0 wave increased with notch depth, while a mid-thickness notch

resulted in the largest amplitude of the reflected A0 mode. However, it was noted that the two

displacement directions (x and y) indicated different trends for the A0 mode. This behaviour

couldn’t be investigated more closely due to time limitations. Further investigation using a

much wider scope is required to validate the observations. It is recommended that future works

focus on this interaction using a more sensitive approach to notch depth variation and a variety

of notch geometries, plate thicknesses and material properties.

104

The interaction between the A0 mode and a horizontal void was also explored in this thesis.

The amplitude of the reflected A0 mode increased consistently for void lengths between

1 – 5 mm, while there was no discernible trend for void lengths between 5 – 30 mm. This trend

couldn’t be investigated more closely due to time limitations. Hence, it is recommended that

future works also focus on this interaction using a more sensitive approach to void length

variation and over a range of void thicknesses. Additionally, the through-thickness location of

the void could be varied to validate the conclusion behind the pure A0 reflection from the void.

The influence of spatial resolution on the obtained wavenumber frequency domain was

investigated. Above 1 node-per-wavelength, no discernible trend was observed between spatial

resolution and accuracy of the obtained phase velocity. It was hypothesised that this was due to

the limited scope of the tested resolution range (0.39 – 6.2 nodes-per-wavelength). It is therefore

recommended that this behaviour is investigated more closely to validate this hypothesis. The

range of tested spatial resolutions should cover a much wider scope and the analysis should

involve a range of frequencies to ensure the conclusions are rigorous.

One of the key difficulties during signal processing was distinguishing the incident Lamb

wave from the boundary reflections. This was particularly challenging in the 3D model where

scattering occurred from the side boundaries and the far edge of the plate. One useful extension

to the project would be the incorporation of a perfectly matched layer (PML) boundary. A PML

boundary is an absorbing layer which dissipates the energy of the incident Lamb wave, hence

preventing the occurrence of reflected waves [41]. This is achieved through successive layers

of matching impedance which result in no reflections between the adjacent layers [41]. This is

a technique often used to model the open boundaries of a particular section of a larger structure.

The main advantage of implementing PML in an FEM model is that complex boundary

reflections, particularly those arising due to the presence of structural damage, are not seen the

signal response. This allows smaller FE models to replace the larger models which are typically

necessary to distinguish the incident wave pulse from its reflections.

Shen discussed a methodology for implementing non-reflective boundaries in an ANSYS

finite element model of Lamb wave propagation [41]. The methodology involved an absorbing

boundary which takes advantage of the multiple reflections of the P and S components of the

Lamb wave [41]. The incident Lamb wave is absorbed through the multiple absorptions of the

individual P and S components [41]. The interaction between the P and S waves with the

absorbing boundary is shown in Figure 97 below [41]. This research could be used as a

reference for future works focused on implementing a PML in ANSYS.

Figure 97: Shen discussed a methodology for modelling non-reflective boundaries in ANSYS [41].

105

Finally, it would be highly useful to investigate and compare the advantages and

disadvantages of the many element types available in ANSYS. In this investigation, PLANE42

and SOLID45 elements were used in the 2D and 3D analyses respectively. These were selected

based on their use in prior research, however it would be valuable to compare the accuracy and

computational efficiency of other element types in ANSYS. This could be beneficial in future

research to aid in element selection, and to ensure that simulations are performed efficiently

and accurately.

13 Conclusion

The aim of the thesis was to investigate the simulation capabilities of ANSYS in modelling

the fundamental antisymmetric Lamb wave. The primary motivation was to deliver a validated

methodology for modelling the A0 mode in ANSYS, which would form a baseline for future

research topics. The results obtained from the ANSYS simulations were to be validated against

analytical solutions to evidence the viability of the software in modelling the A0 mode.

A two-dimensional aluminium 2024-T6 plate was modelled in ANSYS Explicit Dynamics.

An out-of-plane nodal displacement constraint was applied to the surface of the plate to activate

the A0 mode via a 5-cycle 100 kHz sinusoidal tone burst, modulated by a Hanning window

function. The frequency of the excitation was selected at 100 kHz to ensure that only the

fundamental Lamb modes were activated. The 2D finite element analysis assumed plane strain

conditions and the model was meshed using 4-node solid elements. Mesh element length was

varied between 0.15 mm – 1.50 mm to investigate the influence of characteristic element length

on the simulation results. The accuracy of the numerical model was quantified through

comparison of the incident wave pulse group velocity and phase velocity to analytical solutions.

Through qualitative analysis of the results it was found that wave dispersion was accurately

captured in the ANSYS simulation of the A0 mode. Wave-pulse widening was observed as the

Lamb wave propagated along the plate as a consequence of the dispersive relationship between

frequency and velocity. Two methodologies for calculation of group velocity were explored; a

reference-amplitude approach for ToA and an energy-distribution approach for ToA. The

reference-amplitude approach was highly sensitive to the threshold at which time of arrival was

defined. This resulted in calculation of high group velocities when the amplitude threshold was

low, due to the amplitude response of high frequency components being measured. Through a

sensitivity analysis it was shown that the energy-distribution of the wave pulse provides a

significantly more reliable methodology for determination of wave pulse group velocity.

The energy-distribution approach for ToA was used to calculate group velocity of the

incident A0 Lamb mode in each FE model. The simulation with the least numerical error was

obtained using a 0.15 mm mesh element length at only 0.13% error. The maximum error was

obtained in 1.5 mm model at 5.64%, which was relatively low considering the plate was meshed

with only 2 elements in the thickness direction. Accurate results were obtained in all the tested

FE models, which provided evidence of the viability of ANSYS for simulation of Lamb waves.

Simulation accuracy improved as characteristic element length was reduced, which was

consistent with expected outcomes.

The phase velocity of the incident Lamb wave was calculated using the 2D Fast Fourier

Transform. Separation distance between mesh nodes was varied from 2.5 mm – 40 mm to

106

investigate the influence of spatial resolution on the wavenumber-frequency domain. Phase

velocity was calculated at 1552.9 m/s, which corresponded to only 0.19% error compared with

the theoretical value. It was therefore concluded that ANSYS accurately captured the phase

properties of the propagating A0 Lamb wave. It was established that at least 1 node-per-

wavelength is required to obtain physical results in the wavenumber-frequency domain.

However, no discernible trend between spatial resolution and model accuracy was observed

above this minimum threshold. It was hypothesised that the range of tested spatial resolutions

was too narrow to largely affect the numerical analysis.

A three-dimensional model of the aluminium plate was developed using 8-node brick

elements with characteristic mesh element length of 1 mm. The A0 mode showed excellent

agreement with both the 2D and analytical results. Group velocity of the incident Lamb wave

was calculated at 2503 m/s, which showed excellent agreement with the analytical figure at just

3.47% error. Considering good accuracy was obtained using a relatively coarse mesh, the results

indicated ANSYS is highly capable in modelling three-dimensional propagation of the A0 mode.

The rigorousness of the 3D model across the low frequency regime was then explored by

varying f0 from 25 kHz – 400 kHz. An experimental dispersion curve was generated by plotting

the A0 mode group velocity determined at each frequency. There was excellent agreement

between the simulation and analytical dispersion curves and numerical error was consistent

across the frequency range. The results of the analysis indicated that the spatial criterion of 10

– 20 nodes-per-wavelength is not critical for simulation of the A0 mode, with accurate data

obtained using just 6 nodes-per-wavelength. ANSYS was shown to be rigorous across the low-

frequency regime and was therefore recommended as a viable tool for future work involving

three-dimensional propagation of the A0 mode within the tested frequency range.

A surface notch was modelled in the 2D aluminium plate. ANSYS accurately captured mode

conversion between the A0 and S0 modes when the incident wave pulse encountered the damage.

The observed mode conversion showed excellent agreement with results published in the

literature. The errors in group velocity of the reflected S0 and A0 modes were calculated at just

1.18% and 1.44% respectively. The results of this analysis evidenced the capabilities of ANSYS

in accurately modelling interactions of the A0 mode with material damage. Notch depth was

varied from 0.5 – 2.5 mm to investigate its effect on the amplitude response of the reflected

wave modes. Amplitude response of the reflected S0 mode increased consistently with notch

depth, while a mid-thickness notch caused the largest amplitude of the reflected A0 mode.

A mid-thickness void was modelled in the 2D plate. Only the A0 Lamb mode was reflected

when the incident wave pulse encountered the void. This was primarily attributed to the zero

shear stress through-thickness location of the damage. Void length was varied from 1 – 30 mm

to investigate its influence on the amplitude of the reflected A0 mode. Amplitude of the reflected

wave increased consistently with void lengths between 1 – 5 mm. However, no discernible

relationship between void length and amplitude response was identified for voids between 5 –

30 mm. It was recommended that future works focus on this trend to provide more conclusive

evidence of the relationship between void length and amplitude response.

The objectives of the thesis were achieved in this investigation. ANSYS Explicit Dynamics

was proven highly capable in simulating the A0 Lamb mode and its interactions with material

107

damage. This report delivered a detailed methodology for simulating the A0 mode in ANSYS

and it is recommended as a baseline reference for further works. ANSYS Explicit Dynamics is

recommended as a viable and accurate numerical tool for modelling the propagation of the A0

mode in future research projects focused on SHM design and the propagation of Lamb waves.

108

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111

15 Appendices

Appendix A: Signal processing Python code

Signal processing Python code used to analyse the raw data and calculate incident wave

pulse group velocities.

120

Appendix B: Python implementation of the 2D FFT

The Python implementation of the 2D FFT is provided in the code below.

125

Appendix C: Group velocity results for the 3D simulation

The calculated group velocities across the length of the 3D plate are presented in Figure 98

(a) along with the error relative to the theoretical value of 2621 m/s in (b). The 2D simulation

results are also presented for comparison. It can be seen that there is good overall agreement

between the two models, however the 3D FE model sees an increasing trend in error across the

plate. This resulted in a slightly higher overall numerical error for the 3D model compared with

the 2D model.

a) b)

Figure 98: The group velocity calculated across the 3D FE model shows good agreement with the 2D model (a), however

the numerical error showed an increasing trend across the length of the plate (b).

Appendix D: Reflected wave pulse group velocities

The group velocities of the A0 mode reflected from the surface notch are provided in Table

16. The group velocities were averaged to arrive at the final figure of 2583 m/s.

Table 16: Calculated group velocities of the reflected A0 mode from the surface notch.

x t1 t2 tavg cg

90 566 571 568

120 571 559 557 2542

150 555 549 543 2381

180 559 538 532 2483

210 537 525 522 2601

240 549 515 508 2542

270 526 502 498 2643

300 538 488 485 2446

330 519 479 475 2719

360 525 466 464 2612

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390 500 464 453 2839

420 515 444 440 2626

450 494 433 429 2628

480 502 424 418 2517

Average 2583

The group velocities of the S0 mode reflected from the surface notch are provided in Table

17. The group velocities were averaged to arrive at the final figure of 5321 m/s.

Table 17: Calculated group velocities of the reflected S0 mode from the surface notch.

x t1 t2 tavg cg

90 421 442 431

120 423 427 425 4878

150 417 422 419 5106

180 411 416 414 5187

210 406 411 408 5389

240 400 405 403 5357

270 395 400 397 5422

300 389 394 392 5397

330 384 388 386 5455

360 378 383 380 5389

390 373 377 375 5414

420 367 372 369 5389

450 361 366 364 5405

480 356 361 358 5389

Average 5321

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