LEARNING PARADIGMS F
OR THE 
IDENTIFICATION OF EL
ASTIC PROPERTIES 
OF COMPOSITES USING 
ULTRASONIC 
GUIDED WAVES
 
 
By
 
 
Karthik Gopalakrishnan
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
A THESIS
 
 
Submitted to 
Michigan State University
 
in partial fulfillment of the requirements 
for the 
degree of
 
 
Electrical Engineering

Master of Science
 
202
0
 
 
          
ABSTRACT
 
LEARNING PARADIGMS FOR THE
 
IDENTIFICATION
 
OF ELASTIC 
PROPERTIES OF COMPOS
ITES USING ULTRASONI
C GUIDED WAVES
 
 
 
By
 
Karthik Gopalakrishnan
 
Identification of elastic properties of composite
s
 
is
 
relevant for both 
n
ondestructive material
s
 
characterization as well as for in
-
situ condition monitoring to 
assess and 
predict an
y possible 
material degradation
. Learning 
p
aradigms have been well explored when it comes to detect
ion
 
and characteriz
ation of
 
defects in 
safety
-
critical
 
structure
s
, but are relatively unexplored when 
it comes to 
structural 
material
s
 
characterization. In this 
thesis
 
we propose a learning paradigm 
that includes the potential use of Machine Learning
 
(ML)
 
and Deep Learning
 
(DL)
 
algor
ithms 
to solve the inverse problem
 
of material propert
ies
 
identification using ultrasonic guided 
waves. The propagation of guided waves in a composite laminate is modelled using two 
different 
modelling techniques 
as part of the forward problem. 
Here, w
e u
se the two 
fundamental modes of guided wave
s,
 
i.e. the anti
-
symmetric (A0) and the symmetric modes 
(S0) as features for the 
proposed 
learning models. As p
art of the inverse problem, different 
learning models 
are used to map feature space to target space th
at consists of the material 
properties of composite
s
. The performance of the algorithms is evaluated based on different 
metrics and it is seen that the networks are able to learn the mapping and generalize well to 
unseen examples even in the presence of no
ise at various levels. Overall, we are able to 
develop a complete framework consisting of many interlinking data processing algorithms 
that can effectively estimate and predict the material pr
operties of any given composite.
 
iii
 
 
ACKNOWLEDGEMENTS
 
 
 
I would like
 
to first acknowledge my advisor, Dr Yiming Deng, for his unwavering support 
and guidance throughout this journey. The amount of freedom he gave me in pursuing 
research in general was fantastic, and I have been able to grow my scientific temper greatly 
wor
king under him. His work ethic and approach to science inspires me and will certainly 
guide me in the future. I would also like to place my deepest thanks to the members of my 
committee: Dr Lalita Udpa and Dr Mi Zhang for always being available and guiding
 
me 
throughout this journey. In particular, Dr Lalita Udpa has been of immense help to me in the 
last couple of years. Apart from the invaluable technical inputs that I have received from her, 
she has always gone out of her way looking out for me and makin
g sure I am doing well. I 
really appreciate and am grateful for the opportunity of working and interacting with her on a 
daily basis.
 
       
I would also to thank all the members of the Nondestructive Evaluation Laboratory 
(NDEL). Having a large research g
roup at NDEL has been very beneficial and has led to 
many productive and fun conversations. In particular, I would like to thank Srijan, Subrata, 
Bharath, Vivian, Guillermo and Vivek for the banter, and the various conversations we have 
had ranging from fo
otball, food and what not. Without them, going to the lab would have 
become routine and mundane. I would also like thank all the faculty members associated with 
the NDEL who have in some way or other enriched my experience here.
 
       
When I initially mov
ed to Michigan State University, I had the opportunity to work with 
Dr Sunil Kishore Chakrapani for a year. I would like to express my deepest of gratitude 
towards him for placing his trust on me when I was just a fresh undergraduate from India. I 
learnt a
 
great deal in conducting rigorous experimental approach under him, and also learnt a 
lot on writing good scientific documents. I am inspired by his approach to science and 
 
iv
 
 
research in general, and I will always carry his teachings and his advices with me.
 
I also want 
to take this opportunity to thank Dr Shankar Balasubramaniam for always being available and 
giving me some valuable advice on choosing my career 
path
.
 
       
I have had the pleasure of being a teaching instructor for two different course
s during my 
time here. I would like thank Dr Gregory Wierzba and Dr Sunil Kishore Chakrapani for being 
wonderful supervisors, and always letting me perform my duties without a lot of interference, 
but at the same time always being available in case of any 
trouble. I also took many 
interesting courses during my time here, and I would like to thank all the teachers who 
thought the courses for their efforts and endeavors.
 
       
Also, big thanks to all the staff at the Dept. of Electrical and Computer Engineer
ing. In 
particular, I am eternally grateful to Meagan, I have troubled her multiple times with multiple 
different requests and she has always solved it for me in a very timely manner and with a 
smile on her face. I also want to thank Laurene, for always pr
ocessing my time sheets on time 
even if I submit it a day late. Big thanks to Brian, for always being available for any technical 
help or requests that I have had over the last two years. People like them are the unsung 
heroes of every department, and a lo

and services.
 
      
Moving on to the academic collaborators, I would like to first thank Magna International 
Inc. I was fortunate to work on a very interesting project funded by them, where I learnt
 
the 
basics of adhesion and adhesion bonding in materials. I also worked on a project funded by 
the US Department of Transportation, where I worked on evaluating the integrity of oil and 
gas pipelines. As part of this work, I also had the opportunity to pr
esent my work in an R&D 
seminar held by them. I would like to thank them for providing me with such an amazing 
opportunity. Finally, I would like to place my biggest thanks to Mr Mahindra Rautela, who 
has greatly helped me with my work that has eventually 
become my dissertation. A lot of the 
 
v
 
 
work that I have presented in my thesis is in collaboration with him, and I have enjoyed 
working with him all through. His drive, passion and zeal towards research and technology 
are something that I am inspired by and 

 
       
Before moving here, I have interacted with lot of different people that have helped me 
shape my career. In particular, I would like to thank Dr G K Ananthsuresh, and Dr Suresh 
Sundaram for taking me under their
 
wing at IISc and NTU respectively, when I was just a 
curious undergraduate trying to understand what research was all about. I also want to thank 
Dr Senthil Jayavelu, who helped me with a lot of my work for my undergraduate thesis that 
really helped me de
velop a firm interest in the area of Deep Learning. I also want to thank all 
my teachers at the Amrita University, Bangalore Campus from where I obtained my 
undergraduate degree.
 
       
Moving outside the academic sphere, I have had the pleasure of meeting
 
some wonderful 
people who have all been a big part of my graduate life. Cricket is a big part of who I am, and 
I have had the opportunity of playing a good level of cricket at Michigan. I would like to 
thank the entire cricket family here at Michigan for 
giving me an opportunity to have 
something else to look forward to in life. I also want to thank the entire 
Desi Boys
 
group: Sai 
Chaintanya, Ankit, Hitesh, Ronak, Bharath, Yash, Abhiroop, Siddartth and Abu Bakr for 
their company through the last two years.
 
A lot of my weekends went by chilling with them 
over food and drinks, or by watching random movies or playing board games.
 
       
Moving to MSU, I was able to meet one of my best friends in my life, Sejal. I am 
thankful for all the amazing times we have s
pent together. From watching movies, cooking 
meals to playing and fighting over board games, I want to thank you for your companionship 
and for always being there no matter what. I would also like to thank Manali, who along with 
Sejal accompanied me on mul
tiple trips all across Michigan. I also want to thank all my 
friends back home in India, who have all indirectly been a part of this wonderful journey.
 
 
vi
 
 
       
Finally, but most importantly I am eternally grateful to my parents, Gopalakrishnan and 
Anuradha.
 
Your support has been unwavering, and I am who I am because of you and your 
nonstop consistent support. I am very fortunate to have a dad who is himself a revered 
professor. I want to thank him for always taking out time to review my presentations, report
s, 
papers and always giving me feedback to improve it. I want to thank my mom for always 
willingly taking in all my rants and being a big emotional support. I would like to thank my 
sister Keerthana, for always being the sibling whose leg I could always pu
ll. Nothing would 
have been possible if not for you guys. Thanks for everything!
 
 
 
 
vii
 
 
TABLE OF CONTENTS
 
 



.
.
 
ix
 

..


.

 
x
 

 
1.1.
 
Over

 
 
1.1.1.
 
Basic Elements
 


 
1.1.2.
 

 
1.2.
 

 
1.3.
 


 
1.4.
 

 
1.5.
 
Mat
erial Property Identification of
 

 
1.6.
 

 
1.7.
 

 


 
2.1.
 


19
 
 
2.2.
 


 
21
 

.
. 25
 
3.1.
 


 
25
 
 
3.2.
 


 
26
 
3.3.
 

 
3.4.
 

 
3.5.
 



 
CHAPTER 4
 

 
4.1.
 

 
 
4.2.
 
Multi
-

 
4.3.

 
4.4.
 
Recurrent Neural Netwo

 
4.5.
 

 
CHAPTER 5
 
MODELLING TECHNIQUES FOR GUIDED WAVES IN COMPOSITES. 53
 
5.1.
 

 
 
5.2.
 


 
5.3.
 
FEM Modelling using COMSOL
®
 

 
5.4.
 

 
5.5.
 
Non
-

 
5.6.
 
Spectral Element Formulation 

 
5.7.

 
 
 
CHAPTER 6
 

 
6.1.
 

 
 
6.2.
 


 
 
viii
 
 
CHAPTER 7 STUDY PARAMETERS FOR IDENTIFICATION OF ELASTIC 

 
7.1.
 

 
 
7.2.
 


 

 
8.1.
 

 
 
8.2.
 

 
8.3.
Training Results with Dense Neural Netwo

 

 
9.1.
 

 
9.2.
 

 
9.3.
 

 
APPENDIC
ES

 
106
 


 


 

 

  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ix
 
 
LIST OF TABLES
 
 
Table 5.1
 
Material Properties of a transversely isotropic lamina

 
Table 5.2
 
Wave velocities: SFEM vs FEM

..

 
Table 6.1 Secant Sensitivity Values

 
Table 7.1 Material Properties of the composites 
used for the ML based study

 
Table 8.1
 
1DCNN Architecture

 
Table 8.2
 
Prediction Results from the 1DCNN

 
Table 8.3
 
LSTM Architecture

 
Table 8.4
 
DNN (MLP) Architecture


 
Table 8.5
 
P
rediction Results using DNN

 
 
 
 
 
 
 
 
x
 
 
LIST OF FIGURES
 
 
 
Figure 1.1 Elements of Structural Health Monitoring

 
   
4
 
Figure 1.2 (a) Forward Process (b) Inverse Process

.

    
7
 
Fi
g
ure
 
1.3
 
Ultrasonic Guided Wave re
sponse for a healthy pipeline v
s 
a faulty pipeline with a
 
single 
1mm deep corrosion pit

.




.
 
 
 
9
 
Figure 1.4
 
Interdependence of the forward and inverse process

.

     
10
 
Figure 1.5 Active SHM Setup o
f a CFRP Composite Plate

     
11
 
Figure 1.6: Overall Workflow of the Composite Property Estimation Framework

     
17
 
Figure 2.1 (a) RV for longitudinal material property determination (b) RV for transverse 
material property determination


...
 
 
2
0
 
Figure 2.2
 
Co
-
ordinate system for a laminate

.


      
2
1
 
Figure 2.3
 
Principal axes of the laminate and global x
-
y axes

.

..
.
    
2
3
 
Figure 2.4
 
An 8 layered unidirectional composite laminate

.

.
.
  
  
  
 
24
 
Figure 3.1 Bulk 
Wave Testing vs Guided Wave Testing

    
26
 
Figure 3.2 2D infinite plate of thickness 2d

 
     
3
0
 
Figure 3.3 Phase Velocity Dispersion Plot for 1mm thick Steel Plate

 
  
 
   
33
 
Figure 3.4 Group Velocity Dispersion Plot for 1mm thick 
Steel Plate

    
 
3
3
 
Figure 3.5
 
Elementary laminate Composite Waveguide with co
-
ordinate system

..
 
 
  
3
5
 
Figure 4.1
 
Machine Learning and Deep Learning

.

 
 
 
3
9
 
Figure 4.2
 
Pipeline of a Neuron

.

 
  
40
 
Figure 4.3
 
Three 
Layered MLP Network

.

     
4
1
 
Figure 4.4
 
Overfitting and Underfitting
 
[31]

.

.
    
4
1
 
Figure 4.5
 
Example of Data Augmentation
 
[32]

.

.

     
45
 
Figure 4.6
 
Working of a Convolutional Layer

.

    
 
46
 
Figu
re 4.7
 
Example of weight sharing in Convolutional Layers

.

  
     
4
7
 
 
xi
 
 
Figure 4.8
 
Working of a Max Pooling Layer (stride=2)

.

  
 
 
4
8
 
Figure 4.9 (a) Simple Feed Forward Network (b) A RN
N with previous hidden states a
s 
input


 
   
49
 
 
Figure 4.10 Architecture of a simple LSTM Network
 
[48]

   
 
51
 
Figure 4.11 Vanilla based LSTM Cell
 
[48]

.

  
 
 
52
 
Figure 5.1
 
(a) Overall domain of the problem (b) Discretized domain set of the problem with 
boundary co
nditions


    
  
54
 
Figure 5.2
 
Finite Element 
Method Procedure [34]

.

.

    
56
 
Figure 5.3 Cross sectional view of the sample geometry

 
   
5
7
 
Figure 5.4 (a) Time domain representation of the input tone burst (b) Frequ
ency domain 
representation of the input tone burst

     
58
 
Figure 5.5 (a) Finite Element Mesh for the composite laminate

    
59
 
Figure 5.6 (a) Received response at 500 mm from the actuator (b) Filtered received response 
to isolate
 
the fundamental wave modes

  
  
59
 
Figure 5.7
 
SFEM Procedure of solving PDE

  
 
6
1
 
Figure 5.8
 
Spectral Isotropic Rod Element [19]

.



.
    
62
 
Figure 5.9 
(a) and (b):
 
A0 and S0
 
response using the SFEM method
 
(c)
 
Overall 
raw 
 
waveform obtained from the COMSOL FEM simulation (d) Filtered waveform isolating the 
i
ndividual A0 and S0 reflections
 
(filtering done using knowledge of group 
velocities
 
of these
    
 
   
two modes
)




.
.
  
 
67
 
 
Figure 6.
1
 
A0 wa
veforms for two different sets of material properties (
Blue
: 

= 1000 kg/m
3
, 
E
1
= 50 GPa, 
E
2
= 5 GPa, 
v
12
= 0.25, 
v
23
= 0.25,
G
12
= 2 GPa,
 
Orange:
 

= 1750 kg/m
3
, 
E
1
= 
111.53 GPa, 
E
2
= 5 GPa, 
v
12
= 0.25, 
v
23
= 0.25, 
G
12
= 2 GPa)

    
70
 
 
Figure 6.2
 
Sensitivity of density (rho) on A0 and S0 wave velocities

.

     
72
 
Figure 6.3
 

 
(b) 

S0 wave velocities


       
73
 
 
Figure 6.4
 

v
12
 
and 
v
23
 
on A0 and S0 wave velocities (b) 
Sen
sitivity of Shear Modulus 
G
12
 
on A0 and S0 wave velocities

     
73
 
Figure 6.5
 
Uniqueness check for a dataset that had 512 samples originally but trimmed down 
to 88 after the check



.
 
 
 
75
 
Figure 7.1
 
Group Velocity Dispersion 
Curve for a random composite with three modes (A0
, 
SH0
 
and S0) highlighted


.
   
78
 
Figure 
8
.1
 
Research Flow for the two Deep Learning Models (1DCNN and LSTM)

...
.
.
..
.
    
7
9
 
 
xii
 
 
Figure 
8
.2 
Loss
 
curve: MSE vs number of epochs for 1DCNN

.

.


    
8
1
 
Figure 
8
.3
 
MAE curve: MAE vs number of epochs for 1DCNN

.

  
 
 
 
82
 
Figure 
8
.4
 
R
2 
coefficient curve: R
2 
coefficient vs number of epochs for 1DCNN

.

    
82
 
Figure 
8
.5
 
Prediction Results: True vs Predicted Plot for 1DCNN

.

       
83
 
Figure 
8
.
6
 
(a) Original Signal (b) Original signal corrupted with a noise of SNR5

.

      
8
4
 
Figure 
8
.7
 
Prediction results on data with noise of SNR20 for 1DCNN

.

       
85
 
Figure 
8
.
8
 
Prediction results on data with noise of SNR10 for 1DCNN

.

       
86
 
Figure 
8
.
9
 
Prediction results on data with noise of SNR5 for 1DCNN

.

       
86
 
Figure 
8
.10 NOWP for the predictions on noisy and noiseless data
 
for 1DCNN

.

     
87
 
Figure 
8
.11
 
Loss curve: MSE vs number of epochs for LSTM Figure

.

    
89
 
 
Figure 
8
.12
 
MAE curve:
 
MAE vs number of epochs for LSTM

.

      
89
 
Figure 
8
.13
 
R
2
 
coefficient curve: R
2
 
coefficient vs number of epochs for LSTM

.

    
90
 
Figure 
8
.14
 
Prediction Results: True vs Predicted Plot for LSTM

.

      
91
 
Figure 
8
.15
 
Prediction results on data w
ith noise of SNR20 for LSTM

.

     
91
 
Figure 
8
.16
 
Prediction results on data with noise of SNR10 for LSTM

.

 
 
 
  
92
 
Figure 
8
.1
7
 
Prediction results on data with noise of SNR5 for LSTM

.

     
92
 
Figure 
8
.18
 
NOWP for the predictions on noisy and noise
less data for LSTM network

.
..
       
93
 
Figure 
8
.19
 
NOWP for the two deep architectures in the presence and absence of noise in the 
signal


.
  
 
94
 
 
Figure 
8
.20
 
Training time per epoch for the two deep architectures (1DCNN and LSTM)
.
.
.
     
95
 
Figure 8.21 Research Flow for the ML based approach

.

       
96
 
Figure 
8
.22
 
Loss curve: MSE vs Epochs for the DNN

.

      
97
 
Figure 
8
.23
 
MAE curve: MAE vs Epochs for the DNN

.

      
98
 
Figure 
8
.24
 
R
2
 
coefficient curve: R
2
 
coeff
icient vs Epochs for the DNN

.

        
98
 
Figure 
8
.25
 
Prediction results for the DNN

.

       
100
 
Figure 
8
.26
 
NOWP using DNN

.

       
10
1
 
 
xiii
 
 
Figure 
8
.2
7
 
Training time per epoch using DNN

.

    
101
 
 
 
 
 
 
 
1
 
 
CHAPTER
 
1
 
INTRODUC
TION
 
 
The idea of 
structural design
 
has undergone a radical change. Due to the continuous 
technological advances achieved by the human mankind, lighter and sleeker structures have 
replaced the conventional bulky and heavy structures. But newer solutions al
ways bring with
 
it
 
a newer set of problems that requires solving. The new age structures have introduced 
severe constraints on design methodologies that are currently in 
practice [1], which requires 
newer techniques to monitor and assess the integrity of s
tructures. 
Structural Health 
Monitoring
 
(SHM) is one such technology, which can provide vital information on the state 
of the structure at any given time.
 
Combined with the powerful computational and signal 
processing tools available, one can not only prec
isely determine the health of any structure, 
and classify and characterize defects that affect the structural integrity, but also predict the 
future perform
ance of the structure over time.
 
 
1.1.
 
Overview on Structural Health Monitoring
 
 
Structural Health Monit
oring is the process of evaluating the 
health state 
of a structure and 
predicting its remaining life. This process often involves the constant observation of a system 
over time using periodically sampled response measurements from a sensor actuator system,
 
extraction of healthy and damage sensitive features from these measurements and the 
statistical analysis of the these features to establish the state of the system health, and then 
further predict the remaining life of the structure.
 
       
All structures
 
like bridges, aircrafts, and pipelines have a finite lifetime, and begin to 
deteriorate when put into service. Due to a combination of environmental, material and other 
effects, processes such as corrosion, fatigue, erosion, overloads and general wear and
 
tear 
 
2
 
 
degrade them until they are no longer useful. In worse cases, they can lead to fatal damage 
that can endanger people

, their livelihoods and the environment in general. It is 
therefore essential to examine structures of importance periodically
, and determine whether 
or not remedial action is needed. SHM serves as an early warning system and helps in 
resolving the health of the structure before they can progress to cause potential damage.
 
       
Structural design has undergone many changes acros
s years, where stringent restrictions 
are placed on design parameters to produce the most efficient structures having superior 
structural integrity. Such structures are generally geometrically optimized to guarantee their 
resistance to sustain the high des
ign loads. However, they are more susceptible to small 
damages such as horizontal, vertical or inclined cracks, corrosion in metallic structures, and 
delamination, fibre breakages in the case of composites. These defects severely affect the 
struct
ural heal
th, and therefore need
 
consistent monitoring.
 
       
SHM can be viewed 
as 
a generalized tool box that has the objective of providing 
necessary techniques for the constant/periodic monitoring of structures.
 
These techniques are 
specifically designed for the
 
various materials used in critical structures like buildings 
(concrete), bridges (metal) and aircrafts (composites/metals) etc.
 
SHM therefore has 
innumerable applications in varied disciplines including aerospace, mechanical and civil 
engineering [2]
 
[3].
 
SHM potentially offers increased safety, since faults cannot grow to a 
dangerous level, and avoids the vagaries of human behaviour. The benefits of SHM can be 
mainly categorized as:
 

 
To use structures to their optimal best, a minimized downtime, and avoidi
ng fatal 
failures.
 

 
To give designers an improvement on their products.
 

 
Minimization of human involvements, and therefore possible human errors. This 
directly improves safety and reliability.
 
 
3
 
 
1.1.1.
 
Basic Elements of SHM Systems
 
A typical system contains both hard
ware and software elements. The hardware elements 
basically are the sensor actuator setup and its associated instrumentation, while the software 
components can vary but generally compromises of damage modelling and damage 
characterization algorithms. Senso
rs may be active or passive. Passive sensor like strain 
gauges only sense (receive) while active sensors transmit and receive. Commonly used 
sensors include 
Poly Vinyl Di
-
Fluoride
 
(PVDF) sensors, Piezoceramic sensors made from 
Lead Zirconate Titanate
, comm
only known as PZT sensors or fibre optic sensors. The 
responses obtained from the sensors vary sensor to sensor, but typically, they are all time 
histories of a certain variable. It is important to note that SHM is a time dependent process, 
and any sensor 
should be able to monitor a parameter/variable over time. The most common 
response often received from a PZT or a PVDF sensor, is the voltage history. These 
responses
 
are post processed and manipulated to extract healthy and damage features to 
effectively 
characterize 
the health state of the structure.
 
      
SHM compromises of two main components, the 
Diagnosis
 
and the 
Prognosis.
 
Diagnosis
 
aims to, at every instant of time give a procedure to determine the health of the structure 
while 
Prognosis
 
involves co
mputation of the severity of the defects detected during
 
diagnosis
 
in terms of fracture mechanics parameters, and derive 
the structures
 
residual life. 
Diagnosis
 
normally gives informa
tion about the onset of damages 
like cracks, its location and geometric 
p
arameters. Considering only the diagnosis
 
component
, SHM can be described as a new and 
improved way of performing 
Nond
estructive Evaluation
 
(NDE). By combining integrati
on of 
sensors, smart materials,
 
computational modelling
, data transmission
 
and processi
ng ability, 
it extends the traditional NDE approach to reconsider the structure design and its lifetime 
management. 
An alternate view of SHM is that of a discipline combining the following four 
 
4
 
 
subjects 
[3] (see Fig
ure
 
1.1). In general, an SHM system encom
passes the following 
components:
 

 
Test structure/ simulation Model
 

 
Sensors
 

 
Data acquisition systems
 

 
Signal processing algorithms
 

 
Damage modelling and classification algorithms
 

 
Data transfer
, handling, management
 
and storage mechanism
 
 
 
 
Figure 1.1 Elements
 
of Structural Health Monitoring
 
       

if not for powerful and robust post processing algorithms that converts raw data to 
meaningful information. Common modelling techn
iques employ the 
Finite Element Method 
(FEM), which is very adept at model
ling complex geometries. FEM requires the use of very 
fine meshes to detect and c
haracterize very small defects, which results 
in very high 
computation time and cost. An example of u
sing FEM for SHM can be found in [4].
 
 
5
 
 
Therefore, for characterizing smaller defects more efficiently, a suitable mathematical model 
should be based on the physics of wave propagation, and one of the most well exploited 
models
 
is the 
Spectral Finite Element
 
Method
 
(SFEM) [5].
 
       
Modelling has mainly two components, namely the flaw modelling, and damage 
detection algorithms.  In metallic structures, some of the most common types of defects 
are
 
caused by pitting corrosion. Modelling pitting corrosion is re
latively easy, and has been 
extensively studied in [6][7][8]. Another common type of defect in metallic structures is the 
horizontal/vertical cracks which are sometimes through thickness. Damage 
modelling
 
in 
composites is whereas a challenging task due to 
many different types of failure modes. Some 
of the commonly occ
urring modes are delamination, fibre breakage, matrix cracks and 
d
ebonds. In addition, composites are prone to moisture absorption due to its highly porous 
nature 
because of 
unavoidable errors 
during manufacturing. Hence, one needs to be able to 
come up with simplified mathematical models to describe the various types of flaws that 
commonly occur in critical structures. The second aspect of modelling is in devising robust 
damage characterization
 
algorithms. These algorithms should be able to easily detect defects, 
distinguish healthy to faulty samples, and be able to clearly extract the damage features from 
a SHM response. The features should directly or indirectly give information about the stat
e of 
the health of the structure.
 
There is always a lot of noise present when conducting field 
experiments and these algorithms should be able to work well even in the presence of noise. 
 
 
1.1.2.
 
Levels of Structural Health Monitoring
 
SHM can be thought as a syst
em identification problem. Through diagnosis, one can get 
information about any defect or anomaly present and their characteristics, and the prognosis 
uses the information obtained from diagnosis and determines the residual life of the structure.
 
We can br
oadly divide SHM to five levels [2].
 
 
6
 
 

 
Level 1
: Detecting the d
amage i.e. being able to distinguish healthy and faulty responses.
 

 
Level 2
: Defect location and geometry
 

 
Level 3
: Severity of the damage
 

 
Level 
4
: Damage c
ontrol i.e. possibility of controlling or
 
delaying the growth of damage.
 

 
Level 5
: Determining the residual life.
 
       
The first four levels effectively constitute the diagnosis component of SHM, while Level 
5 is effectively prognosis. Level 1 SHM is relatively easy to achieve, and it can be ach
ieved 
by using passive sensors to monitor parameters such as strain energy, fundamental natural 
frequency, phase information, stiffness reduction over time. The most common method 
though is using natural frequencies. As damage reduces stiffness, it induces
 
changes in the 
natural frequencies. Comparing to the baseline fundamental frequency should confirm the 
presence of damage, and act as a reliable feature to isolate faulty samples.
 
       
Level 2 SHM is relatively harder the level 1, as from the known inpu
t and the measured 
SHM response, it is necessary to determine the location and orientation of the flaw. A simple 
way is capturing the response at some known location. This response will contain the 
reflected energy packet coming from the flaw, knowing the 
speed of the wave in the medium, 
and the time of arrival of the reflected pulse, we can effectively locate the flaw. This is easier 
said than done, as the reflection from the flaw will mostly be very small in amplitude, and 
can easily be buried as noise or
 
as part of the reflected wave packets. This makes the 
designing of the detection algorithms challenging. An ideal algorithm would be one which 
works accurately without a baseline response.
 
       
Once the damage characteristics are evaluated in Level 2, i
t is important to determine the 
severity of the damage. For example, if the damage is a crack, then we need to estimate the 
Stress Intensity Factor
 
(SIF) or 
Strain Energy Release rate
 
(SERR).  If these parameters 
reach a pre
-
defined threshold value, these 
cracks will grow. Now, if the defect is found to be 
 
7
 
 
severe, Level 4 SHM deals with the immediate measures to be taken to arrest the growth of 
cracks.
 
Level 5 SHM is closely related with level 4 SHM, wherein the estimation of fracture 
parameters is used to 
perform fatigue life analysis to determine the residual life. The analysis 
is mainly statistical in nature and generally encompasses novel data processing techniques.
 
       
It is to be noted, the knowledge of material properties of the structure under tes
t is critical 
to perform or achieve any level of SHM. For example, in the most basic and probably easiest 
level i.e Level 1 involves detecting possible defects. This is normally done by monitoring the 
material properties of the structure periodically over 
time. This requires an user to know the 

following levels of SHM are all based on the results of the detection process in Level 1. 
Therefore, the material char
acterization of the structure is the fundamental and most 
important task that needs to be carried out in any SHM framework or technique.
 
 
1.2.
 
Forward and Inverse Problems in NDE/SHM
 
Forward problems typically use known models of the system of interest along w
ith a known 
input t
o establish the characteristics
 
of the output response. 
Inverse problems 
 
in most cases 
meanwhile, use an observed output, along with a known input to estimate the properties of 
the model. Figure 1.2 below shows the block diagrams of a b
asic forward and inverse solver.
 
 
 
Figure 1.2 (a) Forward Process (b) Inverse Process
 
 
8
 
 
       
In NDE, and more generally in SHM, physics based mathematical models are utilized to 
describe systems of interest. Some of the common methods used are the FEM and
 
SFEM. 
W
aves excited by different actuating mechanism 
act
 
as
 
the
 
inputs, and as described before
,
 
different type of responses like voltage histories, displacement histories, strain rates are the 
outputs of a typical NDE system.
 

ly solvable in this field, 
because it is very hard to know the exact physical properties of the system or 
model;
 
hence 
one has to reverse engineer using the output response or measurement to gain information 
about the system in general.
 
In SHM particularly
, known inputs and observed outputs needs 
to be used to establish the physical and material state of the critical structure in question. This 
essentially is the inverse process, and is a fundamental part of any SHM/NDE framework.
 
       
A problem should be
 
well posed in order to be solved. In general, one can describe three 
conditions for an inverse problem to be well posed [9]. The systems used in SHM typically 
have distinctive physical and material properties that can be mathematically well defined, and 
t
his is necessary to ensure that the problem in hand has a realistic solution. As is the case in 
solving any engineering related challenges, the solution need
s to be unique. For example in 
F
igure 1.3, when 
Ultrasonic Guided Waves
 
(UGW) is used to study pitt
ing corrosion
 
in long 
pipelines
, it is clearly visible that the 
responses for a healthy and faulty pipe with a very small 

 
very different. The differences are minimal, and a defect response can easily be 
misclassified as a healthy one.
 
This me
ans that in order to obtain a unique accurate 
representation of the system, one needs to collect more data where more informative features 
can be extracted. Also, the solution must be stable for small deviations in the measured data, 
or in the presence of 
noise. In other words, the solution must be robust and should incorporate 
factors that affect systems in practical situations. But by 
incorporating
 
more information or 
measurements, one risks the stability of the system as more data brings in more stochast
ic 
variations.
 
 
9
 
 
 
Fig
ure
 
1.3
 
Ultrasonic Guided Wave 
responses
 
for
 
healthy pi
peline v
s a faulty pipeline with a 
single 1mm deep corrosion pit
 
 
       
Damage detection and characterization is a complex procedure, and it is impossible to 
solve these kinds of i
nverse problems independent of the forward process.  
Typically, 
materials characterization forms the crux of almost all damage characterization techniques. 
Knowledge of the material properties of the structure before being put into service and during 
its s
ervice period is very critical for condition monitoring. 
Therefore, such problems 
generally involve both forward predictors and inverse detectors. 
SHM techniques normally
 
combines the forward solver and the inverse solver to establish robust models, where 
both 
forward predictor and inverse detector models maximize their sensitivity to different 
features 
of the signals
, and at the same time minimize their sensitivity to cofounding factors caused 
due to the variations in the mechanical, physical and material 
properties of the system. It can 
be effectively described as a symbiotic relationship, where information from the forward 
process is used to improve the models established in the inverse process, and these improved 
models are then used in the forward proce
ss to obtain better measurements. Fig
ure
 
1.4 shows 
the inter relationship between the forward and inverse process, and how it can be used to 
establish efficient models.
 
 
 
10
 
 
 
Figure 1.4
 
Interdependence of the forward and inverse process
 
 
 
1.3.
 
SHM for Composites
 
Composites are materials made by combining two or more natural or artificial elements with 
different physical and chemical properties that results in a mechanically superior material 
compared to its constituent material. Generally, fibre reinforced polymer
 
composites consist 
of a polymer matrix reinforced with a man
-
made or natural fibre. 
Compared with traditional 
metallic materials, the main advantages of composites are: a) low de
nsit
y and high specific 
strength and stiffness b) good vibration damping abil
ity, long fatigue life and high wear, 
creep, corrosion and temperature resistances; b) strong tailoring ability in both 
microstructures and properties that make them design efficient to satisfy different application 
needs; c) since detail accessories can b
e combined into a single cured assembly, the number 
of required fasteners and the amount of assembly labour can be significantly reduced 
[12][13]. 
The most common types of damages in composites are 
fibre breakage, matrix 
cracking, fibre
-
matrix debonding an
d delamination between plies, most of them which occur 
through the thickness, and are barely visible. They can severely degrade the performance of a 
structure and can cause fatal damages at the worst. 
Due to the above advantages, composites 
are used in man
y critical structures across different industries like aviation, automobile etc. 
and therefore require constant periodic monitoring to guarantee optimum performance.
 
Figure 1.5 below shows a typical setup to perform active SHM of composites.
 
 
11
 
 
 
 
Figure 1.5 
Active SHM Setup of a CFRP Composite Plate
 
 
1.4.
 
Machine Learning in SHM
 
Machine Learning
 
(ML) is considered to be the natural evolution of statistical learning. It is 
one of the major dominant subset of 
Artificial Intelligence
 
(AI). To summarize in a single 
s
entence, ML is the science of developing statistical models that learn with time, and can 
perform a function/task of interest without necessary human intervention. Most of the current 
SHM frameworks aim to perform fast in
-
situ testing. NDE progressively ha
s required a 
higher level of automatization to handle the huge amount of data generated. This in term 
means quick 
processing of the signals obtained after an experiment
. 
Conventional data 
analysis techniques require expertise and involve
 
manual labour, but
 
with the advent of better 
computational facilities
 
like GPUs
, powerful machine learning based classification and 
characterization algorithms can be developed. 
The complexity involved in defect and 
materials characterization algorithms have been described 
before
. This is mainly due to the 

measured output response space. Neural Networks are in other words known as universal 
function approximators [10[[11]
[26][27]
. Qu
ite simply, they help in mapping the input 
feature space to the output response space. 
In general, SHM is an inverse problem that is data 
intensive and which does not have a unique solution.
 
Hence, ML plays a very important role 
 
12
 
 
in mapping output signals t
o input features to discover important details of the structure. 
Overall, the benefits of Machine Learning in SHM can be described as:
 

 
Automatization of the SHM framework i.e. performs various levels of SHM without 
any human intervention.
 

 
Improves the robu
stness of the algorithms. These algorithms are powerful and work 
very well even in presence of noise. 
 

 
Reliability. Once trained well, the networks can sustain and perform the described 
process multiple times with the same accuracy.
 

 
Mapping input to output
 
space, thereby gaining vital information about the system 
parameters
 

 
Efficient and fast post processing of the observed outputs.
 
 
1.5.
 
Material Property Identification
 
of Composites
 
 
In any Structural Health Monitoring framework, the knowledge of the material
 
properties of 
the structure under inspection is fundamental. The knowledge of material properties is 
required before the structure is put into service.
 
Because of the inherent heterogeneity of 
composites in general, coupled with the large uncertainties ex
pected in the manufacturing 
process, it becomes extremely difficult to predict the material properties of the final part from 
those of the individual constituents. 
Once in service, it is also essential to continuously 
compute the material properties of the
 
structure periodically to capture possible material 
degradation. Not only that,  by monitoring the material properties over time, potential defects 
can be identified in the structure by detecting the changes in the material properties over time.
 
Material 
degradation can cause a structure to fail if the strength falls below a certain 
threshold.
 
Having efficient material property identifications schemes can potentially help fine 
tune the manufacturing process to reduce the amount of error that occurs normall
y. 
Therefore 
 
13
 
 
identification of material properties is not only required for materials characterization but also 
for in
-
situ monitoring. The ideal requirements for any material property identification scheme 
are 
that the method is non
-
destructive in nature,
 
and is able to be deployed for real time 
online in
-
situ monitoring. The algorithm or framework should also be able to generalize well 
on any composite.
 
      
Any framework involving guided waves requires a frequency selection procedure. Guided 
waves are m
ulti
-
modal waves, and multiple modes often occur at higher frequencies 
relatively. Therefore, dispersion curves are normally used to study the dispersion 
phenomenon of guided waves in any material. Based on these curves, an optimal frequency is 
selected fo
r the problem in hand. The dispersion curves which normally is phase/group 
velocity plotted as a function of frequency is dependent on the material properties of a 
composite.  Hence 
uncertainties
 
or differences in the material properties of the composite c
an 
lead to dispersion curves that are not accurate, and therefore can result is selecting a 
frequency that is probably not best suited for the problem in hand. Hence due to the above 
mentioned reason, it is important to have an automated material property 
identification 
scheme that is real time and can be potentially used for in
-
situ monitoring.
 
 
1.6.
 
Literature Review
 
Researchers have used
 
both vibration
-
based and ultrasonic guided wave (UGW)
-
based 
techniques 
to estimate material properties of a composite 
[1
4
]
. 
The vibration
-
based technique 
is global in nature and is sensitive to boundary conditions which 
are not suitable 
for
 
in
-
situ 
condition monitoring [15
].
  
The ultrasonic range of the mechanical guided waves is highly 
sensitive to 
composite laminate
 
propert
ies which makes them useful f
or Non
-
Destructive 
Evaluations 
and Structural Health Monit
oring applications [16][17][18]
.
 
Traditionally, 
considerable research has been done on using Deep Learning for damage detection, and 
 
14
 
 
damage characterization. Rautela et.
al.  proposed different deep learning frameworks for 
damage detecti
on in 1D composite waveguides [48
]. Here, they have modelled cracks of 
various sizes using the Spectral Finite Element formulation, and have used the responses 
obtained from this model to t
rain on different architectures. But the field of material
s
 
characterization using Deep Learning is relatively less explored. Rather, the inverse problem 
of property identification is investigated using different inversion schemes
 
that in some way 
or other
 
uses global optimization algorithms. 
Krishnan Balasubramaniam [7] has explored the 
feasibility of estimating the stiffness constants in a single layer unidirectional composite 
laminate, and the ply
-
up sequence of layered laminates. It is solved as a multi
-
parameter 
optimisation function, and genetic algorithm was used to find the optimal solution.
 
Bernard 
Hosten et.al. [8] have used the phase velocities of lamb wave modes along with a Newton
-
Raphson scheme to estimate the elastic properties of composites
 
b
y minimizing the 
Thomson/Haskell matrix.
. In this work, they have used air coupled transducers to generate 
lamb waves that are sensitive to material properties. J Vishnuvardhan et. al. [9] have utilized 
a genetic algorithm to measure the elastic properties
 
of an orthotropic plate using ultrasonic 
velocity data. They have used slowness curves to verify the quality of the reconstruction. 
Ranting Cui et. al. [2] have investigated a property inversion scheme based on matching 
phase velocity dispersion curves of
 
relevant guided modes (A0, S0, SH0) using Simulated 
Annealing optimization algorithm along with a metropolis criteria. In this process, a Semi 
Analytical Finite Element method is formulated to solve the forward problem. This inversion 
scheme is used to id
entify the elastic properties of a unidirectional, quasi
-
isotropic and 
anisotropic composite laminates. 
 
       
All the above mentioned inversion schemes are limited in terms of large scale 
automation, generalization ability, computational time, in
-
situ pr
edictions and robustness 
towards the noise. 
Also there is a research gap when it comes to using Deep Learning 
 
15
 
 
Algorithms for material property identification. In this thesis, we aim to address this research 
gap by developing learning models within an overa
ll framework that can identify and 
estimate material properties of composites.
 
 
1.7.
 
Thesis
 
Objectives
 
The scope of this 
thesis
 
is to establish a comprehensive framework 
with different learning 
models and interlinking data analysis algorithms 
that can be utili
zed to characterize 
composite laminates. The propagation behaviour of the UGW in the structure is utilized to 
reverse engineering and estimate the material properties. Powerful Deep Learning algorithms 
like 
Convolutional Neural Networks
 
(
CNN)
 
and 
Long Shor
t Term Memory Networks 
(LSTM), along with multi
-
layer 
Dense Neural Networks
 
(DNN)
 
are
 
utilized for this purpose.
 
The performance of these models is to be evaluated on data both in presence and absence of 
noise of various levels to verify its robustness.
 
We
 
use slightly different methodologies to 
collect data and train the deep networks (1DCNN and LSTMs) compared to 
that of the 
DNNs. The approach involving the deep networks will be classified as 

 
approach, while the approach involving DNNs will be 
classified as 

 
approach 
hereon in this thesis. 
Indeed, Deep Learning or DL is a subset of Machine Learning (ML), 
and the demarcation is made to easily differentiate the two different approaches. 
Once the 
framework is well established for complex 
structures like composites, it can be potentially 
extended to other materials used in critical structures like metals, concrete etc. 
To summarize, 
the main objecti
ves of this thesis can be defined based on the two approaches used i.e. DL 
based approach and
 
ML based approach.
 
The main objectives in the DL based approach are 
as follows:
 
 
16
 
 

 
Develop
 
(Forward Process)
: Develop Numerical Models (FEM and SFEM) to simulate 
the two fundamental modes of a UGW (A0 and S0) in a unidirectional multi
-
layered 
composite lamin
ate. 
 

 
Compare
: Compare and comment on the two different numerical methods used.
 

 
Creating the dataset
: Use the numerical models in the forward process to obtain the A0 
and S0 responses for differ
ent sets of material properties representing different 
composi
tes.
 

 
Dataset Check
: Sensitivity Analysis, Uniqueness check
 

 
Training (Inverse Process)
: Use the generated dataset, with A0 and S0 time histories as 
inputs and the material properties as the 
output 
labels to perform supervised 
regression
 
using differ
ent deep
 
learning architectures (1DCNN and LSTM)
 

 
Prediction
: Once trained, for any uns
een A0 and S0 waveforms
, the networks should be 
able to identify and estimate the material properties.
 

 
Robustness
:
 
Use a model trained on noiseless data and predict on datasets w
ithout and 
with noise of various levels.
 
The main objectives of the ML based approach are as follows:
 

 
Dataset
 
(Forward Process)
: Use Dispersion Calculator
® 
software to generate group 
velocity dispersion curves for different sets of material properties repr
esenting different 
composites.
 

 
Training (Inverse Process)
: Use the generated dataset, with group velocity curves for 
A0, S0 and the Shear Horizontal mode SH0 and the material properties as the output 
labels to perform supervised regression using a simple m
ulti
-
layer Dense Neural 
Network.
 
 
17
 
 
 
Figure 1.6
 
Overall Workflow of the Composite Property Estimation Framework
 
       
Figure 1.6 shows a flowchart describing the overall workflow of the composite material 
property identification framework.  The organization
 
of the thesis is as follows, Chapter 2 
gives a basic introduction to composites and the Composite Laminate Theory (CLT). Chapter 
3 then discusses guided waves in general before looking at specific cases of guided wave 
propagation in composite laminates. C
hapter 4 covers the basics of Machine Learning, and 
describes in detail the Convolutional Neural Network, Long Short Term Memory Networks 
and the Dense Neural Networks. Chapter 5 describes in detail the two modelling techniques 
used in this thesis i.e. the
 
Finite Element Method and the Spectral Finite Element Method. 
Chapter 6 outlines the data analysis algorithms used in this thesis, while Chapter 7 mentions 
the study parameters for the two different methods used. The results are presented and 
discussed in
 
Chapter 8, and the thesis is concluded in Chapter 9.
 
 
 
 
18
 
 
CHAPTER 2
 
CLASSICAL LAMINATE THEORY
 
 
Composites can be broadly classified into three different types: 
Fibrous Composites
, 
Particulate Composites
 
and 
Laminated Composites
. For structural applications,
 
only the 
laminated composites are extensively used, and therefore the whole material property 
identification framework 
is
 
built considering composite laminates.
 
A composite laminate 
consists of many layers which are commonly referred to as laminae or plie
s. They are stacked 
together in particular order to form structures. Based on the strength
 
and stiffness
 
requirements, the number of plies is decided. Typically, each lamina consists of fibres 
oriented in a direction where maximum strength is required. The
 
fibres are bonded by a 
matrix material which is mechanically inferior compared to the fibres. The laminate derives 
all its strength from the fibres. Some of the commonly used fibres are: Carbon Fibre, Kevlar, 
and Glass Fibre. The most commonly used matrix
 
material is the Epoxy Resin. Laminated 
composites are assumed to have orthotropic properties at the lamina level, but when at the 
laminate level, they exhibit anisotropy. The anisotropic behaviour results in stif
fness 
coupling, such as bending
-
axial
-
shear
 
coupli
ng in beams and plates, bending
-
axial
-
torsion 

identification of these 
types
 
of composites 
is
 
complex compared to isotropic materials.
 
The 
material properties of the
se composites typically can be determined at the lamina level 
(Micro Mechanics) 
or
 
at the laminate level (Macro Mechanics). Section 2.1 below outlines 
the micro mechanical analysis while 
Section 
2.2 
describes
 
the macro mechanical analysis.
 
 
 
 
 
 
19
 
 
2.1. Micro M
echanical Analysis
 
A 2D transversely isotropic lamina require
s determination of 6 properties 
namely, Elastic 
modulus in two coordinate directions (E
1
, E
2

12
 

23
), shear modulus 
(G
12
) and density
 

. These properties are determined using the properties of t
he fibre and 
the binding matrix. 
The lamina and the derived laminate strength are strongly influenced by 
the type of fibres an
d their orientation. Another important parameter that has a direct 
influence on the laminate strength is
 
the
 
Volume Fraction.
 
Volume fraction is the volume of 
fibre present with respect to the overall volume of the lamina. According to Jones [20], 
micro
-
me
chanics is the study of composite material behaviour, wherein the interaction of the 
constituent materials is examined in detail as a part of the definition of the behaviour of the 
heterogeneous composite material. The overall elastic moduli of a composite
 
material are 
expressed in relation to that of the fibres and the matrix. The properties of a lamina can be 
mathematically expressed as shown in Equation 
(
2.1
)
.
 
























   
                        
       
(2.1)
 
       
Where 
E
, 

 
and 
V
 

fraction respectively, while
 
f 
and 
m
 
in the subscript denotes the fibre and matrix respectively. 
The volume fraction of fibre and the matrix is given bel
ow in 
Equations (
2.2
)
 
and 
(
2.3
)
.
 







                                                                   
(2.2)
 







                                                                  
(2.3)
 
       
Where 
vol
f
 
and 
vo
l
com
 
are the volume of fibre and the volume of the whole composite 
laminate respectively. In order to determine material properties of a lamina, some 
fundamental assumptions are required. The main fundamental one is that the fibre is 
mechanically superior 
compared to the matrix, and therefore is the main load bearing member 
in the constituent matrix. Also, the strains in the fibre and the matrix are assumed to be the 
same.
 
In this 
work
, a transversely isotropic laminate is considered
 
for material property 
 
20
 
 
i
dentification
. In doing so, the analysis can be limited to a small representative volume. Such 
a volume is called 
Representative Volume (RV)
. 
A simple RV would be a fibre surr
ounded by 
a matrix as shown in F
igure 2.1.
 
 
Figure 2.1 (a) RV for longitudinal m
aterial property determination (b) RV for transverse 
material property determination
 
 
       
By applying a stress 

1
 
along direction 1, 
t

direction 1 are determined in [19] to be:
 












                                                       
(2.4)
 
       
Where 
E
f
 
and 
E
m
 

fibre and matrix respectively. Equation 
(
2.4
)
 
is more commonly well known as the 
Rule of Mixtures
. 
Similarly, a stress 

2
 
is applied 
along direction 2, and the elastic modulus in direction 2 is computed to be:
 
















                                                            
(2.5)
 
       

 












                                                      
(2.6)
 
       
The shear modulu
s is express
ed in E
quation 
(
2.7
)
, while the 
lamina density is expressed 
in E
quation 
(
2.8
)
.
 
 
21
 
 
















                                                          
(2.7)
 













         
                                                    
(2.8)
 
       
Once the material properties have been determined at the lamina level, one can proceed 
then to perform a macro mechanical analysis of the lamina to effectively characterize the 
overall consti
tutive model of the laminate.
 
 
2.2. Macro Mechanical Analysis
 
Macro mechanical analysis involves the determin
ation of the overall constitutive
 
model i.e. 
the overall 
stress strain relations of the 
composite laminate consisting of individual laminas 
stacked
 
together. 
This is done using 
Classical Laminate Plate Theory (CLPT)
. As mentioned 
earlier, only necessary equations have been provided here. The theoretical detail of these 
equations can be found in [19]. Main assumptions made in CLPT are the following:
 
1.
 
T

superposition principle are valid.
 
2.
 
At the lamina level, the composite is homogeneous and orthotropic and has exactly two 
planes of symmetry, one along the direction of the f
ibre and other perpendicular to it.
 
3.
 
The state of stress at the lamina level is predominantly plane stress.
 
 
Figure 2.2
 
Co
-
ordinate system for a laminate
 
 
22
 
 
       
Looking at F
igure 2.2, the principal axes are denoted as 1
-
2
-
3, where direction 1 is along 
the 
fibres, and direction 2 is transverse to it. 
. The lamina is assumed to 
be in 3
-
D state of 
stress with six
 
stress components. For a transversely isotropic laminate, strain and stress is 
related by a
 
compliance matrix as shown in E
quation 
(
2.9
)
.
 
All bold fa
ce variables denoted 
here are matrices.
 










                          
 
                                   
(2.9)
 
       
Where 

 
is the strain developed in the composite due to the applied stress 

. T
he 
compliance matrix 
S
 
is given by [21]
:
 




























































































                     
         
(2.10)
 
       
The stiffness matrix 
Q
 
is expressed in equation in 
Equation (
2.11
)
:
 







                
  
                                                
(2.11)
 
       
On solving, the entries of matrix 
Q 
is derived in terms of the individua
l lamina properties 
determined in the micro mechanical analysis explained in section 2.1 [19] to be:
 











                                                           
(2.12)
 







                                  
                          
(2.13)
 











                                                           
(2.14)
 





                                                                  
(2.15)
 
       
Through symmetry, the stiffne
ss matrix 
Q
 
can be reduced to a 3*3 matrix, and is 
expressed below 
in Equation (
2.16
)
.
 


















           
 
  
                                
(2.16)
 
 
23
 
 
       
 
       
The above relations are for a unidirectional co
mposite with fibres along direction 1 at 0
°
. 
Typically fibres of arbitrary orientations are used in structural applications.
 
In most cases, the 
orien
tations of the global axes do not
 
coincide with the principal axes of the composite. 
Hence a combination of
 
translation
 
(
T
)
 
and rotation
 
(
R
)
 
operations is required to obtain the 
stiffness matrix of such composites. The reduced stiffness matrix is expressed in
 
Equation 
(2.17) while F
igure 2.3 shows the difference in the orientations of the principal axes and the
 
global co
-
ordinate system.
 





















                                         
(2.17)
 
 
Figure 2.3
 
Principal axes of the laminate and global x
-
y axes
 
       
The matrix 

 
is fully populated. Hence, although the lamina in its own prin
cipal 
direction is orthotropic, in the transformed coordinate, it represents complete anisotropic 
behaviour. The elements of 

 
is given by
 







+2(
















            
   
           
(2.18)
 




























    
      
       
(2.19)
 
































  
(2.20)
 





















+




                         
(2.21)
 
































  
(2.22)
 
































    
(2.23)
 
 
24
 
 
*











 
The reduced stiffness matrix 


 
can
 
be full expres
sed as shown in E
quation 
(
2.24
)
.
 






















                                              
(2.24)
 
       
The axial stiffness for the whole constituent model is given in 
Equation (
2.25
)
, while the 
flex
ural (bending) stiffness is mathematically expressed in 
Equation (
2.26
)
. Equation 
(
2.27
)
 
expresses the axial bending coupling i.e. 
it relates the in
-
plane forces with mid
-
plane 
curvatures.
 





















                          
             
(2.25)
 

























                                     
(2.26)
 

























                                     
(2.27)
 
       
Where 
z
 
is the vertical position of the p
articular ply from the mid
-
plane of the laminate 
as sho
wn in F
igure 2.4 which shows a typical 
8 
layered composite laminate. The constituent 
equation of a comp
osite laminate is concluded in E
quation 
(
2.28
)
.
 
The constituent equation 
relates the forces 
N
 
and 
the moments 
M
 
with the strains 

 
and curvatures 
k
.
 




























                                       
(2.28)
 
 
Figure 2.4
 
An 8 layered unidirectional composite laminate
 
 
 
 
25
 
 
CHAPTER
 
3
 
FUNDAMENTALS OF GUIDED WAVES
 
 
3
.1. Introduc
tion to Guided Waves
 
Guided waves are typically elastic waves carrying energy which is confined between the 
boundaries separating two mediums. Guided waves travel along the length of the sample. 
They are created by the interactions of bulk waves and the bo
undaries which results in 
several mode changes. These interactions also create standing wave modes normal to the 
boundaries which propagate parallel to the boundaries. 
In essence, guided wave modes are a 
constructive interference pattern created by the int
eraction of many bulk waves with the 
boundaries. 
They can be broadly classified into three types: 
Rayleigh waves (Surface 
Acoustic Waves),
 
Shear Horizontal Waves (SH)
 
and 
Lamb waves
. 
Rayleigh waves
 
are surface 
waves that
 
can propagate in a half space i.e. 
it can propagate when there is only a single 
boundary present, while SH waves and Lamb waves need two boundaries to propagate
.
 
Hence, a lot of studies involve analysing the propagation of Guided Waves in plate like 
structures or pipelines. 
 
       
Guided W
aves can travel along surfaces, interfaces or even throughout the volume 
structures. This can be achieved by appropriately applying an incident pulse, and making sure 
the 
proper 
boundary conditions exist 
on 
the structures in question. Though the direction 
of 
propagation is always parallel to the boundaries, the particle motion can be longitudinal, 
transverse or elliptical depending on the mode that is generated. Because of the 
boundary 
effects, which induces the dispersiveness of the wave, analysis of guide
d waves are much 
more complex compared to Bulk waves. However, this complexity can be utilized with great 
effect in SHM. Compared to Bulk waves, Guided waves can travel long distances, and 
therefore it can be used to propagate along long structures (pipeli
nes), and large areas can be 
 
26
 
 
monitored from a single location.
 
Inspecting using b
ulk waves are
 
cumbersome and time 
consuming
 
and sometimes c
annot
 
inspect st
ructures that are about few 

 
thick
 
due 
to 
impedance
 
issues
. 
Guided waves can also be use
d to detect very small defects depending 
on the frequency and the mode selection [22]. Figure 3.1 below roughly approximates the 
inspection areas of Bulk Wave Testing and Guided Wave Testing.
 
 
 
Figure 3.1 Bulk Wave Testing vs Guided Wave Testing
 
       
Th
is section gives a general introduction to Guided waves, and describes the analysis of 
propagation of guided waves in isotropic plate
 
(metal)
 
like structures. It then talks about 
dispersion curves, before concluding with the analysis of guided waves in a 1
D Laminated 
Composite. Since this is very well 
researched area and also well 
documented in literature, 
 
only the important equations relevant to this thesis is given
. The in depth derivations can be 
found in the references cited.
 
 
3.2
. Lamb Waves
 
Lamb wave
s are elastic stress waves that propagate between two traction free plates. They are 
also commonly known as 
plate guided waves
. 
The governing equation of 
Lamb
 
waves can be 
approximated to elastic wave propagation 
in doubly
 
bounded
 
media. 
The 

 
equat
ion 
 
27
 
 
[23] can be used to st
udy 
L
amb waves in general in a three dimensional solid. Equation 
(
3.1
)
 
describes it in a condensed form.
 























                                  
(3.1)
 
Where 

 
is the density of the material,
 

and 

 
are known as 

 
constants, 

 
is the 
del
 
operator and 

2
 
is the Laplacian. 
u
 
is the particle displacement vector and is defined as 
u
=iu
x
+ju
y
+ku
z
. 
 
      
The 

fundamental differential equations governing wave propagation are 
based upon 
three fundamental relationships from Linear Elasticity Theory.  These 
include strain
-
displacement relation shown in Equation 
(
3.2
)
, generalized equation of 
motion by Newton in Equation 
(
3.3
)
 
and constitutive stress
-
strain relations described 
previously in 
Chapter 2 and in Equation 
(
3.4
)
.
 
















                                                             
(3.2)
 















                                                                           
(3.3)
 







                                                                        
(3.4)
 
       

kl
 
is the 
second order 
strain tensor, while 
c
ijkl
 
is the 
fourth order 
stiffness tensor and 

ij
 
the 
second order 
stress tensor. The indices indicate summation and
 
commas indicate 
partial derivatives where 
i,j
 
and
 
k
=1,2,3. Basically, the 

equation combines 
three equilibrium equations, six stress displacement equations and six constituent 
stress strain relations. For an isotopic medium, 
the 

 
equation 
can be solved 
using 
Helmholtz
 
vector
 
by splitting the equation into two partial differential equations 
based on
 
scalar potentials 
H
 
and 

. For an isotropic medium, the governing equation 
can be effectively solved to give separate relations for the longitudinal (
P waves
) and 
the transverse waves (
S waves
). In essence, a single propagating guided wave can be 
decoupled to two of its fundament
al components, the P wave and the S wave. Equation
s
 
 
28
 
 
(
3.5
)
 
and 
(
3.6
)
 
describe the 
Helmholtz 
decomposition, where Equation (3.6) is known 
as Gauge invariance.
 








                                                                 
(3.5)
 





            
                                                                     
(3.6)
 
Substituting Equation 
(
3.5
)
 
and 
(
3.6
)
 
in Equation 
(
3.1
)
, we obtain
 
two separate 
governing partial differential equation, which are given by
:
 














=0                 
                                  
(3.7)
 










                                                                 
(3.8)
 
       
Rearranging Equation 
(
3.7
)
 
and 
(
3.8
)
, and then expressing it in the rectangular co
-
ordinate system, we obtain the classi
c equations of wave propagation in a 3D 
homogeneous solid in terms of scalar and vector potentials.
 












                                                                        
(3.9)
 












    
                                                                  
(3.10)
 
       
Where 
c
L
 
is the longitudinal wave speed
 
(or P wave speed)
 
and is equal to 
(2


 

, and 
c
S
 
is the shear or transverse wave speed and is equal to 


 
 
A complete 
guide to t
hi
s derivation can be found in [17
].
 
To find the longitudinal mode solutions, 
classic separation of variables is performed using the expressions of potentials 
presented in [24] where the continuity condition in circumferential direction is applied. 
For tor
sional or transverse mode solutions, alternative solutions are presented by [25]. 
The last step is the application of boundary conditions, and eigenvalues can be used to 
trace dispersion curves in general.
 
       
For any non 1D solid, the total displacemen
t vector can be expressed as shown in 
Equation 
(
3.11
)
.
 










                                                          
(3.11)
 
 
29
 
 
       
Where 
u
L
 
is the displacement due to the longitudinal or Pressure wave (P wave). 
The particle motion i
s parallel with respect to the direction of wave propagation in this 
case. 
u
S
 
and 
u
SH
 
are the displacements due to the transverse (S wave) and the
 
Shear 
Horizontal 
modes, where the particle motion is normal to that of the direction of wave 
propagation.
 
The
 
interaction of shear and pressure waves in thin plates and shells give rise to 
guided waves.
 
      
In principle, there exist infinite guided wave modes. 
The 
first 
two fundamental 
of these 
wave modes are the 
Symmetric (S0)
 
mode, and the 
Anti
-
Symmetric (A0)
 
mode. These modes 

frequency range where the wavelength is greater than the plate thickness, these modes are 
called the extensional mode and the flexural mode res
pectively. The names are derived based 
on the particle motion and the elastic 

 
that govern the propagation of these waves.
 
These characteristics change as frequency increases.
 
For simplicity, we refer to these modes 
as A0 and S0 in this work. Th
ese modes are particularly important as (a) they exist at all 
frequencies and (b) they in most cases carry more energy than the higher order modes. Point 
(a) is particularly important as they are the only modes to exist at lower frequencies until a 
certain
 
cut off frequency. Therefore, these modes can be exploited for various applications in 
SHM
, for if we use higher modes, multiple reflections from different wave modes will bury 
the defect signature thereby making the analysis of data more cumbersome and e
rror prone.
 
In the following section, we briefly show the symmetric and anti
-
symmetric solution for lamb 
wave propagation in two dimensional plates of infinite length.
 
 
3.3. Lamb Wave Propagation in 2D Plates
 
Let us consider an infinite 2D plate as shown i
n Figure 3.2. For simplicity of analysis it is 
possible to assume that the wave potentials are invariant to the z
-
direction along the wave 
 
30
 
 
front [23]. The direction of propagation of the wave is considered to be in the x direction. The 
plate here is consid
ered to be isotropic in nature, as only then the 

decomposition 
principle can be effectively applied to decouple the governing equation to its constituent 
longitudinal and transverse wave components. 
 
 
Figure 3.2 2D infinite plate of thickness 
2d.
 
       
In the above case, since the wave potentials are assumed to be invariant along the z 

0
 
and 
u
SH
 
has only the shear displacement component 
u
z
. The displacement vector for the longitudinal mode and torsional m
ode 
u
L
 
and 
u
S
 
have 
both 
u
x
 
and 
u
y
 
components. These displacements only depend on the scalar potential 

 
and z
-
component of a vector potential 
H
z
. 
Therefore the 

 
equations can be rewritten as:
 












                    
                                          
(3.12)
 














                                                           
(3.13)
 
      
By substituting simple notations for 


 
and 
H
z
=


, 
and assuming plane wave solution 
o
f the form 





 
and 





we can get a classic formulation for 
L
amb wave 
propagation in the potential form:
 




















                                                 
(3.14)
 




















                                                 
(3.15)
 
       
Assuming a harmonic solution 
e
-

x
, the above equations reduce to:
 
 
31
 
 



















                           
                    
(3.16)
 



















                                               
(3.17)
 
       
In equations 
(
3.16
)
 
and 
(
3.17
)
, 

=

/c
 
is defined as the wavenumber. To simpl
if
y the 
analysis further and express the e
quations in a more compact form for convenience, the 
following substitutions are made:
 
 











 
                                                             
(3.18)
 











                                   
                           
(3.19)
 
       
Therefore, Equations (3.16) and (3.17) can be expressed as:
 











                                                           
(3.20)
 











                               
                            
(3.21)
 
       
A general solution for the above governing equations can be expressed as shown below:
 











                                        
(3.22)
 











         
                               
(3.23)
 
       
Where, 
A
1
,
 
A
2
, B
1
 
and 
B
2
 
can be established by applying the appropriate boundary 
conditions.
 
This can be done by applying symmetric or antisymmetric boundary conditions 
which gives rise to the symmetric and anti
symmetric mode solution respectively.
 
Equations 
(3.24) and (3.25) show the symmetric and antisymmetric mode solutions respectively. 
An
 
in 
depth derivation of this can be found in 
Appendix A
 
and B
 
based on 
[25].
 

















                                                         
(3.24)
 

















                                                         
(3.2
5
)
 
       
Where 
p
 
and 
q
 
are described in Equations (3.18) and (3.19) respectively. Therefore, 
using the above relations, one can trace dispersion curves.
 
 
32
 
 
3.4. Dispersion Principles
 
Dispersion in general signifies a changing relationship between velocity and frequency. 
Bulk waves
 

wave does not change. 
The concept of phase and group velocity is critical to explain 
dispersion. In simple terms, group velocity is the velocity of the whole wave packet, 
or the 
whole wave front. A wave packet typically contains multiple sine waves of different 
frequency content packed within. The 
velocity
 
of the sine waves that make up the envelope 
or wave mode is the phase velocity, while the velocity of the whole packet 
itself is the 
group velocity.
 


















                                                
(3.26)
 
      
Since both 
p
 
and 
q
 
both depend on 

, the phase and group velocities should be 
evaluated numerically at each frequency step as a multiple of plate 
thickness. Figures 3.3 
and 3.4 below show the phase and group velocity plots of a 1mm Steel Plate respectively. It 
follows that the solution is not unique, and
 
at higher frequencies, multiple higher order 
modes of the Symmetric and Anti
-
Symmetric modes exi
st. Group velocity dispersion plots 
are more vital, for we can determine how fast the wave front propagates which helps us 
establish the 
Time of Flight
 
(TOF). This information is extremely useful for damage 
detection and damage characterization in structur
es. Not only 
that, the group velocities is
 
useful in establishing the material properties of a structure. Group velocity is derived from 
phase velocity, and the mathematical expression for it is expressed in Equation (3.26) [17].
 
       
Dispersion curves a
re critical in guided wave inspection. Depending on the wave mode 
that is suitable for testing, it helps in finalizing an operating frequency based on the 
requirements. It also gives an engineer the cut
-
off frequency up to which he can operate 
without invo
lving higher order modes, which invariable complicates the structural analysis.
 
The first step of any guided wave experiment or simulation is to plot the dispersion curves. 
 
33
 
 
Performing any guided wave setup without doing so would result in wasted efforts, l
ost 
time and energies.
 
 
          
Figure 3.3 Phase Velocity Dispersion Plot for 1mm thick Steel Plate
1
 
 
          
Figure 3.4 Group Velocity Dispersion Plot for 1mm thick Steel Plate
1
 
       
Metals are mostly isotropic in nature. Though they contain a ver
y small degree of 
anisotrop
y
, it is neglected for most of the analysis in Wave Mechanics. Helmholtz 
decomposition principle can be used to solve the governing wave equations only when the 
                                        
             
 
1
 
Dispersion Plots are plotted with the help of D
C software  This footnote can go in Reference section
 
  
Link: 
https://www.dlr.de/zlp/en/desktopdefault.aspx/tabid
-
14332/24874_read
-
61142/#/galler
y/33485
 
 
34
 
 
material is Isotropic. By applying it for any non 1D solid, we can b
asically decouple the 
equation into its constitutive longitudinal and transverse mode. But when it comes to 
composites, they are highly anisotropic in nature. 
There is not true P wave or true S Wave. 
Hence, anisotropy cannot be neglected in the analysis, a
nd 
hence 
Helmholtz principal can 
no longer be used to
 
solve a system that has more than one dimension. The analysis of 2D 
composites is achieved by using the 
Partial Wave Theory
 
which is explained in depth in 
[19]. Though this concept is critical to unders
tand in SHM, the scope of this thesis is to 
establish a reliable framework to estimate and identify material properties. Therefore, we 
consider a 1D composite, and the next section outlines briefly the guided wave propagation 
in 1D Laminated Composite.
 
It 
is well known that using higher order 1
-
D theories, we can 
get to great extent the dispersion relations obtained from 2
-
D Lamb wave theory
 
[19][49]
. 
For example, if we use First Order Shear deformation theory (Timoshenko theory) and 
higher order Mindlin
-
He
r
mann Theory [19}, we can accurately get the dispersion relations 
obtained by the 
2
-
D Lamb wave equations. Hence, we pursue this approach in this thesis.
 
A key point to note is that in a simple 
higher order
 
1D model, there will be only one wave 
mode presen
t at a time i.e. the axial and flexural mode will not be coupled. The type of 
mode generated will depend on the direction of application of the incident tone burst signal.
 
 
3.5. Guided Waves in 1D Laminated Composite
 
In this derivation, we have considered 
a 1D composite waveguide that satisfies both 
elementary rod and beam theories. 
We begin with defining the displacement vectors fields. 
 



























                              
(3.27)
 

















                                                
(3.28)
 
       
Where 
u
o
 
is the axial displacement along the mid plane, and 
w
 
is the transverse 
d
isplacement, and 
z
 
is measured from the middle plane
 
as shown in Figure 3.5.
 
 
35
 
 
 
    
     
Figure 3.5
 
Elementary laminate Composite Waveguide with co
-
ordinate system


 
 
        
A constitutive 3D model for laminated composites was explained in Chapter 2 
(
Equat
ions 2.18
-
2.23). The analysis of 1D composites can be achieved if we consider that the 
composite waveguide in in a 1
-
D state of stress, and therefore the layer
-
wise constitutive law 
is expressed in Equation (3.29).
 








                                                              
(3.29)
 
       
Where 

xx
 
and 

xx
 
are the stress and strain applied in the x direction. The expression for 

11
 
as a function of ply angle 

 
is shown below.
 


























          
(3.30)
 
       
Where 

and 

. 
 
Q
ij
 
is the orthotropic elastic coefficients of the individual 
lamina, and its constitutive relations can be found in Chapter 2.
 
The strain energy is defi
ned 
in Equation (3.31).
 










                                                       
(3.31)
 
       
The kinetic energy is then defined as,
 















                                              
(3.32)
 
       
Where 

 
is the layer wise density, 

differential equations are obtained, and they can be expressed as
 





























             
                
(3.33)
 




























                              
(3.34)
 
 
36
 
 
       
The corresponding boundary conditions are,
 


















 
   
                   
                       
(3.35)
 





















                                             
(3.36)
 



















             
              
   
            
(3.37)
 
       
Where 
A
11
 
is the axial stiffness, 
B
11
 
the axial
-
bending coupled stiffness and 
D
11
 
the 
bending stiffness. They are mathematically expressed in Equation (3.38). 
 





























 
                           
(3.38)
 
       
Where 
h
 
is the depth of the beam, 
b
 
is the layer width. 
A
 
is the cross sectional area while 
d
2
u
o
/dt
2
 
and 
d
2
w/dt
2
 
are the longitudinal and transverse accelerations. 
N
x
 
is the force acting in 
the axial direction; 
V
x
 
i
s the shear force while 
M
x
 
is the bending moment. In our studies here, 
we primarily consider only a symmetric laminate. Symmetric Laminates are composite 
laminates where the lay up above the mid plane is the mirror image of that of the layup below 
the mid 
plane. For a symmetric laminate, the axial
-
bending stiffness 
B
11
 
is equal to 0. 
As a 
consequence, axial and flexural modes become uncoupled. 
Therefore for a symmetric 
laminate, we can rewrite equations (3.33) to (3.37) as
 





















                                              
(3.39)
 


















                                                
(3.40)
 
       
The boundary conditions can be rewritten as,
 









 
                                      
                       
(3.41
)
 












       
               
                                      
(3.42
)
 











                                    
                         
(3.43)
 
 
37
 
 
       
From Equations (3.39) and (3.40), it is visible that the governing equations are 
uncoupled, i.e. solution to Equation (3.39) will give rise to the axial mode, while solution to 
the equation (3.40) will give rise to th
e flexural mode.
 
An in depth derivation of the solution 
based on computation of wavenumbers is available in [19].
 
 
 
38
 
 
CHAPTER
 
4
 
OVERVIEW OF MACHINE LEARNING
 
Deep Neural Networks
 
are an important subgroup of Machine Learning. 
Machine Learning
 
(ML) is consider
ed to be
 
the natural evolution of statistical learning.
 
It is one of the major 
dominant subset of 
Artificial Intelligence
 
(AI). To summarize in a single sentence, ML is the 
science of developing statistical models that learn with time, and can perform a fu
nction/task 
of interest without necessary human intervention. Previously, the field of AI relied heavily on 
hard coded rules and pre fixed algorithms. This required enormous computation
al
 
resources. 
Also, the 
systems depended on programmed intelligence, an
d were
 
not
 
capable of learning on 
their own. ML, whereas relies on learning using real time data instead of relying on hard 
coded rules. This is done by building a model that best describes patterns between the input 
and output data. In a nutshell, they ca
n be seen as function 
approximators
 
that fits the best 
possible function for a set of input and output vectors [
26]
 
[27].
 
ML algorithms have the 
ability to predict the output even on unseen data. It greatly reduces the computational effort, 
while the accur
acy keeps improving as more real data is available to train. There are different 

Decision Trees, Support Vector 
Machines, Radial Basis Functions
 
etc. Deep Learning Algorithms meanwhile are a subset
 
of 
ML 
techniques that
 
basically mimics a human brain
. A deep neural network basically 
contains multiple layers that have nodes that resemble a neuron in the brain with associated 
weights. The weights can be learned using many optimization algorithms to pr
edict the 
desired output. They are very flexible, and can learn complex and often nonlinear 
relationships. Though the amount of training data required is large, the type of tasks it can 
achieve is 
unparalleled
, and is pretty much driving the field of AI th
ese days.
 
Figure 4.1 
summarizes how these different techniques fit in. To summarize, Deep Learning techniques 
are one of the many different ML techniques, that is part of the broader AI field.
 
 
39
 
 
 
Figure 4.1
 
Machine Learning and Deep Learning
 
       
In this 
chapter, we introduce the fundamentals of an 
Artificial Neural Network
 
(ANN) 
that include the basic theory, math and the algorithms behind it. 
Following this
,
 
more 
advanced deep learning models like 
Convolutional Neural Networks
 
(CNN) are discussed in 
deta
il.
 
 
4.1 Artificial Neural Networks
 
An artificial neuron w
as formalized as initially the 
Threshold Logic U
nit
 
(TLU) by 
McCulloch and Pitts in 1943 [28]. 
For a single neuron shown in Figure 4.2, the output signal 
can be written mathematically as:
 



















                
 
                            
(4.1)
 
       
Where, 
x
i
 
is the input to a neuron, 
y
j
 
is the output from the neuron. 
W
ji
 
is the associated 
weight matrix and 
b
 
is the bias term, which is generally used t
o shift the decision boundary 
line or any hyper
-
plane in multidimensional problems [29]. The function 
f
 
is the activation 
function which decides in which manner the neuron will be excited or fired for a particular 
input signal. As the name suggests, a heav
eside step function is used in the TLU
. The 
function is expressed mathematically in Equation (4.2).
 
 
40
 
 





























                                                 
(4.2)
 
 
Figure 4.2
 
Pipeline of a Neuron
 
        
This activation is actually not used these days because it has a constant derivative, and 
more importantly t
he derivative is not defined at 
x=0.
 
It must be noted 
that
 
it is similar to 
how a biological neuron works. Another important point of note here is that, the weight 
matrix initially designed in the TLU does not have the capability to learn.
 
       
 
A TLU th
en evolved into the famous perceptron networks. Perceptron networks were 
similar to TLU, but the major difference was that there was now a learning rule to adjust the 
weight matrix. The delta rule, which uses the Vanilla gradient descent approach to minimi
ze 
the error, is used as the learning rule. Multiple neurons stacked together in layers are now 
commonly known as 
Multi
-
layer Perceptron Networks 
(MLP) or as an 
Artificial Neural 
Network 
(ANN). Since each neuron is fully connected with neurons from the nex
t layer, they 
are also referred to as 
Fully Connected Networks
 
(FCN)
 
or 
Dense Neural Networks
 
(DNN)
.
 
Perceptron Networks were first reported by Rosenblatt in 1958 [30]
.
 
 
 
 
 
41
 
 
4.
2
 
Multi Layered Perceptron (MLP) 
 
Figure 4.3 shows a typical MLP network with one 
input layer, one output layer and 
one
 
hidden layers. The MLP is a feed forward network, where the flow of information is forward, 
while the flow of error information is backward. Each node 
i
 
is connected another node 
j
 
in 
the previous and following layers 
with an associated weight 
w
ij
. In layer 
k
, the weighted sum 
of is performed at each node 
i
 
of all the signals 
x
j
(k
-
1)
 
from the preceding layer 
k
-
1
 
giving the 
sum 
z
i
(k)
 
of the node. The sum is passed through a nonlinear activation function
 
f
, which gives 
th
e output of the MLP network. Mathematically, it can be expressed as,
 
































                              
(4.3)
 
 
Figure 4.3
 
Three Layered MLP Network
 
       
There are different types of activati
on functions used. Each one has its own advantages 
and disadvantages which is not discussed in this thesis. The main nonlinear activation 
functions extensively used are the sigmoid
, tanh, softmax and ReLu
 
activation functions.  The 
ReLu
 
function
 
i.e. the R
ectified Linear Unit
 
is computationally very efficient due to its 
simplicity. Equation
s
 
(4.4) 
and (4.5) describe
 
the 
ReLu
 
function. The 
ReLu
 
function 
introduces nonlinearity in the system that greatly improves the performance in most cases. 
One disadvantag
e of
 
ReLu
 
is that, when the inputs approach zero or negative, the gradient of 
the function becomes zero, therefore the network then cannot perform backpropagation and 
 
42
 
 

Dying ReLu

introd
ucing a small positive slope in the negative area, and is known as the 
Leaky ReLu.
 











                                                         
(4.4)                                                                                 
 






























                                      
          
(4.5)
 
       
Machine Learning can be broadly classified into three categories
. Supervised Learning
 
is 
where the model is task driven in order to predict/classify a desired output.
 
Unsupervised 
Learning
 
is data driven and includes clustering algor
ithms. The final type is the 
R
einforcement L
earning
 
where the model learns to react to an environment. In this work, we 
predominantly use Supervised Learning Algorithms. The core concept of Supervised 
Learning involves two passes, the first being the forwa
rd pass where an input signal is 
propagated through multiple layers to obtain a predicted output. The second being the 
backward pass, where the error obtained from the predicted and true output is propagated 
back up to the first hidden layer. Using the err
or information, the weights are adjusted based 
on some rules to move the predicted output close to the true output i.e minimize the error. 
 
       
This is often achieved through the backpropagation algorithm. The beauty of the 
algorithm is that the weights
 
are changed accordingly depending on how much they 
contribute to final error. Suppose we consider a Loss function as defined in Equation (4.6), 
where 
n
 
is the number of samples, 
y
 
and 

 
are the true and predicted outputs.
 






















                                             
(4.6)
 
       
A point to note is that this type of evaluating the loss using all samples at the end of 
every 
epoch
 
is called batch or online training
, where an epoch is defined as 
a measure of the 
number o
f times all of the training vectors are 
used once to update the weights. 
By back 
propagating through the network, we can obtain information on how the previous outputs 
influence the error using the chain rule. Assuming there are 
k
 
hidden layers, for any pa
rticular 
neuron, we compute:
 
 
43
 
 

























                                              
(4.7)
 
w
here 
net
ik
 
is the total value entering the activation function in Equation (4.
7
). By looking at 
the individual derivative
s, it becomes obvious to calculate all of them from the final output 
layer to the first input layer.  By repeating the chain rule process, we have all the partial 
derivatives to update th
e weights according to Equation
 
(4.8
).
 
This is typically the Vanilla 
version of back propagation. There are other versions that have been developed
, which
 
are 
more advanced, and computationally more efficient. The overall process of driving the loss to 
a minimum by means of backpropagation is known as gradient descent
, whic
h is given by
 

















 
                                                     
(4.8)
 
     
Here,
 

 
is called the learning rate, which basically regulates the rate of the learning 
process. A general methodology to find 
an optimum learning rate is 
a 
very well posed 

this; rather 
mostly 
by trial and error
.
 
a suitable learning rate can be established. 
However, it is 
t
o 
be
 
noted
 
that 
establishing a suitable learning rate is fundamental to any ML study. 
Choosing too small a value can lead to premature convergence, while a big value can cause 
oscillations near the global minima which
,
 
is commonly known as overshooting.
 
       
The main pu
rpose of a neural network is to classify or predict on unseen data, after being 
trained on seen data. In order to do that, a dataset is broken down in to three main subsets.
 

 
Training dataset
: This is the dataset that is used to train the model which basica
lly 
fits 
for 
all the weights to get the desired output.
 

 
Validation dataset
: In order to verify how well the network is training, some part of 
the dataset is fed in at pre fixed time intervals and the validation loss is monitored. If 
the validation loss inc
reases during training, then it is generally advisable to stop 
training. This is normally known as overfitting
.
 
 
44
 
 

 
Test dataset
: Once training is completed, the weights are all updated and stored as a 
model. This is used on a dataset that consists of previous
ly unseen data to predict or 
classify based on the task.
 
       
Typically, there is again no pre
-
set rule to determine the ratio in which these datasets are 
split from the main dataset. Having too many training examples will result in over 
generalization, 
while having a small training dataset will result in insufficient learning. 
Another important concept to consider is the problem of overfitting and underfitting. If the 
model is complex with multiple layers, and if a line is fit to pass through all the tra
ining 
samples, this results in 
under 
generalisation of the model, where the model has learnt so well 
on the training network 
and yet 
it fails to predict on any unobserved data. This is known as 
overfitting
. The exact opposite occurs when the model is too s
imple, where the model suffers 
from high bias and no matter how many observations is fed into it; the model always 
produces similar results. This is known as 
underfitting
. When the training loss is far lesser 
than the validation loss, this indicates overfi
tting while the opposite indicates underfitting. 
Therefore, achieving the right trade
-
off is one of the most challenging tasks in ML. 
 
 
Figure 4.4 Overfitting and Underfitting [31]
 
       
By nature, 
deep neural n
etworks
 
are always complex models. Hence th
e problem of 
underfitting occurs much lesser than that of overfitting. Therefore, there is a need to avoid 
overfitting when designing deep neural networks. Typically this is achieved by many 
different approaches. The main methods 
use 
regularizers, which re
strict the sudden changes 
 
45
 
 
in weights by adding a penalty term in the loss function. Another way to tackle overfitting is 
by adding dropout layers,
 
which randomly prunes some neurons in the hidden layers thereby 
favouring some features over others. This gre
atly helps in making sure the model doe
s not
 
just generalize on the training dataset. Another commonly exploited technique is data 
augmentation. This simply refers to distorting some features of the data to create more data 
that can be fed in to the model 
to achieve better performance. Deep Neural Networks are 
always data hungry, and more the 
data,
 
better 
is the 
performance in most cases.
 
Typically the 
MLP/DNNs presented this section involves a time consuming feature selection process. The 
features are then
 
fed to the neural network to perform an associated task. But deeper networks 
like Convolutional Neural Networks and Long Short Term Memory Networks eliminate this 
step by having a feature selection process incorporated within the network itself i.e. one 
w
ould have to just feed in the raw signal, and the network would pick out features that best 
describe the input signals and complete an associated task.
 
 
Figure 4.5
 
Example of Data Augmentation [32]
 
 
 
 
 
46
 
 
4.3 
Convolutional Neural Networks
 
(CNN)
 
CNNs take
 
biol
ogical inspiration from the visual cortex. The visual cortex has small regions 
of cells that are sensitive to specific regions of the visual field. This idea was expanded upon 
by a fascinating experiment by Hubel and Wiesel in 1962 where they showed that s
ome 
individual neuronal cells in the brain responded (or fired) only in the presence of edges of a 
certain orientation [33]. A C
NN
 
is a network that performs the Convolution operation (rather 
Correlation) in at least one of its layers instead of the genera
l matrix multiplication and 


contain images for training. 
Some of the most 
salient features of a CNN are spatial down 
sampling, shared weights and local reception. 
 
       
A typical CNN has convolutional layers, pooling layers and fully connected layers. The 
convolutional layer is the first layer that is used to extract features 
from the input that best 
describes the input data.
 
This is done by convoluting the input image by convolutional filters 
called as 
convolutional 
kernels
. The resulting output is typically known as feature maps.
 
Figure 4.6 shows how a convolutional layer wor
ks.
 
 
Figure 4.6
 
Working of a Convolutional Layer
 
 
4
7
 
 
       
Assuming an input image that is a 6
 
x 
6 matrix, a convolutional filter of size 2 is applied 
to the input matrix. A simple multiplication of the elements in the filter and the input images 
is done to 
establish the first entry (@ (1, 1) of the feature matrix. The size of the output 
feature map depends on an entity called
 
stride
. The stride gives the spatial distance between 
the central pixels (both vertically and horizontally) on which the convolution o
peration takes 
place. The feature map in Figure 4.6 is computed with a stride of 1. 
 
Mathematically, the 
convolution operation can be defined as:
 






































             
(4.9)
 
       
The output 
G
 
is the feature map produced by the convolution operation, while
 
f 
is the 
input and 
h
 
is the kernel/filter. 
m 
and 
n 
are the indices of the row and column of th
e output 
feature map. A convolutional layer enforces the idea of weight sharing. If every pixel in the 
image is imagined to be a neuron in the first layer, and if the 
output feature map 
is treated as 
the second layer where each pixel once again is treated 
as a neuron, it is clearly evident that 
at any single given computation, only few neurons from the first input layer is connected to 
the second 
output
 
layer. As the kernel is moved across the image, different neurons are 
connected and this therefore ensure
s shared weights. Shared weights reduce the number of 
parameters to be learnt, and it becomes computationally more efficient.
 
 
 
Figure 4.7
 
Example of weight sharing in Convolutional Layers
 
 
48
 
 
       

s. These layers are used to 
sub sample the feature maps spatially. They reduce the dimensions that we are working with 
and make it computationally more efficient. Similar to convolutional layers, the input image 
is stride over by a kernel, wherein operatio
ns like max pooling, average pooling takes place. 
A point to note is that pooling layers have no learnable parameters. Figure 4.8 below shows 
the working of a max poolin
g layer for a stride length 2.
 
 
Figure 4.8
 
Working of a Max Pooling Layer (stride=2)
 
 
      
The final architectural idea involved in a CNN is the fully connected layer (FCN). This is 
very similar to the MLP networks discussed before, and is basically a classifier/predictor 
model that uses the features extracted by the kernels in previous la
yers. All the parameters of 
this layer 
are
 
learnable and is generally the last layer of a CNN. 
Convolutional Neural 
Networks are generally used only on images, but a lot of recent research has gone into using 

-
CNN) for time series 
data. 
1D
-

inherent features from long time signals. 
1D
-


only difference being the operations is
 
done in single dimension.
 

of kernel size is very critical and can
 
be treated as a hyper parameter. 
 
 
49
 
 
4.4 
Recurrent Neural Networks (RNN)
 
The traditional neural networks and convolutional neural networks all work with fixed input 
and output lengths. But that is never the case in many practical applications (for example, 
f
inding the number of vowels in a sentence). Problems like speech recognition and time 
series predictions or forecasting require a system to store and use context information.
 
More 
so, if we consider a human brain, one of the most salient features is the pe
rsistence present in 
our system i.e. we remember something we lear
nt/saw days or weeks before. Recurrent 
Neural network (RNN)
 
typically brings about persistence in a traditional neural network.
 
Recurrent Neural Networks take the previous output or hidden s
tates in most cases as the 
inputs for the current computation
 
[44]
. The composite input at time 
t
 
has some historical 
information about the happening at time 
T<t
. 
 
RNNs are useful as their intermediate values 
(state) and can store information about past in
puts for a time that is not fixed a priori. 

in general hold information over a certain time that is not fixed. 
Figure 4.9 shows how a RNN 
is different from a typical feed forward network
.
 
 
Figure 4.9 (a) Simple Feed Forward Network (b) A RN
N with pr
evious hidden states a
s input
 
       
Recurrent Neural Networks though in theory are capable of handling long
-
term 
dependencies fall short when it comes to practical applications. This problem was very well 
explored in depth by Hochreiter (1991) and Bengio,
 
et al. (1994) [45]. It was seen that 
remembering information over long periods requires calculating the distances between distant 
 
50
 
 
nodes that involves multiple multiplications of the Jacobian Matrix. Problems with the more 
commonly occurring vanishing grad
ients and lesser frequent exploding gradients caused the 
performance of these models to be not satisfactory. It was seen that a trade of between 
gradient descent based learning and the time over which the information is held was required.
 
In order to overc
ome this, Hochreiter and Schmidhuber (1997) [46] introduced the Long 

-
term 
relationships between distant nodes by designing weight coefficients between connections. 
These networks
 
have shown unbelievable applications in speech processing, Natural 
Language Processing and image captioning among other applications.
 
 
4.
5 Long Short Term Memory Networks (LSTM)
 
A 
LSTM
 




is useful or not and a gating mechanism that contains three non
-
linear gates: (i) an input 
(denoted by 


), (ii) an output (denoted by 


) and (iii) a forget gate (denoted by 


). The gates 
regulate the fl
ow of signals into and out of the cell to adjust long
-
term dependencies 
effectively and achieve successful RNN training. The standard equations for 
LSTM
 
mem
ory 
blocks are given as follows.
 
       
The three gates:
 
 


=

 
(




-
1
+




+


)         
            
                 
           
(
4.
1
0
)
 


=

 
(




-
1
+




+


)     
                 
            
             
(4.11
)
 
 


=

 
(



-
1
+



+


)   
                 
                    
          
(4.12
)
 
 
       
Memory Cells and hidden uni
ts: 
 
 


=





-
1 
+
 
tanh 
(

c
h
t
-
1
+
W
c
x
t
+


)                               (4.13
) 
 
 


=



tanh
 
(


)                                                             (4.14
) 
 
 
51
 
 
 
       
Where 


 
is the external input vector, 
tanh
 
is the hyperbolic tangen
t function, and the 
parameters are the matrices 
W
 
and 
U
, and vector bias 
b
, with appropriate sizes for 
compatibility.
 
      
The output layer: 
 
 


=



+

 
                             
 
                                     
(4.15
) 
 
       
The internal dyn

-
history 
profile. In any time series signal processing scheme in recurrent systems, repeated 

avo
ided. Figure 4.10 shows a LSTM 
network
. Due to the higher number of trainable 
parameters in a LSTM network, the training time is typically very high. Therefore, LSTMs 
though very powerful somet
imes are very painful to train in many cases. CuDNNLSTM is a 
in
built tensorflow API that almost speeds up training by 10 times while using LSTM layers. 
This is achieved by skipping repeated activation function at each cell. A vanilla based LSTM 
cell is shown in Figure 4.11. Alternative methods called Slim LSTMs have b
een investigated 
where the 
number of adaptive parameters at each gate is
 
reduced [47].
 
 
 
Figure 4.10 Architecture of a simple LSTM Network
 
[48]
 
 
52
 
 
 
 
Figure 4.11 Vanilla based LSTM Cell
 
[48]
 
       

act inherent features in 
long temporal signals. This allows us to directly feed raw waveforms (A0 and S0 time 
histories) to the neural networks, thereby eliminating the time consuming feature engineering 
process. This is exploited in the DL based approach,
 
while in the ML based approach, we 
pick out three features (group velocities of three wave modes as a function of frequency) and 
feed it into a multi
-
layer DNN. This is done to address one of the limitations of the DL based 
approach that will be discussed
 
in Chapter 6 in detail.
 
 
53
 
 
CHAPTER
 
5
 
MODELLING TECHNIQUES FOR GUIDED WAVES IN COMPOSITES
 
 
5.1
 
Why do we need modelling?
 
The behaviour of any wave in any dynamic system is generally governed by a set of 
Partial 
Differential Equations
 
(PDE), which is normally cal


 
in 
mechanics
. Apart from the 

 
have an exact analytical 
solution, or at best have solutions that can be reached after an enormous computational effort. 

lve without alternate solution strategies. This 
typically involves assuming a solution involving many constants and we then determine these 
constants such that the governing PDE is satisfied in an approximate sense
. 
To put it in 
simpler words, it is like f
inding one of the solutions for a problem that has infinite solutions.
 
However, the assumed solution should satisfy certain conditions called the boundary 
conditions and initial conditions. This can be thought of as conditions that drive the particular 
sol
ution to the closest possible approximation. Typically there are two types of boundary 
conditions: the natural boundary conditions and the kinematic boundary conditions. The 
former is also called sometimes as the force boundary conditions or in general Nue
mann 
boundary conditions that give the value of the derivatives of the unknown variable at the 
boundary. Meanwhile, the kinematic boundary conditions are also called the essential 
boundary conditions or in general Dirche
l
et boundary condition that gives th
e value of the 
unknown variable at any boundary [34]. The governing PDE is not amenable for this solution 
philosophy due to higher order continuity requirements of the assumed solutions. Therefore, 
one requires an alternate statement of the governing PDE t
hat is generally most suited for the 
numerical solution. This is normally provided by the variational statement of the problem. 
Finite Element 
M
ethod
 
(FEM)
 
is one such method of numerical solution which is
 
further 
 
54
 
 
explained in this chapter. 
In this chapter
 
we also discuss yet another semi
-
analytical method 
called Spectral Finite Element Method (SFEM), which can be described as FEM but 
formulated in frequency domain. There are similarities and differences between these 
methods which will be di
s
cussed in deta
il in this chapter. 
 
 
5.2
 
Finite Element Method
 
 
The finite element method is one of the most common numerical methods, which essentially 
assumes a solution that only solves the kinematic boundary conditions. It is a stiffness based 
method, wherein the domain 
of the problem is broken down to its constituent elements that 
are connected to each other 
by nodes
.
 
 
Figure 5
.1
 
(a) Overall domain of the problem (b) Discretized domain set of the problem with 
boundary conditions
 
 
       
The FEM procedure begins with the
 
assumption of the displacement field in each of the 
elements, and the assumed field is then substituted into the weak form of the differential 
equations, which is basically the integral form of the original differential form of the 
governing PDE.
 
The numb
er of elements is driven directly by the element size, or also known 
as mesh size. Changing the mesh size changes the number of elements. The mesh size is a 
very important parameter when it comes to the accuracy of the solution. For example, if the 
 
55
 
 
stress 
and strain gradients are high at a particular location (this is especially true at the edges 
of a crack or a defect)
;
 
the mesh discretization should be very fine. Also, when 
performing 
high frequency studies
 
(wave propagation problems)
 
where the wavelength
 
is very small, 
mesh sizes that are 
in the
 
scale of the wavelength can act as potential boundaries and can 
reflect waves, which complicates the problem in hand even further.
 
Suppose 
u(x,y,z,t)
 
is the 
displacement vector (in mechanics), the objective here i
s to estimate a solution of the 
dependent variable 
u
 
in the form:
 






























                           
(5.1)
 
       
Where 
a
n
(t)
 
are the unknown time dependent coefficients that are to be determined, and 


 
are the spatial dependent functions that normally satisfy the kinem
atic boundary 
conditions.
 
The displacement vector 
u(x,y,z,t)
 
is a function of all three co
-
ordinates and time. 
The formulation of the different approximate methods is the 
Weighed Residual Technique
 
(WRT), where in the error or the residual obtained by subs
tituting the assumed approximate 
solution in the governing PDE is weighted with a weight function and integrated
 
over the 
domain. This process has 
similarities to the backpropagation algorithm in neural networks 
discussed 
previously.
 
The
 
error obtained by 
the difference in the approximate target solution 
to the true solution is propagated back to each neuron (node) and the 
weights (weighted 
function) are
 
updated accordingly. 
The final integration is basically the adding up of the 
contribution of each neuron
 
(node).
 
       
In complex structures, the different approximation techniques are very difficult to use. 
The main difficulty lies in determining functions



, which are called the 
Ritz functions
. 
Typically methods like 
Rayleigh
-
Ritz
 
method [35] cannot be used in complex structures. 
Hence 
by 
dividi
ng the domain into sub domains (
which constitutes an elem
ent in the finite 
element mesh), 
one can apply diff
erent approximation techniques over all the elements which 
are then pieced together to obtain the total approximate solution. This in essence is the Finite 
 
56
 
 
Element Method.
 
To summarize, the flow chart in Figure 5.2 gives an overall perspective of 
the diffe
rent steps involved in the FEM.
 
       
The Figure 5.2 can be explained as follows: FEM method 
fits the approximate solution 
for the dependent variable as given in 
Equation (5.1)
 
over each element. Using this equation 
under the WRT framework, reduces the go
verning PDE into a set of coupled 
Ordinary 
Differential Equations (
ODE) of the form as shown below
 


















                                               
(5.2)
 
       
Where 
[M]
 
is the assembled mass matrix, 
[K] 
is the assembled stiffness ma
trix, 




 
is 
the nodal acceleration vector 
{
u
}
 
is
 
the nodal displacement vector and 
{
F
}
 
is the nodal force 
vector
. After enforcing displacement boundary conditions, the above equation is time 
integrated by replacing the differential operators with 
finit
e difference
 
coefficients. The 
details of the assembly of elemental matrices to obtain the assembled 
[M]
 
and 
[K]
 
is given in 
[34] and not repeated here. Although there are many time integration scheme is reported in 
the 
literature
, 
Newmark
-

 
central differ
ence method
 
[34]
 
is the most common method 
and 
the 
Figure 5.2 explains this method of obtaining dynamic 
displacements
 
and velocities.
 
 
 
Figure 5.2 Finite Element Method Procedure [34]
 
 
57
 
 
       
For modelling guided waves using FEM on th
in plates as used here in this thesis, we 
need to enforce zero stress (both normal and shear stresses) on the top and bottom surface. 
This can be accomplished by not enforcing any 
displacement boundary conditions on these 
surface nodes. The details of how 
this is modelled using commercial software COMSOL
®
 
is 
explained in the next section.
 
 
5.3
 
FEM Modelling using COMSOL ® Multiphysics
 
COMSOL ® Multiphysics is a software package that among its many 
functionalities
 
lets us 
model guided waves in structures. The wa
ves are modelled using the FEM described in the 
last section, where the governing equations to be solved are discussed previously in Chapter 
3. In this section, the details of the simulation model are discussed. A 2D FEM model is 
developed to mimic guided 
wave propagation in thin composite plates. A 16 layered 
unidirectional symmetric laminate has been considered in this case. A zoomed in version of 
the cross section of the geometry is shown in Figure 5.3.
 
The material properties of the 
composite used here 
is given in table 5.1.
 
The length of the composite is 2.5 
m,
 
while it is 2 
mm thick, with each layer 0.125 mm thick.
 
 
Figure 5.3 Cross sectional view of the sample geometry
 
Direction of wave propagation
 
 
58
 
 
       
As seen in Figure 5.3, an input disturbance is given at the top right of 
the sample. 
The 
disturbance is given normal to the direction of wave propagation. 
The input signal is a 7 cycle 
cosine function passed through a hanning window. The central frequency of the signal is 25 
KHz. Figure 5.4 shows the time domain and frequency d
omain representation of the input 
disturbance
 
e
, while 
Equation (5.2)
 
describes the disturbance mathematically.
 
 
















                  
                
      
(5.3
)
 
Figure 5.4 (a) Time domain representation of the input tone burst (b) Frequency domain 
representation of the input tone burst
 
 
       
A
s discussed before, the finite elemen
t mesh and its related parameters are important 
parameters to any numerical study. In this particular study, we have used the quadrilateral 
(QUAD) to mesh the whole domain [36]. A fine mesh is used where each mesh element is of 
the size 
0.75 mm.
 
 
        
F
igure 5.5
 
shows the meshing of the whole domain for solving the governing PDE using 
FEM. A time dependent analysis using the direct linear solver MUMPS available in 
COMSOL ® is used to simulate the ultrasonic guided wave in the composite. The total time 
in
terval 
for the simulation is 1200 
µ
s
 
while the time step is 2 
µ
s
. 
The stress free boundary on 
the top and bottom surfaces are enforced as explained in the previous section.
 
 
59
 
 
 
Figure 
5.5 Finite
 
Element Mesh for the composite laminate
 
Table 5.1
 
Material Prop
erties of a transversely isotropic lamina
 
 

 
(kg/m
3
)
 
E
1
 
(GPa)
 
E
2
 
(GPa)
 

12
 
v
23
 
G
12 
(Gpa)
 
1750
 
173.1
 
12.5
 
0.31
 
0.38
 
2
 
        
The responses are received at a location 
500 mm 
away from the actuator
. When the 
composite laminate is excited, the two fundamental modes A0 and S0 are generated and 
propag
ate through the structure. The edges of the composite are artificially loaded with low 
reflecting conditions to avoid boundary reflections to the best extent possible.
 
Figure 5.6a 
shows the received waveform and Figure 5.6b shows the filtered waveform wher
e the 
fundamental wave modes are isolated. The filtering is done using the knowledge of the group 
velocities of the fundamental wave modes in the composite laminate.
 
 
Figure 5.6 (a) Received response at 500 mm from the actuator (b) Filtered received respo
nse 
to isolate the fundamental wave modes
 
 
60
 
 
       
Through multiple research advancements and increase in computational capabilities, the 
numerical FEM approach can produce very accurate solutions that are close to the ground 
truth. They indeed find multiple
 
uses in various applications. But due to the way FEM is built 
upon, the computational requirement are way too high. For example, if one is dealing with a 
very fine mesh
 
(for problems involving in high frequency excitations)
, thereby increasing the 
number 
of elements which directly increases the computation time and memory. Therefore, it 
was important to work a way around this and Spectral Finite Element Method (SFEM) 
provides very accurate 
alternative but
 
with 
very small 
computational requirements. The nex
t 
section describes the basic working
 
principles
 
of SFEM and some fund
amental derivations 
based on it
 
 
5.4
 
Spectral Finite Element Method
 
The 
SPECTRAL
 
FINITE
 
ELEMENT
 
METHOD
 
(
SFEM
) is 
a method 
based on FEM principle. 
This method is 
typically used to solve dynam
ic
 
wave propagation problems. In this method, 
the field variable in the governing 
PDE
 
is transformed into frequency domain using 
Fast 
Fourier Transforms
 
(FFT), which is
 
a
 
numerical version of the 
Discrete Fourier Transforms
 
(DFT). 
The D
F
T is performed by a
 
FFT algorithm popularly known as the 
Cooley Turkey
 
algorithm [37] and is expressed as described in Eq
uation
 
(5.3). 
When transformed to the 
frequency domain, 
for a 1D realization, 

Differential Equations) having fre
quency (sampling frequency in FFT) as the primary 
variable
 
and in most cases 
the resulting equations are ODEs with 
constant coefficients.
 
The 
finite element procedure is then applied to the transformed governing equations
 
by solving 
them in the wavenumber 
space (k space)
 
to find the exact solution for the ODEs
.
 
SFEM 
employs this exact solution as an interpolating function for element formulation. The 
constants of integration are driven to satisfy the boundary conditions, and therefore all the 
 
61
 
 
requirements a
re satisfied at every discrete frequency. Using the Inverse Fast Fourier 
Transform (IFFT), the time domain representation of the solution can be obtained. 
The SFEM 
procedure is shown as flow chart shown in Figure 5.7
 
       
There is more to SFEM to than ju
st transforming the governing PDE into frequency 
domain and solving the ODE. In essence, it can be categorized as a type of FEM as methods 
like the Ritz method 
that 
can be employed in frequency domain to obtain the structural 
stiffness matrix similar to th
e stiffness matrix obtained in FEM. In essence, it can be 
categorized
 
as a type of FEM [38
].
 

































                      
(5.4
)
 
       
Where N is the number of FFT points, 

n
 
is the discrete circular f
requency. The 
formulation of SFEM for 2D structural waveguides is more complex, because the governing 
PDEs can no longer be reduced to a set of simple ODEs when converting to the frequency 
domain. 
Here we will need the second Fourier Transform in one of th
e spatial direction to 
reduce the dimens
ionality of the problem to one dimension
.
 
In the next two sections, the 
formulation of spectral elements for a isotropic rod and a higher order Timoshenko beam is 
outlined, as these are the two formulations used in t
he thesis.
 
 
Figure 5.7
 
SFEM Procedure of solving PDE
 
 
62
 
 
       
In the previous section we described time domain 2
-
D FEM procedure where in we got 
the low frequency guided waves (A
0
 
and S0 modes) in a single analysis. SFEM procedure is 
normally used to analys
e 1
-
D wave propagation (both A0 and S0 propagation). The question 
is how the equivalence of these two methods can be used to estimate guided wave modes and 
thereby use these results for material property identification. According to [34], the 2
-
D 
guided wa
ve modes can be accurately obtained using higher order 1
-
D wave theories. Hence, 
we use here higher order Timoshenko beam theory to obtain accurate estimate of S0 mode, 
while we use elementary theory to obtain A0 mode. Reference [34] also shows that the 
ac
curacy of the A0 mode does not suffer much by using elementary rod theory. Hence, in the 
next section we will discuss in detail the formulation of Spectral 
Elementary
 
Rod and 
Timoshenko beam element.
 
 
5.5
 
Non
-
dispersive Isotropic Rod: FFT Based Spectral Elemen
t Formulation
 
A rod element of length 
L
 
model has two degrees of freedom u
1 
and u
2
 
which define the axial 
the two axial boundary conditions at 
x=0
 
and 
x=L
. The corresponding forces at the two nodes 
are 
F
1
 
and 
F
2
. The governing differential (PDE) equation f
or an isotropic homogeneous rod 
of density 

 

E
 
is
 
 
 
Figure 5.8
 
Spectral Isotropic Rod Element
 
[19
]
 
 















                                     
                            
(5.5
)
 
 
63
 
 
       
Where 
u=u(x,t)
 
is the axial displacement and 
c
2
=E/

is 
the square of the wave speed of 
the material. The force boundary condition is given by
 










                                
                            
(5.6
)
 
       
Where 
A
 
is the cross sectional area of the rod and 
F(x,t)
 
is the axial 
force due to the axial 
displacement. The displacement (essential) boundary condition is the specification of the 
displacement 
u 
at the boundaries. Assuming a solution of the form
 


























       
                 
           
(5.7
)
 
       
Substituting Equation (5.7) in E
quation (5.5
), the reduced governing ODE becomes
 
















                             
                            
(5.8
)
 
       
Whose solution is of the form 
u
o
e
-
jkx
. U
po
n substitution in E
quation (5.8
), the discretized 
form of the ODE becomes
 















                                                 
  
(5.9
)
 
       
The wavenumber can be computed trivially in this case as 
k
n
=
±w
n
/c 
and for both mod
s, 
wave amplitude can be taken as 1. Thus the complete solution is,
 






























                           
(5.10
)
 
       
Where 
c
1
 
and c
2
 
are coefficients to be determined using the boundary condition
s, and L 
being the length of the element (isotropic rod). Following a derivation as shown in [1], we 
obtain a relationship between the nodal forces and nodal displacements as given by
 
 























             
             
                           
(5.11
)
 
       
Where 
T
2
 
and 
T
1
 
are defined as 
 










































 

































                      
                 
(5.12
)
 
 
64
 
 
       
Hence, the dynamic stiffness matrix (DSM) for the rod at frequency 
w
n
 
is
 









                               
                           
(5.13
)
 
       
FFT based SFEM requires the formulation of one noded infinite segment called the 
throw off element for good time resolution. This is obtained by removing the reflected 
coefficients from the solution. The solution to the throw off elements and the d
ynamic 
stiffness for the throw off elements is given in Equa
tions (5.14) and (5.15
)
 
 

















                       
                           
(5.14
)
 










                                 
                           
(5.15
)
 
 
5.
 
6
 
Spectral
 
Element Formulation of Timoshenko Beams
 
The guid
ed wave propagation in a 1D composite laminate was shown in Section 3.5. There, 
we assumed a 1D waveguide that satisfies both the elementary rod and beam theory. In the 
beam theory, the slope 

 
is equal to 
dw/dx
. However, in the Timoshenko theory, 

 
is ass
umed 
as an independent variable. Timoshenko beam theory captures the beam dynamics more 
accurately with the introduction of shear deformations. It is therefore preferred to generate 
responses that mimic the S0 mode. 
Spectral element formulation requires go
verning PDE, 
which for Timoshenko beam is given by
 
                  













                                            
(5.16
)
 



















                   
                
(5.17)
 
       
where 
w
(x,t)
 
is the transverse deformation, 

 
is the rot
ation of the mid
-
plane of the 
beam, 
G
 
is the shear modulus, 
A
 
and 
I
 
are the area and area moment of inertia of the cross 
section, 
K
 
is the shape factor that takes care of anomalies in the shear stress deformation 
across the depth of the cross section. For 
rectangular cross section, 
K=0.85. 
All the quantities 
with two dots on them
 
represent the acceleration. These equations are supplemented by the 
following force boundary conditions
 
 
65
 
 
















                          
      
(5.18
)
 
       
Where 
V
 
is the shear force and 
M
 
is the bending moment. Assuming the solution of the 
form
 


































 
    
         
(5.19
)
 
       
Substituting Equation
 
(
5.19
)
 
in Eq
uatio
n 
(
5.18
)
, we get fourth order equation for 
wavenumber 
k
, which is given by 
 













































       
   
           
(5.20
)
 
       
Where, 
















 








 
       
Solving Equation
 
(
5.20
)
, we get four modes of the form 













 
and the 
complete solution at frequency 


 
is given by
 






































             
       
       
(5
.21
)
 
       
Where 


 
is an unknown coefficient to be determined with four boundary conditions at 
the two nodes of the Ti
moshenko bea
m. Evaluating Equation
 
(
5.21
)
 
at the two ends of the 
beam situated at 
x=0
 
and 
x=L
, where 
L
 
is the length of the beam with 



and 



 
being the 
amplitude ratios at four 
wavenumbers













, we get
 




 
                 
    
                
(5.22
)
 
       
This matrix gives relationship between unknown coefficients 


 
and nodal degrees of 
freedom, 














. Similarly, For
ces and Moments given by Equation (5.18)
 
is 
evaluated at two nodes
 
located 
at x
=0
 
and 
x=L
 
as 













































 
 
66
 
 
        
Evaluating the forces, we can express unknown coefficients 


in terms of nodal degrees 
of freedom as 
 




















































                           
   
 
(5.23)
 
       
Where 
[
T
2
]
 
matrix is given by
 




















































 


















































   
(5.24
)
 
       
The Dynamic stiffness matrix



] of the Timoshenko beam is then given by 
 



] =










     
                              
                         
(5.25
)
 
 
5.7
 
FEM vs SFEM: A comparison
 
A thin 2D composite plate was modelled using the FEM approach, while 1D waveguides 
were modelled using the SFEM approach. By using the elementary rod element in SFEM, we 
could mimic the A0 mode
 
similar to the ones generated in the FEM approach, while the 
higher order Timoshenko Beam was used to generate the S0 mode. 
Since the FEM model was 
in two dimensions, both the modes are coupled. Using the SFEM on 1D waveguides, the 
modes are decoupled, an
d A0 and S0 modes can be obtained individually depending on the 
direction of incident loading (axial loading or transverse loading). This can be seen in the 
Figure 5.9. The velocities of the two modes obtained using the two different methods are 
calculated
 
and is outline in Table 5.2.
 
Table 5.2
 
Wave
 
velocities: SFEM vs FEM
 
Method
 
S0 Velocity (m/s)
 
A0 velocity (m/s)
 
SFEM
 
8696
 
1048
 
FEM
 
8602
 
1064
 
Error %
 
1.07
 
1.52
 
       
 
 
67
 
 
       
From Table 5.2, it is observed that the velocities of the two wave modes match
 
very well 
for both the methods. The
refore, we can conclude that, at this particular frequency and by 
using just the fundamental wave modes A0 and S0, it is possibly to scale down a thin 2D 
laminate plate to a 1D waveguide. 1D waveguides can easily be mode
lled in SFEM to 
generate wave modes that mimic the A0 and S0 modes generated in the FEM scheme. This is 
particularly important as SFEM is computationally many orders better than FEM. The 
computation time for an FEM simulation in this case is about 30 mins,
 
compared to a SFEM 
simulation that takes seconds to execute.
 
 
Figure 5.9 
(a) and (b): A0 and S0 response using the SFEM method. (c) Overall raw 
waveform obtained from the COMSOL FEM simulation (d) Filtered waveform isolating the 
individual A0 and S0 refl
ections (filtering done using knowledge of group velocities of these 
two modes)
 
 
68
 
 
       
There are of course methods to optimize the FEM model to reduce the computation time, 
but even at the best performing level, any COMSOL simulation takes couple of minute

which is still more compared to that of SFEM. It is hence important to be able scale down 
geometrically, in order to model guided waves using SFEM. This enables us to generate a 
much larger dataset that in turn can be used to train data hungry netw
orks like 1DCNN and 
LSTMs to achieve much improved performance levels. Therefore, for the DL based 
approach, we use the SFEM approach to generate a large dataset to solve the inverse problem 
of materials characterization.
 
 
 
                                
  
                          
             
                        
                                                                                                                             
                                                                  
                                                                                                                             
                                                                   
 
69
 
 
CHAPTER
 
6
 
SENSITIVITY ANALYSIS AND DATA UNIQUENESS
 
 
In this the
sis, one of the main objectives was to build an overall framework
 
for material 
property identification in composite structures
, complete with signal processing, and other 
data handling algorithms. In this section, the different methodologies used are discu
ssed. The 
overall flow of the research is also described in detail.
 
Figure 1.6 (see Section 1.7)
 
shows a 
flowchart detailing the flow of 
analysis methodology
. Initially as part of the forward problem
 
for the deep learning based approach (
DL based
)
, we mode
l guided waves in a composite 
laminate using two different methods (Numerical FEM and Semi Analytical SFEM). For this 
process, the priori knowledge of the material properties of different composites used typically 
in the industry was considered. In the pre
vious chapter, the advantages and disadvantages of 
the two di
fferent modelling techniques were
 
discussed. 
 
        
It was concluded that, for the problem of material characterization, it is most advisable 
to use SFEM to model guided waves. This allows us t
o collect or generate a large database of 
A0 and S0 waveforms for different combinations of material properties
 
at a very small time
.
 
A large training dataset is always preferred to train deep neural networks in order to achieve 
better performance levels. 
Figu
re 6.1
 
shows how the 
A0 wave mode 
change for two different 
sets of material properties.
 
        
Note that unlike in typical wave propagation and structural health monitoring problems, 
which necessarily deals with high frequency loading (which in turn r
equires very very fine 
finite element mesh for analysis), material property identification analysis can be performed 
using only two guided wave modes (A0 and S0), which are low frequency modes. Hence, in 
all the analysis performed in this thesis, we use an
 
input tone burst signal of 25 kHz. The 
 
70
 
 
main advantage of using such low frequency signals is that the FE analysis does not require 
too fine a mesh to capture the two guided wave modes.
 
 
Figure 6.1
 
A0 waveforms for two different sets of material propertie
s (
Blue
: 

= 1000 kg/m
3
, 
E
1
= 50 GPa, 
E
2
= 5 GPa, 
v
12
= 0.25, 
v
23
= 0.25,
G
12
= 2 GPa,
 
Orange:
 

= 1750 kg/m
3
, 
E
1
= 
111.53 GPa, 
E
2
= 5 GPa, 
v
12
= 0.25, 
v
23
= 0.25, 
G
12
= 2 GPa)
 
 
       
Using the two numerical models, and at the frequency range of 25 KHz, it is not poss
ible 
to estimate or identify the shear modulus 
G
12
 
(Section 6.1). Therefore, we propose an 
alternative strategy to predict shear modulus 
G
12
. This is done by using 
software
 
called 
DC
®
 
to generate 
and create a dataset using 
group velocity dispersion curves 
for different sets of 
material properties
 
[39]
. The dispersion plot data for three modes A0, S0 and 
the 
shear 
horizontal mode SH0
 
are
 
directly fed into a Dense Neural Network to predict the material 
properties. By incorporating the SH0 mode, the framework 
extends its capability to predict 
shear modulus 
G
12
. 
Similarly the deep models like the 1DCNN and LSTMs are trained on a 
large dataset collected using the SFE method to predict on the lamina properties. Two key 
challenges that come into picture in the proc
ess of data collection are:
 

 
What material properties to change every forward pass?
 

 
How unique is each set of A0 and S0 waveform? (SFEM Method)
 
 
71
 
 
       
In the next few se
ctions
, we try to answer the two above questions by using Sensitivity 
Analysis, and data
 
uniqueness checks.
 
 
6.1 Sensitivity Analysis
 
Sensitivity analysis is an important part of any inverse problem. Several previous work 
revolving around solving materials characterization as an inverse problem have used 
sensitivity analysis to different effe
cts [15][40][41][42]. The primary reason of this analysis 
with respect to this thesis is:
 
1.
 
To verify whether the five material properties and density 
independently 
have any 
effect on the A0 and S0 wave velocities.
 
2.
 
If they do have an effect, quantify it usin
g some means to gain a better understanding.
 
       
This is important because the training datasets are collected by perturbing material 
properties every forward pass to collect the respective A0 and S0 wave forms. By this 
analysis, we 
aim to establish whi
ch parameter changes wave velocities the most, thereby 
adjusting our strategy to collect data accordingly. This is useful f
rom
 
the point of view of 
training a Convolutional 
N
eural 
N
etwork to learn the inverse problem effectively. The wave 
velocities are fo
cussed as the central part of this study, as they would be the most likely 
feature to change for different sets of material properties.
 
       
One of the parameter is varied in a range and the effect on the 
velocities
 
is calculated 
while taking the mean, m
inimum or maximum of other non
-
involved parameters. For 
example, density is varied from 400
-
3200 kg/m
3
 
while other parameters are set at 
E
1
 
= 220 
GPa (mean of the range 40
-
400 GPa), 
E
2
 
= 41.5 GPa (mean of the range 3
-
80 GPa), 

12
 
= 
0.35 (mean of the range 0.2
-
0.5),
 

23
 
= 0.35 (mean of the range 0.2
-
0.5), 
G
12
 
= 6 GPa (mean 
of the range 2
-
10 GPa). Secant Sensitivity is used to quantify the sensitivity of a parameter to 
the wave velocities. It is described as:
 
 
72
 
 












                                                             
(6.1)
 
       
Where, 
C
e
 
is end value of velocity in m/s for lamina property 
P
e
 
and 
C
s
 
in the start value 
velocity for
 
P
s
.  
 
 
       
Figure 6.2
 
Sensitivity of density (rho) on A0 a
nd S0 wave velocities
 
       
Figure 6.2
 
shows the effect of density on A0 a
nd S0 wave velocities. Figure 6.3
 
shows 

E
1
 
and 
E
2
 
and Figure 6.4
 

 
ratio and the shear modulus 
G
12
 
on the wave velocit
ies. 
The sensitivity values are 
summarized in Table 6.1.
 
Figure 6.3
 


 
 
 
73
 
 
Figure 6.4
 

v
12
 
and 
v
23
 
on A0 and S0 wave velocities (b) 
Sensitivity of Shear Modulus 
G
12
 
on A0 and S0 wave velocities
 
 
Table 6.1 Secant Sensitivity Values
 
Parameter
 
Secant Sensitivity
 

 

S0
=5.61
 

A0
=
-
0.215
 
E
1
 

S0
=24.8
 

A0
=1.20
 
E
2
 

S0
=10.61
 

A0
=0.47
 
v
12
 

S0
=2569
 

A0
=1
13.7
 
G
12
 

S0
=0
 

A0
=0
 
 
       
The following conclusions can be made from the sensitivity analysis conducted above:
 
1.
 
The parameters are more sensitive to the S0 mode compared to the A0 mode. This can 
be seen bot
h in the above figures, and in T
able 6.1.
 
2.
 
The 
wave speeds are more sensitive to E
1
 
compared E
2
.
 
3.
 

12
 
seems to 
have the biggest effect on the wave speeds
 
4.
 
G
12
 
has no effect on the wave velocities and therefore, it is conceptually no
t possible 
to predict G
12
 
from a neural network using S0 and A0 time histories as inputs. 
 
 
74
 
 
       
Therefore, after this analysis it was decided to not perturb shear modulus G
12
 
as it ha
s no 
effect on the speeds. The four
 
material properties and density 
wer
e
 
perturbed and
 
its 
respective A0 and S0 wave modes were collected using
 
the SFEM model outlined in the 
previous chapter.
 
 
6.2 Data Uniqueness
 
The forward system is a non
-
unique system. For different sets of material properties, it is 
possible to get simil
ar A0 and S0 waveforms. This can adversely affect the performance of 
any Deep Learning algorithm, for there exists similar A0 and S0 waveforms for different sets 
of material properties. This is similar to one when there is a single equation but two differe
nt 
solutions to it. In 
these cases
, it becomes impossible to find one unique solution. Therefore, it 
is important to clean the training dataset. This is achieved by having a uniqueness check 
before training. Once again, the velocity of the wave modes are a
nalysed and training se
ts 
(A0 and S0 waveforms) where
 
A0 and S0 wave
s with same velocities
 
are same are deleted. 
This then ensures that every resulting training dataset is unique to a particular set of material 
properties.
 
        
Originally, more than 100
00 training sets were generated for more than 10000 different 
sets of material properties using the SFEM model. After the uniqueness check, only 2719 
unique training sets were found. Figure 6
.5
 
shows the A0 and S0 velocities of a
n example
 
training set that
 
is 512 samples big. 
 
After the uniqueness check, the training dataset reduced 
to a size of 88 unique A0 and S0 pairs.
 
 
75
 
 
 
Figure 6.5
 
Uniqueness check for a dataset that had 512 samples originally but trimmed down 
to 88 after the check
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
76
 
 
CHAPTER
 
7
 
STUDY PARAMETERS FOR THE IDENTIFICATION OF ELASTIC PROPERTIES 
OF A
 
COMPOSITE
 
 
In this thesis, a complete framework with inter linking data analysis algorithms and different 
learning models are 
explored
 
for solving the inverse problem of ma
terial property 
identification. In particular, two deep architectures i.e. there 1D Convolutional Neural 
Network and Long Short Term Memory Networks are used, while a much shallower ML 
based Dense Neural Network is also used in the framework. The data generation and 
colle
ction, and the study parameters for the DL based and ML based approach are different 
and 
are
 
highlighted in the next two sections.
 
 
7.1 Study Parameters: DL Based Approach
 
The DL based approach uses a dataset created by the Spectral Finite Element Method. 
The 
SFEM method was finalized based on the advantages it has as discussed in 
section 5.7
.
 
Sensitivity Analysis (S
ection 6.1) is used to di
ctate the way data is generated. A dataset 
consisting of more than 10,000 pairs of A0 and S0 histories are generated f
or 10,000 different 
sets of material properties. The 
da
ta uniqueness check
 
is 
used to filter the created dataset o
f 
repeated and redundant data (S
ection 6.2).
 
After this process, the size of the dataset is 
reduced to 2719 unique pairs of A0 and S0 time his
tories. 
The composite considered is a 16 
layered unidirectional laminate that is transversely isotropic. The thickness of the composite 
laminate is 2 mm and each of its 16 layers 
has
 
the same material properties and same 
thickness.
 
The created dataset is u
sed to train the two main deep architectures commonly 
ex
plored in the field i.e. the 1D
CNN (Section 4.3) and the LSTM (Section 4.5). The model 
 
77
 
 
trained on noiseless data is used to predict the five material properties 
(

, E
1
, E
2
, v
12
, v
23
)
 
on 
data both in t
he presence and absence of noise of different levels.
 
 
7.2 Study Parameters
: ML Based Approach
 
In the previous chapter (section 6.1), the limitation of the current DL based models in 
predicting Shear Modulus 
G
12 
is highlighted. Hence, we adopt an alternati
ve Machine 
Learning based (ML based) approach that utilizes group velocity dispersion data directly as 
inputs to a multi layered Dense Neural Network (DNN). To obtain the group velocity 
dispersion data, an open source software called the DC
®
 
is used. This 
software is very similar 
to the very popular 
DISPERSE
®
 
software, which are both fundamentally developed to study 
the dispersion phenomena of guided waves in different materials and configurations. The 
basics of the dispersion phenomena and group velocity a
re introduced in Chapter 3 (Section 
3.4). 
 
       
In order to increase the complexity of the whole framework, three different composites 
layups
 
([0 0]
s
, [0 0 0 0]
s
, [0 0 0 0 0 0 0 0]
s
)
 
of three different thicknesses
 
(1mm, 2mm and 
3mm)
 
are considered. All t
he composite layups considered are symmetric, unidirectional and 
transversely isotropic in nature (similar to the DL based approach). 
Eight commonly used 
composites are considered
. The material properties of the eight composites are given in Table 
7.1.
 
   
    
The data collection is done using the DC
®
 
software as previously mentioned.
 
In 
particular the group velocity data as a function of frequency is taken for three different wave 
modes. The first two modes are the fundamental wave modes of a lamb wave i.e 
the A0 and 
S0 mode as used in the DL based approach. The third new mode incorporated in this method 
is the Shear Horizontal (SH0) mode. Including the physics of this particular mode enables us 
to also predict Shear Modulus 
G
12
. The shear horizontal mode i
s another type of guided wave 
 
78
 
 
similar to lamb waves. A lamb wave acts in plane while the SH0 mode acts out of plane i.e. if 
the lamb 
waves act 
in x
-
y direction, the SH0 mode acts in the x
-
z direction. 
The shear mode 
is typically generated due
 
to a transver
se loading that br
ing
s about a bending motion, thereb
y 
affecting the shear modulus of the composite
.
 
A typical group velocity dispersion curve 
involving the three different modes is shown in Figure 7.1.
 
Table 7.1 Material Properties of the composites used 
for the ML based study
 
Composite
 

 
(g/cm
3
)
 
E
1
 
(Gpa)
 
E
2
 
(Gpa)
 
G
12
  
(Gpa)
 
v
12
 
v
23
 
AS4M8502
 
1.55
 
144.65
 
9.636
 
6
 
0.299
 
0.289
 
SAERTEX7006919
 
1.454
 
119.99
 
7.25
 
6.02
 
0.322
 
0.447
 
SigrafilCE125023039
 
1.5
 
128.555
 
6.873
 
6.1
 
0.330
 
0.375
 
T300M914
 
1.56
 
139.817
 
10.052
 
5.7
 
0.313
 
0.478
 
T700M21
 
1.571
 
125.5
 
8.7
 
4.135
 
0.37
 
0.45
 
T700PPS
 
1.6
 
149.956
 
9.988
 
4.5
 
0.291
 
0.368
 
T800M913
 
1.55
 
152.137
 
6.653
 
4.2
 
0.252
 
0.543
 
T800M924
 
1.5
 
161
 
9.25
 
6
 
0.34
 
0.41
 
 
Figure 7.1 Group Velocity Dispersion Curve for a random composite with three modes (A0
, 
SH0
 
and S0) high
lighted
 
 
 
79
 
 
CHAPTER
 
8
 
RESULTS AND DISCCUSION
 
 
As previously described in the previous chapters, a large dataset containing sets of A0 and S0 
waveforms were collected by perturbin
g the material properties every 
iteration. This 
constituted the forward problem,
 
and in the inverse problem we reverse engineer to obtain the 
material properties using the collected dataset. In this chapter, the training and testing results 
of three different ML architectures (DNN, 1DCNN and LSTM) are discussed. The benefits 
and disad
vantages of the models are 
highlighted 
and related comparisons are also established. 
The performance of these models in the presence of noise of various levels in the data is also 
studied. Finally, the limitations of the frameworks used are also discussed 
in detail. Figure 
8
.1 summarizes the research flow for the deep learning architectures (1DCNN and LSTM) 
used in this thesis. In 
the 
case of DNN, the data collection process is different as we have 
used the DC
®
 
software to generate group velocity curves for
 
different sets of material 
properties (or different composites).
 
 
 
Figure 
8
.1
 
Research Flow for the two Deep Learning Models (1DCNN and LSTM)
 
 
80
 
 
 
The metrics used to evaluate the performance of the learning algorithms are the Mean 
Absolute Error (MAE), and 
the coefficient of determination (R
2
 
coefficient). Since the 
problem in hand can be viewed as a regression task, where
in
 
we are trying to regress two 
inputs with reflections of certain amplitude to five absolute material property values. 
We are 
training ou
r models on Google Colaboratory, which is a free cloud
-
based Python platform to 
train ML models. We are provided with Tesla T80 GPUs, Intel
-
 
Xeon CPU with 
2.30 GHz 
clock speed (2 Cores),
 
RAM 
of 12 GB and storage of 312 GB.
 
 
8
.1 Training Results with 1D Con
volutional Neural Network
 
The dataset collected using the SFEM consisted of 2
8
19 unique A0, S0 waveforms for 2
8
19 
different composites.  
Training and validation samples are split in 9:1 
ratio 
which gives 244
8
 
training examples and 2
8
2 test 
(validation) 
exa
mples. S0 and A0 are fed as two different 
input 
features
 
(Figure 
8
.1)
 
into the network and 5 lamina properties 
act as the output labels.  
A supervised regression based 3
-
 
Conv. + 2
-
Dense layered 1D
-
CNN (see Table 
8
.1
) is trained 
to learn the relationship b
etween ultrasonic guided waves and 
corresponding lamina 
properties. 
 
Table 
8
.1
 
1DCNN Architecture
 
Layer
 
Units
 
Ou
t
put
 
Parameters
 
Conv1D (LRelu)
 
(3,16)
 
(4096,16)
 
112
 
Maxpool
 
(2,)
 
(204
8
,16)
 
0
 
Conv1D (LRelu)
 
(3,32)
 
(2095,32)
 
1568
 
Maxpool
 
(2,)
 
(1022,32)
 
0
 
Conv1D (LRelu)
 
(3,64)
 
(1020,64)
 
6208
 
Maxpool
 
(2,)
 
(510,64)
 
0
 
Flatten
 
-
 
-
 
32,640
 
Dense (LRelu)
 
64
 
-
 
522,256
 
Dropout (0.05)
 
-
 
-
 
0
 
Dense (Linear)
 
5
 
-
 
85
 
Total parameters
 
 
 
530,229
 
 
 
81
 
 
       
The hyper parameters like learning rate, batch size, number of e
pochs etc. were set after 
some trial and error testing where a trade
-
off between the best performance and quick 
training time was desired. A learning rate of 0.01 and a batch size of 4 were found most 
suitable for this architecture.
 


function is used. The network is trained for 500 epochs
. 
The training is performed keeping in 
mind the factors like:
 
1.
 
Under fitting and overfitting. 
 
2.
 
A smoother loss curve
 
3.
 
Generalization on test set.
 
       
The loss curve, 
MAE and R
2 
coefficient curves are given in Figures 
8
.2, 
8
.3 and 
8
.4 
respectively. 
It is seen from 
the above figures 
that the network is able to obtain a MAE of 32 
and a R2 coeffi
cient of 0.98. The model takes 4
s/epoch to train and 1ms/sample for 
prediction
. The MAE and the loss is a high value, because the target labels (material 

11
 
to 10
-
1
.  If 
normalized, the loss and MAE values will be low and closer to zero.
 
 
Figure 
8
.2
 
Loss
 
curve: MSE vs number of epochs for 1DCNN
 
 
82
 
 
 
Figure 
8
.3 MAE curve: MAE vs number of epochs for 1DCNN
 
 
Figure 
8
.4 R
2 
coefficient curve: R
2 
coefficient vs number of epochs for 1DCNN
 
 
       
Once the 1DCNN is trained, a dataset containing previously unseen ex
amples are fed in 
to the network to predict its respective material properties.
 
The prediction results are 
presented in the Figure 
8
.5 and Table 
8
.2.
 
Looking at Figure 
8
.5, the ideal scenario would be 
all scatter points lying on the linear 
y=x
 
line (green 
dotted). The network performs very well 
in predicting the material properties, but they are certain outliers or poor predictions. One 
such prediction is highlighted in red (see Table 
8
.2 for values).
 
For 
the material property 
E
2
 
in 
the 2
nd
 
example, 
the pre
dicted
 
value 
is 2
8
.8
 
GPa but the true value is 9 GPa. 
We have tried to 
 
83
 
 
use 
the model
 
to predict on samples with almost similar material properties (only one/two 
parameter(s) are different) to verify the robustness of our architecture.
 
All the prediction 
re
sults are plotted on normalized values (to accommodate bigger modulus values and the 

 
 
 
Figure 
8
.5 Prediction Results: True vs Predicted Plot for 1DCNN
 
Table 
8
.2
 
Prediction Results from the 1DCNN
 
Example
 
 

 
(kg/m
3
)
 
E
1
(Gpa)
 
E
2
(Gpa)
 
v
12
 
v
23
 
Example 1
 
True
 
1500
 
160
 
25
 
0.28
 
0.36
 
 
Predicted
 
1316
 
160
 
2
8
.
8
 
0.31
 
0.35
 
Example 2
 
True
 
1500
 
180
 
9
 
0.28
 
0.36
 
 
Predicted
 
1362
 
160
 
2
8
.8
 
0.30
 
0.34
 
Example 3
 
True
 
2200
 
160
 
25
 
0.28
 
0.36
 
 
Predicted
 
199
8
 
196
 
31.8
 
0.28
 
0.34
 
Example 4
 
True
 
2200
 
180
 
25
 
0.28
 
0.36
 
 
Predicted
 
2205
 
181
 
2
8
.5
 
0.2
8
 
0.35
 
 
       
The poor predictions can mainly be due to two reasons:
 

 
The model is trained on about 2500 samples approximately, and as per ou
r
 
understanding
,
 
it is still a small dataset considering how data hungry c
onvolutional 
neural networks are. The waveforms change very little physically for different 
 
84
 
 
composites (i.e. different material properties) and a larger dataset might be a viable 
solution to avoid poor predictions.
 

 
One of the facts which are not highlighte
d in deep learning literature is the limitation 
of regression
-
based neural networks in extrapolation. It is observed that neural 
networks do not predict well on a data point that lies outside the range of training or 
towards the extreme points of the train
ing range. For example, the range of 
E
2
 
is 
between 5
-
50 GPa in training and the predictions are not good on 9 GPa (near to one 
the extremes of the range i.e. 5 GPa) as compared to the prediction on 25 GPa.
 
       
Currently, the training and predictions hav
e been on noiseless data. But in no practical 
consideration, one can expect no noise. Experimental data from guided wave testing typically 
have many uncertainties that manifest itself into noise and corrupt the 
original signal
. 
Therefore, any framework tha
t aims to work with such signals should be robust to noise. 
There are two ways to make a framework/model robust. 
 
 
Figure 
8
.6 (a) Original Signal (b) Original signal corrupted with a noise of SNR5
 
       
The first method should be to use a model trained o
n noiseless data to predict on noisy 
data. If the performance is satisfactory, the trained model is already robust to noise. If not, 
one would have to include noisy data in the training process itself. In this thesis, we use the 
former and try to leverage 
a model trained on noiseless data to predict on data with different 
 
85
 
 
levels of noise. 
Additive White Gaussian Noise (AWGN) of different levels (SNR5
, SNR10
 
and SNR20) 
was
 
added to the original signal to create three separate datasets on which the 
trained mo
del will predict on.
 
Figure 
8
.6 shows how the original signal is corrupted with a 
noise of SNR5.
 
Figures 
8
.
7
, 
8
.8 and 
8
.9 show the prediction results o
n data with noise levels 
of SNR20, SNR10 and SNR5
 
respectively.
 
 
Figure 
8
.
7
 
Prediction results on data w
ith noise of SNR20 for 1DCNN
 
 
86
 
 
 
Figure 
8
.8 Prediction results on data with noise of SNR10 for 1DCNN
 
 
Figure 
8
.9 Prediction results on data with noise of SNR5 for 1DCNN
 
        
The noise levels are very high in these cases in order to test the limit of the 
algorithms, 
but such high 
noise 
levels (i.e. SNR5) typically do no occur in guided wave testing.
 
From 
 
87
 
 
F
igures 
8
.
8
 
to 
8
.9, we observe that the predictions are much poorer on noisy data compared to 
noiseless data. This was expected, but as long as the predic
tions are within a certain error 
range, they can be acceptable. In order to quantify the prediction performance of the 1DCNN 
architecture, a new parameter called 
Number of Wrong Predictions (NOWP)
 
is introduced. 
Any prediction is termed a wrong prediction 
if the predicted value lies outside the 15% error 
range of the true value.
 
Figure 
8
.10 shows the NOWP for the prediction on noiseless 
(baseline) and different levels of noisy data. It is clearly visible that the performance 
deteriorates in the presence of 
noise, but an interesting observation can be made for the SNR5 
dataset 
 
. SNR5 signifies that the noise level is highest compared to SNR10 and SNR20, but 
the predictions on the SNR5 dataset is better compared to that of SNR10 and SNR20. In fact, 
the predic
tions on the SNR5 dataset are second best after the predictions on the noiseless 
data. There is certainly no linear trend in terms of performance
 
of the model
 
and increasing 
noise levels.
 
 
 
                     
Figure 
8
.10 NOWP for the predictions on noisy and noiseless data
 
for 1DCNN
 
 
88
 
 
 
8
.2
 
Training Results with 
Long Short Term Memory Networks (LSTM)
 
A LSTM network is used to map the A0 and S0 waveforms to the target material properties 
in this case. The most important di

of the model is that the LSTM layer takes way too long to train. Therefore, in this thesis an 
inbuilt tensorflow API called CuDNNLSTM (in place of LSTMs) is used which accelerates 
the training process
 
of LSTMs. The CuDNNLSTM layer can easily ramp up the training 
speed by more than 10 times. This is done by skipping repeated activation functions at every 
gate. They have shown to be equally accurate and have been used in various applications in 
signal pr
ocessing, biomedical image processing etc. The architecture of the LSTM model 
used is shown in Table 
8
.3.
 
A learning rate of 0.9e
-
3 and a batch size of 8 are found suitable. 
The model is run over 50 epochs.
 
Table 
8
.3
 
LSTM Architecture
 
Layer
 
Activation
 
Outp
ut (Feature map, Units)
 
Parameters
 
Batch Normalization
 
-
 
(4096,2)
 
8
 
CuDNNLSTM
 
-
 
(4096,10)
 
560
 
Dropout (0.2)
 
-
 
(4096,10)
 
0
 
Flatten
 
-
 
40960
 
0
 
Dense
 
ReLU
 
16
 
6553
8
6
 
Batch Normalization
 
-
 
16
 
64
 
Dropout (0.05)
 
-
 
16
 
0
 
Dense
 
Linear
 
5
 
85
 
Total parameters
 
 
 
656,093
 
 
  
     
 
       
Figure
s
 
8
.11, 
8
.12 and 
8
.13 show the loss curve, MAE and R
2
 
curves respectively. The 
training performance of the LSTM model is very similar to that of the 1DCNN. It reaches a 
MAE of 3
7
 
and a R
2 
coefficient of 0.98. The network tak
es 120 ms per 
epoch to train which 
is almost 3
0 times more than that of the 1DCNN. This increase is even after using the 
CuDNNLSTM layer.
 
 
89
 
 
 
Figure 
8
.11 Loss curve: MSE vs number of epochs for LSTM
 
 
 
 
 
Figure 
8
.12
 
MAE curve
: 
MAE vs number of epochs for LST
M
 
 
90
 
 
 
Figure 
8
.13 R
2
 
coefficient curve: R
2
 
coefficient vs number of epochs for LSTM
 
       
The trained model is then fed with previously unseen examples. The data here is 
noiseless and will serve as the baseline for evaluation of the performance on noisy dat
a. 
Figure 
8
.14 outlines the prediction results of the LSTM network. Similar
 
to the 1DCNN, the 
trained LSTM network was utilized to predict on noisy data of three different levels. The 
performance of the network on noisy and noiseless data is quantified usi
ng the NOWP 
parameter. Figures 
8
.15, 
8
.16 and 
8
.1
7
 
show the performance of the trained model on noisy 
data of SNR20, SNR10 and SNR5 respectively.
 
 
91
 
 
 
 
Figure 
8
.14 Prediction Results: True vs Predicted Plot for LSTM
 
 
 
Figure 
8
.15 Prediction results on data w
ith noise of SNR20 for LSTM
 
 
92
 
 
 
Figure 
8
.16 Prediction results on data with noise of SNR10 for LSTM
 
 
Figure 
8
.1
7
 
Prediction results on data with noise of SNR5 for LSTM
 
 
93
 
 
       
The limitations of the LSTM network are similar to that of the 1DCNN architecture.
 
Therefore the reasons might be similar to the ones explained for the 1DCNN architecture. 
The performance of the network on noisy and noiseless data is quantified as mentioned 
before using the NOWP parameter. The NOWP plot is shown in Figure 
8
.18.
 
The 
perf
ormance of the LSTM network on noisy data is seen to be similar to the performance on 
noiseless data. Unlike the 1DCNN, the predictions do not deteriorate in the presence of noise 
of various levels.
 
 
 
Figure 
8
.18 NOWP for the predictions on noisy and nois
eless data for LSTM
 
network
 
       
The performance of the two deep models i.e. 1DCNN and LSTM on noiseless data is 
comparable as seen in Figure 
8
.19
 
where the NOWP is plotted for both in presence and 
absence of noise in data
. But the LSTM model has the edg
e when it comes to predictions on 
noisy data. It comfortably outperforms the 1DCNN model in the presence of noise. For the 
1DCNN model, the performance clearly deteriorates as noise is added, but the performance 
of the LSTM network is consistent even as th
e noise levels are increased. The downside of 
the LSTM architecture is that the training time is very high compared to the 1DCNN 
 
94
 
 
network. This is evident in Figure 
8
.20 where the training time per epoch for the LSTM is 
almost 30 times that of the 1DCNN net
work.
 
Even though convergence is relatively quicker 
in a LSTM network, the overall training time is 
naturally 
quite high when compared to 
1DCNN. Higher accuracy of LSTM comes with longer training time which may not be 
feasible with big datasets. There shou
ld be a trade
-
off between training time and accuracy of 
the results depending upon the complexity of the problem in hand. It may be useful to start 
with both 1DCNN and LSTM with small subsets of 
the 
training set and compare the results 
and decide the trade
-
off. Another way of solving this dilemma is to 
develop 
a hybrid network 
like 
the 
Convolutiona
l Recurrent Neural Network
 
(CRNN).
 
[43
]
 
 
 
Figure 
8
.19 NOWP for the two deep architectures in the presence and absence of noise in the 
signal
 
       
 
 
95
 
 
 
Figure 
8
.2
0
 
Training time per epoch
 
for the two deep architectures (1DCNN and LSTM)
 
 
8.3
 
Training Results with 
Dense Neural Network
 
In the previous two sec
tions, we presented the training and prediction 
using two Deep 
Learning architectures i.e. the 1D Convolutional
 
Neural Network and the Long Short Term 
Memory Networks. Though the models perform very well even in presence of high noise 
levels, the current framework is incapable of predicting the Shear Modulus (
G
12
)
 
(
See S
ection 
6.1). In this section we present the r
esults of a DNN trained on data collected using the DC
®
 
software. The data collection process is explained in detail in 
Chapter 7
. This framework is 
capable of predicting the s
hear modulus due to the incorpo
ration of the SH0 mode physics in 
our dataset.
 
As
 
mentioned in 
Chapter 7
, we have considered 8 composites widely used in the 
industry. We have generated and collected group velocity (
C
g
)
 
dispersion curves for the eight 
unidirectional 
composites with three different layups ([0 0]
s
, [0 0 0 0]
s
, [0 0 0 0 0 
0 00]
s
) and 
three different thickness ( 1mm, 2
 
mm and 3
 
mm). This results in generating a total of 
7
2
 
(8*3*3)
 
different 
dispersion plots. Different layups and different thickness of the composites 
were introduced to increase the complexity of the problem i
n hand.
 
The training procedure is 
 
96
 
 
illustrated in the 
Figure 
8
.21
. We use the dispersion curve data (i.e. C
g 
v/s frequency) for the 
A0, S0 and SH0 mode as our input features, while the output labels are the six material 
properties (

1
, E
2, 
v
12
, v
23, 
G
12
)
.
 
The 
C
g
 
information for all A0 modes of all the 
7
2 
combinations
 
as a function of the frequency vector is concatenated into a single input vector. 
Similarly, all S0 and SH0 modes are concatenated respectively to form three long input 
feature vectors.
 
 
   
    
Figure 
8
.21 
Research Flow for the ML based approach
 
       
The architecture of the dense neural network used is shown in Table 
8
.4.
 
 
The training 
and testing dataset is split in a 39:1 ratio. The training dataset is further split in to a 
separate
 
train
ing and validation dataset with a 4:1 ratio. 
A learning rate of 1e
-
3
 
and a batch size of 8 
are found suitable for this architecture. The network is run over 100 epochs.
 
We have used 
similar metrics and loss functions to the ones used for 1DCNN and LSTM to 
evaluate the 
performance.
 
 
 
 
97
 
 
 
Table 
8
.4
 
DNN (MLP) Architecture
 
Layer
 
Activation
 
Output (Units)
 
Parameters
 
Dense
 
ReLU
 
512
 
2048
 
Dropout (0.0001)
 
-
 
-
 
0
 
Dense
 
ReLU
 
256
 
131328
 
Dense
 
ReLU
 
64
 
16448
 
Dropout (0.0001)
 
-
 
-
 
0
 
Dense
 
Linear
 
6
 
590
 
Total parameters
 
 
 
150,214
 
 
       
The loss (MSE) curve is shown in Figure 
8
.22, while the MAE curve is shown in Figure 
8
.23. The R
2 
coefficient is shown in Figure 
8
.24. It is seen that the model trains very well and 
attains a MAE value of only 0.84 and a very high R
2
 
coe
fficient of 0.98
. 
The spikes 
in the 
loss and MAE curve 
are an unavoidable consequence of Mini
-
Batch Gradient Descent in 
Adam (
batch 
size=
8
). Some mini
-
batches have 'by chance' unlucky data for the optimization, 
inducing those spikes 
that are 
seen 
in
 
the co
st function.
 
 
 
Figure 
8
.22
 
Loss curve: MSE vs Epochs for the DNN
 
 
98
 
 
 
Figure 
8
.23
 
MAE curve: MAE vs Epochs for the DNN
 
 
 
 
Figure 
8
.24
 
R
2
 
coefficient curve: R
2
 
coefficient vs Epochs for the DNN
 
 
 
99
 
 
       
For the testing or predictions process, group velocity d
ispersion curves of previously 
unseen composites are fed in to the trained model. The prediction results are shown in Figure 
8
.25 and in Table 
8
.5.
 
It is seen that the predictions are very satisfactory for almost all 
cases. 
Unlike the prediction by 1DCNN a
nd LSTM, the DNN can predict any value of 
E
2
 
accurately. Moreover it is also able to accurately predict the shear modulus 
G
12
. The 

v
23
 
(highlighted in red in the Table 
8
.5 and Figure 
8
.25)
. In
 
most c

v
23
) ,
 
the network 
was able to 
 
predict 
the material properties 

ratio 
v
23
 
is only about 55% compared to the accuracies of the other material properties that 
are
 
m
ore than 95%.
 
       
 
Figure 
8
.26 shows the 
NOWP for
 
all three models we have used in the study. A0 and S0 
waveforms were used as inputs for the deep models, while velocity dispersion data was used 
for the DNN.
 
1DCNN has the least NOWP and DNN the most. Th
ough, all the NOWP for 

v
23. 
In terms of 
predicting other material properties
,
 
the DNN seems to do a much better job as it can 
effectively predict 
E
2
 
and 
G
12
 
accurately. 
But the dispersi
on data has been collected using the 
DC
®
 
software that is not developed for the purpose of generating lots of dispersion curves. It 
is rather 
a software
 
to generate a couple of dispersion curves to study the dispersion 
phenomenon of the guided wave. Hence 
data collection and then training using that data is 
not probably a solution that would be viable for real time use. We have only used it in this 
thesis, for we wanted our framework to be capable of predicting all the material properties 
including the Shea
r Modulus. 
 
 
100
 
 
 
 
Figure 
8
.25
 
Prediction results 
using
 
DNN
 
Table 
8
.5
 
P
rediction Results using DNN
 
Example
 
 

 
(kg/m
3
)
 
E
1
(Gpa)
 
E
2
(Gpa)
 
G
12
(GPa)
 
v
12
 
v
23
 
Example 1
 
True
 
1500
 
161
 
9.25
 
6
 
0.
34
 
0.
41
 
 
Predicted
 
1520
 
1
50.28
 
9.35
 
6.2
8
 
0.31
 
0.5
8
 
Example 2
 
True
 
1
450
 
1
1
9.99
 
8.25
 
6.02
 
0.
32
 
0.
45
 
 
Predicted
 
1530
 
122.84
 
8.32
 
6.1
8
 
0.34
 
0.65
 
Example 3
 
True
 
15
8
1
 
125.5
 
8.
8
 
4.13
 
0.
34
 
0.
45
 
 
Predicted
 
15
8
2
 
129.3
 
8.
8
5
 
4.40
 
0.
33
 
0.61
 
 
     
The training tim
e per epoch for the DNN is 3 ms
. 
Figure 
8
.2
7
 
shows the training time per 
ep
och for the three models. 
The overall training time is the least for the DNN, while 1DCNN 
also takes similar computational time. The LSTM lags behind the other two architectures in 
this regard as it takes about 120 ms per epoch which is 30 times more than 
the 1DCNN and 
40 times more than the DNN.
 
Do note that the comparisons are made for the deep 
architectures using only the noiseless data. It does not make sense to add noise to the 

f uncertaininity. 
 
 
101
 
 
 
Figure 
8
.26
 
NOWP using DNN
 
 
Figure 
8
.2
7
 
Training time per epoch using DNN
 
102
 
 
CHAPTER
 
9
 
CONCLUSIONS AND FUTURE WORK
 
 
9
.1. Conclusions
 
In this thesis, we have explored different learning models to estimate and predict the elastic 
propertie
s of composites using Ultrasonic Guided Waves.
 
We have evaluated the 
performance of the models by predicting on data in the presence and absence of noise. Some 
of the major conclusions that can be drawn from this thesis are:
 
1.
 
We used two different 
modelling
 
methods to simulate guided waves in a composite, and 
sho
wed how at the operating frequency range, and by using fundamental modes we can 
geometrically scale down a 2D model to a 1D waveguide.
 
2.
 
It is seen that with A0 and S0 time histories as inputs and five
 
properties as outputs
, and
 
in this frequency range, the inverse problem of property identification is limited by its 
uniqueness. Due to this, it is not possible to measure shear modulus using fundamental 
modes. 
 
3.
 
This being a conceptual work, and with a li
mited amount of dataset, the predictions from 
the 1DCNN and the LSTM models on unseen examples are very satisfactory. Further, the 
performance of these models on noisy data is also satisfactory. It can be concluded that 
the two deep neural networks can lea
rn the inherent relationship between ultrasonic 
guided wave signal and material properties. It can also be concluded that the deep models 
are robust to noise of various levels. 
 
4.
 
It is seen that the LSTM network is better in terms of performance when it com
es to 

-
off 
required depending on the complexity of the problem in hand. 
 
 
103
 
 
5.
 
ML based DNN performs very 
well
 
generally but is 
limited
 
on its predictions on 

. Moreo
ver, the 
data collection 
for this particular method is 
time 
consuming 
and not 
viable
 
for real time use.
 
6.
 
Some of the other research works have demonstrated the use of a single transmitter and 
multiple sensors. Using multiple sensors may not be a feasible op
tion for a lot of 
applications. On the other hand, we have demonstrated the use of UGW for property 
identification using a single transmitter and a single sensor.
 
7.
 
Previous research is more inclined to solve the inverse problem of materials 
characterization
 
using global optimisation schemes like GA, SA etc. With this research, 
we have obtained good results with deep learning networks trained using Adam 
optimization (variant of stochastic gradient descent with added momentum and adaptive 
learning rate) on mea
n square loss function. 
 
8.
 
Heuristics based GA and SA takes a long time with very high computation resources. 
With neural networks and stochastic gradient descent, the training can be accelerated 
with the use of GPUs. 
 
9.
 
The learning is necessary to automate a
 
materials characterization problem. Neural 
networks provide that automation since it learns the inverse problem by minimising the 
loss function on the dataset. 
 
10.
 
It is also required the estimation is real time and quick. Unlike previously explored 
methods 
like GA or SA, any new prediction here takes only 1 ms which enables the 

is no literature available on the application of deep learning techniques for the 
identificat
ion of materials properties. Our research demonstrates the application of this 
concept clearly, and even in the absence of experimental results, the performance is very 
 
104
 
 
good even with high noise content. Our future research will be focused towards 
implemen
tation on real experimental data
.
 
 
9
.2
. 
Future Work
 
The results presented in this thesis can be looked at as a sort of starting step towards a bigger 
problem that is using Deep Learning for Material Property Identification. Though our models 
perform very w
ell and predict with accuracies more than 85%, they are still subject to 
improvements. One approach to improve the existing framework would be to further refine 
our FEM and SFEM models. For example, if one can optimize the FEM model better to 
reduce the co
mputational time, a dataset can be created using the FEM model instead of the 
SFEM model. The FEM model incorporates more physics compared to the SFEM model as 
the wave modes are coupled. This might lead to better predictions. Another possible direction 
to
 
pursue would be to incorporate experimental data in the framework. There are two way to 
do this. The first being using a model trained on simulation data to predict on experimental 
data. If the performance is not satisfactory, some of the experimental dat
a can even be 
included in the training process itself.
 
       
The current work presented is limited to composite laminates. The framework can be 
extended easily to other materials like metals, soft polymers etc. Experimental determination 
of material prope
rties of soft polymers is specifically hard due to the flimsy nature of the 
materials which make it sometimes impossible to conduct experiments on.
 
 
9
.3
. 
Thesis Contributions
 












 
105
 
 












 
 
 
106
 
 
 
 
 
 
 
 
 
 
 
 
 
 
APPENDICES
 
 
 
 
 
 
 
 
 
 
 
 
 
 
107
 
 
APPENDIX A
 
 
Symmetric Solution
 
For a symmetric solution, the 
u
x
 
component of the particle displacement vector and shear 
stresses are symmetric about the mid
-
plane across the thickness of the 
2D infinite plate.
 
                    

















                         


















 
                    


















                      

















               
 
(
A
.1
)
 
       
Symmetric boundary conditions also include traction free surfaces, therefore a part of 
Equation (
A
.2
) becomes:
 
                                                     




















   
                               
(
A
.2
) 
                                                                     
 
  

















=0
 
       
Using the general solution for potentials in Equations (
3.
22) and (
3
.2
3
), and expressions 
for stress and displacements, Equation (
A
.2
) can 
be expressed as:
 





















         
 
        
(
A
.3
)
 




















         
  
        
(
A
.4
)
 
       
A non
-
trivial solution for the above linear system of equations exists i
f the determinant 
in the above Equation vanishes.
 
























       
(
A
.5
)
 
       
Rearranging, the dispersion relation for the symmetric mode is obtained.
 

















                                                    
  
     
(
A
.6
)
 
       
Where 
p
 
and 
q
 
have been described in Equation (
3
.18) and (
3
.19) respectively.
 
 
 
 
 
108
 
 
APPENDIX B
 
 
Anti
-
Symmetric Solution
 
For an anti
-
symmetric solution, the 
u
x
 
component of 
the particle displacement vector and 
shear stresses are 
anti
-
symmetric about the mid
-
plane across the thickness of the 2D infinite 
plate.
 
                    


















                         

















 
     
               

















                      


















                
  
(
B.1
)
 
       
The anti
-
s
ymmetric boundary conditions also include traction free surfaces, therefore a 
part of Equation (
B.1
) becomes:
 
 
                                                    



















                                  
      
(
B.2
)                                                                      
 
    


















=0
 
  
 
     
  
Using the general solution for potentials in Equations (
3
.22) and (
3
.2
3
), and expressions 
for stress and displacements, Equation (
B.2
) can be expressed as:
 




















   
          
   
    
(
B.3
)
 




















      
 
   
           
(
B.4
)
 
       
A non
-
trivial solution for the above linear system of equations exists if the determinant 
in the above Equation vanishes.
 


























       
(
B.5
)
 
       
Rearranging, the dispersion relation for the symmetric mode is obtained.
 

























               
                 
                   
(
B.6
)
 
       
Whe
re 
p
 
and 
q
 
have been described in Equation (
3
.18) and (
3
.19) respectively.
 
 
 
 
109
 
 
APPENDIX 
C
 
 
LIST OF PUBLICATIONS
 
1.
 
Karthik Gopalakrishnan, Mahindra Rautela and Yiming Deng ( submitted and accepted), 

Deep Learning Based Identification of Elastic Properties Usin
g Ultrasonic Guided 
Waves

 
 
2.
 
Rautela M, Karthik Gopalakrishnan, Yiming Deng and Gopalakrishnan S (in press). 
Ultrasonic Guided Waves Based Identification of Elastic Properties Using 1D
-
Convoluti
onal Neural Networks. 2020 IEEE International Conference on Prognostics and 
Health Management (ICPHM).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
110
 
 
 
 
 
 
 
 
 
 
 
 
 
 
BIBLIOGRAPHY
 
 
 
 
 
 
 
 
 
 
 
 
 
 
111
 
 
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https://doi.org/10.1007/978
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0
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1
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Structural Health Monitoring
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