A genetic algorithm based back propagation network for simulation
of stress–strain response of ceramic-matrix-composites
H. Sudarsana Rao
a
a,*
, Vaishali G. Ghorpade a, A. Mukherjee
b
Department of Civil Engineering, JNTU College of Engineering, Anantapur, AP 515 002, India
b
Indian Institute of Technology, Bombay, Mumbai 400 076, India
Abstract
Ceramic-matrix-composites (CMCs) are fast replacing other materials in many applications where the higher production costs can be
offset by significant improvement in performance. In applications such as cutting and forming tools, wear parts in machinery, nozzles,
valve seals and bearings, improvement in toughness and hardness translate into longer life. However, the recent resurgence in the field of
development of CMCs has been due to their potential use for the Space Transport systems, Combustion engines and other energy conversion systems. The CMCs are ideal structural material for these applications. However, due to their lack of toughness, they are prone
to brittle fractures. Therefore, the main consideration in the development of CMCs has been to toughen them. To achieve this, the bimaterial interface should be weak and must allow debonding, resulting in crack deflection. In the present work, the stress–strain response
of Al2O3 (matrix)/SiC (whisker) ceramic composite has been simulated using a back propagation neural network (BPN), which incorporates the effect of interface shear strength (IFS) in the analysis. For efficient and quick training, the weights for the BPN have been
obtained by using a genetic algorithm (GA). The GA has been modelled with 150 genes and a chromosome string length of 750. The
network simulation is based on the stress–strain response obtained from the finite element analysis. A three noded isoparametric interface
element has been employed to model the whisker/matrix interface in finite element analysis. The finite element analysis has been carried
out only for a limited number of specimens. However, the simulation model is capable of predicting the stress–strain relationship for a
new interface shear strength even with this limited information. Thus, the robustness and the generalisation capability of the neural network model is demonstrated. The development stages of the GA/BPN model such as the preparation of training set, selection of a network configuration, training of the net and a testing scheme, etc., have been addressed at length in this paper.
Keywords: Genetic algorithms; Neural networks; Ceramic-matrix-composites; Back propagation; Hybrid networks; Interface elements
1. Introduction
Ceramic composites due to their very attractive thermomechanical properties such as the high strength and high
stiffness at elevated temperatures, chemical inertness, low
density, high thermal insulation, and so forth, have become
an ideal choice for a variety of important engineering applications. They are fast replacing other materials in applica-
tions where the higher production cost is offset by a
significant improvement in the performance. In applications such as cutting and forming tools, wear parts in
machinery, nozzles, valve seals and bearings, etc., an
improvement in toughness and hardness translate into a
longer life. The recent resurgence in the field of development of CMCs, however, has been mainly due to their
potential use for the Space Transport Systems, combustion
engines and other energy conversion systems.
Although the CMCs have a wide application range, it is
still difficult to design CMCs for the desired application
due to a single flaw in them, namely lack of toughness.
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This flaw makes them prone to catastrophic failures, and
hampers their usefulness severely. Therefore, the most
important consideration in the development of ceramicmatrix-composites has been to toughen them so as to
exploit their aforementioned properties without risking a
catastrophic failure.
Recent experimental studies of whisker-reinforced
ceramics have shown that significant improvements in fracture toughness can be achieved via mechanisms of whisker
bridging, whisker pull-out and crack deflection [1–4]. These
mechanisms and hence, the thermo-mechanical properties
of ceramic-matrix-composites depend on various parameters such as the volume fraction of constituents, shape,
diameter, length and aspect ratio of whiskers and their orientation. In addition to the above geometrical factors, the
elastic and mechanical properties of constituents as well as
the properties of the interface are critical in dictating the
nature of the fracture. It has been shown [5] that oxidation-resistant interface control is necessary to obtain high
fracture toughness in ceramic-matrix-composites. Thus, it
can be noted that active research is needed in the field of
developing interface-coating materials, resulting in developing CMCs with desired mechanical properties. Consequently, the design of interface coatings would become a
highly engineered task in future. The ability to successfully
design such coatings could be greatly enhanced by a thorough understanding of the effect of interface shear strength
(IFS) on toughening behaviour of ceramic-matrix-composites. As a result, it is necessary for any analysis of CMCs to
model the whisker/matrix interface realistically and study
its effect on the stress–strain response.
Analytical studies of wake toughening mechanisms in
ceramic-matrix-composites are rather complicated due to
the involvement of a huge number of parameters. Such
analytical studies [3,6,7] necessitates incorporation of many
simplifying assumptions (e.g., steady state cracking, one
dimensional strain analysis, etc.) to make the problem amenable to solution. Therefore, to alleviate the use of unrealistic assumptions in the model, a numerical technique for
the analysis of ceramic-matrix-composites, which presents
a scope for a more realistic model, is needed. The analysis
of CMCs using finite element models, although more realistic, warrants large number crunching and is computationally expensive. This is also a very time taking process. To
overcome this shortcoming, a novel approach to simulating
the stress–strain relationship of CMCs has been presented
in this paper using genetic algorithm (GA) based back
propagation neural networks (GA/BPN). Genetic algorithms have been successfully used in the field of structural
engineering particularly in structural optimisation [8,9]. In
the present work, the applicability of genetic algorithms for
modelling the stress–strain behaviour of CMCs has been
explored. A back propagation (feed forward) neural network (BPN) [10] has been developed here for simulating
stress–strain behaviour of CMCs. The neural network simulates the stress–strain behaviour in an adaptive fashion
through learning the effect of interface shear strength on
mechanical behaviour of CMCs. This effect of interface
strength on the stress–strain response is taught to the network through training examples. The weights for the
BPN have been obtained by using a genetic algorithm. This
alleviates the large number of iterations needed for training
a BPN. The finite element analysis has been carried out for
generation of exemplar patterns (training examples) for the
network. The network configuration has been developed
after selecting the input/output parameters of the network.
The network has been trained on an IBM compatible Pentium-4 machine using an artificial neural network simulator, ANNS [11]. The network is capable of predicting the
stress–strain behaviour of CMCs for a new interface
strength after successful training. This capability of the network has been demonstrated in this paper. The reliability
of the GA/BPN is thus established for its practical use in
the field of design of CMCs.
2. The finite element model of CMC
Finite element studies of wake toughening mechanisms
in CMCs are limited [12,13]. The mechanics of whisker/
matrix interface are complex. The complexity arises due
to the possible relative displacements at the bi-material
interface. Further, if the interface is in compression, a frictional force between the two materials exists. If this
frictional force is considered then the system, becomes
non-conservative and the global stiffness matrix in finite
element analysis becomes un-symmetric. Such un-symmetric matrices pose difficulties in the solution routines. On the
other hand, if the interface is in tension, the frictional force
disappears. Hence, it has been rather difficult to model the
whisker/matrix interface in the finite element analysis of
ceramic-matrix-composites. Laird II and Kennedy [14]
have presented a study of crack wake toughening mechanisms in a whisker reinforced ceramic using finite element
analysis. In their study, the interface between the whisker
and matrix has been modelled using spring slider elements.
However, this approach models the connectivity between
the whisker and matrix at discrete nodal points only. As
a result, the chances of overlapping of the nodes along
the interface is not eliminated. Further, this type of idealisation of interface considers only the shear failure of the
interface and cannot consider the tensile (normal) failure
of the interface which is also possible in CMCs.
In the present work, a finite element model for the analysis of CMCs suggested by Mukherjee and Rao [15] has
been used for the development of exemplary patterns for
the network model. In this approach, the mechanics of
behaviour of the interface in CMCs is treated similar to
that of a rock joint as the relative displacements occur
across a thin discontinuity in both the cases. The model
uses six noded isoparametric interface elements to idealise
the whisker/matrix interface (Fig. 1). The detailed formulation of the interface element can be found elsewhere
[15,16]. However, for completion sake, a brief formulation
is presented below.
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where, Dus and Dun are relative displacements, tangential
and normal, respectively, to the element axis. The transformation matrix, T in the above equation can be obtained as
2
3
dx
dy
1 6 dn dn 7
7;
T¼ 6
ð11Þ
s 4 dy dx 5
dn dn
where,
" 2 #12
2
dx
dy
s¼
þ
.
ð12Þ
dn
dn
y,v
Element A
ξ
3
ξ=1
2
1
ξ = -1
Element B
x,u
Fig. 1. Six noded isoparametric interface element.
The shape functions associated with each node i are
given as
At the interface, the tangential force per unit length rs
and rn at any point in the element can be related to the corresponding relative displacements as
Ks 0
Dus
rs
¼
.
ð13Þ
0 Kn
rn
Dun
Symbolically,
N 1 ¼ 12ð1 nÞn;
ð1Þ
re ¼ De ee ;
N 2 ¼ ð1 n2 Þ;
ð2Þ
where, Ks and Kn are the interface stiffnesses tangential and
normal to the interface, respectively.
Using Eqs. (10) and (14),
N3 ¼
1
nð1
2
þ nÞ.
ð3Þ
The global co-ordinates of any point (x, y) are defined in
terms of nodal co-ordinates as
x¼
3
X
N i xi ;
y¼
i¼1
3
X
ð4Þ
N iyi.
i¼1
Similarly, the relative displacements at any point are expressed in terms of nodal relative displacements as
Du ¼
3
X
N i Dui ;
Dv ¼
i¼1
3
X
ð5Þ
N i Dvi .
re ¼ De TBDq.
ð15Þ
The relative nodal displacements can be expressed in terms
of nodal displacements of the adjacent elements A and B
(see Fig. 1) through a difference matrix G as
Dq ¼ Gq;
ð16Þ
where,
qt ¼ uB1 vB1 uB2 vB2 uB3 vB3 uA
vA
uA
vA
uA
vA
3
3
2
2
1
1
ð17Þ
i¼1
The relative displacement e in global co-ordinates at any
point of the element can then be expressed in terms of nodal values of relative displacements as
X
3 Du
Dui
Ni 0
ð6Þ
e¼
¼
.
Dv
0 Ni
Dvi
i¼1
Or symbolically,
e ¼ BDq ¼
ð14Þ
3
X
ð7Þ
Bi Dqi ;
i¼1
where,
"
N1 0 N2 0 N3
B¼
0 N1 0 N2 0
Dqt ¼ f Du1 Dv1 Du2 Dv2
0
2
1 0
0
0
0
0
6 1 1 0
0
0
0
6
6
6 1
0
1
0
0
0
G¼6
6 1
0
0 1 0
0
6
6
4 1
0
0
0 1 0
1
0
0
0
0 1
ð8Þ
;
Dv3 g.
ð9Þ
Denoting the strains (relative displacements) in the element
co-ordinates as ee, the relation between the global and local
strains can be expressed as
es
Dus
e
e ¼
¼
¼ Te ¼ TBDq;
ð10Þ
en
Dun
0
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
3
0
17
7
7
07
7.
07
7
7
05
0
ð18Þ
Combining Eqs. (15) and (16), the following relation can
be obtained:
re ¼ De TBGq.
#
N3
Du3
and
ð19Þ
The curved length of the element can be transformed to
dðlengthÞ ¼ s dn.
ð20Þ
Using energy approach, the interface stiffness matrix is obtained as
Z 1
T
T T e
k¼G
B T D TBs dn G
ð21Þ
1
where k, is the interface element stiffness matrix; B, is the
matrix connecting strain and relative displacements at
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any point in the interface element; T, is the transformation
matrix for relating local and global strains; D, is the constitutive stress–strain matrix for the interface element; G, is
the difference matrix to express relative nodal displacements in terms of nodal displacements of the adjacent
elements.
Eight noded isoparametric quadrilateral elements are
used for modelling the matrix as well as the whisker. As elements of compatible shape functions are used to model all
the components, i.e., whisker, matrix and the interface, the
interaction between them is modelled realistically. The
whisker/matrix interface is modelled all along the interface
and the model is capable of considering both shear and
normal failures of the interface. The possibility of overlapping is also effectively alleviated.
A two-whisker quarter symmetry finite element model of
Al2O3 (matrix)/SiC (whisker) ceramic-matrix-composite is
used to study the effect of interface shear strength on the
stress–strain response (Fig. 2). This model is analysed in
2-D plane stress. Though this 2-D analysis is an approximation of the real three-dimensional situation (random orientation of whiskers) present in these composites;
nevertheless this analysis should still provide meaningful
insights into the effect of interface strength on the stress–
strain response of whisker reinforced CMCs (i.e., a case
in which whiskers are aligned in one direction). Material
properties used in this study are shown in Table 1.
The finite element mesh used for the present problem is a
rectangular square-based grid having 9 · 30 squares, where
9 represents the number of squares in the rectangleÕs width
and 30 represents the number of squares in the rectangleÕs
length. The use of quarter symmetry results in locking of
the nodes along the axis of symmetry in x direction. The
model consists of 120 matrixes and 60 whisker 8 node
quadrilateral elements along the whisker/matrix interface.
y
Applied Force
15 µm
Matrix
Interface elements
Axis of
Symmetry
Matrix
Whisker
Matrix
Whisker
30 µm
x
0.9 µm
0.6 µm
0.9 µm
0.3 µm
Fig. 2. Quarter symmetry FE model of the Al2O3/SiC ceramic composite.
Table 1
Material properties used for the present CMC
Property
Matrix (Al2O3)
Whisker (SiC)
YoungÕs modulus
PoissonÕs ratio
Tensile strength
Radius
Length
400.0 GPa
0.22
700 MPa
–
–
600.0 GPa
0.22
6.89 GPa
0.3 lm
60 lm
The whisker has been modelled as having a half-length of
30 lm. It is connected to a matrix of 0.9 lm at regular
intervals. These dimensions have been used so as to provide
a 33.3% of whisker reinforcement. The connectivity
between the whisker and matrix has been modelled using
60 six-noded isoparametric interface elements. Nine interface elements are used at the bottom axis of symmetry
(y = 0 in Fig. 2) as control elements. When the normal
stress in these control elements exceed the matrix tensile
strength (700 MPa), the stiffness of these elements is
reduced to zero indicating the onset of matrix cracking.
In the other 60 interface elements used to model the whisker/matrix interface, the shear stiffness (ks) is set to a high
value initially. This alleviates the possibility of any non-slip
relative displacement. Once the interface shear stress
exceeds the interface shear strength, ks is set to zero allowing a slip to occur. Load has been applied incrementally.
Iterative finite element analysis is carried out at each load
step and the stress–strain response is obtained. The mesh
convergence has been investigated by refining the finite element mesh from the present 60 to 120 interface elements.
The displacements of the selected nodes have been found
to differ by 3–5%, indicating that the present mesh is a stable configuration.
3. Results of the finite element analysis
Fig. 3 shows the stress–strain curves obtained from the
finite element analysis of the above ceramic-matrix-composite for different interface shear strength parameters.
The matrix cracking occurred at an applied stress of 1142
MPa, transferring the load onto the whisker. When the
interface shear strength is 800 MPa, the interface started
debonding readily, thus deflecting the crack parallel to
the whisker (crack deflection mechanism). At this point,
the stress–strain curve deviates from the linear path (see
Fig. 3). With further increase in the applied stress, the
crack continues to propagate along the whisker/matrix
interface. This is indicated in the present model by the failure of the interface elements along the interface. The deviation of the stress–strain curve from the linear path
indicates the initiation of the toughening exhibited by the
CMC.
On the other hand, the CMCs with IFS ranging from
1200 to 2400 MPa, debonding of the interface could not
be observed even when the matrix crack reached the whisker. The whisker momentarily halts the further propagation of the crack, thus, bridging the matrix crack (i.e., the
334
putational support in the form of mainframe computers
is also necessary for carrying out the analysis. This further
makes it difficult to carry out a comprehensive parametric
study. On the other hand, a comprehensive parametric
study has to be carried out for a number of CMCs to
develop a set of guidelines for their design with desired
toughening behaviour. This recognizes the necessity of
models, which allow efficient interpolation between the
well-conceived numerical experiments. The use of artificial
neural networks as a tool to provide such a model has been
presented in the next section.
4. A neural net for simulating stress–strain response of
CMCs
Fig. 3. FEA derived stress–strain responses for different interface shear
strengths.
whisker bridging mechanism). The whisker bridging continues till the interface starts debonding or the whisker
fractures, depending on the relative strengths of the interface and the whisker. The matrix crack has been bridged
by the strong whisker, till the interface debonding initiated
at a stress of 1305 MPa, for an IFS of 1200 MPa and at a
stress of 1690 MPa for an IFS of 1600 MPa, thus resulting
in the non-linearity in the stress–strain response (see
Fig. 3). In all these cases, the failure of the CMC occurred
due to the whisker pulling out of the matrix. On the other
hand, in the cases of CMCs with an IFS of 2000 MPa and
2400 MPa, whisker bridging continued till the stress in the
whisker became nearly equal to its fracture value. Finally,
the CMC failed due to whisker fracture, with very little
debonding of the interface. This is depicted in the stress–
strain response by its near linear nature. With increasing
interface shear strength (IFS), it can be observed that the
stress–strain curve deviates from the linear path at a higher
stress level. This is due to relatively stronger whisker/
matrix interfaces, not debonding readily, thus contributing
to lesser toughening. It can be observed that the interface
should not be too weak to effect a total whisker pull out
at a low stress, nor it should be too strong to cause a brittle
fracture. Therefore, the interface shear strength must be
engineered to an optimum level to achieve maximum
toughness for the CMCs.
From the foregoing discussion, it is clear that the FE
analysis provides the designer with a tool to carry out the
analysis more realistically. However, this type of finite element iterative solution requires considerable computational time. Further, the finite element analysis involves
extensive number crunching and requires graphical
pre- and post-processors for data preparation and visual
interpretation of the long tabular output. Expensive com-
As discussed above, the interface shear strength has a
significant effect on the nature of the stress–strain response
of CMCs. As mentioned earlier, a comprehensive parametric study has to be carried out for a number of samples to
develop a set of guidelines for their design with desired
toughening behaviour. Therefore, to develop the guidelines
for design of CMCs, a computationally economical, fast
and efficient tool is needed for conducting extensive sensitivity studies. In this paper, we demonstrate the application
of GA/BPN networks as one such tool.
The artificial neural networks are basically nonparametric, neuro-biological computational models. An
artificial neuron is a very approximately simulated mathematical model of the biological neuron. The artificial neuron can carry out only a very simple mathematical function
(such as adding two values) and/or can compare the two
values. The processing of data within the neuron is based
on a typical function associated with it, which is often
called as threshold function, or squashing function. The
input and the output of a neuron is typically in an analog
form and all computations are carried out using this analog
signals only. A typical artificial neuron gets an input either
from other neurons or directly from the environment (i.e.,
input nodes). The paths connecting the input nodes to the
neurons and the connections between the various neurons
are associated with a certain variable weight, which represents a multiplying factor for the incoming signal representing the synaptic strength of the connection. These
weights are initially set to some random values and are
later adjusted in the process of training of the net. The artificial neuron then sums this input, which is actually a
weighted sum of all the input signals. The input so obtained
is used to calculate a node value according to the Squashing function of the neuron. This node value is then compared with the threshold value of the neuron and if the
node value is higher, then the neuron goes to the ‘‘higher
state’’ (excitation state) and a signal is passed on to the next
layer of neurons.
The artificial neural network is developed by using a
number of such artificial neurons as described above by
trial and error method. Many different configurations of
the artificial neuron can be made to develop different net-
335
work configurations. The configuration so developed is
trained using a predetermined learning algorithm. A
detailed information on neural networks, training, performance, etc., can be found elsewhere [11,17–20]. In the present work, a multilayer feed forward network has been used
to simulate the stress–strain response of CMCs. The general scheme of a multiplayer feed forward network is
depicted in Fig. 4(a). It may be noted that all the neurons
between two successive layers are fully connected, i.e., each
neuron of a layer is connected to each neuron of the neighbouring layers. However, there is no connection between
neurons of the same layer or the neurons, which are not
in successive layers. The input layer receives input information and passes it onto the neurons of the hidden layer(s),
which in turn pass the information to the output layer. The
output from the output layer is the prediction of the net for
the corresponding input supplied at the input nodes. In a
feed forward network, the knowledge is stored in a distributed manner in the form of synaptic strengths and thresholds. Thus, it can be generalised, i.e., it may be used for the
situations for which the net has not been trained. Initially,
the synaptic strengths (weights) and the threshold values
are allocated randomly. To train the network for a specific
knowledge, a set of training examples is prepared. A training example consists of a set of values for the input neurons
and the corresponding values for the output neurons. Several of such input–output pairs are to be prepared carefully
to reflect all the aspects that the net needs to learn. All the
training examples together form the training set.
A multilayer back propagation network chosen for the
present problem is shown in Fig. 4(b). The network consists of one input, one output and two in-between hidden
layers. Each layer consists of one or more number of neurons. The input nodes receive the input signals in the form
of normalised excitation signals. This input is further processed within the network by passing it to the higher (hidden or the output) layers. The input to nodes of each layer
is passed by the neurons in the lower layer through the
respective connection strength. These connection strengths
are called weights. In the general approach, these weights
OUTPUT NODES
HIDDEN
LAYERS
Feedforward
Backpropagation
INPUT NODES
Fig. 4(a). A typical feed forward neural network architecture.
O1
OUTPUT
(1 NODE)
SECOND
HIDDEN
LAYER
(10 NODES)
FIRST
HIDDEN
LAYER
(10 NODES)
INPUT
(4 NODES)
I1
I2
I3
I4
LEGEND
O1 : OUTPUT: STRESS LEVEL
I1 : INPUT : INTERFACE STRENGTH
I2 : INPUT : SQUARE TERM OF INTERFACE STRENGTH (derived term)
I3 : INPUT : FAILURE MODE
I4 : INPUT : STRAIN LEVEL
Fig. 4(b). The neural network model.
are set initially to some random values. These values are
then modified automatically according to the learning algorithm during the process of learning. This type of learning
requires huge number of training cycles depending on the
size of the training data, non-linearity present in the problem and size of the network. On the other hand, use of
genetic algorithms for deriving the initial weights to the
BPN alleviates this problem and reduces the number of
training cycles considerably. In the present work, a hybrid
network consisting of a back propagation net guided by a
genetic algorithm has been used. This type of networks are
known as genetic algorithm based back propagation networks. The network develops the capability to adaptively
learn the stress–strain response of CMCs through a set of
select training examples obtained from FE analysis. The
trained network can then be used for conducting sensitivity
studies. The learning phase in network development
addresses the modelling phase for stress–strain response.
The architecture of the feed forward network developed
for the present work has been shown in Fig. 4(b). The network consists of an input layer of four nodes, two hidden
layers of ten nodes each and one output node (i.e., 4–10–
10–1 configuration). Though a few methods like Genetic
Programming and Adaptive Algorithms are available for
designing the neural network architecture, a trial and error
method is popularly used due to its simplicity. The present
network configuration has been arrived at by a trial and
error method. A sigmoidal threshold function has been
used as a nodal property. It may be noted here that the
configuration selected for the network is symmetric. The
symmetric configurations may lead to a network paralysis
[11]. In the present work, the problem of possible network
paralysis has been overcome by using a genetic algorithm
336
(GA) for deriving the weights to the BPN. The process of
development and validation of the network consisted of
the following stages:
1. Selection of input/output vectors;
2. Selection of network configuration and training; and
3. Validation of the resulting network.
4.1. Selection of input/output vectors
The input for the present network has been selected so
as to facilitate fast mapping of the stress–strain response.
The input vector for the network is
T
x ¼ sS s2S fm eL ;
where sS, is the interface shear strength; ‘‘fm’’ represents
the failure mode, i.e., either whisker pull-out or whisker
fracture; and eL, represents strain level, at which the stress
level is obtained from the network as output.
The term s2S has been included in the input vector as the
effect of the interface shear strength on the stress–strain
response of CMCs is highly non-linear. This term is a
derived input term and it is provided for fast network development [11]. The input for the interface shear strength is
provided as continuous valued input parameter in a normalised range of (0, +1). For this, all the interface strengths
are divided by the maximum value of the parameter coming to this node.
The failure mode (fm) has been provided as a binary
input for this network. Two binary values, i.e., 0 and +1
are associated with the two possible failure modes, i.e.,
when the failure is by whisker pull-out, the input to this
node is 0, else the input is fed as +1. It would be extremely
useful to obtain the type of failure mode also as output
from the network. However, as an initial attempt to use
neural networks in CMC analysis, the failure mode has
been provided only in the input vector for the present network. The task of development of the network, which can
predict the failure mode, becomes more complicated due to
a large number of singularities present on the training surface. However, the present authors are engaged in the
development of a network, which is able to predict the failure mode also. The input vector also includes the term eL,
which represents the strain level, at which the stress level is
obtained from the network. The response of the network
has been obtained as output stress value at some predetermined strain levels. For this purpose, the entire strain range
is sub-divided into a number of strain levels. This strain
level is fed as input to the network. The output obtained
from the network at this stage represents the corresponding
stress level.
4.2. Training of the network
The training of present network has been accomplished
using the back propagation algorithm (BP). Convention-
ally, A BPN determines its weights based on a gradient
search technique and hence runs the risk of encountering
the local minimum problem[18]. GA on the other hand is
found to be good at finding ‘‘acceptably good’’ solutions.
The idea to hybridise the two networks has been successful
to enhance the speed of training [21]. In the present work,
the initial weights for the BPN have been obtained by using
a GA. Genetic algorithms (GAs) which use a direct analogy of natural behaviour work with a population of individual strings, each representing a possible solution to the
problem considered. Each individual string is assigned a fitness value, which is an assessment of how good a solution
is to a problem. The high-fit individuals participate in
‘‘reproduction’’ by crossbreeding with other individuals
in the population. This yields new individual strings as offspring, which share some features with each parent. The
least-fit individuals are kept out from reproduction and
so they ‘‘die out’’. A whole new population of possible
solutions to the problem is generated by selecting the
high-fit individuals from the current generation. This new
generation contains characteristics, which are better than
their ancestors. The parameters, which represent a potential solution to the problem, genes, are joined together to
form a string of values referred as a chromosome. A decimal
coding system has been adopted for coding the chromosomes in the present work. The network configuration of
the BPN for the present work is 4–10–10–1. Therefore,
the number of weights (genes) that are to be determined
are 4 · 10 + 10 · 10 + 10 · 1 = 150. With each gene being
a real number, and taking the gene length as 5, the string
representing the chromosomes of weights will have a length
of 150 · 5 = 750. This string represents the weight matrices
of the input-hidden layers–output layers. An initial population of chromosomes is randomly generated. Weights from
each chromosome have been extracted then using the procedure suggested in reference [22]. The fitness function has
been devised using FITGEN algorithm [21]. The pseudocode for the implementation of the GA/BPN is presented
in Appendix A.
The network has been trained using a supervised learning strategy. The input and the corresponding learning output has been presented to the network till it learnt the
desired relationship. A constant learning rate of 0.7 and a
constant momentum factor of 0.9 has been used throughout the learning process for the BPN. The training data
have been normalised to be in the binary form for speedy
training of the network. About 90% of the data has been
used in the training set and the rest of the data has been
used for validation of the network model. Satisfactory
learning has been obtained after just 2000 training epochs
in comparison to 32000 training cycles required when
BPN alone was used. This indicates the efficiency of the
genetic algorithm based BPN. The stress–strain curves for
different interface strength as obtained from the network
after learning have been shown in Fig. 5. These stress–
strain curves have been obtained from the FE analysis,
and were included in the training set. It can be seen from
337
2500
2500
IFS=2400 MPa
FEA
GA/BPN FEA
2000
IFS=800 MPa
IFS=1400 MPa
GA/BPN
1500
GA/BPN
1500
Stress, MPa
Stress, MPa
FEA
FE A
GA/BPN
2000
IFS=1600 MPa
IFS=2000 MPa
1000
500
IFS=1200 MPa
1000
FEA
0
0.000
0.002
0.004
0.006
0.008
0.010
Strain
GA/BPN
500
Fig. 5. Learning of the network model.
the figure that the network has learnt the desired stress–
strain response of CMCs very satisfactorily. The network
has been trained for the stress–strain paths of CMCs having interface strength of 800 MPa, 1200 MPa, 1600 MPa,
2000 MPa and 2400 MPa.
4.3. Validation of the network
The neural network model must be validated for its
practical use after training. As it can be seen from Fig. 5,
the network simulates the stress–strain response of the
CMCs having IFS value which is included in the training
set. However, the network must be tested by asking it to
predict the stress–strain paths for samples having new
IFS values, which are not included in the training set.
The network model validation in the present case has been
carried out by observing the difference between the stress–
strain paths obtained from FE and ANN models for new
interface shear strengths. This study has been presented
in Fig. 6. The network has been validated for two new
interface shear strengths within the application domain of
the network. The application domain for the network
model is defined automatically from the lower and upper
bounds of the training set data. The stress–strain response
of CMCs with IFS values of 1200 MPa and 1400 MPa have
been obtained from the trained network. These stress–
strain responses have been compared with those obtained
from the FE model. It can be seen from Fig. 6 that the
stress–strain paths obtained from the network are in good
agreement with those obtained from FE analysis.
Although, a good agreement between the FE and GA
based BPN model has been obtained, the advantages of
the present model over the FE model are as follows:
1. The stress–strain path for a number of new samples
from the GA/BPN model can be obtained very quickly,
with a significantly less computational efforts. On the
0
0.000
0.002
0.004
0.006
0.008
0.010
Strain
Fig. 6. Validation of the network model for new IFS.
other hand, the FE analysis is carried out separately
for each interface strength. Thus, a lot of time saving
is effected using GA/BPN model without sacrificing
appreciable computational accuracy.
2. The need to prepare large input data for the finite element analysis for each interface shear stress value is
alleviated using the present GA/BPN model. Thus, the
possibility of human error coming into picture without
a robust pre-processor is automatically reduced.
3. The present GA/BPN model after training can be implemented on a simple personal computer (PC) having minimum configuration requirements. On the other hand,
the present FE model requires large RAM as well as
disk space for implementation on a simple personal
computer.
5. Conclusion
In this paper, the application of genetic algorithm based
neural networks for simulating the stress–strain response of
CMCs is demonstrated. The results obtained from the
micromechanical finite element analysis have been synthesised into a GA/BPN model. The stress–strain response of
whisker reinforced Al2O3/SiC CMC has been simulated
by training the feed forward form of neural architecture
using genetic algorithm for weight extraction. From the
training examples provided, the network model captures
the stress–strain behaviour of the composite automatically.
This relationship is automatically incorporated into the
network model in an implicit manner during the training
process. Further, during the training process, the degree
of non-linearity present in the problem is also automatically
338
established. This alleviates the necessity to have an
a-priori knowledge of the degree of non-linearity in the relationship. The various stages in the development of the GA/
BPN model viz. selection of network type, configuration,
input and output vectors, training and testing have been
presented in detail. Use of genetic algorithms for efficient
and fast training of the network has been demonstrated.
Based on the work presented, the following conclusions
are drawn.
• A novel approach to simulation of stress–strain response
of CMCs has been demonstrated using the genetic algorithm based back propagation neural networks.
• Isoparametric interface elements have been used to
model the whisker/matrix interface in CMCs in the finite
element analysis. The results obtained from this finite
element model have been useful in understanding the
effect of interface strength on the stress–strain response
of CMCs.
• The network is able to predict the stress–strain response
for a new interface strength of CMC, after its successful
training. The reliability of the GA/BPN network is thus
established for its practical use in design of CMCs.
• It is clear from the advantages of the present network
model presented here that neural networks provide a
tool for conducting sensitivity studies of CMCs to derive
their optimum property profiles. This would enable the
designer to come up with CMCs having desired mechanical properties.
• The GA based neural networks provide a very fast and
computationally efficient economical analysis tool to
analyse the CMCs considering the interface strength
effect on the stress–strain response.
• Future scope exists for using neural networks for arriving at the guidelines to design the CMCs with desired
mechanical properties.
Appendix A
A.1. Pseudocode for implementation of GA/BPN
Algorithm FITGEN ()
{Let (Ii,Ti), i = 1, 2, . . . , N where Ii = (I1i, I2i, . . . , Ili)
and Ti = (T1I, T2I, . . . , Tni) represent the input–output
pairs of the problem to be solved by BPN with a configuration 1 m n.
For each chromosome Ci, i = 1, 2, . . . , p belonging to
the current population Pi whose size is p
{Extract weights Wi from Ci;
Keeping Wi as a fixed weight setting, train the
BPN for the N input instances;
Calculate error Ei for each of the input instances
using the formula,
X
2
Ei ¼
ðT ji Oji Þ ;
j
where Oji is the output vector calculated by BPN;
Find the root mean square E of the errors Ei,
i = 1, 2, . . . , N;
Calculate the fitness value Fi for each of the individual string of the population as
F i ¼ 1=E;
}
Output Fi for each Ci, i = 1, 2, . . . , p;
}
END FITGEN
A.2. Pseudocode for GA based weight determination
{i 0;
Generate the initial population Pi of real-coded chromosomes C Ji each representing a weight set for the BPN;
While the current population Pi has not converged
{Generate fitness values F JI for for each C JI 2 P i using
the Algorithm FITGEN ();
Get the mating pool ready by terminating worst fit
individuals and duplicating high fit individuals;
Using the cross over mechanism, reproduce offspring
from the parent chromosomes;
i
i + 1;
Call the current population PI;
Calculate fitness values F Ji for each C iJ 2 P i ;
}
Extract weights from Pi to be used by the BPN;
}
END
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