Biomedical Engineering Reference
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( i ) FE fatigue cracks simulations of
different trabecular bone specimens for
different combinations of loading inputs.
Equations (7 and 8)
( ii ) Averaging the specimens outputs
(Crack.Dn and Cr.Le).
( iii
) Training data preparation for the NN.
Steps (i) and (ii) supply training data
in the form of a inputs-outputs patterns.
Preparation of Design of Experiment table
based on steps (i) and (ii).
Application of in-house NN code (Hambli et
al.,2009b).
( iv ) Training the NN: Building the
architecture of NN.
Implemented into Abaqus code using UMAT
subroutine.
( v ) Incorporation of the NN into macro FE
model to link meso-to-macro scales.
Fig. 3 Building and incorporation of the trained NN in FE code in five steps (i)-(v). The
interdependencies of steps (i)-(v) is expressed in terms of cascades of step execution sequences
3 Neural Network for Approximation and Interpolation
NNs have recently been widely used for the analysis of an increasing number of
problems in science and technology [ 17 - 21 , 23 , 28 , 33 , 48 , 51 , 53 ]. NNs can be used
for the mapping of input to output data without knowing ''a priori'' the relationship
between the data. One of the distinct characteristics of the NN is its ability to learn
and generalize from experience and examples and to adapt to changing situations.
Once the NN has been satisfactorily trained and tested, it is able to generalize rules
and will be able to respond very rapidly (a few seconds) to input data to predict the
required output, within the domain covered by the training examples [ 27 , 29 , 44 ]. NN
architecture is composed of an input layer, a certain number of hidden layers and an
output layer in forward connections (Fig. 4 ). Each neuron in the input layer repre-
sents a single input parameter. These values are directly transmitted to the subsequent
neurons of the hidden layers. The neurons of the last layer represent the NN outputs
[ 18 , 27 , 29 ].
The output y i
of a neuron i in a layer m is calculated as [ 18 , 29 , 44 ]:
y i ¼ fv i
ð 1 Þ
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