A Neural Network Material Model for the
Analysis of Fabric Structures
N J Bartle1*, P D Gosling1, B N Bridgens1
1
School of Civil Engineering & Geosciences, Newcastle University, Newcastle upon
Tyne, NE1 7RU, UK.
* Author for correspondence
Abstract
Neural networks are an artificial intelligence concept. Through a process of training
they are capable of capturing the relationship between experimental input and output
data. With the addition of historical inputs and internal variables it is demonstrated
that neural network models are capable of representing the complex history
dependant behaviour of architectural fabrics.
Keywords: architectural fabric, neural network, constitutive modelling, hysteresis.
1
Introduction
Architectural fabrics are effectively woven composites with yarn-based cores and
outer layer coating substrates. In the analysis of fabric structures it is common
practice to represent membrane material as a homogeneous continuum described by
a plane stress stress-strain relationship, where the material characteristics are
represented by Young's modulus and Poisson's ratio. Fitting a plane stress model to
biaxial test data for typical architectural fabrics leads to inconsistencies between the
physical and theoretical descriptions, with values of Poisson's ratio in excess of the
compressibility limit of 0.5, and for some fabrics approaching 2.0 [1]. An alternative
to the plane stress framework is therefore required to more correctly represent fabric
behaviour.
Neural networks offer an exciting solution for the constitutive modelling of
architectural fabrics as they are capable of capturing highly non-linear response.
Furthermore, neural network material models effectively ‘learn’ the material
response directly from experimental data and therefore no assumptions regarding the
response are required.
1
2
Scope and context
The aim of this study is to develop a novel approach to represent the relationship
between fabric stress and strain which surpasses the capabilities of the current best
practice.
Through a process of training neural networks are capable of capturing the
relationship between sets input and output data. The trained network may then be
used to generate outputs from previously unseen inputs.
To date limited attempts have been made incorporate the effects of load history and
residual strain within fabric material models. It has been shown that relatively
simple neural network models are capable of representing hysteretic behaviour
through the use of ‘internal variables’ [2]. It is proposed here to use this method to
capture the hysteretic behaviour of architectural fabric.
3
Neural Networks Architecture and Training
The architecture of a neural network takes on a layered form with each layer
containing neurons (Figure 2) which in turn are connected to the neurons of the next
layer via weighted connections (Figure 1).
Forward propagation of input signals to be
converted to outputs
Input
Signals
𝑤1𝑗
Inpu
⋮
Out
Output
Signal
Bias
𝑤2𝑗
⋮
𝑏𝑗
∑
f
𝑤𝑛𝑗
Input Layer
Hidden Layer Output Layer
Back propagation of error signals to update
connection weights
Figure 1. Network
Architecture
Figure 2. General
Neuron from hidden or
output layer.
Each neuron sums the weighted output signals from each neuron of the previous
layer, adds a bias signal and passes the result through an activation function. This
function may be any differentiable function. In this study (and many similar studies)
a tan sigmoid transfer function is used in the hidden layer and a linear transfer
function is used in the output layer. The non-linear transfer function in the hidden
layer enables this kind of network to capture non-linear relationships between inputs
and outputs.
A multilayer feed forward neural network with a single tan sigmoid hidden layer
may be represented by the following set of equations. The form of these equations is
2
similar to that used by Hashash et al. in [3], and represent a strain controlled neural
network material model.
𝜀𝑖𝑁𝑁 =
𝜀𝑖
𝑆𝑖𝜀
𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 − 1 < 𝜀𝑖𝑁𝑁 < 1
(1)
𝐵𝜀 𝑁𝑁
𝐵𝑖 = tanh(𝛽[∑𝑁𝜀
+ 𝑏𝑗 ]) 𝑤ℎ𝑒𝑟𝑒 𝛽 = 1
𝑗=1 𝑤𝑖𝑗 𝜀𝑗
(2)
𝜎𝐵
𝜎𝑖𝑁𝑁 = (𝛽[∑𝑁𝐵
𝑗=1 𝑤𝑖𝑗 𝐵𝑗 + 𝑏𝑗 ]) 𝑤ℎ𝑒𝑟𝑒 𝛽 = 1
(3)
𝜎𝑖 = 𝑆𝑖𝜎 𝜎𝑖𝑁𝑁 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 − 1 < 𝜎𝑖𝑁𝑁 < 1
(4)
A number of training algorithms have been developed but the most commonly used
fall under the general term of back-propagation. In this study the Matlab Neural
Network Toolbox [4] is used for the development of the neural networks. The
Levenberg-Marquardt training method ‘trainlm’ is used. A detailed description of
the method is available in [5].
A key consideration when designing Neural Networks is the issue of over fitting
where a network is trained to a point where it no longer possesses the ability to
generalise. It is therefore vital that the network is tested using previously ‘unseen’
data in order to identify where over fitting has occurred.
4
Response surface style network
Minami [6] extends the use of multi-step linear approximation through the use of
response surfaces. A pair of response surfaces made up of triangular and
quadrilateral elements are developed from biaxial stress-strain curves obtained for
0:1, 1:2, 1:1, 2:1 and 1:0 load ratios. A plane stress material stiffness matrix is
derived for each element of the surface to be used in analysis.
Bridgens and Gosling took the response surface concept further suggesting the
complete removal of any plane stress assumption. The use of either direct look up
tables [7] or spline curves used to interpolate between measured stress strain points
within the surface [8] were proposed.
When a neural network is trained using data in the form of load ratio arms similar to
those used in [6] and [7] it may be used to interpolate between those arms thus
creating a response surface style model. As this model is easy to visualise it is
selected as a good starting point for the initial development of an architectural fabric
neural network material model.
4.1
Training Data Collection and Pre-Processing
A biaxial testing profile was developed which included additional arms between the
standard 0:1, 1:2, 1:1, 2:1 and 1:0 load ratios (Figure 3). These ratios offer the
opportunity to further investigate the shape of the response surface (Figure 4) and
also provide ‘unseen’ testing data for network validation.
3
Figure 3. Load ratio
arms investigated during
PVC biaxial test
Figure 4. Response
surface derived from PVC
biaxial test data.
Biaxial testing and data processing closely follows the protocol laid out in [1]. In
post processing residual strain is removed from the experimental data, this is done to
remove the skew from the final response surface.
4.2
Network Testing and Validation
Within typical finite element analysis the current strain state is used to determine the
current level of stress. The current strain will therefore be used as input and the
stress as output. This leads to a network comprising two inputs (warp and fill strain),
a single hidden layer containing 10 neurons and two outputs (warp and fill stress).
Three individual sets of data are used to train the networks. Network 1 is trained
using a data set which contains the full range of load ratios. Network 2 is trained
using a data set which contains only the standard 0:1, 1:2, 1:1, 2:1 and 1:0 load
ratios. Network 3 is trained using a data set that is derived from a reversed stress
controlled version of the neural network described above. This set is also used as a
testing set along with the data set containing all load ratios. In this way the gaps
between loading arms may be investigated and cases of over-fitting may be
identified.
Because neural networks are initialised using random numbers for all weights and
biases each trained network produces a different functional mapping despite being
presented with identical training data. In this study a total of 15 independent
networks have been trained and tested. The resulting coefficients of determination
are set out in Table 1.
The highest coefficients of determination are observed when a network is tested
using the same set of data it was trained with. The network trained using the partial
data set achieves the lowest performance, this is expected as this is the least
comprehensive data set.
4
Network 1: Trained using full data set
Testing
data set
Network Version
1
2
3
4
5
Full
0.9978
0.9991
0.9978
0.9990
0.9993
Network derived
0.9936
0.9919
0.9640
0.9912
0.9814
Network 2: Trained using partial data set
Testing
data set
Network Version
1
2
3
4
5
Full
0.9886
0.9933
0.9855
0.9938
0.9784
Network derived
0.9772
0.9900
0.9598
0.9872
0.9649
4
5
Network 3: Trained using network derived data set
Testing
data set
Network Version
1
2
3
Full
0.9964
0.9951
0.9904
0.9950
0.9952
Network derived
0.9993
0.9975
0.9954
0.9990
0.9993
Table 1. Coefficients of determination derived from testing of
networks trained using 3 different data sets. Coefficients indicating
best performance are highlighted.
Version 5 of Network 1 (Figure 5) shows the best performance when tested with the
full range of experimental load ratios. However, when tested with the more
comprehensive network derived data set a considerably lower performance is
observed, the effects of overfiitting are clearly evident in the warp surface.
Figure 5. Output from Network 1, trained using full set of
experimental load ratios and tested with both experimental and
synthetic data.
5
Version 1 of Network 3 (Figure 6) demonstrates the best performance overall and
may be selected as having the best ability to generalise the fabric response. The use
of synthetic training data reduces the impact of over-fitting without the need for
additional expensive physical testing. It is however of extreme importance to gather
sufficient experimental data to thoroughly train and validate any network model to
be used in structural analysis.
Figure 6. Output from Network 3, trained using synthetic data and
tested with both experimental and synthetic data.
The same procedure has been undertaken for PTFE glass fabric with reduced
success. The response surface of PTFE glass is much steeper than that of PVC
polyester fabric, this is due to the fabric’s very stiff yarns and high level crimp
interchange [1]. This effect leads to the requirement of the network to map multiple
pairs of strains to a single value of stress. This is impossible for the network to
achieve and therefore additional inputs are required. This same issue is encountered
when attempting to train a network to capture the hysteretic behaviour of
architectural fabrics.
5
Uniaxial network including load history
5.1
Internal variables
The use of internal variables is proposed by Yun et al [2] as a solution to transform a
many to one mapping to a single valued mapping in order to model materials which
exhibit hysteretic behaviour. These internal variables (Equations 5 and 6) have been
adopted to capture the uniaxial hysteric behaviour of architectural fabric. For a
detailed explanation of the internal variables and a proof of single valuedness for a 1
dimensional mapping please refer to [2]. Previous strain is denoted by 𝜀𝑛−1 and
previous stress by 𝜎𝑛−1 , Δ𝜀𝑛 denotes the strain step.
𝜉𝑛 = 𝜎𝑛−1 𝜀𝑛−1
(5)
Δ𝜂𝑛 = 𝜎𝑛−1 Δ𝜀𝑛
(6)
6
5.2
Training Data Collection and Pre-Processing
The uniaxial testing protocol is based on the British Standard [9] (BS EN
ISO1421:1998). Testing is completed using an Instron constant rate extension
machine. Three different profiles were used to provide full data sets for training and
testing and to investigate the ideal training profile (Figure 7).
Figure 7. Uniaxial cyclic load profiles 1,2 and 3, load is shown in
terms of percentage of ultimate tensile strength for use with a range of
fabrics
5.3
Network Testing and Validation
As in the previous study a basic Matlab fitting neural network is used. The network
model input comprises current strain, previous strain, previous stress, and the
internal variables combined into a single input via addition. The output is the current
level of stress.
A preliminary study into the effect of training profile, training data density and
network architecture was undertaken. It was found that a reasonably sparse data
density should be used in order to obtain the best generalisation and profile three
was demonstrated to be the most successful training profile.
7
Figure 8. Network output tested in recurrent mode using training
data gathered using Profile 3.
Figure 9. Network output tested in recurrent mode using previously
unseen data generated from profile 2.
8
The final network has 7 hidden nodes and is trained using Profile 3. Each loading or
unloading cycle is described by 10 data points, a total of 480 input sets are used in
training. Once trained the network is tested in recurrent mode, where previous
network outputs are used to generate the current network input. This is important as
it is the mode in which the network would be used in analysis
When tested in recurrent mode using Profile 3 the network model is shown to
successfully capture the complex hysteretic fabric response (Figure 8). Deviation
observed in the latter load cycles may be due to accumulated error caused by the
recurrent mode of testing. The powerful generalisation capability of the network is
demonstrated when it is tested with ‘unseen’ data from Profile 2 (Figure 9). The
network achieves the lowest levels of accuracy at maximum stress levels this
highlights the importance of capturing data beyond the bounds of stress range
anticipated in analysis.
6
Conclusions
It has been demonstrated by the networks described above that neural network
material models are capable of capturing the complex non-linear stress-strain
response of architectural fabrics. The response surface style network captures the
strain stress relationship in a similar manner to the plane stress framework however
removed the need for plane stress assumptions. However, this network has
limitations when employed to model the much steeper response surface of PTFE
coated glass fabric and neglects the effects of load history.
The uniaxial network demonstrates the capability of neural networks to model the
effects of load history. This initial study offers a promising proof of concept which
aims to lead to a biaxial response network which includes the effects of load history.
7
Further work
The development of a load history biaxial response network will require careful
design of biaxial profiles to explore the full response envelope and provide
comprehensive testing and training data sets. The effects of load step size will also
require further consideration.
The modelling of shear has been omitted from this study due to its complexity. A
novel approach would be required to capture the interaction between shear and
biaxial stress which is currently outside the scope of this body of work. However,
once training data is available it is likely that neural networks will have the ability to
successfully model the shear response of architectural fabrics.
The final step in the development of neural network material models for
architectural fabrics will be to implement them within a custom fabric analysis finite
element programme. This will be achieved using a method proposed by Hashash et
al [3] whereby the network equations (Equations 1 to 4) are used to calculate an
‘implied’ stiffness matrix. Initial studies into this process show promising results.
9
Acknowledgements
This research is funded by the Engineering and Physical Sciences Research Council
(EPSRC), with material contributions from Serge Ferrari and Verseidag Indutex.
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