Gaussian random field model

Statistical description of the twill weave textile obtained in Chapter 3 showed that the maximum deviation of the yarn paths from a nominal design was approximately 0.7 mm. Additionally, it was shown that the deviation from the average yarn path tends to be normally distributed with standard deviations for different samples of up to 0.1 mm. It was also shown that adjacent yarns are highly correlated which was explained by tight weaving of the textile where a deviation of a yarn causes similar deviations in neighbouring yarns. The correlation length in the transverse direction was found to be larger than the size of the studied textile. The autocorrelation of the yarns was found to decay much faster and vanish at length of 100 mm. The presence of the spatial correlations implies that deviations of yarn paths in neighbouring section are not independent and hence a model which can capture this feature is required.

Chapter 2 identified several existing approaches to model yarn path variability: description as a series of sine waves with random frequencies [121], description as Markov chains [124] and as a Gaussian random field [77]. The first, whilst useful and simple, was not developed to fit the frequency and amplitude domains to existing data and relied on the identification of minimal and maximal frequencies and amplitudes. Other approaches were based on a similar theoretical basis but differed in realisations: the Markov chain approach was not constrained by the form of the transition matrix while the Gaussian field approach is based on a priori definition of the correlation matrix. In this chapter the Gaussian random field approach is employed for modelling of deviations of yarn paths due to its simple and robust realisation for statistical studies.

The random field theory is based on the theory of random processes [123]. In general, a random process is a collection of random variables distributed in a certain domain and ordered in the ―time‖-domain . A random process can also be viewed as a state of a physical system (e.g. temperature at a point) which changes with time. In this sense, a random field is a natural extension of random process theory where a time variable is a position in space

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. The condition of variables being ordered still holds for a random field. Following the introduced notation, a Gaussian random field is a field where random variables follow a Gaussian distribution.

A textile structure dictates the choice of space domain . In fact, the domain has already been implicitly introduced in Chapter 3 during the experimental studies. For each of the yarn directions the x-axis was chosen to coincide with the nominal yarn direction and the other axis was chosen to be orthogonal to it in the plane of the textile. For the random field the - coordinate was chosen to be discrete in the same way that the data from real textiles were measured i.e. with a regular spacing of 10.0 mm. The y-coordinate was chosen to coincide with nominal yarn position and hence have a spacing of 2.5 mm as shown in Figure 6.2. The choice of the space domain was equivalent for both yarn directions. It can be noted that data points on neighbouring yarns have an offset of -2.5 mm relative to each other. The measured data were transformed into a rectangular grid by an affine transformation:

(6.1) where = –2.5 mm is the offset coefficient.

Figure 6.2 Coordinates of space domain of the Gaussian random field

Since variables in a random field are strictly ordered in a space domain, it becomes necessary to describe their spatial correlations (if there are any). It was assumed that

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there is no correlation between warp and weft yarns. It was assumed that the Gaussian field is the Ornstein-Uhlenbeck (OU) sheet [123] as defined by Skordos and Sutcliffe [77]. The covariance matrix of the OU sheet is:

(6.2)

where is the standard deviation, and and are the inverse correlation lengths.

The parameters of the correlation matrix can be approximated using the maximum log-likelihood estimator as described by Ying [170]. Three samples of data obtained in Chapter 3 were used for the parameter estimation. Depending on the size of the used data set it was found that the parameters stabilise with an increase of the data set (including additional data points into analysis). The minimal textile size required to estimate the parameters is approximately 5050 yarns in each direction i.e. about 1212 unit cells. The dependence of the parameters on the size of the used textile is given in Figure 6.3 – Figure 6.5. Average values of the parameters at length of 50 yarns (125 mm) are =0.2 mm, 1=0.09 cm-1, 2=0.01 cm-1

.

Figure 6.3 Estimated value of when different sizes of sample are used

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 20 40 60 80 100 120 . mm

Yarns taken for estimation

Sample 1, warp Sample 2, warp Sample 3, warp Sample 1, weft Sample 2, weft Sample 3, weft

143 Figure 6.4 Estimated values when different sizes of sample are used

Figure 6.5 Estimated values when different sizes of sample are used

The generation of the Gaussian random field with given covariance matrix was achieved by calculating the Cholesky decomposition of the covariance matrix and then multiplying it by a random vector of normally distributed numbers.

(6.3)

(6.4)

The matrix of the Gaussian field is then reshaped into a matrix accordingly to the chosen size of the textile and coordinates are transformed back to

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 20 40 60 80 100 120 1 , 1/cm

Yarns taken for estimation

Sample 1, warp Sample 2, warp Sample 3, warp Sample 1, weft Sample 2, weft Sample 3, weft

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 20 40 60 80 100 120 2 , 1/cm

Yarns taken for estimation

Sample 1, warp Sample 2, warp Sample 3, warp Sample 1, weft Sample 2, weft Sample 3, weft

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which the define textile geometry as shown in Figure 6.2. A listing of the MatLab code can be found in Appendix H. Samples of generated yarn paths are shown in Figure 6.6. Generated Gaussian field are employed for creating models of textile reinforcements.

Figure 6.6 Samples of generated weft yarns