1. Introduction
Recently, Radial Basis Function Networks (RBFN) based
meshless methods have been used to approximate complex functions to deal with
engineering problems (Zerroukat et al., 2000; Kansa, 1990; Mai-Duy and
Tran-Cong, 2001; Tran and Phillips, 2006). In this work, the Integral RBFN
method (Mai-Duy and Tran-Cong, 2001) is applied to a complex non-linear textile
problem; the shape of the yarn spinning balloon.
2. The non-linear governing equation for the yarn balloon
shape
The identification of the shape of the yarn balloon in
ring-spinning has been an important topic for several years (He, 2004; Stump
and Fraser, 1996; Batra et al., 1989 and the governing equation describing the
yarn balloon shape (Fig. 1) is given by the following non-linear function
(e.g., He, 2004):
where y'' and y' are the second and first derivatives of the
function y with respect to x and a2 =mΩ2
/Tx (m is
the yarn mass, Ω is the rotational frequency, Tx is
the yarn tension in the x direction). The boundary conditions are given by y (0)
= 0; y(r) = h. (2a, b)
3. The basis of the Integral RBFN approach
In principle, any function y(x) can be approximated by a linear combination of m fixed RBFs (Haykin, 1999) as follows 
where {w j, j = 1.m} is the set of network
weights, hj the chosen radial basis function
corresponding to the jth neuron, and m the number of RBFs.
In this work, the more accurate multi-quadric MQRBF h(x) (Hardy, 1971) is used
as defined by: 
where c is the center and a is the set of RBF widths. Using
this h(x) in Eq. (3), the unknown weights {wi}mi=1 are found via the general linear least squares principle based on a set of n data points {xp,yp}n where xp is the coordinate of the pth
input data point and yp is the desired value of function y at the
point xp, usually m<=n (Haykin, 1999).
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