Prediction of double layer grids' maximum deflection using neural network

American Journal of Applied Sciences, Nov, 2008 by Reza Kamyab Moghadas, Kok Keong Choong

INTRODUCTION

Many thousands of impressive space structures have been built all over the world for covering sport stadiums, gymnasiums, leisure centers, aircraft hangars, railway stations and many other purposes. A number of these structures have spans of well over 200m. Due to their three dimensional action, space structures are very efficient structural systems to carry heavy loads as well as to span large distances. As the use of space structures becomes more and more popular, it is essential to evolve strategies for their convenient analysis and design. In the present investigation, neural network techniques are employed for this purpose. Over the last decade, artificial intelligence techniques have emerged as a robust tool to replace time consuming procedures in many scientific or engineering applications. The use of neural networks to predict finite element analysis outputs has been studied previously in the context of many engineering applications (1-5). The principal advantage of a properly trained neural network is that it requires a trivial computational effort to produce an approximate solution.

The main aim of the present study is to train neural networks for predicting the maximum deflection of double layer grid space structures for static loadings. The double layer grids considered are two-way on two-way and the bar elements are connected by MERO type of joints. The length of the spans, L and the height, H, of the space structures are variable. Due to the practical demands, members are grouped. In this work, three groups are considered. That is, all the top, web and bottom layer elements are grouped in groups 1 to 3, respectively. Cross-sectional areas of the all groups, A, are selected from a list of available tube sections in STAHL. In the present study, the inputs of neural networks are the length of the spans, L, the height, H and cross-sectional areas of the all groups, A, while the outputs are maximum deflections of the corresponding double layer grids. In the present study, backpropagation (BP) and Radial Basis Function (RBF) networks are employed. Fundamental concepts of BP and RBF networks are briefly explained as follows.

Backpropagation neural network: The most popular and successful learning method for training the multilayer neural networks is the backpropagation algorithm. The development of the Backpropagation learning was reported by Rumelhart Hinton and Williams (6). The algorithm employs an iterative gradient-descent method of minimization which minimizes the mean squared error between the desired output and network output. The backpropagation training procedure is presented below.

E = [1/2][N.summation over (j = 1)][M.summation over (i = f)][e.sub.i.sup.2](n) (1)

N, M and n are number of training input patterns, dimension of output space and number of iterations, respectively.

[e.sub.i](n) = [d.sub.i] (n)-[[y.sub.i].sup.(L)] (n) (2)

where L represent the output layer.

[v.sub.i.sup.l](n) = [N.summation over (j = 1)][w.sub.ij.sup.[(1)]][y.sub.j.sup.[(1 - l)]](n) (3)

where [y.sub.j.sup.(1-1) (n) is the function signal of neuron j in the previous layer (l-1) at iteration n. [w.sub.ij.sup.(1)](n) is the weight of neuron i in layer l that is fed from neuron j in layer (l-1).

Then the output signal of neuron i in layer l is

[y.sub.i.sup.l](n) = f([v.sub.i.sup.l](n)) (4)

where, f (*) is the activation function.

If neuron i is in the first hidden layer (l = 1), then set [y.sub.i.sup.0](n) = [x.sub.i] (n).

Backward computation (local gradients)

[.sub.i](n) = - [[[partial derivative][E.sub.i]]/[[[partial derivative][v.sub.i]] (5)

is called the local error or local gradients. Equation (5) can be simplified to for neuron i in output layer L:

[.sub.i.sup.L](n) = [e.sub.i.sup.L](n)f'([v.sub.i.sup.L](n)) (6)

for neuron i in hidden layer l:

[.sub.i.sup.l](n) = f'([v.sub.i.sup.l]))[summation over (k)][.sub.k.sup.(l 1)](n) [w.sub.ki.sup.(l 1)](n) (7)

where, f'(*) is the derivative of the activation function with respect to v(n).

If the activation function is chosen to be the hyperbolic tangent function, then f (*) is:

f'([v.sub.i]) = [[df([v.sub.i])]/[d[v.sub.i]] = (1 - [f.sup.2]([v.sub.i])) (8)

Hence, adjust the weights of the network in layer l according to the generalized delta rule:

[w.sub.ij.sup.(1)](n 1) = [w.sub.ij.sup.(l)](n) [.sub.i.sup.(1)](n)[y.sub.j.sup.[l - 1)](n) (9)

where, [mu] is the positive constant learning rate, usually equals to 0.01.

If after updating the weights, the error E is not minimized, new iterations are required.

Radial basis function neural network: The backpropagation algorithm for the design of a multilayer neural network described earlier may be viewed as a form of stochastic approximation. Radial Basis Functions (RBFs) take a different approach by viewing the design of a neural network as a curve-fitting problem by finding a best fit to the training data in a multidimensional space. The use of RBF in the design of neural networks was first introduced by Broomhead and Lowe (7). The RBF network basically involves three entirely different layers; an input layer, a hidden layer of high enough dimension and an output layer. The transformation from the hidden unit to the output space is linear. Each output node is the weighted sums of the outputs of the hidden layer. However, the transformation from the input layer to the hidden layer is nonlinear. Each neuron or node in the hidden layer forming a linear combination of the basis (or kernel) functions which produces a localized response with respect to the input signals. This is to say that RBF produce a significant nonzero response only when the input falls within a small localized region of the input space. The most common basis of the RBF is a Gaussian kernel function of the form: