Keywords: dynamic analysis, genetic algorithm, fast wavelet transform, wavelet neural network, reverse wavelet transform.

Wavelet transform is a tool that cuts up data or functions into different frequency components, and then studies each component with resolution mathematics. In wavelet analysis the use of a fully scalable window solves the signal-cutting problem. The window is shifted along the signal and for every position the spectrum is calculated. Then this process is repeated with varying windows length for every new cycle. The final resolution will be a collection of time-frequency representation of the signal, all with different resolutions.

For problems with large number of degrees of freedom, the structural analysis is time consuming. This makes the optimal design process very inefficient, especially when a time history analysis is considered. To overcome this difficulty, a fast wavelet transform (FWT) [1] is used to transfer the ground acceleration record of the specified earthquake into a function with very small number of points.

There is a relationship between the FWT and digital filter banks. It turns out that the wavelet transform can be simply achieved by a tree of digital filter banks. The main idea behind the filter bank is to divide a signal into two parts; one is the low frequency part and the other is the high frequency part. This idea can be achieved by a set of filters. A filter banks, consists of a low pass and a high pass filters, which separates a signal into different frequency bands. By applying a low pass filter to a signal, the high frequency bands of the signal are removed and an approximate version of the original signal is obtained. A high pass filter removes the low frequency components of the original signal, and the result is a signal containing the details of the main signal.

By FWT, the number of points of the earthquake record is reduced. The decomposition starts from the original record, and produces two sets of new records and the process is repeated. In each stage, the record is divided into two parts, the first part contains the high frequencies and the other contains the low frequencies. By down sampling, the number of points of each part is almost half of the record in the last stage. In this paper this process is repeated in three stages, and the number of points of the original record is reduced to 0.125 of the primary points. The low frequency part of the record is used for dynamic analysis. Thus the time history dynamic analysis is carried out at a fewer points. The error is negligible, in particular in the first three stages of decomposition [2].

Despite substantial reduction in the dynamic analysis, optimization process requires a great number of time history dynamic analyses; thus the overall time of the optimization process against earthquake is excessive. To reduce the computational time of optimization, a wavelet neural network (WNN) [3], is used for approximating the dynamic responses of the structures. By such approximation, the dynamic analysis of the structure is not necessary during the optimization process. This network is inspired by both feedforward neural networks and wavelet decompositions. An algorithm of backpropagation type is proposed for training the network. In this network the input is FWT points of the earthquake record, and the output is the responses of the structures against these reduced points. After training the network, using reverse wavelet transform (RWT) the results of the dynamic analysis is obtained for the original earthquake accelerograph record from the output of the network.

The proposed wavelet neural networks to calculate the dynamic responses of the structure are designed as a three-layer with a wavelet layer, weighting layer, and summing layer. Each layer has one or more nodes. The activation functions of the wavelet nodes in the wavelet layer are derived from a mother wavelet. Each output of the weighting nodes in the weighting layer is multiplied by an appropriate weight value determined by the weighting node. The weighted sum of the output of weighting nodes in the weighting layer produces the final output of the summing layer.

In the full paper, the details of the optimization approach with approximation concepts will be discussed and some numerical examples for optimum design of structures will be presented. The details of the FWT, WNN and RWT will also be outlined. The computational time is compared for the exact optimization method with those of the approximate results. The numerical results of the dynamic analysis show that this approximation is a powerful technique and the required computational work can be reduced greatly. The error involved in this transformation is small.

1

S. Mallat, "A Theory for Multiresolution Signal Decomposition: the Wavelet Representation", IEEE Transactions on Pattern Analysis Machine Intelligence, 11, 674-693, 1989. doi:10.1109/34.192463

2

A. Heidari, "Optimum Design of Structures Against Earthquake by Advanced Optimisation Methods", PhD Thesis, University of Kerman, Kerman, Iran (under preparation), 2004.

3

M. Thuillard, "Wavelets in Soft Computing", New York: World Scientific Publishing Co. Pte. Ltd., 2001.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £95 +P&P)