(266a) The Application of Wavelet Transforms to Many-Body Particle Interactions

The flow of particulate materials is a critical part of many industrial and mining processes. For the simulation of granular flows, the Discrete Element Method (DEM) has been well established for simple flows for many years. In the simulation model, at every time step the positions and orientations of particles are updated, so there is a wealth of data available on a time referenced basis which has the potential to allow a quantitative analysis of dynamics of assembly evolution. Fourier transforms have been used as a tool for extracting local-frequency information from the signal but the Fourier representation only provides the spectral content with no indication about time localization of the spectral component. Since the wavelet transform can be used to analyse time series that contain non-stationary power at many different frequencies. The wavelet transform is a method of converting a signal or function into another form which either makes features of the original signal more amenable or allows the original data set to be described more succinctly. To perform a wavelet transform we need a wavelet known as a mother wavelet or analysing wavelet. A wavelet or mother wavelet is a function satisfying certain mathematical criteria. The wavelet transform decomposes a discrete signal into a set of translated and dilated mother wavelets, where these various scales and shifts in the mother wavelet are based on powers of two.

This paper will present the preliminary results from our data analysis. The data sets are results from a simulation study of packing of uniform fine-spherical particles received from UNSW, Australia. A simulation began with filling the particles in a rectangular box and then a vibration along the z direction was applied. The first data set was composed of the X, Y, Z positions of 2500 particles for 21 time-steps with 0.2 s increment of time. The second data set used the same conditions as the first one except a smaller time step with 0.05 s increment of time and a longer time running. We have analysed both data sets using the idea of making particle layer bands. The layers were made at the first time step and particles in those layers were followed as  t increased.

A movie of the particles flow was made from the first data set by using MATLAB in order to examine the particles motion. The program shows movies of 2500 particles in 2D and 3D. In order to understand the motion of the particles, at t = 0 the particles are divided into 4 layers in the z direction: the top layer, the upper middle layer, the lower middle layer and the bottom layer. Figure 1(a) shows a snapshot of the first time step. The snapshots of time step 21(4.2 s) are shown in figure 1(b). We have found that there is little diffusion of particles along the z direction after the third time step in every layer, but there is a greater horizontal motion of particles along x and y directions with most motion in the top layer as it can be observed the particle flows in the horizontal (x ,y) plane in figure 1(c) and figure 1(d). Figure 1(c) shows the first time step of flows consisting of four sections in the (x ,y) plane. In figure 1(d) we show the analogous pictures at time step 21.

In the analysis of the second data set, different thickness layers in z direction are made at  t = 0 and then keep tracing in those layers. A box is divided into two halves consisting of 1250 particles for each half, then divided again into quarters, consisting of 625 particles for each quarter and finally subdivided the bottom layer into two halves consisting of 312 particles. Mean and standard deviation of particle positions in each layer are calculated. We focused on the bottom layer. In order to perceive an increase of fluctuation of mean when the layers are made thinner, mean of different thickness bottom layers are plotted together as shown in the figure 2. Noticeably, mean of position in thin layers are more fluctuated than in thick layers. We found that if we make the layer thinner, we see more fluctuation than statistically expected.

                                    (a)                                                                    (b)

                                    (c)                                                                    (d)

Figure 1: Snapshot pictures

These results suggest that because mono-sized particles are used, at the bottom layer where they are subject to most force from the upper layers, they lock into crystallize state consisting of group of particles. Net results in that motion is not for individual particle moving but for a small number of groups of particles.

Figure 2: Mean of particle positions

Furthermore, multi resolution decompositions are applied to the second data set. Different thickness of bottom layers and different directions are investigated. A member of Daubechies, db2 is selected as a mother wavelet. Mean of x and y positions are decomposed into different bands of frequencies, but there are no significant events.

So far, two data sets have been analysed. We have planned for further investigation. The plan is to focus on three areas. These are: applying cross-correlation and wavelet cross correlation to the simulation data sets, further wavelet analysis and analysis of further simulation data sets.