The water industry is facing increased pressure to produce higher quality treated water at a lower cost. The efficiency of a treatment process closely relates to the operation of the plant. To improve the operating performance, an Artificial Neural Network (ANN) paradigm has been applied to a water treatment plant. An ANN which is able to learn the non-linear performance relationships of historical data of a plant, has been proved to be capable of providing operational guidance for plant operators. A back-propagation network is used to determine the alum and polymer dosages. The results show that the ANN model is most promising. The correlation coefficients (r) between the actual and predicted values for the alum and polymer dosages were both 0.97 and the average absolute percentage errors were 4.09% and 8.76% for the alum and polymer dosages respectively. The application of the ANN model is illustrated using data from Wyong Shire Council's Wyong Water Treatment Plant on the Central Coast of NSW.
SUMMARYThe performances of various numerical schemes used to model hyperbolic/parabolic equations have been studied by the calculation of their numerical group velocities. Numerical experiments conducted with one dimensional linear and quadratic Lagrangian finite elements with a Crank-Nicolson finite differencing in time confirm the results of the analysis. The group velocity analysis supplements the well-known amplitude and phase portraits introduced by Leendertse' and helps explain the occurrence and behaviour of numerical oscillations in both finite difference and finite element schemes.
SUMMARYThe purpose of this paper is to investigate the effect of a non-uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model.The domain consists of two conterminous regions discernible only by their different nodal spacings in the xdirection. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank-Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and transmitted waves shows that energy is conserved across the interface between the two regions.KEYWORDS: spurious wave refraction; total internal reflection INTRODUCTKlNAn appreciation of the effects of a non-uniform mesh is fundamental to a good understanding of the processes occumng in numerical models with varying mesh sizes. This is particularly relevant to time dependent finite element (FE) models and interactively nested finite difference (FD) models (in contradistinction to passively nested FD models). Practioners are often faced with questions such as 'how much can I change the mesh size and what are the consequences?'In 1D linear systems, these questions have been partly addressed.'-3 The analyses were completed for the simplified situation of a 1D domain consisting of two semi-infinite regions abutting at a common interfacial node. The two regions were discernible on numerical grounds (e.g. different mesh spacings or different numerical algorithms) or physical grounds (effected by an abrupt change in coefficient in the governing equation). The latter case is not considered herein and hence the inclusion of the word 'spurious' in the title of the present paper.The magnitude of the effects of a change in 1D mesh size were quantified by determining the amplitudes of the transmitted and (unwanted) reflected waves, and their associated energy fluxes due to an incident wave. In a more general sense, the incident wave may be interpreted as one of many Fourier components which can be superimposed to make up a general waveform.In ID, a change in mesh size results in both wave rejection and wave transmission. In 2D however, there is an additional process at work viz wave refraction. The interface between the two conterminous
SUMMARYThis is the first of a series of three related papers dealing with some of the consequences of non-uniform meshes in a numerical model. In this paper the accuracy of the Crank-Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths A AX, 4Ax, 8Ax) in three 'hot-start' numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.
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